{{About|equations over semigroups|semigroups compact with respect to a topology|Topological semigroup|semigroups of compact operators|C0-semigroup#Compact semigroups}} In mathematics, a '''compact semigroup''' is a semigroup in which the sets of solutions to equations can be described by finite sets of equations. The term "compact" here does not refer to any topology on the semigroup.

Let ''S'' be a semigroup and ''X'' a finite set of letters. A system of equations is a subset ''E'' of the Cartesian product ''X''<sup>∗</sup> × ''X''<sup>∗</sup> of the free monoid (finite strings) over ''X'' with itself. The system ''E'' is satisfiable in ''S'' if there is a map ''f'' from ''X'' to ''S'', which extends to a semigroup morphism ''f'' from ''X''<sup>+</sup> to ''S'', such that for all (''u'',''v'') in ''E'' we have ''f''(''u'') = ''f''(''v'') in ''S''. Such an ''f'' is a ''solution'', or ''satisfying assignment'', for the system ''E''.<ref name=LotII444>Lothaire (2011) p.&nbsp;444</ref>

Two systems of equations are ''equivalent'' if they have the same set of satisfying assignments. A system of equations if ''independent'' if it is not equivalent to a proper subset of itself.<ref name=LotII444/> A semigroup is ''compact'' if every independent system of equations is finite.<ref name=LotII458>Lothaire (2011) p.&nbsp;458</ref>

==Examples== * A free monoid on a finite alphabet is compact.<ref name=LotII447>Lothaire (2011) p.&nbsp;&nbsp;447</ref> * A free monoid on a countable alphabet is compact.<ref name=LotII461>Lothaire (2011) p.&nbsp;461</ref> * A finitely generated free group is compact.<ref name=LotII462>Lothaire (2011) p.&nbsp;462</ref> * A trace monoid on a finite set of generators is compact.<ref name=LotII461/> * The bicyclic monoid is not compact.<ref name=LotII459>Lothaire (2011) p.&nbsp;459</ref>

==Properties== * The class of compact semigroups is closed under taking subsemigroups and finite direct products.<ref name=LotII460>Lothaire (2011) p.&nbsp;460</ref> * The class of compact semigroups is not closed under taking morphic images or infinite direct products.<ref name=LotII460/>

==Varieties== The class of compact semigroups does not form an equational variety. However, a variety of monoids has the property that all its members are compact if and only if all finitely generated members satisfy the maximal condition on congruences (any family of congruences, ordered by inclusion, has a maximal element).<ref name=LotII466>Lothaire (2011) p.&nbsp;466</ref>

==References== {{reflist}} * {{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}

Category:Semigroup theory Category:Formal languages Category:Combinatorics on words