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". 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 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). 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. 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.
Formal definition Let S be a semigroup with its binary operation written multiplicatively. An involution in S is a unary operation * on S (or, a transformation * : S → S, x ↦ x*) satisfying the following conditions:
For all x in S, (x*)* = x. For all x, y in S we have (xy)* = y*x*. 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."
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 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*.
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