Semigroup with involution: Difference between revisions

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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 (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]].

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 bookharvcoltxt|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>

==Formal definition==

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Semigroups that satisfy only the first of these axioms belong to the larger class of [[U-semigroup]]s.

In some applications, the second of these axioms has been called [[antidistributive]].<ref name="BrinkKahl1997">{{cite bookharvcoltxt|author1=Chris Brink|author2=Wolfram Kahl|author3=Gunther Schmidt|title=Relational Methods in Computer Science|date=1997|publisher=Springer|isbn=978-3-211-82971-4|page=4}}</ref> Regarding the natural philosophy of this axiom, [[Harold Scott MacDonald Coxeter|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."<ref>H.S.M. {{harvcoltxt|Coxeter, ''Introduction to Geometry'', p. |1961|page=33}}</ref>

==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 bookharvcoltxt|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|publisherpages=Springer Science & Business Media|isbn=97887-1-4612-1128-0|pages=87–8888}}</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>{{harvcoltxt|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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==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'', 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)

* {{cite book|last=Coxeter|first=H.S.M.|author-link=Donald Coxeter|title=Introduction to Geometry|year=1961}}

* 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.

* {{citation|last1=Yamada|first1=Miyuki|date=December Yamada, ''1982|title=P-systems in regular semigroups'', |journal=[[Semigroup Forum]], |volume=24(|issue=1), December 1982, pp.&nbsp;173–187|pages=173-187}}

* {{citation|last1=Easdown|first1=David|last2=Munn|first2=Walter Douglas|year=1993|title=On semigroups with involution|journal=Bulletin of the Australian Mathematical Society|volume=48|issue=1|doi=10.1017/S0004972700015495|pages=93-100}}

* {{cite book|last1=Brink|first1=Chris|last2=Kahl|first2=Wolfram|last3=Schmidt|first3=Gunther|title=Relational Methods in Computer Science|date=1997|publisher=Springer|isbn=978-3-211-82971-4}}

* Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". [[World Scientific]] {{isbn|981-02-3316-7}}

* {{cite book|last=Lawson|first=Mark|year=1998|title=Inverse semigroups: the theory of partial symmetries|publisher=[[World Scientific]]|isbn=981-02-3316-7}}

* 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.

* {{cite book|first1=C.|last1=van den Berg|first2=J. P. R.|last2=Christensen|first3=P.|last3=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}}

* {{cite book|first=Christopher|last=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}}

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

{{Use dmy dates|date=September 2019}}