Semigroup with involution: Difference between revisions

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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—roughlywhich—roughly speaking—bringsspeaking—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 {{nowrap|1=(''AB'')^T = ''B''^T''A''^T}}, 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''^T 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 [[String (computer science)#Reversal|reverse]]s the [[linear order]] of the letters in a string. A third example, from basic [[set theory]], is the set of all [[binary relation]]s between a set and itself, with the involution being the [[converse relation]], and the multiplication given by the usual [[composition of relations]].

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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''^&minus;1−1 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.<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''_''n''('''C''') of ''n''-by-''n'' [[matrix (mathematics)|matrices]] over '''[[Complex number|C]]''', with the [[conjugate transpose]] as involution.

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==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.<ref>Easdown, David, and W. D. Munn. "On semigroups with involution." Bulletin of the Australian Mathematical Society 48.01 (1993): 93-10093–100.</ref>

Certain basic concepts may be defined on *-semigroups in a way that parallels the notions stemming from a [[Regular semigroup|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'').<ref>Lawson, p. 116</ref> 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''.<ref name="L117">Lawson, p. 117</ref>

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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 {{nowrap|1=''x''′''xx′'' = ''x''′}}, {{nowrap|1=''xx''′''x'' = ''x''}}, {{nowrap|1=(''xx''′)* = ''xx''′}}, {{nowrap|1=(''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&ndash;PenroseMoore–Penrose inverse of ''x''. This agrees with the classical definition of the [[Moore–Penrose inverse]] of a square matrix.

In the [[Matrix multiplication|multiplicative]] semigroup ''M''_''n''(''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''_''n''(''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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The construction above is actually the only way to extend a given map <math>\theta\,</math> from <math>X\,</math> to <math>X^\dagger\,</math>, to an involution on <math>Y^+\,</math> (and likewise on <math>Y^*\,</math>). The qualifier "free" for these constructions is justified in the usual sense that they are [[universal algebra|universal construction]]s. In the case of the free semigroup with involution, given an arbitrary semigroup with involution <math>S\,</math> and a map <math>\Phi:X\rightarrow S</math>, then a [[semigroup homomorphism]] <math>\overline\Phi:(X\sqcup X^\dagger)^+\rightarrow S</math> exists such that <math>\Phi = \iota \circ \overline\Phi</math>, where <math>\iota : X \rightarrow (X\sqcup X^\dagger)^+</math> is the [[inclusion map]] and [[composition of functions]] is taken in [[Function composition#Alternative notations|diagram order]].<ref name="L172"/> The construction of <math>(X\sqcup X^\dagger)^+</math> as a semigroup with involution is unique up to [[isomorphism]]. An analogous argument holds for the free monoid with involution in terms of [[monoid homomorphism]]s and the uniqueness up to isomorphism of the construction of <math>(X\sqcup X^\dagger)^*</math> 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 <math>xx^\dagger</math> or <math>x^\dagger x</math>. 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.<ref name="L51"/> (Otherwise put, these rules define a [[Confluence (abstract rewriting)|confluent]] rewriting system.) Equivalently, a free group is constructed from a free monoid with involution by taking the [[quotient (universal algebra)|quotient]] of the latter by the [[Congruence relation|congruence]] <math>\{ (yy^\dagger, \varepsilon) : y\in Y\}</math>, which is sometimes called the '''Dyck congruence'''&mdash;in—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 <math>()=)(=\varepsilon</math>; the one-sided congruence that appears in the Dyck language proper <math>\{ (xx^\dagger, \varepsilon) : x\in X\}</math>, which instantiates only to <math>()=\varepsilon</math> 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 <!--with involution?-->; nevertheless it has been called the '''free half group''' by its first discoverer&mdash;discoverer—[[Eli Shamir]]&mdash;although—although more recently it has been called the '''involutive monoid''' generated by ''X''.<ref name="Sakarovitch"/><ref name="DrosteKuich2009">{{cite book|editors=Manfred Droste, Werner Kuich, Heiko Vogler|title=Handbook of Weighted Automata|year=2009|publisher=Springer |isbn=978-3-642-01492-5|page=271|authors=Ion Petre and [[Arto Salomaa]]|chapter=Algebraic Systems and Pushdown Automata}}</ref> (This latter choice of terminology conflicts however with the use of "involutive" to denote any semigroup with involution&mdash;ainvolution—a practice also encountered in the literature.<ref name="Neeb2000">{{cite book|author=Karl-Hermann Neeb|title=Holomorphy and Convexity in Lie Theory|year=2000|publisher=Walter de Gruyter|isbn=978-3-11-015669-0|page=21}}</ref><ref name="BeltramettiCassinelli2010">{{cite book|author1=Enrico G. Beltrametti|author2=Gianni Cassinelli|title=The Logic of Quantum Mechanics|year=2010|origyear=1981|publisher=Cambridge University Press|isbn=978-0-521-16849-6|page=178}}</ref>)

== Baer *-semigroups ==

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The projection ''e'' is in fact uniquely determined by ''x''.<ref name="BeltramettiCassinelli2010"/>

More recently, Baer *-semigroups have been also called '''Foulis semigroups''', after [[David James Foulis]] who studied them in depth.<ref name="Blyth2006"/><ref>Harding, John. “Daggers, Kernels, Baer *-Semigroups, and Orthomodularity.” ''Journal of Philosophical Logic''. April 6, April 2013. {{doi|10.1007/s10992-013-9275-5}}</ref>

=== Examples and applications ===

The set of all binary relations on a set (from [[#ex5|example 5]]) is a Baer *-semigroup.<ref name="Foulis63">Foulis, D. J. Relative inverses in Baer *-semigroups. Michigan Math. J. 10 (1963), no. 1, 65--8465–84. {{doi|10.1307/mmj/1028998825}}.</ref>

Baer *-semigroups are also encountered in [[quantum mechanics]],<ref name="BeltramettiCassinelli2010"/> in particular as the multiplicative semigroups of [[Baer *-ring]]s.

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* 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.&nbsp;173–187

* 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), 7-477–47.

* {{PlanetMath attribution|id=8283|title=Free semigroup with involution}}

{{Use dmy dates|date=SeptemberAugust 20102019}}

{{DEFAULTSORT:Semigroup With Involution}}