as defined by Nordahl & Scheiblich, if and only if it has a p-system F(S). In this case F(S) is the set of projections of S with respect to the operation ° defined by F(S). In an inverse semigroup the entire semilattice of idempotents is a p-system. Also, if a regular semigroup S has a p-system that is multiplicatively closed (i.e. subsemigroup), then S is an inverse semigroup. Thus, a p-system may be regarded as a generalization of the semilattice of idempotents of an inverse semigroup. *-regular semigroups (Drazin) A semigroup S with an involution * is called a *-regular semigroup (in the sense of Drazin) if for every x in S, x* is H-equivalent to some inverse of x, where H is the Green's relation H. This defining property can be formulated in several equivalent ways. Another is to say that every L-class contains a projection. An axiomatic definition is the condition that for every x in S there exists an element x′ such that , , , . Michael P. Drazin first proved that given x, the element x′ satisfying these axioms is unique. It is called the Moore–Penrose inverse of x. This agrees with the classical definition of the Moore–Penrose inverse of a square matrix. One motivation for studying these semigroups is that they allow generalizing the Moore–Penrose inverse's properties from and to more general sets. In the multiplicative semigroup Mn(C) of square matrices of order n, the map which assigns a matrix A to its Hermitian conjugate A* is an involution. The semigroup Mn(C) is a *-regular semigroup with this involution. The Moore–Penrose inverse of A in this *-regular semigroup is the classical Moore–Penrose inverse of A. Free semigroup with involution As with all varieties, the category of semigroups with involution admits free objects. The construction of a free semigroup (or monoid) with involution is based on that of a free semigroup (and respectively that of a free monoid). Moreover, the construction of a free group can easily be derived by refining the construction of a free monoid with involution. The generators of a free semigroup with involution are the elements of the union of two (equinumerous) disjoint sets in bijective correspondence: . (Here the notation emphasized that the union is actually a disjoint union.) In the case were the two sets are finite, their union Y is sometimes called an alphabet with involution or a