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href="https://eu.wikipedia.org/wiki/Erlazio_hirutar" title="Erlazio hirutar – Basque" lang="eu" hreflang="eu" data-title="Erlazio hirutar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Relation_ternaire" title="Relation ternaire – French" lang="fr" hreflang="fr" data-title="Relation ternaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Relazione_ternaria" title="Relazione ternaria – Italian" lang="it" hreflang="it" data-title="Relazione ternaria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl 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.ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Ternary_relation" title="Special:EditPage/Ternary relation">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Ternary+relation%22">"Ternary relation"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Ternary+relation%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Ternary+relation%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Ternary+relation%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Ternary+relation%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Ternary+relation%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>ternary relation</b> or <b>triadic relation</b> is a <a href="/wiki/Finitary_relation" title="Finitary relation">finitary relation</a> in which the number of places in the relation is three. <a href="https://en.wiktionary.org/wiki/ternary" class="extiw" title="wikt:ternary">Ternary</a> relations may also be referred to as <b>3-adic</b>, <b>3-ary</b>, <b>3-dimensional</b>, or <b>3-place</b>. </p><p>Just as a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> is formally defined as a set of <i>pairs</i>, i.e. a subset of the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <span class="nowrap"><i>A</i> × <i>B</i></span> of some sets <i>A</i> and <i>B</i>, so a ternary relation is a set of triples, forming a subset of the Cartesian product <span class="nowrap"><i>A</i> × <i>B</i> × <i>C</i></span> of three sets <i>A</i>, <i>B</i> and <i>C</i>. </p><p>An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are <a href="/wiki/Collinearity" title="Collinearity">collinear</a>. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are <a href="/wiki/Incidence_(geometry)" title="Incidence (geometry)">incident</a> with) the line. </p> <meta property="mw:PageProp/toc" /> <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=Ternary_relation&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Binary_functions">Binary functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=2" title="Edit section: Binary functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a> and <a href="/wiki/Binary_function" title="Binary function">Binary function</a></div> <p>A function <span class="nowrap"><i>f</i> : <i>A</i> × <i>B</i> → <i>C</i></span> in two variables, mapping two values from sets <i>A</i> and <i>B</i>, respectively, to a value in <i>C</i> associates to every pair (<i>a</i>,<i>b</i>) in <span class="nowrap"><i>A</i> × <i>B</i></span> an element <i>f</i>(<i>a</i>, <i>b</i>) in <i>C</i>. Therefore, its graph consists of pairs of the form <span class="nowrap">((<i>a</i>, <i>b</i>), <i>f</i>(<i>a</i>, <i>b</i>))</span>. Such pairs in which the first element is itself a pair are often identified with triples. This makes the graph of <i>f</i> a ternary relation between <i>A</i>, <i>B</i> and <i>C</i>, consisting of all triples <span class="nowrap">(<i>a</i>, <i>b</i>, <i>f</i>(<i>a</i>, <i>b</i>))</span>, satisfying <span class="nowrap"><i>a</i> in <i>A</i></span>, <span class="nowrap"><i>b</i> in <i>B</i></span>, and <span class="nowrap"><i>f</i>(<i>a</i>, <i>b</i>) in <i>C</i>.</span> </p> <div class="mw-heading mw-heading3"><h3 id="Cyclic_orders">Cyclic orders</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=3" title="Edit section: Cyclic orders"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cyclic_order" title="Cyclic order">Cyclic order</a></div> <p>Given any set <i>A</i> whose elements are arranged on a circle, one can define a ternary relation <i>R</i> on <i>A</i>, i.e. a subset of <span class="nowrap"><i>A</i><sup>3</sup> = <i>A</i> × <i>A</i> × <i>A</i></span>, by stipulating that <span class="nowrap"><i>R</i>(<i>a</i>, <i>b</i>, <i>c</i>)</span> holds if and only if the elements <i>a</i>, <i>b</i> and <i>c</i> are pairwise different and when going from <i>a</i> to <i>c</i> in a clockwise direction one passes through <i>b</i>. For example, if <span class="nowrap"><i>A</i> = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }</span> represents the hours on a <a href="/wiki/Clock_face" title="Clock face">clock face</a>, then <span class="nowrap"><i>R</i>(8, 12, 4)</span> holds and <span class="nowrap"><i>R</i>(12, 8, 4)</span> does not hold. </p> <div class="mw-heading mw-heading3"><h3 id="Betweenness_relations">Betweenness relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=4" title="Edit section: Betweenness relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Betweenness_relation" class="mw-redirect" title="Betweenness relation">Betweenness relation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Ternary_relation&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">May 2011</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ternary_equivalence_relation">Ternary equivalence relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=5" title="Edit section: Ternary equivalence relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ternary_equivalence_relation" title="Ternary equivalence relation">Ternary equivalence relation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Ternary_relation&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">August 2020</span>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Congruence_relation">Congruence relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=6" title="Edit section: Congruence relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Congruence_modulo_m" class="mw-redirect" title="Congruence modulo m">Congruence modulo m</a></div> <p>The ordinary congruence of arithmetics </p> <dl><dd><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\equiv b{\pmod {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv b{\pmod {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad3dfcf6e055fce344fa935fbed43aacd9bcc613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.05ex; height:2.843ex;" alt="{\displaystyle a\equiv b{\pmod {m}}}"></span></dd></dl> <p>which holds for three integers <i>a</i>, <i>b</i>, and <i>m</i> if and only if <i>m</i> divides <span class="nowrap"><i>a</i> − <i>b</i></span>, formally may be considered as a ternary relation. However, usually, this instead is considered as a family of <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a> between the <i>a</i> and the <i>b</i>, indexed by the <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulus</a> <i>m</i>. For each fixed <i>m</i>, indeed this binary relation has some natural properties, like being an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>; while the combined ternary relation in general is not studied as one relation. </p> <div class="mw-heading mw-heading3"><h3 id="Typing_relation">Typing relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=7" title="Edit section: Typing relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Simply_typed_lambda_calculus#Typing_rules" title="Simply typed lambda calculus">Simply typed lambda calculus § Typing rules</a></div> <p>A <i>typing relation</i> <span class="nowrap">Γ ⊢ <i>e</i>:<i>σ</i></span> indicates that <i>e</i> is a term of type <i>σ</i> in context Γ, and is thus a ternary relation between contexts, terms and types. </p> <div class="mw-heading mw-heading3"><h3 id="Schröder_rules"><span id="Schr.C3.B6der_rules"></span>Schröder rules</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=8" title="Edit section: Schröder rules"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given <a href="/wiki/Homogeneous_relation" title="Homogeneous relation">homogeneous relations</a> <i>A</i>, <i>B</i>, and <i>C</i> on a set, a ternary relation <span class="nowrap">(<i>A</i>, <i>B</i>, <i>C</i>)</span> can be defined using <a href="/wiki/Composition_of_relations" title="Composition of relations">composition of relations</a> <i>AB</i> and <a href="/wiki/Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">inclusion</a> <span class="nowrap"><i>AB</i> ⊆ <i>C</i></span>. Within the <a href="/wiki/Calculus_of_relations" class="mw-redirect" title="Calculus of relations">calculus of relations</a> each relation <i>A</i> has a <a href="/wiki/Converse_relation" title="Converse relation">converse relation</a> <i>A</i><sup>T</sup> and a complement relation <span style="text-decoration:overline;"><i>A</i></span>. Using these <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutions</a>, <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> and <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a> showed that <span class="nowrap">(<i>A</i>, <i>B</i>, <i>C</i>)</span> is equivalent to <span class="nowrap">(<span style="text-decoration:overline;"><i>C</i></span>, <i>B</i><sup>T</sup>, <span style="text-decoration:overline;"><i>A</i></span>)</span> and also equivalent to <span class="nowrap">(<i>A</i><sup>T</sup>, <span style="text-decoration:overline;"><i>C</i></span>, <span style="text-decoration:overline;"><i>B</i></span>)</span>. The mutual equivalences of these forms, constructed from the ternary relation <span class="nowrap">(<i>A</i>, <i>B</i>, <i>C</i>),</span> are called the <a href="/wiki/Composition_of_relations#Schröder_rules" title="Composition of relations">Schröder rules</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </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=Ternary_relation&action=edit&section=9" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFSchmidtStröhlein1993" class="citation cs2"><a href="/wiki/Gunther_Schmidt" title="Gunther Schmidt">Schmidt, Gunther</a>; Ströhlein, Thomas (1993), <i>Relations and Graphs</i>, <a href="/wiki/Springer_books" class="mw-redirect" title="Springer books">Springer books</a>, pp. <span class="nowrap">15–</span>19</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relations+and+Graphs&rft.pages=%3Cspan+class%3D%22nowrap%22%3E15-%3C%2Fspan%3E19&rft.pub=Springer+books&rft.date=1993&rft.aulast=Schmidt&rft.aufirst=Gunther&rft.au=Str%C3%B6hlein%2C+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ternary_relation&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMyers1997" class="citation cs2">Myers, Dale (1997), "An interpretive isomorphism between binary and ternary relations", in Mycielski, Jan; Rozenberg, Grzegorz; Salomaa, Arto (eds.), <i>Structures in Logic and Computer Science</i>, Lecture Notes in Computer Science, vol. 1261, Springer, pp. <span class="nowrap">84–</span>105, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-63246-8_6">10.1007/3-540-63246-8_6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-63246-8" title="Special:BookSources/3-540-63246-8"><bdi>3-540-63246-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+interpretive+isomorphism+between+binary+and+ternary+relations&rft.btitle=Structures+in+Logic+and+Computer+Science&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E84-%3C%2Fspan%3E105&rft.pub=Springer&rft.date=1997&rft_id=info%3Adoi%2F10.1007%2F3-540-63246-8_6&rft.isbn=3-540-63246-8&rft.aulast=Myers&rft.aufirst=Dale&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNovák1996" class="citation cs2">Novák, Vítězslav (1996), "Ternary structures and partial semigroups", <i>Czechoslovak Mathematical Journal</i>, <b>46</b> (1): <span class="nowrap">111–</span>120, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F127275">10338.dmlcz/127275</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Czechoslovak+Mathematical+Journal&rft.atitle=Ternary+structures+and+partial+semigroups&rft.volume=46&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E111-%3C%2Fspan%3E120&rft.date=1996&rft_id=info%3Ahdl%2F10338.dmlcz%2F127275&rft.aulast=Nov%C3%A1k&rft.aufirst=V%C3%ADt%C4%9Bzslav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNovákNovotný1989" class="citation cs2">Novák, Vítězslav; Novotný, Miroslav (1989), "Transitive ternary relations and quasiorderings", <i>Archivum Mathematicum</i>, <b>25</b> (<span class="nowrap">1–</span>2): <span class="nowrap">5–</span>12, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F107333">10338.dmlcz/107333</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archivum+Mathematicum&rft.atitle=Transitive+ternary+relations+and+quasiorderings&rft.volume=25&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E5-%3C%2Fspan%3E12&rft.date=1989&rft_id=info%3Ahdl%2F10338.dmlcz%2F107333&rft.aulast=Nov%C3%A1k&rft.aufirst=V%C3%ADt%C4%9Bzslav&rft.au=Novotn%C3%BD%2C+Miroslav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNovákNovotný1992" class="citation cs2">Novák, Vítězslav; Novotný, Miroslav (1992), "Binary and ternary relations", <i>Mathematica Bohemica</i>, <b>117</b> (3): <span class="nowrap">283–</span>292, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F126278">10338.dmlcz/126278</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematica+Bohemica&rft.atitle=Binary+and+ternary+relations&rft.volume=117&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E283-%3C%2Fspan%3E292&rft.date=1992&rft_id=info%3Ahdl%2F10338.dmlcz%2F126278&rft.aulast=Nov%C3%A1k&rft.aufirst=V%C3%ADt%C4%9Bzslav&rft.au=Novotn%C3%BD%2C+Miroslav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNovotný1991" class="citation cs2">Novotný, Miroslav (1991), "Ternary structures and groupoids", <i>Czechoslovak Mathematical Journal</i>, <b>41</b> (1): <span class="nowrap">90–</span>98, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F102437">10338.dmlcz/102437</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Czechoslovak+Mathematical+Journal&rft.atitle=Ternary+structures+and+groupoids&rft.volume=41&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E90-%3C%2Fspan%3E98&rft.date=1991&rft_id=info%3Ahdl%2F10338.dmlcz%2F102437&rft.aulast=Novotn%C3%BD&rft.aufirst=Miroslav&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŠlapal1993" class="citation cs2">Šlapal, Josef (1993), "Relations and topologies", <i>Czechoslovak Mathematical Journal</i>, <b>43</b> (1): <span class="nowrap">141–</span>150, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F128381">10338.dmlcz/128381</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Czechoslovak+Mathematical+Journal&rft.atitle=Relations+and+topologies&rft.volume=43&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E141-%3C%2Fspan%3E150&rft.date=1993&rft_id=info%3Ahdl%2F10338.dmlcz%2F128381&rft.aulast=%C5%A0lapal&rft.aufirst=Josef&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATernary+relation" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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