semigroup with involutionSemigroupWithInvolution2006-08-23 07:01:512006-08-24 11:42:03Definitioninvolutionsemigroup with involutionmonoid with involutionsemigroup% this is the default PlanetMath preamble. as your knowledge
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An \emph{involution} on a semigroup $S$ [on a monoid $M$] is a unary operation $x\mapsto x^*$ defined on $S$ [resp. on $M$] such that for each $x,y\in S$ [resp. for each $x,y\in M$] $$(x^*)^*=x,\ \ \ (xy)^*=y^*x^*.$$
With this added internal operation, the semigroup $S$ [resp. the monoid $M$] becomes a \emph{semigroup with involution} or \emph{$*$-semigroup} [resp. a \emph{monoid with involution} or \emph{$*$-monoid}].
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\bibitem{b:howie} J.M. Howie, \emph{Fundamentals of Semigroup Theory}, Oxford University Press, Oxford, 1991.
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