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Compact semigroup
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<h2 id="examples">Examples</h2> <ul><li>A free monoid on a finite alphabet is compact.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li>A free monoid on a countable alphabet is compact.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>A finitely generated <a href="/facts/Free_group/zPo2NIGb">free group</a> is compact.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li>A <a href="/facts/Trace_monoid/Y7ShnaSE">trace monoid</a> on a finite set of generators is compact.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li> <li>The <a href="/facts/Bicyclic_monoid/GehG9bj5">bicyclic monoid</a> is not compact.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li></ul> <h2 id="properties">Properties</h2> <ul><li>The class of compact semigroups is closed under taking subsemigroups and finite direct products.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li> <li>The class of compact semigroups is not closed under taking morphic images or infinite direct products.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li></ul> <h2 id="varieties">Varieties</h2> <p>The class of compact semigroups does not form an <a href="/facts/Equational_variety/vXWYFyDI">equational variety</a>. However, a variety of monoids has the property that all its members are compact if and only if all finitely generated members satisfy the <a href="/facts/Maximal_condition_on_congruences/rgSdk2kt">maximal condition on congruences</a> (any family of congruences, ordered by inclusion, has a maximal element).<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <ul><li><a href="/facts/M._Lothaire/pC43ftpJ">Lothaire, M.</a> (2011). <i>Algebraic combinatorics on words</i>. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-18071-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1221.68183">1221.68183</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lothaire (2011) p. 444 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lothaire (2011) p. 444 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Lothaire (2011) p. 458 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lothaire (2011) p. 447 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Lothaire (2011) p. 461 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lothaire (2011) p. 462 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Lothaire (2011) p. 461 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Lothaire (2011) p. 459 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Lothaire (2011) p. 460 <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Lothaire (2011) p. 460 <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Lothaire (2011) p. 466 <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> </ol>