# semigroup with involution

An 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 semigroup with involution or $*$-semigroup [resp. a monoid with involution or $*$-monoid].

## References

• 1 J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1991.
 Title semigroup with involution Canonical name SemigroupWithInvolution Date of creation 2013-03-22 16:11:24 Last modified on 2013-03-22 16:11:24 Owner Mazzu (14365) Last modified by Mazzu (14365) Numerical id 7 Author Mazzu (14365) Entry type Definition Classification msc 20M10 Synonym *-semigroup Related topic ISemigroup Related topic ThueSystem Defines involution Defines semigroup with involution Defines monoid with involution