Semigroup with involution
In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.
An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). 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). However, for an arbitrary matrix, AAT does not equal the identity element (namely the diagonal matrix). 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. A third example, from basic set theory, is the set of all binary relations between a set and itself, with the involution being the converse relation, and the multiplication given by the usual composition of relations.
The semigroup S with the involution * is called a semigroup with involution.
Semigroups that satisfy only the first of these axioms belong to the larger class of U-semigroups.
In some applications, the second of these axioms has been called antidistributive. Regarding the natural 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."
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 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*.
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.
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).
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 commute. 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.
It is a simple calculation to establish that a regular *-semigroup is also a regular semigroup because x* turns out to be an inverse of x. The rectangular band from Example 7 is a regular *-semigroup that is not an inverse semigroup. It is also easy to verify that in a regular *-semigroup the product of any two projections is an idempotent. In the aforementioned rectangular band example, the projections are elements of the form (x, x) and [like all elements of a band] are idempotent. However, two different projections in this band need not commute, nor is their product necessarily a projection since (a, a)(b, b) = (a, b).
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.
The problem of characterizing when a regular semigroup is a regular *-semigroup (in the sense of Nordahl & Scheiblich) was addressed by M. Yamada (1982). He defined a P-system F(S) as subset of the idempotents of S, denoted as usual by E(S). Using the usual notation V(a) for the inverses of a, F(S) needs to satisfy the following axioms:
A regular semigroup S is a *-regular semigroup, 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.
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 x′xx′ = x′, xx′x = x, (xx′)* = xx′, (x′x)* = x′x. 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.
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.
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.
A Baer *-semigroup is a *-semigroup with (two-sided) zero in which the right annihilator of every element coincides with the right ideal of some projection; this property is expressed formally as: for all x ∈ S there exists a projection e such that