I-semigroup

An $I$-semigroup [resp. $I$-monoid] is a semigroup $S$ [resp. a monoid $M$] with a unary operation $x\mapsto x^{-1}$ defined on $S$ [resp. on $M$] such that for each $x,y\in S$ [resp. for each $x,y\in M$]

$${(x^{-1})}^{-1}=x,x=x{x}^{-1}x.$$

Notice that

$$x^{-1}x{x}^{-1}=x^{-1}{(x^{-1})}^{-1}{x}^{-1}=x^{-1},$$

so $x^{-1}$ is an inverse of $x$.

The class of $I$-semigroups [resp. $I$-monoids] strictly contains the class of inverse semigroups [resp. inverse monoids]. In fact, the class of inverse semigroups [resp. inverse monoids] is precisely the class of $I$-semigroups with involution [resp. $I$-monoids with involution], i.e. the class of $I$-semigroups [resp. $I$-monoids] in which the unary operation ${}^{-1}$ is also an involution.