Finite lattice representation problem

<h2 id="background">Background</h2>
<p>A <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a> is called <a href="/facts/Algebraic_lattice/1aLoXrfR">algebraic</a> if it is <a href="/facts/Complete_lattice/pQbphWeR">complete</a> and <a href="/facts/Compact_element/1aLoXrfR">compactly generated</a>.  In 1963, Grätzer and Schmidt proved that every algebraic lattice is isomorphic to the <a href="/facts/Congruence_lattice/8uQzYkm7">congruence lattice</a> of some <a href="/facts/Universal_algebra/uMJAL6ER">algebra</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  Thus there is essentially no restriction on the shape of a congruence lattice of an algebra. The finite lattice representation problem asks whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a <i>finite</i> algebra?
</p><p>In 1980, Pálfy and Pudlák proved that this problem is equivalent to the problem of deciding whether every finite lattice occurs as an interval in the <a href="/facts/Subgroup_lattice/hdIChdWA">subgroup lattice</a> of a finite <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  For an overview of the group theoretic approach to the problem, see Pálfy (1993)<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> and Pálfy (2001).<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>This problem should not be confused with the <a href="/facts/Congruence_lattice_problem/KAHqg197">congruence lattice problem</a>.
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<h2 id="significance">Significance</h2>
<p>This is among the oldest unsolved problems in <a href="/facts/Universal_algebra/uMJAL6ER">universal algebra</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Until it is answered, the theory of finite algebras is incomplete since, given a finite algebra, it is unknown whether there are, <i>a priori</i>, any restrictions on the shape of its congruence lattice.
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<h2 id="further-reading">Further reading</h2>
<ul><li>DeMeo, William J (2012). "Congruence lattices of finite algebras". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1204.4305">1204.4305</a> [<a href="https://arxiv.org/archive/math.GR">math.GR</a>].</li>
<li>Dowling, T A (February 1973). <a href="https://doi.org/10.1016%2FS0095-8956%2873%2980007-3">"A class of geometric lattices based on finite groups"</a>. <i>Journal of Combinatorial Theory, Series B</i>. 14 (1): 61–86. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0095-8956%2873%2980007-3">10.1016/S0095-8956(73)80007-3</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://garden.irmacs.sfu.ca/?q=op/finite_congruence_lattice_problem">Finite Congruence Lattice Problem</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>G. Grätzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Pálfy and Pudlák. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 11(1), 22–27 (1980). DOI <a href="https://dx.doi.org/10.1007/BF02483080" target="_blank">https://dx.doi.org/10.1007/BF02483080</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Péter Pál Pálfy. Intervals in subgroup lattices of finite groups. In Groups ’93 Galway/St. Andrews, Vol. 2, volume 212 of London Math. Soc. Lecture Note Ser., pages 482–494. Cambridge Univ. Press, Cambridge, 1995. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Péter Pál Pálfy. Groups and lattices. In Groups St. Andrews 2001 in Oxford. Vol. II, volume 305 of London Math. Soc. Lecture Note Ser., pages 428–454, Cambridge, 2003. Cambridge Univ. Press. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Joel Berman. Congruence lattices of finite universal algebras. PhD thesis, University of Washington, 1970, ProQuest 302550051. <a href="/wiki/ProQuest" target="_blank">/wiki/ProQuest</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Bjarni Jónsson. Topics in universal algebra. Lecture Notes in Mathematics, Vol. 250. Springer Verlag, Berlin, 1972. <a href="https://books.google.com/books?id=2S57CwAAQBAJ&q=%22Finite+lattice+representation+problem%22&pg=PA1" target="_blank">https://books.google.com/books?id=2S57CwAAQBAJ&q=%22Finite+lattice+representation+problem%22&pg=PA1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Ralph McKenzie. Finite forbidden lattices. In: Universal algebra and lattice theory
(Puebla, 1982), Lecture Notes in Math., vol. 1004, pp. 176–205. Springer, Berlin
(1983). DOI <a href="https://dx.doi.org/10.1007/BFb0063438" target="_blank">https://dx.doi.org/10.1007/BFb0063438</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
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Finite lattice representation problem open-in-new

Finite lattice representation problem