However, there are significant natural examples of semigroups with involution that are not groups.
An example from linear algebra is a set of real-valued n-by-n square matrices with the matrix-transpose as the involution.
The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB)T = BTAT, which has the same form of interaction with multiplication as taking inverses has in the general linear group (which is a subgroup of the full linear monoid).
Another example, coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a string.
Semigroups with involution appeared explicitly named in a 1953 paper of Viktor Wagner (in Russian) as result of his attempt to bridge the theory of semigroups with that of semiheaps.
Semigroups that satisfy only the first of these axioms belong to the larger class of U-semigroups.
Coxeter remarked that it "becomes clear when we think of [x] and [y] as the operations of putting on our socks and shoes, respectively.
[7] Certain basic concepts may be defined on *-semigroups in a way that parallels the notions stemming from a regular element in a semigroup.
[9] Equivalently, s ≤ t if and only if s = et and e = ett* for some projection e.[9] In a *-semigroup, PI(S) is an ordered groupoid with the partial product given by s⋅t = st if s*s = tt*.
[13] As mentioned in the previous examples, inverse semigroups are a subclass of *-semigroups.
In 1963, Boris M. Schein showed that the following two axioms provide an analogous characterization of inverse semigroups as a subvariety of *-semigroups: The first of these looks like the definition of a regular element, but is actually in terms of the involution.
Likewise, the second axiom appears to be describing the commutation of two idempotents.
It is known however that regular semigroups do not form a variety because their class does not contain free objects (a result established by D. B. McAlister in 1968).
This line of reasoning motivated Nordahl and Scheiblich to begin in 1977 the study of the (variety of) *-semigroups that satisfy only the first these two axioms; because of the similarity in form with the property defining regular semigroups, they named this variety regular *-semigroups.
The rectangular band from Example 7 is a regular *-semigroup that is not an inverse semigroup.
Semigroups that satisfy only x** = x = xx*x (but not necessarily the antidistributivity of * over multiplication) have also been studied under the name of I-semigroups.
He defined a P-system F(S) as subset of the idempotents of S, denoted as usual by E(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.
Thus, a p-system may be regarded as a generalization of the semilattice of idempotents of an inverse semigroup.
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.
Michael P. Drazin first proved that given x, the element x′ satisfying these axioms is unique.
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.
As with all varieties, the category of semigroups with involution admits free objects.
[15] The generators of a free semigroup with involution are the elements of the union of two (equinumerous) disjoint sets in bijective correspondence:
in this choice of terminology is explained below in terms of the universal property of the construction.)
is the inclusion map and composition of functions is taken in diagram order.
The additional ingredient needed is to define a notion of reduced word and a rewriting rule for producing such words simply by deleting any adjacent pairs of letter of the form
[17][20] (This latter choice of terminology conflicts however with the use of "involutive" to denote any semigroup with involution—a practice also encountered in the literature.
If H is a Hilbert space, then the multiplicative semigroup of all bounded operators on H is a Baer *-semigroup.