Semigroup with involution: Difference between revisions

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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''⁻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.

# {{anchor|ex5}} If ''X'' is a set, the set of all [[binary relation]]s 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.

# {{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==

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): 93–100.</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’sGreen'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′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′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 [[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 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"Daggers, Kernels, Baer *-Semigroups, and Orthomodularity".” ''Journal of Philosophical Logic''. 6 April 2013. {{doi|10.1007/s10992-013-9275-5}}</ref>

=== Examples and applications ===

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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.PhD 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|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

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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}}

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