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In mathematics, an 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''∗ × ''X''∗ 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''+ 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''.〔Lothaire (2011) p. 444〕 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.〔 A semigroup is ''compact'' if every independent system of equations is finite.〔Lothaire (2011) p. 458〕 ==Examples== * A free monoid on a finite alphabet is compact.〔Lothaire (2011) p. 447〕 * A free monoid on a countable alphabet is compact.〔Lothaire (2011) p. 461〕 * A finitely generated free group is compact.〔Lothaire (2011) p. 462〕 * A trace monoid on a finite set of generators is compact.〔 * The bicyclic monoid is not compact.〔Lothaire (2011) p. 459〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compact semigroup」の詳細全文を読む スポンサード リンク
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