philosophy of this axiom, H.S.M. Coxeter remarked that it "becomes clear when we think of [x] and [y] as the operations of putting on our socks and shoes, respectively." Examples If S is a commutative semigroup then the identity map of S is an involution. If S is a group then the inversion map * : S → S defined by x* = x−1 is an involution. Furthermore, on an abelian group both this map and the one from the previous example are involutions satisfying the axioms of semigroup with involution. If S is an inverse semigroup then the inversion map is an involution which leaves the idempotents invariant. As noted in the previous example, the inversion map is not necessarily the only map with this property in an inverse semigroup. There may well be other involutions that leave all idempotents invariant; for example the identity map on a commutative regular, hence inverse, semigroup, in particular, an abelian group. A regular semigroup is an inverse semigroup if and only if it admits an involution under which each idempotent is an invariant. Underlying every C*-algebra is a *-semigroup. An important instance is the algebra Mn(C) of n-by-n matrices over C, with the conjugate transpose as involution. If X is a set, the set of all binary relations on X is a *-semigroup with the * given by the converse relation, and the multiplication given by the usual composition of relations. This is an example of a *-semigroup which is not a regular semigroup. If X is a set, then the set of all finite sequences (or strings) of members of X forms a free monoid under the operation of concatenation of sequences, with sequence reversal as an involution. A rectangular band on a Cartesian product of a set A with itself, i.e. with elements from A × A, with the semigroup product defined as (a, b)(c, d) = (a, d), with the involution being the order reversal of the elements of a pair (a, b)* = (b, a). This semigroup is also a regular semigroup, as all bands are. Basic concepts and properties An element x of a semigroup with involution is sometimes called hermitian (by analogy with a Hermitian matrix) when it is left invariant by the involution, meaning x* = x. Elements of the form xx* or x*x are always hermitian, and so are all powers of a hermitian