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class="vector-toc-numb">2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordinal_type" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ordinal_type"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ordinal type</span> </div> </a> <ul id="toc-Ordinal_type-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructing_new_wpo's_from_given_ones" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constructing_new_wpo's_from_given_ones"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Constructing new wpo's from given ones</span> </div> </a> <ul id="toc-Constructing_new_wpo's_from_given_ones-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wqo's_versus_well_partial_orders" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wqo's_versus_well_partial_orders"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Wqo's versus well partial orders</span> </div> </a> <ul id="toc-Wqo's_versus_well_partial_orders-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_increasing_subsequences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Infinite_increasing_subsequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Infinite increasing subsequences</span> </div> </a> <ul id="toc-Infinite_increasing_subsequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_wqos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties_of_wqos"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Properties of wqos</span> </div> </a> <ul id="toc-Properties_of_wqos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> 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searchaux" style="display:none">Mathematical concept for comparing objects</div> <style data-mw-deduplicate="TemplateStyles:r1034237262">.mw-parser-output .stack{box-sizing:border-box}.mw-parser-output .stack>div{margin:1px;overflow:hidden}@media all and (min-width:720px){.mw-parser-output .stack-clear-left{float:left;clear:left}.mw-parser-output .stack-clear-right{float:right;clear:right}.mw-parser-output .stack-left{float:left}.mw-parser-output .stack-right{float:right}.mw-parser-output .stack-margin-clear-left{float:left;clear:left;margin-right:1em}.mw-parser-output .stack-margin-clear-right{float:right;clear:right;margin-left:1em}.mw-parser-output .stack-margin-left{float:left;margin-right:1em}.mw-parser-output .stack-margin-right{float:right;margin-left:1em}}</style><div class="stack mw-stack stack-right"><div> <table class="wikitable mw-collapsible floatright mw-collapsed" style="float:right; margin-top:0; padding-top:0; text-align:center;"> <tbody><tr> <th style="text-align:center;padding-left:0.2em;padding-right:0.2em;font-size:90%;"><span style="font-size:120%"><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a> <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a></span> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Connected_relation" title="Connected relation">Connected</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Join_and_meet" title="Join and meet">Has joins</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Join_and_meet" title="Join and meet">Has meets</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Reflexive_relation#Irreflexive" title="Reflexive relation">Irreflexive</a></td><td style="padding-left:0.3em; padding-right:0.3em;"> <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"></td><td></td><td></td><td> Total, Semiconnex</td><td></td><td></td><td></td><td></td><td> Anti-<br />reflexive</td><td></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <span class="nowrap"><a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a></span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <span class="nowrap"><a href="/wiki/Preorder" title="Preorder">Preorder <span style="font-size:85%;">(Quasiorder)</span></a></span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total preorder</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Total_order" title="Total order">Total order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Prewellordering" title="Prewellordering">Prewellordering</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <span class="nowrap"><a class="mw-selflink selflink">Well-quasi-ordering</a></span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Well-order" title="Well-order">Well-ordering</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Join-semilattice" class="mw-redirect" title="Join-semilattice">Join-semilattice</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Meet-semilattice" class="mw-redirect" title="Meet-semilattice">Meet-semilattice</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict partial order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Weak_ordering#Strict_weak_orderings" title="Weak ordering">Strict weak order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> <a href="/wiki/Strict_total_order" class="mw-redirect" title="Strict total order">Strict total order</a></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span style="color:red;" title="Red X">✗</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td><td> <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"></td><td> <a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></td><td> <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></td><td> <a href="/wiki/Connected_relation" title="Connected relation">Connected</a></td><td> <a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></td><td> <a href="/wiki/Join_and_meet" title="Join and meet">Has joins</a></td><td> <a href="/wiki/Join_and_meet" title="Join and meet">Has meets</a></td><td> <a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></td><td> <a href="/wiki/Reflexive_relation#Irreflexive" title="Reflexive relation">Irreflexive</a></td><td> <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></td></tr><tr style="vertical-align:middle;"><td style="text-align:right;padding-left:0.6em; padding-right:1.2em; font-weight:bold;"> Definitions, for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\neq \varnothing :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\neq \varnothing :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a538fab804c9428c4f7d4ca3ed214a97483c4260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.698ex; height:2.676ex;" alt="{\displaystyle S\neq \varnothing :}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18cac1ed3115c87d45b4d751de57124233a6e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.712ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> </mtd> <mtd> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cc3f46e9f0b647ce6771396a975ef8d364674d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.643ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi>b</mi> <mo stretchy="false">⇒</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> </mtd> <mtd> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c73cadd7ab7bf3b5257f5c505c756ec098902f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.97ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">min</mo> <mi>S</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>exists</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ec5a418cceeb68b9945ca75d31604719db4660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.513ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>∨</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>exists</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3386379a9406067b50b99e12241e9d7fab3369b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.395ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>∧</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>exists</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abff244e064dd72fd16781277ad3440b35bd767d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.395ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRa}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRa}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7fc1d9d50c65105d5edcb3478b5ca4172c54d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.224ex; height:2.176ex;" alt="{\displaystyle aRa}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{not }}aRa}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>not </mtext> </mrow> <mi>a</mi> <mi>R</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{not }}aRa}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8263f0c706367e5306eae1b9353034024639da23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.164ex; height:2.176ex;" alt="{\displaystyle {\text{not }}aRa}"></span></td><td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mi>R</mi> <mi>b</mi> <mo stretchy="false">⇒</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>not </mtext> </mrow> <mi>b</mi> <mi>R</mi> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aec245f8776556d3ad3fcdc18d4ba61929eccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.683ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}"></span></td></tr></tbody></table> </td></tr> <tr> <td style="text-align:center;"><span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> indicates that the column's property is always true for the row's term (at the very left), while <span style="color:red;" title="Red X">✗</span> indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by <span typeof="mw:File"><span><img alt="Green tick" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/20px-Green_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/26px-Green_check.svg.png 2x" data-file-width="600" data-file-height="600" /></span></span><span style="display:none">Y</span> in the "Symmetric" column and <span style="color:red;" title="Red X">✗</span> in the "Antisymmetric" column, respectively. <br /> <p>All definitions tacitly require the <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> be <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>: for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6021b6ac503535d74098454c2a870a1b5c187d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.949ex; height:2.509ex;" alt="{\displaystyle a,b,c,}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRb}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRb}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b0b52168739fd16b254298771ec07b900e5a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.991ex; height:2.176ex;" alt="{\displaystyle aRb}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bRc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>R</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bRc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a62ff0d4c99429cb3465edecb65d2b1a53cf30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.768ex; height:2.176ex;" alt="{\displaystyle bRc}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aRc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>R</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aRc.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e3d14d72455c36aab129326d6dceefafb3f12a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.648ex; height:2.176ex;" alt="{\displaystyle aRc.}"></span> <br /> A term's definition may require additional properties that are not listed in this table. </p> </td></tr></tbody></table></div></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, specifically <a href="/wiki/Order_theory" title="Order theory">order theory</a>, a <b>well-quasi-ordering</b> or <b>wqo</b> on a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a <a href="/wiki/Quasi-ordering" class="mw-redirect" title="Quasi-ordering">quasi-ordering</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> for which every <a href="/wiki/Infinity" title="Infinity">infinite</a> sequence of elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},x_{1},x_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1},x_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995c6e8f2f8c807b3c82daa28a3765e026d2fd8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.977ex; height:2.009ex;" alt="{\displaystyle x_{0},x_{1},x_{2},\ldots }"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> contains an increasing pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\leq x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\leq x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2becf5bb3e25d7f611840fdec4e25728f9cc8a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.676ex;" alt="{\displaystyle x_{i}\leq x_{j}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<j.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<j.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280b30be032a77d024b24f18128d7b0dce685596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.506ex; height:2.509ex;" alt="{\displaystyle i<j.}"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=1" title="Edit section: Motivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Well-founded_induction" class="mw-redirect" title="Well-founded induction">Well-founded induction</a> can be used on any set with a <a href="/wiki/Well-founded_relation" title="Well-founded relation">well-founded relation</a>, thus one is interested in when a quasi-order is well-founded. (Here, by abuse of terminology, a quasiorder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span> is said to be well-founded if the corresponding strict order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y\land y\nleq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>∧</mo> <mi>y</mi> <mo>≰</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y\land y\nleq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/513ceade76b978ab3049885a5c376c6f178c5c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.75ex; height:3.176ex;" alt="{\displaystyle x\leq y\land y\nleq x}"></span> is a well-founded relation.) However the class of well-founded quasiorders is not closed under certain operations—that is, when a quasi-order is used to obtain a new quasi-order on a set of structures derived from our original set, this quasiorder is found to be not well-founded. By placing stronger restrictions on the original well-founded quasiordering one can hope to ensure that our derived quasiorderings are still well-founded. </p><p>An example of this is the <a href="/wiki/Power_set" title="Power set">power set</a> operation. Given a quasiordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span> for a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> one can define a quasiorder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e750321d0d76e1cee9cb865050aecda733ae9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.319ex; height:2.676ex;" alt="{\displaystyle \leq ^{+}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>'s power set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf94167ebc67db1573e6be9e261cd0518e49e160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.535ex; height:2.843ex;" alt="{\displaystyle P(X)}"></span> by setting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leq ^{+}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leq ^{+}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2aa51ea59d9dde1fc053068f492773042e94def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.116ex; height:2.676ex;" alt="{\displaystyle A\leq ^{+}B}"></span> if and only if for each element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> one can find some element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> that is larger than it with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span>. One can show that this quasiordering on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf94167ebc67db1573e6be9e261cd0518e49e160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.535ex; height:2.843ex;" alt="{\displaystyle P(X)}"></span> needn't be well-founded, but if one takes the original quasi-ordering to be a well-quasi-ordering, then it is. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=2" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>well-quasi-ordering</b> on a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is a <a href="/wiki/Quasi-ordering" class="mw-redirect" title="Quasi-ordering">quasi-ordering</a> (i.e., a <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a>, <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a> <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a>) such that any <a href="/wiki/Infinity" title="Infinity">infinite</a> sequence of elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},x_{1},x_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1},x_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995c6e8f2f8c807b3c82daa28a3765e026d2fd8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.977ex; height:2.009ex;" alt="{\displaystyle x_{0},x_{1},x_{2},\ldots }"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> contains an increasing pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\leq x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\leq x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2becf5bb3e25d7f611840fdec4e25728f9cc8a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.676ex;" alt="{\displaystyle x_{i}\leq x_{j}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60ff2d1b23e30fb2979e8c1536da03493f943cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i<j}"></span>. The set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is said to be <b>well-quasi-ordered</b>, or shortly <b>wqo</b>. </p><p>A <b>well partial order</b>, or a <b>wpo</b>, is a wqo that is a proper ordering relation, i.e., it is <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>. </p><p>Among other ways of defining wqo's, one is to say that they are quasi-orderings which do not contain infinite <i>strictly decreasing</i> sequences (of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}>x_{1}>x_{2}>\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}>x_{1}>x_{2}>\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5147800404304364f461fb35768bf2bb5dd6dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.17ex; height:2.176ex;" alt="{\displaystyle x_{0}>x_{1}>x_{2}>\cdots }"></span>)<sup class="plainlinks nourlexpansion citation" id="ref_a"><a class="external autonumber" href="https://en.wikipedia.org/wiki/Well-quasi-ordering#endnote_a">[1]</a></sup> nor infinite sequences of <i>pairwise incomparable</i> elements. Hence a quasi-order (<i>X</i>, ≤) is wqo if and only if (<i>X</i>, <) is <a href="/wiki/Well-founded_relation" title="Well-founded relation">well-founded</a> and has no infinite <a href="/wiki/Antichain" title="Antichain">antichains</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Ordinal_type">Ordinal type</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=3" title="Edit section: Ordinal type"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> be well partially ordered. A (necessarily finite) sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d8b3ec62633086002489cad8df4214e9585880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.337ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}"></span> of elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> that contains no pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\leq x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\leq x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2becf5bb3e25d7f611840fdec4e25728f9cc8a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.676ex;" alt="{\displaystyle x_{i}\leq x_{j}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60ff2d1b23e30fb2979e8c1536da03493f943cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i<j}"></span> is usually called a <i>bad sequence</i>. The <i>tree of bad sequences</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e80072681ec0d850d5d10601d58032619803e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}"></span> is the tree that contains a vertex for each bad sequence, and an edge joining each nonempty bad sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n-1},x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n-1},x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef88413e177df7a8b5b551578ae98c7f368d097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.602ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n-1},x_{n})}"></span> to its parent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113a81afd50ceaed8205e63ee1e6fe87ea1f907a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.02ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n-1})}"></span>. The root of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e80072681ec0d850d5d10601d58032619803e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}"></span> corresponds to the empty sequence. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> contains no infinite bad sequence, the tree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e80072681ec0d850d5d10601d58032619803e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}"></span> contains no infinite path starting at the root.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This is far from being obvious, and requires a (reference to a) proof. (February 2024)">citation needed</span></a></i>]</sup> Therefore, each vertex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e80072681ec0d850d5d10601d58032619803e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}"></span> has an ordinal height <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19085f019ea50c0c059ae2705a8a412b7e4381b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.065ex; height:2.843ex;" alt="{\displaystyle o(v)}"></span>, which is defined by transfinite induction as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(v)=\lim _{w\mathrm {\ child\ of\ } v}(o(w)+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">d</mi> <mtext> </mtext> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">f</mi> <mtext> </mtext> </mrow> <mi>v</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(v)=\lim _{w\mathrm {\ child\ of\ } v}(o(w)+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ae44a79a484b54e9ce1eb5daf96330d35d9928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.199ex; height:4.009ex;" alt="{\displaystyle o(v)=\lim _{w\mathrm {\ child\ of\ } v}(o(w)+1)}"></span>. The <i>ordinal type</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2209bd3f7740e3be286c9878d37e06014fd63f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.917ex; height:2.843ex;" alt="{\displaystyle o(X)}"></span>, is the ordinal height of the root of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94e80072681ec0d850d5d10601d58032619803e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle T_{X}}"></span>. </p><p>A <i>linearization</i> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is an extension of the partial order into a total order. It is easy to verify that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2209bd3f7740e3be286c9878d37e06014fd63f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.917ex; height:2.843ex;" alt="{\displaystyle o(X)}"></span> is an upper bound on the ordinal type of every linearization of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. De Jongh and Parikh<sup id="cite_ref-dejongh_parikh_1-0" class="reference"><a href="#cite_note-dejongh_parikh-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> proved that in fact there always exists a linearization of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> that achieves the maximal ordinal type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2209bd3f7740e3be286c9878d37e06014fd63f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.917ex; height:2.843ex;" alt="{\displaystyle o(X)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=4" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Integers-line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Integers-line.svg/200px-Integers-line.svg.png" decoding="async" width="200" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Integers-line.svg/300px-Integers-line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Integers-line.svg/400px-Integers-line.svg.png 2x" data-file-width="706" data-file-height="58" /></a><figcaption><b>Pic.1:</b> Integer numbers with the usual order</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Infinite_lattice_of_divisors.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Infinite_lattice_of_divisors.svg/200px-Infinite_lattice_of_divisors.svg.png" decoding="async" width="200" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Infinite_lattice_of_divisors.svg/300px-Infinite_lattice_of_divisors.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Infinite_lattice_of_divisors.svg/400px-Infinite_lattice_of_divisors.svg.png 2x" data-file-width="354" data-file-height="407" /></a><figcaption><b>Pic.2:</b> <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of the natural numbers ordered by divisibility</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:N-Quadrat,_gedreht.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/N-Quadrat%2C_gedreht.svg/200px-N-Quadrat%2C_gedreht.svg.png" decoding="async" width="200" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/N-Quadrat%2C_gedreht.svg/300px-N-Quadrat%2C_gedreht.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/N-Quadrat%2C_gedreht.svg/400px-N-Quadrat%2C_gedreht.svg.png 2x" data-file-width="260" data-file-height="200" /></a><figcaption><b>Pic.3:</b> Hasse diagram of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578dc62c2bb6d5e2b2624c6b58b02787df469372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {N} ^{2}}"></span> with componentwise order</figcaption></figure> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d3fc9234cbfa1d0932c5f6205b81b3fa65dc05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ,\leq )}"></span>, the set of natural numbers with standard ordering, is a well partial order (in fact, a <a href="/wiki/Well-order" title="Well-order">well-order</a>). However, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} ,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5264b7f134e415f47c698541423131f85de3e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.202ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,\leq )}"></span>, the set of positive and negative integers, is <b>not</b> a well-quasi-order, because it is not well-founded (see Pic.1).</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ,|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ,|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e27e519e9a67c927d3c383da8b9bf1a73d948099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ,|)}"></span>, the set of natural numbers ordered by divisibility, is <b>not</b> a well-quasi-order: the prime numbers are an infinite antichain (see Pic.2).</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ^{k},\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ^{k},\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40fb442311359b0d622a005806fcd4b9511678b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.418ex; height:3.176ex;" alt="{\displaystyle (\mathbb {N} ^{k},\leq )}"></span>, the set of vectors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> natural numbers (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is finite) with <a href="/wiki/Product_order" title="Product order">component-wise ordering</a>, is a well partial order (<a href="/wiki/Dickson%27s_lemma" title="Dickson's lemma">Dickson's lemma</a>; see Pic.3). More generally, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is well-quasi-order, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X^{k},\leq ^{k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X^{k},\leq ^{k})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e492b58447a1daed979dca617c44f45c33d4e6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.826ex; height:3.176ex;" alt="{\displaystyle (X^{k},\leq ^{k})}"></span> is also a well-quasi-order for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.</li> <li>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> be an arbitrary finite set with at least two elements. The set <a href="/wiki/Kleene_star" title="Kleene star"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span></a> of <a href="/wiki/Formal_language#Words_over_an_alphabet" title="Formal language">words over</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> ordered <a href="/wiki/Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographically</a> (as in a dictionary) is <b>not</b> a well-quasi-order because it contains the infinite decreasing sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,ab,aab,aaab,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mi>a</mi> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,ab,aab,aaab,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075e4b1c0c5397f6102159cf29c3396795275fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.228ex; height:2.509ex;" alt="{\displaystyle b,ab,aab,aaab,\ldots }"></span>. Similarly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span> ordered by the <a href="/wiki/Prefix_(computer_science)#Prefix" class="mw-redirect" title="Prefix (computer science)">prefix</a> relation is <b>not</b> a well-quasi-order, because the previous sequence is an infinite antichain of this partial order. However, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span> ordered by the <a href="/wiki/Subsequence" title="Subsequence">subsequence</a> relation is a well partial order.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has only one element, these three partial orders are identical.)</li> <li>More generally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X^{*},\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X^{*},\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e02858303d38b85505a1674c758e5f2ee94291f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.702ex; height:2.843ex;" alt="{\displaystyle (X^{*},\leq )}"></span>, the set of finite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-sequences ordered by <a href="/wiki/Embedding" title="Embedding">embedding</a> is a well-quasi-order if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is a well-quasi-order (<a href="/wiki/Higman%27s_lemma" title="Higman's lemma">Higman's lemma</a>). Recall that one embeds a sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> into a sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> by finding a subsequence of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> that has the same length as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> and that dominates it term by term. When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,=)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>=</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,=)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce4f91df888a03cc6c3c495dcf2fbe120588bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,=)}"></span> is an unordered set, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\leq v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>≤</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\leq v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5730c3f2fd9ab2b37f23c0f5ae368e254aeb9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.556ex; height:2.176ex;" alt="{\displaystyle u\leq v}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> is a subsequence of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X^{\omega },\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X^{\omega },\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11c698202839702de0385a3621e8d57a1626b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.903ex; height:2.843ex;" alt="{\displaystyle (X^{\omega },\leq )}"></span>, the set of infinite sequences over a well-quasi-order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>, ordered by embedding, is <b>not</b> a well-quasi-order in general. That is, Higman's lemma does not carry over to infinite sequences. <a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better-quasi-orderings</a> have been introduced to generalize Higman's lemma to sequences of arbitrary lengths.</li> <li>Embedding between finite trees with nodes labeled by elements of a wqo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is a wqo (<a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a>).</li> <li>Embedding between infinite trees with nodes labeled by elements of a wqo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is a wqo (<a href="/wiki/Crispin_St._J._A._Nash-Williams" class="mw-redirect" title="Crispin St. J. A. Nash-Williams">Nash-Williams</a>' theorem).</li> <li>Embedding between countable <a href="/wiki/Scattered_order" title="Scattered order">scattered</a> <a href="/wiki/Linear_order" class="mw-redirect" title="Linear order">linear order</a> types is a well-quasi-order (<a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a>).</li> <li>Embedding between countable <a href="/wiki/Boolean_algebras" class="mw-redirect" title="Boolean algebras">boolean algebras</a> is a well-quasi-order. This follows from Laver's theorem and a theorem of Ketonen.</li> <li>Finite graphs ordered by a notion of embedding called "<a href="/wiki/Graph_minor" title="Graph minor">graph minor</a>" is a well-quasi-order (<a href="/wiki/Robertson%E2%80%93Seymour_theorem" title="Robertson–Seymour theorem">Robertson–Seymour theorem</a>).</li> <li>Graphs of finite <a href="/wiki/Tree-depth" title="Tree-depth">tree-depth</a> ordered by the <a href="/wiki/Induced_subgraph" title="Induced subgraph">induced subgraph</a> relation form a well-quasi-order,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> as do the <a href="/wiki/Cograph" title="Cograph">cographs</a> ordered by induced subgraphs.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Constructing_new_wpo's_from_given_ones"><span id="Constructing_new_wpo.27s_from_given_ones"></span>Constructing new wpo's from given ones</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=5" title="Edit section: Constructing new wpo's from given ones"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"></span> be two disjoint wpo sets. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X_{1}\cup X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∪</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=X_{1}\cup X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1059ea0c8de4bb720551ab20691a2e7d204661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.411ex; height:2.509ex;" alt="{\displaystyle Y=X_{1}\cup X_{2}}"></span>, and define a partial order on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> by letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}\leq _{Y}y_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}\leq _{Y}y_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ea268fde36d3265187600c7629508fd9543360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.972ex; height:2.343ex;" alt="{\displaystyle y_{1}\leq _{Y}y_{2}}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1},y_{2}\in X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1},y_{2}\in X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721fa7d124e0ab00d8960ad35a7563adaddca52b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.985ex; height:2.509ex;" alt="{\displaystyle y_{1},y_{2}\in X_{i}}"></span> for the same <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \{1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebe15d55dc04257a7634ea5447371b6f8c1c205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.327ex; height:2.843ex;" alt="{\displaystyle i\in \{1,2\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}\leq _{X_{i}}y_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}\leq _{X_{i}}y_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35a1e5e14686e64f91e666bfccb7f209c6bbbcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.703ex; height:2.676ex;" alt="{\displaystyle y_{1}\leq _{X_{i}}y_{2}}"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is wpo, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(Y)=o(X_{1})\oplus o(X_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊕</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(Y)=o(X_{1})\oplus o(X_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94cca022ae10583576a9f8bec04c6e73c09a8aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.48ex; height:2.843ex;" alt="{\displaystyle o(Y)=o(X_{1})\oplus o(X_{2})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></span> denotes <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">natural sum</a> of ordinals.<sup id="cite_ref-dejongh_parikh_1-1" class="reference"><a href="#cite_note-dejongh_parikh-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Given wpo sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"></span>, define a partial order on the Cartesian product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X_{1}\times X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=X_{1}\times X_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf71a1b2c8dedce95f53baf836d41bde8856063b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.669ex; height:2.509ex;" alt="{\displaystyle Y=X_{1}\times X_{2}}"></span>, by letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{2})\leq _{Y}(b_{1},b_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{2})\leq _{Y}(b_{1},b_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5534fd487b0a45b0ce4762b45db71c02f1a6c97c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.943ex; height:2.843ex;" alt="{\displaystyle (a_{1},a_{2})\leq _{Y}(b_{1},b_{2})}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\leq _{X_{1}}b_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}\leq _{X_{1}}b_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd4e610555bb0fb17292c35214a8f4242e3fccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.859ex; height:2.843ex;" alt="{\displaystyle a_{1}\leq _{X_{1}}b_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}\leq _{X_{2}}b_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}\leq _{X_{2}}b_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/736ba6be2badb3712e120393aca26b1166b81501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.859ex; height:2.843ex;" alt="{\displaystyle a_{2}\leq _{X_{2}}b_{2}}"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is wpo (this is a generalization of <a href="/wiki/Dickson%27s_lemma" title="Dickson's lemma">Dickson's lemma</a>), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(Y)=o(X_{1})\otimes o(X_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(Y)=o(X_{1})\otimes o(X_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2715a3714566a9667cbd00e19b61b5c6e44560a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.48ex; height:2.843ex;" alt="{\displaystyle o(Y)=o(X_{1})\otimes o(X_{2})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \otimes }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }"></span> denotes <a href="/wiki/Ordinal_arithmetic" title="Ordinal arithmetic">natural product</a> of ordinals.<sup id="cite_ref-dejongh_parikh_1-2" class="reference"><a href="#cite_note-dejongh_parikh-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Given a wpo set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span> be the set of finite sequences of elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, partially ordered by the subsequence relation. Meaning, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})\leq _{X^{*}}(y_{1},\ldots ,y_{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n})\leq _{X^{*}}(y_{1},\ldots ,y_{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625f1749b83825804182be1fe240a1ec24930c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.489ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n})\leq _{X^{*}}(y_{1},\ldots ,y_{m})}"></span> if and only if there exist indices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i_{1}<\cdots <i_{n}\leq m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i_{1}<\cdots <i_{n}\leq m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c1c86f700099590d8e1abc3a6f8e2452871ba1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.198ex; height:2.509ex;" alt="{\displaystyle 1\leq i_{1}<\cdots <i_{n}\leq m}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}\leq _{X}y_{i_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}\leq _{X}y_{i_{j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f201e111896d7372b7e624af415f51acf60ee5e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.623ex; height:2.843ex;" alt="{\displaystyle x_{j}\leq _{X}y_{i_{j}}}"></span> for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq j\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq j\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829066d1e818107baff629179da368c45fcb8883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.712ex; height:2.509ex;" alt="{\displaystyle 1\leq j\leq n}"></span>. By <a href="/wiki/Higman%27s_lemma" title="Higman's lemma">Higman's lemma</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span> is wpo. The ordinal type of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01924e6e5570e2631081fea6c6981b4872d3e04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.051ex; height:2.343ex;" alt="{\displaystyle X^{*}}"></span> is<sup id="cite_ref-dejongh_parikh_1-3" class="reference"><a href="#cite_note-dejongh_parikh-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-schmidt_5-0" class="reference"><a href="#cite_note-schmidt-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(X^{*})={\begin{cases}\omega ^{\omega ^{o(X)-1}},&o(X){\text{ finite}};\\\omega ^{\omega ^{o(X)+1}},&o(X)=\varepsilon _{\alpha }+n{\text{ for some }}\alpha {\text{ and some finite }}n;\\\omega ^{\omega ^{o(X)}},&{\text{otherwise}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> finite</mtext> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>+</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for some </mtext> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and some finite </mtext> </mrow> <mi>n</mi> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(X^{*})={\begin{cases}\omega ^{\omega ^{o(X)-1}},&o(X){\text{ finite}};\\\omega ^{\omega ^{o(X)+1}},&o(X)=\varepsilon _{\alpha }+n{\text{ for some }}\alpha {\text{ and some finite }}n;\\\omega ^{\omega ^{o(X)}},&{\text{otherwise}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588c84cfd3263a9557fee29e75e8aab62239e4b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:66.34ex; height:10.509ex;" alt="{\displaystyle o(X^{*})={\begin{cases}\omega ^{\omega ^{o(X)-1}},&o(X){\text{ finite}};\\\omega ^{\omega ^{o(X)+1}},&o(X)=\varepsilon _{\alpha }+n{\text{ for some }}\alpha {\text{ and some finite }}n;\\\omega ^{\omega ^{o(X)}},&{\text{otherwise}}.\end{cases}}}"></span> </p><p>Given a wpo set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe67aad4eff628fcb5bb28ee6a2213d28ff12e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle T(X)}"></span> be the set of all finite rooted trees whose vertices are labeled by elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Partially order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe67aad4eff628fcb5bb28ee6a2213d28ff12e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle T(X)}"></span> by the <a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">tree embedding relation</a>. By <a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe67aad4eff628fcb5bb28ee6a2213d28ff12e7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle T(X)}"></span> is wpo. This result is nontrivial even for the case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |X|=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97472a4878998de0c77b0d19810ee6262021ab13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.535ex; height:2.843ex;" alt="{\displaystyle |X|=1}"></span> (which corresponds to unlabeled trees), in which case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(T(X))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(T(X))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7552527a699a1a4901381c7c9e01e318debcfec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.362ex; height:2.843ex;" alt="{\displaystyle o(T(X))}"></span> equals the <a href="/wiki/Small_Veblen_ordinal" title="Small Veblen ordinal">small Veblen ordinal</a>. In general, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2209bd3f7740e3be286c9878d37e06014fd63f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.917ex; height:2.843ex;" alt="{\displaystyle o(X)}"></span> countable, we have the upper bound <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(T(X))\leq \vartheta (\Omega ^{\omega }o(X))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ϑ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mi>o</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(T(X))\leq \vartheta (\Omega ^{\omega }o(X))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf1810cee750729aac524dbb002e25e9a5bffb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.494ex; height:2.843ex;" alt="{\displaystyle o(T(X))\leq \vartheta (\Omega ^{\omega }o(X))}"></span> in terms of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϑ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d00eaf197c35bbfa391b9477490a4af955416837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.374ex; height:2.176ex;" alt="{\displaystyle \vartheta }"></span> <a href="/wiki/Ordinal_collapsing_function" title="Ordinal collapsing function">ordinal collapsing function</a>. (The small Veblen ordinal equals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vartheta (\Omega ^{\omega })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϑ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vartheta (\Omega ^{\omega })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49d1dc1173f4037a56ca7b6d044ddfcb268c03b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.116ex; height:2.843ex;" alt="{\displaystyle \vartheta (\Omega ^{\omega })}"></span> in this ordinal notation.)<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Wqo's_versus_well_partial_orders"><span id="Wqo.27s_versus_well_partial_orders"></span>Wqo's versus well partial orders</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=6" title="Edit section: Wqo's versus well partial orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In practice, the wqo's one manipulates are quite often not orderings (see examples above), and the theory is technically smoother<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2017)">citation needed</span></a></i>]</sup> if we do not require antisymmetry, so it is built with wqo's as the basic notion. On the other hand, according to Milner 1985, <i>no real gain in generality is obtained by considering quasi-orders rather than partial orders... it is simply more convenient to do so.</i> </p><p>Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes induced by the kernel of the wqo. For example, if we order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> by divisibility, we end up with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\equiv m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≡</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\equiv m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c20fcef81d76adf3a59a110f98eb9f68c70c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle n\equiv m}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\pm m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mo>±</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\pm m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7bd56e222b3a159f10d1a5025e1e00e193ad960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.342ex; height:2.176ex;" alt="{\displaystyle n=\pm m}"></span>, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} ,|)\approx (\mathbb {N} ,|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,|)\approx (\mathbb {N} ,|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0626fca588e09df4903d4c3c1bd54d557f7f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.307ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,|)\approx (\mathbb {N} ,|)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Infinite_increasing_subsequences">Infinite increasing subsequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=7" title="Edit section: Infinite increasing subsequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is wqo then every infinite sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},x_{1},x_{2},\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1},x_{2},\ldots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455988e97a48251886035a7e5c235efb277b3d08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.011ex; height:2.009ex;" alt="{\displaystyle x_{0},x_{1},x_{2},\ldots ,}"></span> contains an <b>infinite</b> increasing subsequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n_{0}}\leq x_{n_{1}}\leq x_{n_{2}}\leq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n_{0}}\leq x_{n_{1}}\leq x_{n_{2}}\leq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9919b64867fe59a5bab623674b07201e9bc87f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.158ex; height:2.676ex;" alt="{\displaystyle x_{n_{0}}\leq x_{n_{1}}\leq x_{n_{2}}\leq \cdots }"></span> (with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}<n_{1}<n_{2}<\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}<n_{1}<n_{2}<\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753e2fa1997e024945c19d50a99f4f8262807a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.365ex; height:2.176ex;" alt="{\displaystyle n_{0}<n_{1}<n_{2}<\cdots }"></span>). Such a subsequence is sometimes called <b>perfect</b>. This can be proved by a <a href="/wiki/Ramsey_theory" title="Ramsey theory">Ramsey argument</a>: given some sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i})_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0aba2e1dc63976b69db29835f228f1cded9730b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.738ex; height:2.843ex;" alt="{\displaystyle (x_{i})_{i}}"></span>, consider the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> of indexes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> has no larger or equal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span> to its right, i.e., with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60ff2d1b23e30fb2979e8c1536da03493f943cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i<j}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> is infinite, then the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>-extracted subsequence contradicts the assumption that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is wqo. So <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> is finite, and any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> larger than any index in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> can be used as the starting point of an infinite increasing subsequence. </p><p>The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering, leading to an equivalent notion. </p> <div class="mw-heading mw-heading2"><h2 id="Properties_of_wqos">Properties of wqos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=8" title="Edit section: Properties of wqos"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Given a quasiordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> the quasiordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P(X),\leq ^{+})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P(X),\leq ^{+})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a1125bed7b3fc53527a1ccbf0113dd16bdb6ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.697ex; height:3.009ex;" alt="{\displaystyle (P(X),\leq ^{+})}"></span> defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leq ^{+}B\iff \forall a\in A,\exists b\in B,a\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mi>B</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mi mathvariant="normal">∃</mi> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo>,</mo> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leq ^{+}B\iff \forall a\in A,\exists b\in B,a\leq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1588e2484d5e34ea8f6a4c3c283f9446bf474c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.408ex; height:2.843ex;" alt="{\displaystyle A\leq ^{+}B\iff \forall a\in A,\exists b\in B,a\leq b}"></span> is well-founded if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span> is a wqo.<sup id="cite_ref-forster_7-0" class="reference"><a href="#cite_note-forster-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>A quasiordering is a wqo if and only if the corresponding partial order (obtained by quotienting by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y\iff x\leq y\land y\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mi>y</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>∧</mo> <mi>y</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim y\iff x\leq y\land y\leq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94dda79ad9a1998fd56cd30f183a6797ab919104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.231ex; height:2.343ex;" alt="{\displaystyle x\sim y\iff x\leq y\land y\leq x}"></span>) has no infinite descending sequences or <a href="/wiki/Antichain" title="Antichain">antichains</a>. (This can be proved using a <a href="/wiki/Ramsey_theory" title="Ramsey theory">Ramsey argument</a> as above.)</li> <li>Given a well-quasi-ordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>, any sequence of upward-closed subsets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{0}\subseteq S_{1}\subseteq \cdots \subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊆</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊆</mo> <mo>⋯</mo> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{0}\subseteq S_{1}\subseteq \cdots \subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d85e78afe44cd8ec741b529d8972eb2e9ce1638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.957ex; height:2.509ex;" alt="{\displaystyle S_{0}\subseteq S_{1}\subseteq \cdots \subseteq X}"></span> eventually stabilises (meaning there exists <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}=S_{n+1}=\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=S_{n+1}=\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c49af3f3b0a7d3a81c8bb674a543b3bffc579e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.307ex; height:2.509ex;" alt="{\displaystyle S_{n}=S_{n+1}=\cdots }"></span>; a subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44aba72977e43f863dd873b095d1dc0bd3f17608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle S\subseteq X}"></span> is called <i>upward-<a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a></i> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>∧</mo> <mi>x</mi> <mo>∈</mo> <mi>S</mi> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/725fde46f9df5fbb3e74e59c9248f2b81a5f0cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.611ex; height:2.509ex;" alt="{\displaystyle \forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S}"></span>): assuming the contrary <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall i\in \mathbb {N} ,\exists j\in \mathbb {N} ,j>i,\exists x\in S_{j}\setminus S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">∃</mi> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mi>j</mi> <mo>></mo> <mi>i</mi> <mo>,</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo class="MJX-variant">∖</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall i\in \mathbb {N} ,\exists j\in \mathbb {N} ,j>i,\exists x\in S_{j}\setminus S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4df1977f29488db3597b9d5620381a19907142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.561ex; height:3.009ex;" alt="{\displaystyle \forall i\in \mathbb {N} ,\exists j\in \mathbb {N} ,j>i,\exists x\in S_{j}\setminus S_{i}}"></span>, a contradiction is reached by extracting an infinite non-ascending subsequence.</li> <li>Given a well-quasi-ordering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>, any subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> has a finite number of minimal elements with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span>, for otherwise the minimal elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> would constitute an infinite antichain.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better-quasi-ordering</a> – mathematical relation<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Prewellordering" title="Prewellordering">Prewellordering</a> – Set theory concept</li> <li><a href="/wiki/Well-order" title="Well-order">Well-order</a> – Class of mathematical orderings</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=10" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><span class="citation wikicite" id="endnote_a"><b><a href="#ref_a">^</a></b></span> Here <i>x</i> < <i>y</i> means: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637423b227b64dd6b186522e92ec84862b48e074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.23ex; height:2.676ex;" alt="{\displaystyle x\neq y.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Well-quasi-ordering&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-dejongh_parikh-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-dejongh_parikh_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dejongh_parikh_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-dejongh_parikh_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-dejongh_parikh_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFde_JonghParikh1977" class="citation journal cs1">de Jongh, Dick H. G.; Parikh, Rohit (1977). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F1385-7258%2877%2990067-1">"Well-partial orderings and hierarchies"</a>. <i>Indagationes Mathematicae (Proceedings)</i>. <b>80</b> (3): 195–207. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F1385-7258%2877%2990067-1">10.1016/1385-7258(77)90067-1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indagationes+Mathematicae+%28Proceedings%29&rft.atitle=Well-partial+orderings+and+hierarchies&rft.volume=80&rft.issue=3&rft.pages=195-207&rft.date=1977&rft_id=info%3Adoi%2F10.1016%2F1385-7258%2877%2990067-1&rft.aulast=de+Jongh&rft.aufirst=Dick+H.+G.&rft.au=Parikh%2C+Rohit&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F1385-7258%252877%252990067-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWell-quasi-ordering" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGasarch1998" class="citation cs2"><a href="/wiki/William_Gasarch" title="William Gasarch">Gasarch, W.</a> (1998), "A survey of recursive combinatorics", <i>Handbook of Recursive Mathematics, Vol. 2</i>, Stud. 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Order_theory" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Order_theory" title="Template:Order theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Order_theory" title="Template talk:Order theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Order_theory" 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algebra</a></li> <li><a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partial order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li> <li><a href="/wiki/Cantor%27s_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li> <li><a href="/wiki/Dilworth%27s_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li> <li><a href="/wiki/Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li> <li><a href="/wiki/Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li> <li><a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li> <li><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li> <li><a href="/wiki/Laver%27s_theorem" title="Laver's theorem">Laver's theorem</a></li> <li><a href="/wiki/Mirsky%27s_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li> <li><a href="/wiki/Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li> <li><a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<small><a href="/wiki/List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a></small>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li> <li><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a> <ul><li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li> <li><a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li> <li><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></li> <li><a href="/wiki/Covering_relation" title="Covering relation">Covering</a></li> <li><a href="/wiki/Dense_order" title="Dense order">Dense</a></li> <li><a href="/wiki/Directed_set" title="Directed set">Directed</a></li> <li>(<a href="/wiki/Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence</a></li> <li><a href="/wiki/Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li> <li><a href="/wiki/Idempotent_relation" title="Idempotent relation">Idempotent</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a> <ul><li><a href="/wiki/Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented</a></li> <li><a href="/wiki/Complete_lattice" title="Complete lattice">Complete</a></li> <li><a href="/wiki/Distributive_lattice" title="Distributive lattice">Distributive</a></li> <li><a href="/wiki/Join_and_meet" title="Join and meet">Join and meet</a></li></ul></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></li> <li><a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">Partial order</a> <ul><li><a href="/wiki/Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li> <li><a href="/wiki/Graded_poset" title="Graded poset">Graded</a></li> <li><a href="/wiki/Eulerian_poset" title="Eulerian poset">Eulerian</a></li> <li><a href="/wiki/Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li> <li><a href="/wiki/Prefix_order" title="Prefix order">Prefix order</a></li> <li><a href="/wiki/Preorder" title="Preorder">Preorder</a> <ul><li><a href="/wiki/Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Semiorder" title="Semiorder">Semiorder</a></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></li> <li><a href="/wiki/Total_relation" title="Total relation">Total</a></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></li> <li><a href="/wiki/Well-founded_relation" title="Well-founded relation">Well-founded</a></li> <li><a class="mw-selflink selflink">Well-quasi-ordering</a> (<a href="/wiki/Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li> <li>(<a href="/wiki/Prewellordering" title="Prewellordering">Pre</a>) <a href="/wiki/Well-order" title="Well-order">Well-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_of_relations" title="Composition of relations">Composition</a></li> <li><a href="/wiki/Converse_relation" title="Converse relation">Converse/Transpose</a></li> <li><a href="/wiki/Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li> <li><a href="/wiki/Linear_extension" title="Linear extension">Linear extension</a></li> <li><a href="/wiki/Product_order" title="Product order">Product order</a></li> <li><a href="/wiki/Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li> <li><a href="/wiki/Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li> <li><a href="/wiki/Star_product" title="Star product">Star product</a></li> <li><a href="/wiki/Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li> <li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="/wiki/Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li> <li><a href="/wiki/Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a> <ul><li><a href="/wiki/Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li> <li><a href="/wiki/Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li> <li><a href="/wiki/Order_topology" title="Order topology">Order topology</a></li> <li><a href="/wiki/Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a> <ul><li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach</a></li> <li><a href="/wiki/Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li> <li><a href="/wiki/Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li> <li><a href="/wiki/Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antichain" title="Antichain">Antichain</a></li> <li><a href="/wiki/Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li> <li><a href="/wiki/Cofinality" title="Cofinality">Cofinality</a></li> <li><a href="/wiki/Comparability" title="Comparability">Comparability</a> <ul><li><a href="/wiki/Comparability_graph" title="Comparability graph">Graph</a></li></ul></li> <li><a href="/wiki/Duality_(order_theory)" title="Duality (order theory)">Duality</a></li> <li><a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li> <li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li> <li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net</a> <ul><li><a href="/wiki/Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Order morphism</a> <ul><li><a href="/wiki/Order_embedding" title="Order embedding">Embedding</a></li> <li><a href="/wiki/Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li> <li><a href="/wiki/Order_type" title="Order type">Order type</a></li> <li><a href="/wiki/Ordered_field" title="Ordered field">Ordered field</a> <ul><li><a href="/wiki/Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li> <li><a href="/wiki/Ordered_vector_space" title="Ordered vector space">Ordered vector space</a> <ul><li><a href="/wiki/Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li> <li><a href="/wiki/Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz space</a></li></ul></li> <li><a href="/wiki/Partially_ordered_group" title="Partially ordered group">Partially ordered group</a> <ul><li><a href="/wiki/Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li> <li><a href="/wiki/Upper_set" title="Upper set">Upper set</a></li> <li><a href="/wiki/Young%27s_lattice" title="Young's lattice">Young's lattice</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Well-quasi-ordering&oldid=1225106966">https://en.wikipedia.org/w/index.php?title=Well-quasi-ordering&oldid=1225106966</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a 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