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 $x,y\in M$ $$(x^{-1})^{-1}=x,\ \ \ x=xx^{-1}x.$$

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

Bibliography

1

J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1991.