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In [[mathematics]], particularly in [[abstract algebra]], a '''semigroup with involution''' or a '''*-semigroup''' is a [[semigroup]] equipped with an [[Involution (mathematics)|involutive]] [[anti-automorphism]], which—roughly speaking—brings it closer to a [[group (mathematics)|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 [[Matrix multiplication|multiplicative]] [[monoid]] of [[Real number|real]] square [[Matrix (mathematics)|matrices]] of order ''n'' (called the [[full linear monoid]]). The [[Map (mathematics)|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'')<sup>''T''</sup> = ''B''<sup>''T''</sup>''A''<sup>''T''</sup> 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 ''AA''<sup>''T''</sup> 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 (computer science)|alphabet]]), with string [[concatenation]] as the binary operation, and the involution being the map which [[
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 [[semiheap]]s.<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=265}}</ref>
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==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.
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==Examples==
# If ''S'' is a [[commutative]] semigroup then the [[identity function|identity map]] of S is an involution.
# If ''S'' is a [[group (mathematics)|group]] then the inversion map * : ''S'' → ''S'' defined by ''x''* = ''x''<sup>−1</sup> 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.<ref name="BergChristensen2012">{{cite book|author1=C. van den Berg|author2=J. P. R. Christensen|author3=P. Ressel|title=Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-1128-0|pages=87–88}}</ref>
# If ''S'' is an [[inverse semigroup]] then the inversion map is an involution which leaves the [[idempotent]]s [[Invariant (mathematics)|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.<ref>Munn, Lemma 1</ref>
# Underlying every [[C*-algebra]] is a *-semigroup. An important [[C*-algebra#Finite-dimensional C*-algebras|instance]] is the algebra ''M''<sub>''n''</sub>('''C''') of ''n''-by-''n'' [[matrix (mathematics)|matrices]] over '''[[Complex number|C]]''', with the [[conjugate transpose]] as involution.
# {{anchor|ex5}} If ''X'' is a set, the set of all [[binary relation]]s on ''X'' is a *-semigroup with the * given by the [[inverse relation]], and the multiplication given by the usual [[composition of relations]]. This is an example of a *-semigroup which is not a regular semigroup.
# {{anchor|ex6}} If X is a set, then the set of all finite sequences (or [[String (computer science)|strings]]) of members of X forms a [[free monoid]] under the operation of concatenation of sequences, with sequence reversal as an involution.
# {{anchor|ex7}} 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.<ref name="Nordahl and Scheiblich">Nordahl and Scheiblich</ref>
==Basic concepts and properties==
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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 object]]s (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 [[#ex7|example 7]] is a regular *-semigroup that is not an inverse semigroup.<ref
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-semigroup]]s.
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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:
# For any ''a'' in S, there exists a unique a° in V(''a'') such that ''aa''° and ''a''°''a'' are in F(S)
# For any ''a'' in S, and b in F(S), ''a°ba'' is in F(S), where ° is the well-defined operation from the previous axiom
# For any ''a'', ''b'' in F(S), ''ab'' is in E(S); note: not necessarily in F(S)
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.
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{{anchor|Drazin}}
{{expand section|clarify motivation for studying these|date=April 2015}}
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 relations|Green’s relation]] ''H''. This defining property can be formulated in several equivalent ways. Another is to say that every [[Green's relations#The L.2C R.2C and J relations|''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''.
In the [[Matrix multiplication|multiplicative]] semigroup ''M''<sub>''n''</sub> ( ''C'' ) of square matrices of order ''n'', the map which assigns a matrix ''A'' to its [[Hermitian conjugate]] ''A''* is an involution. The semigroup ''M''<sub>''n''</sub> ( ''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.
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As with all varieties, the [[category theory|category]] of semigroups with involution admits [[free object]]s. 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.<ref name="L51">Lawson p. 51</ref>
The [[Generator (mathematics)|generators]] of a free semigroup with involution are the elements of the union of two ([[equinumerous]]) [[disjoint sets]] in [[Bijection|bijective correspondence]]: <math>Y=X\sqcup X^\dagger</math>. (Here the notation <math>\sqcup\,</math> 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 (computer science)|alphabet]] with involution''<ref name="EhrenfeuchtHarju1999">{{cite book|author1=Andrzej Ehrenfeucht|author2=T. Harju|author3=Grzegorz Rozenberg|title=The Theory of 2-structures: A Framework for Decomposition and Transformation of Graphs|year=1999|publisher=World Scientific|isbn=978-981-02-4042-4|pages=13–14}}</ref> or a ''symmetric alphabet''.<ref name="Sakarovitch">{{cite book|title=Elements of Automata Theory|publisher=Cambridge University Press|pages=
: <math>y^\dagger =
\begin{cases}
\theta(y) & \text{if } y \in X \\
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Now construct <math>Y^+\,</math> as the [[free semigroup]] on <math>Y\,</math> in the usual way with the binary (semigroup) operation on <math>Y^+\,</math> being [[concatenation]]:
: <math>w = w_1w_2 \cdots w_k \in Y^+</math> for some letters <math>w_i\in Y.</math>
The bijection <math>\dagger</math> on <math>Y</math> is then extended as a bijection <math>{ }^\dagger:Y^+\rightarrow Y^+</math> defined as the string reversal of the elements of <math>Y^+\,</math> that consist of more than one letter:<ref name="EhrenfeuchtHarju1999"/><ref name="Lipscomb1996"/>
: <math>w^\dagger=w_k^\dagger w_{k-1}^\dagger \cdots w_{2}^\dagger w_{1}^\dagger.</math>
This map is an [[#Formal definition|involution]] on the semigroup <math>Y^+\,</math>. Thus, the semigroup <math>(X\sqcup X^\dagger)^+</math> with the map <math>{ }^\dagger\,</math> is a semigroup with involution, called a '''free semigroup with involution''' on ''X''.<ref name="L172">Lawson p. 172</ref> (The irrelevance of the concrete identity of <math>X^\dagger</math> and of the bijection <math>\theta</math> in this choice of terminology is explained below in terms of the universal property of the construction.) Note that unlike in [[#ex6|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 <math>Y^+\,</math> we use the [[free monoid]] <math>Y^*=Y^+\cup\{\varepsilon\}</math>, which is just the free semigroup extended with the [[empty word]] <math>\varepsilon\,</math> (which is the [[identity element]] of the [[monoid]] <math>Y^*\,</math>), and suitably extend the involution with <math>\varepsilon^\dagger = \varepsilon</math>,
we obtain a '''free monoid with involution'''.<ref name="Lipscomb1996"/>
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Baer *-semigroups are also encountered in [[quantum mechanics]],<ref name="BeltramettiCassinelli2010"/> in particular as the multiplicative semigroups of [[Baer *-ring]]s.
If ''H'' is a [[Hilbert space]], then the multiplicative semigroup of all [[bounded operator]]s on ''H'' is
Baer *-semigroup allow the [[coordinatization]] of [[orthomodular lattice]]s.<ref name="Blyth2006">{{cite book|author=T.S. Blyth|title=Lattices and Ordered Algebraic Structures|year=2006|publisher=Springer Science & Business Media|isbn=978-1-84628-127-3|pages=
==See also==
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==References==
* Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". [[World Scientific]] ISBN 981-02-3316-7
* D J Foulis (1958). ''Involution Semigroups'', Ph.D. Thesis, Tulane University, New Orleans, LA. [http://www.math.umass.edu/~foulis/publ.txt 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
* 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.
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* S. Crvenkovic and Igor Dolinka, "[http://people.dmi.uns.ac.rs/~dockie/papers/031.pdf Varieties of involution semigroups and involution semirings: a survey]", Bulletin of the Society of Mathematicians of Banja Luka Vol. 9 (2002), 7-47.
* {{PlanetMath attribution|id=8283|title=Free semigroup with involution}}
{{Use dmy dates|date=September 2010}}
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