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data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">1990 book on mathematical order theory</div> <p> <i><b>Introduction to Lattices and Order</b></i> is a <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> textbook on <a href="/wiki/Order_theory" title="Order theory">order theory</a> by Brian A. Davey and <a href="/wiki/Hilary_Priestley" title="Hilary Priestley">Hilary Priestley</a>. It was published by the <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> in their Cambridge Mathematical Textbooks series in 1990,<sup id="cite_ref-blyth1_1-0" class="reference"><a href="#cite_note-blyth1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-davidow_2-0" class="reference"><a href="#cite_note-davidow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-niederle_3-0" class="reference"><a href="#cite_note-niederle-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> with a second edition in 2002.<sup id="cite_ref-blyth2_4-0" class="reference"><a href="#cite_note-blyth2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cohen_5-0" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-slavik_6-0" class="reference"><a href="#cite_note-slavik-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The second edition is significantly different in its topics and organization, and was revised to incorporate recent developments in the area, especially in its applications to <a href="/wiki/Computer_science" title="Computer science">computer science</a>.<sup id="cite_ref-blyth2_4-1" class="reference"><a href="#cite_note-blyth2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-slavik_6-1" class="reference"><a href="#cite_note-slavik-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The Basic Library List Committee of the <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a> has suggested its inclusion in undergraduate mathematics libraries.<sup id="cite_ref-maa_7-0" class="reference"><a href="#cite_note-maa-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Topics">Topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Introduction_to_Lattices_and_Order&action=edit&section=1" title="Edit section: Topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both editions of the book have 11 chapters; in the second book they are organized with the first four providing a general reference for mathematicians and computer scientists, and the remaining seven focusing on more specialized material for <a href="/wiki/Mathematical_logic" title="Mathematical logic">logicians</a>, <a href="/wiki/Topology" title="Topology">topologists</a>, and <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> theorists.<sup id="cite_ref-blyth2_4-2" class="reference"><a href="#cite_note-blyth2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The first chapter concerns <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a>, with a fundamental example given by the <a href="/wiki/Partial_function" title="Partial function">partial functions</a> ordered by the <a href="/wiki/Subset" title="Subset">subset</a> relation on their <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphs</a>, and covers fundamental concepts including <a href="/wiki/Greatest_element_and_least_element" title="Greatest element and least element">top and bottom elements</a> and <a href="/wiki/Upper_set" title="Upper set">upper and lower sets</a>. These ideas lead to the second chapter, on <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattices</a>, in which every two elements (or in <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattices</a>, every set) has a <a href="/wiki/Greatest_lower_bound" class="mw-redirect" title="Greatest lower bound">greatest lower bound</a> and a <a href="/wiki/Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</a>. This chapter includes the construction of a lattice from the lower sets of any partial order, and the <a href="/wiki/Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a> constructing a lattice from the <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> of an order-preserving functions on a complete lattice. Chapter three concerns <a href="/wiki/Formal_concept_analysis" title="Formal concept analysis">formal concept analysis</a>, its construction of "concept lattices" from collections of objects and their properties, with each lattice element representing both a set of objects and a set of properties held by those objects, and the universality of this construction in forming complete lattices. The fourth of the introductory chapters concerns special classes of lattices, including <a href="/wiki/Modular_lattice" title="Modular lattice">modular lattices</a>, <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattices</a>, and <a href="/wiki/Boolean_lattice" class="mw-redirect" title="Boolean lattice">Boolean lattices</a>.<sup id="cite_ref-cohen_5-1" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In the second part of the book, chapter 5 concerns the <a href="/wiki/Theorem" title="Theorem">theorem</a> that every finite Boolean lattice is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the lattice of subsets of a finite set, and (less trivially) <a href="/wiki/Birkhoff%27s_representation_theorem" title="Birkhoff's representation theorem">Birkhoff's representation theorem</a> according to which every finite distributive lattice is isomorphic to the lattice of lower sets of a finite partial order. Chapter 6 covers congruence relations on lattices. The topics in chapter 7 include closure operations and <a href="/wiki/Galois_connection" title="Galois connection">Galois connections</a> on partial orders, and the <a href="/wiki/Dedekind%E2%80%93MacNeille_completion" title="Dedekind–MacNeille completion">Dedekind–MacNeille completion</a> of a partial order into the smallest complete lattice containing it. The next two chapters concern <a href="/wiki/Complete_partial_order" title="Complete partial order">complete partial orders</a>, their fixed-point theorems, information systems, and their applications to <a href="/wiki/Denotational_semantics" title="Denotational semantics">denotational semantics</a>. Chapter 10 discusses order-theoretic equivalents of the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, including extensions of the <a href="/wiki/Representation_theorem" title="Representation theorem">representation theorems</a> from chapter 5 to infinite lattices, and the final chapter discusses the representation of lattices with <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>, including <a href="/wiki/Stone%27s_representation_theorem_for_Boolean_algebras" title="Stone's representation theorem for Boolean algebras">Stone's representation theorem for Boolean algebras</a> and the <a href="/wiki/Duality_theory_for_distributive_lattices" title="Duality theory for distributive lattices">duality theory for distributive lattices</a>.<sup id="cite_ref-cohen_5-2" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Two appendices provide background in <a href="/wiki/Topology" title="Topology">topology</a> needed for the final chapter, and an annotated bibliography.<sup id="cite_ref-slavik_6-2" class="reference"><a href="#cite_note-slavik-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Audience_and_reception">Audience and reception</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Introduction_to_Lattices_and_Order&action=edit&section=2" title="Edit section: Audience and reception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This book is aimed at beginning graduate students,<sup id="cite_ref-davidow_2-1" class="reference"><a href="#cite_note-davidow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> although it could also be used by advanced undergraduates.<sup id="cite_ref-slavik_6-3" class="reference"><a href="#cite_note-slavik-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Its many exercises make it suitable as a course textbook,<sup id="cite_ref-davidow_2-2" class="reference"><a href="#cite_note-davidow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-niederle_3-1" class="reference"><a href="#cite_note-niederle-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and serve both to fill in details from the exposition in the book, and to provide pointers to additional topics.<sup id="cite_ref-cohen_5-3" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Although some mathematical sophistication is required of its readers, the main prerequisites are <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a>, <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, and <a href="/wiki/Group_theory" title="Group theory">group theory</a>.<sup id="cite_ref-davidow_2-3" class="reference"><a href="#cite_note-davidow-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cohen_5-4" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Writing of the first edition, reviewer Josef Niederle calls it "an excellent textbook", "up-to-date and clear".<sup id="cite_ref-niederle_3-2" class="reference"><a href="#cite_note-niederle-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Similarly, Thomas S. Blyth praises the first edition as "a well-written, satisfying, informative, and stimulating account of applications that are of great interest",<sup id="cite_ref-blyth1_1-1" class="reference"><a href="#cite_note-blyth1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and in an updated review writes that the second edition is as good as the first.<sup id="cite_ref-blyth2_4-3" class="reference"><a href="#cite_note-blyth2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Likewise, although Jon Cohen has some quibbles with the ordering and selection of topics (particularly the inclusion of congruences at the expense of a <a href="/wiki/Category_theory" title="Category theory">category-theoretic</a> view of the subject), he concludes that the book is "a wonderful and accessible introduction to lattice theory, of equal interest to both computer scientists and mathematicians".<sup id="cite_ref-cohen_5-5" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Both Blyth and Cohen note the book's skilled use of <a href="/wiki/LaTeX" title="LaTeX">LaTeX</a> to create its diagrams, and its helpful descriptions of how the diagrams were made.<sup id="cite_ref-blyth1_1-2" class="reference"><a href="#cite_note-blyth1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cohen_5-6" class="reference"><a href="#cite_note-cohen-5"><span class="cite-bracket">[</span>5<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=Introduction_to_Lattices_and_Order&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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id="cite_note-blyth1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-blyth1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-blyth1_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-blyth1_1-2"><sup><i><b>c</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 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