Clearly a group is both an I-semigroup and a *-semigroup. 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°. There are few concrete examples of such semigroups however; most are completely simple semigroups. In contrast, a subclass of *-semigroups, the *-regular semigroups (in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the Moore–Penrose inverse. In this case however the involution a* is not the pseudoinverse. Rather, the pseudoinverse of x is the unique element y such that xyx = x, yxy = y, (xy)* = xy, (yx)* = yx. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the generalized inverse or Moore–Penrose inverse.
Clearly a group is both an I-semigroup and a *-semigroup. 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°. There are few concrete examples of such semigroups however; most are completely simple semigroups. In contrast, a subclass ,of *-semigroups, the *-regular semigroups (in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the Moore–Penrose inverse. In this case however the involution a* is not the pseudoinverse. Rather, the pseudoinverse of x is the unique element y such that xyx = x, yxy = y, (xy)* = xy, (yx)* = yx. Since *-regular semigroups generalize inverse semigroups the unique element defined this way in a *-regular semigroup is called the generalized inverse or Moore–Penrose inverse.