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 $\mathrm{*}$-semigroup [resp. a monoid with involution or $\mathrm{*}$-monoid].