Original sentence: S is a unital magma) and , then  is called a left inverse of  and  is called a right inverse of
Edited sentence: *-semigroups, in which the interaction axiom is (ab)° = b°a°.A class of semigroups important in semigroup theory are completely regular semigroups; these are I-semigroups in which one additionally has aa° = a°a; in other words every element has commuting pseudoinverse a°.A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (a°)°=a for all a in S; this endows S with a type <2,1> algebra.A semigroup endowed with such an operation is called a U-semigroup.Although it may seem that a° will be the inverse of a, this is not necessarily the case.Clearly a group is both an I-semigroup and a *-semigroup.I-semigroups, in which the interaction axiom is aa°a = a.In contrast, a class of *-semigroups, the *-regular semigroups, yield one of best known examples of a pseudoinverse, the Moore-Penrose inverse.In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation.In this case however the involution a* is not the pseudoinverse.Inverse semigroups are exactly those semigroups that are both I-semigroups and *-semigroups.Rather, the pseudoinverse of x is the unique element y such that xyx = y, yxy = y,   (xy)* = xy, (yx)* = yx.S is an unital magma) and , then  is called a left inverse of  and  is called a right inverse of.Such an operation is called an involution, and typically denoted by a *.There are few concrete examples of such semigroups however; most are completely simple semigroups.Two classes of U-semigroups have been studied: