element. As noted in the examples section, a semigroup S is an inverse semigroup if and only if S is a regular semigroup and admits an involution such that every idempotent is hermitian. Certain basic concepts may be defined on *-semigroups in a way that parallels the notions stemming from a regular element in a semigroup. A partial isometry is an element s such that ss*s = s; the set of partial isometries of a semigroup S is usually abbreviated PI(S). A projection is an idempotent element e that is also hermitian, meaning that ee = e and e* = e. Every projection is a partial isometry, and for every partial isometry s, s*s and ss* are projections. If e and f are projections, then e = ef if and only if e = fe. Partial isometries can be partially ordered by s ≤ t defined as holding whenever s = ss*t and ss* = ss*tt*. Equivalently, s ≤ t if and only if s = et and e = ett* for some projection e. In a *-semigroup, PI(S) is an ordered groupoid with the partial product given by s⋅t = st if s*s = tt*. Examples In terms of examples for these notions, in the *-semigroup of binary relations on a set, the partial isometries are the relations that are difunctional. The projections in this *-semigroup are the partial equivalence relations. The partial isometries in a C*-algebra are exactly those defined in this section. In the case of Mn(C) more can be said. If E and F are projections, then E ≤ F if and only if imE ⊆ imF. For any two projection, if E ∩ F = V, then the unique projection J with image V and kernel the orthogonal complement of V is the meet of E and F. Since projections form a meet-semilattice, the partial isometries on Mn(C) form an inverse semigroup with the product . Another simple example of these notions appears in the next section. Notions of regularity There are two related, but not identical notions of regularity in *-semigroups. They were introduced nearly simultaneously by Nordahl & Scheiblich (1978) and respectively Drazin (1979). Regular *-semigroups (Nordahl & Scheiblich) As mentioned in the previous examples, inverse semigroups are a subclass of *-semigroups. It is also textbook knowledge that an inverse semigroup can be characterized as a regular semigroup in which any two idempotents