# 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 $x,y\in M$]

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

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.

## References

• 1 J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1991.
Title I-semigroup Isemigroup 2013-03-22 16:11:27 2013-03-22 16:11:27 Mazzu (14365) Mazzu (14365) 5 Mazzu (14365) Definition msc 20M10 SemigroupWithInvolution I-semigroup I-monoid