Semigroup with involution

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:

1. For all x in S, (x*)* = x.
2. 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.[2] 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."[3]

1. If S is a commutative semigroup then the identity map of S is an involution.
2. 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.[4]
3. 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.[5]
4. Underlying every C*-algebra is a *-semigroup. An important instance is the algebra M_n(C) of n-by-n matrices over C, with the conjugate transpose as involution.
5. 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.
6. 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.
7. 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.[6]

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.[7]

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).[8] 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.[9]

Partial isometries can be partially ordered by s ≤ t defined as holding whenever s = ss*t and ss* = ss*tt*.[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*.[10]

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).[13]

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.[15]

The generators of a free semigroup with involution are the elements of the union of two (equinumerous) disjoint sets in bijective correspondence: ${\displaystyle Y=X\sqcup X^{\dagger }}$. (Here the notation ${\displaystyle \sqcup \,}$ emphasized that the union is actually a disjoint union.) In the case were the two sets are finite, their union Y is sometimes called an alphabet with involution[16] or a symmetric alphabet.[17] Let ${\displaystyle \theta :X\rightarrow X^{\dagger }}$ be a bijection; ${\displaystyle \theta }$ is naturally extended to a bijection ${\displaystyle {}\dagger :Y\to Y}$ essentially by taking the disjoint union of ${\displaystyle \theta }$ (as a set) with its inverse, or in piecewise notation:[18]

${\displaystyle y^{\dagger }={\begin{cases}\theta (y)&{\text{if }}y\in X\\\theta ^{-1}(y)&{\text{if }}y\in X^{\dagger }\end{cases}}}$

Now construct ${\displaystyle Y^{+}\,}$ as the free semigroup on ${\displaystyle Y\,}$ in the usual way with the binary (semigroup) operation on ${\displaystyle Y^{+}\,}$ being concatenation:

${\displaystyle w=w_{1}w_{2}\cdots w_{k}\in Y^{+}}$ for some letters ${\displaystyle w_{i}\in Y.}$

The bijection ${\displaystyle \dagger }$ on ${\displaystyle Y}$ is then extended as a bijection ${\displaystyle {}^{\dagger }:Y^{+}\rightarrow Y^{+}}$ defined as the string reversal of the elements of ${\displaystyle Y^{+}\,}$ that consist of more than one letter:[16][18]

${\displaystyle w^{\dagger }=w_{k}^{\dagger }w_{k-1}^{\dagger }\cdots w_{2}^{\dagger }w_{1}^{\dagger }.}$

This map is an involution on the semigroup ${\displaystyle Y^{+}\,}$. Thus, the semigroup ${\displaystyle (X\sqcup X^{\dagger })^{+}}$ with the map ${\displaystyle {}^{\dagger }\,}$ is a semigroup with involution, called a free semigroup with involution on X.[19] (The irrelevance of the concrete identity of ${\displaystyle X^{\dagger }}$ and of the bijection ${\displaystyle \theta }$ in this choice of terminology is explained below in terms of the universal property of the construction.) Note that unlike in Example 6, the involution of every letter is a distinct element in an alphabet with involution, and consequently the same observation extends to a free semigroup with involution.

If in the above construction instead of ${\displaystyle Y^{+}\,}$ we use the free monoid ${\displaystyle Y^{*}=Y^{+}\cup \{\varepsilon \}}$, which is just the free semigroup extended with the empty word ${\displaystyle \varepsilon \,}$ (which is the identity element of the monoid ${\displaystyle Y^{*}\,}$), and suitably extend the involution with ${\displaystyle \varepsilon ^{\dagger }=\varepsilon }$, we obtain a free monoid with involution.[18]

The construction above is actually the only way to extend a given map ${\displaystyle \theta \,}$ from ${\displaystyle X\,}$ to ${\displaystyle X^{\dagger }\,}$, to an involution on ${\displaystyle Y^{+}\,}$ (and likewise on ${\displaystyle Y^{*}\,}$). The qualifier "free" for these constructions is justified in the usual sense that they are universal constructions. In the case of the free semigroup with involution, given an arbitrary semigroup with involution ${\displaystyle S\,}$ and a map ${\displaystyle \Phi :X\rightarrow S}$, then a semigroup homomorphism ${\displaystyle {\overline {\Phi }}:(X\sqcup X^{\dagger })^{+}\rightarrow S}$ exists such that ${\displaystyle \Phi =\iota \circ {\overline {\Phi }}}$, where ${\displaystyle \iota :X\rightarrow (X\sqcup X^{\dagger })^{+}}$ is the inclusion map and composition of functions is taken in diagram order.[19] The construction of ${\displaystyle (X\sqcup X^{\dagger })^{+}}$ as a semigroup with involution is unique up to isomorphism. An analogous argument holds for the free monoid with involution in terms of monoid homomorphisms and the uniqueness up to isomorphism of the construction of ${\displaystyle (X\sqcup X^{\dagger })^{*}}$ as a monoid with involution.

The construction of a free group is not very far off from that of a free monoid with involution. 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 ${\displaystyle xx^{\dagger }}$ or ${\displaystyle x^{\dagger }x}$. It can be shown than the order of rewriting (deleting) such pairs does not matter, i.e. any order of deletions produces the same result.[15] (Otherwise put, these rules define a confluent rewriting system.) Equivalently, a free group is constructed from a free monoid with involution by taking the quotient of the latter by the congruence ${\displaystyle \{(yy^{\dagger },\varepsilon ):y\in Y\}}$, which is sometimes called the Dyck congruence—in a certain sense it generalizes Dyck language to multiple kinds of "parentheses" However simplification in the Dyck congruence takes place regardless of order. For example, if ")" is the inverse of "(", then ${\displaystyle ()=)(=\varepsilon }$; the one-sided congruence that appears in the Dyck language proper ${\displaystyle \{(xx^{\dagger },\varepsilon ):x\in X\}}$, which instantiates only to ${\displaystyle ()=\varepsilon }$ is (perhaps confusingly) called the Shamir congruence. The quotient of a free monoid with involution by the Shamir congruence is not a group, but a monoid ; nevertheless it has been called the free half group by its first discoverer—Eli Shamir—although more recently it has been called the involutive monoid generated by X.[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.[21][22])

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

{ y ∈ S | xy = 0 } = eS.[22]

The projection e is in fact uniquely determined by x.[22]

More recently, Baer *-semigroups have been also called Foulis semigroups, after David James Foulis who studied them in depth.[23][24]

• Dagger category (aka category with involution) — generalizes the notion
• *-algebra
• Special classes of semigroups

• Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". World Scientific ISBN 981-02-3316-7
• D J Foulis (1958). Involution Semigroups, PhD Thesis, Tulane University, New Orleans, LA. Publications of D.J. Foulis (Accessed on 5 May 2009)
• W.D. Munn, Special Involutions, in A.H. Clifford, K.H. Hofmann, M.W. Mislove, Semigroup theory and its applications: proceedings of the 1994 conference commemorating the work of Alfred H. Clifford, Cambridge University Press, 1996, ISBN 0521576695. This is a recent survey article on semigroup with (special) involution
• Drazin, M.P., Regular semigroups with involution, Proc. Symp. on Regular Semigroups (DeKalb, 1979), 29–46
• Nordahl, T.E., and H.E. Scheiblich, Regular * Semigroups, Semigroup Forum, 16(1978), 369–377.
• Miyuki Yamada, P-systems in regular semigroups, Semigroup Forum, 24(1), December 1982, pp. 173–187
• S. Crvenkovic and Igor Dolinka, "Varieties of involution semigroups and involution semirings: a survey", Bulletin of the Society of Mathematicians of Banja Luka Vol. 9 (2002), 7–47.
• This article incorporates material from Free semigroup with involution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.