Semigroup with involution: Difference between revisions

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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|Munn|1993}}</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>{{harvcoltxt|Lawson, |1998|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">{{harvcoltxt|Lawson, |1998|p. =117}}</ref>

Partial isometries can be [[partial order|partially ordered]] by ''s'' ≤ ''t'' defined as holding whenever ''s'' = ''ss''*''t'' and ''ss''* = ''ss''*''tt''*.<ref name="L117"/> Equivalently, ''s'' ≤ ''t'' if and only if ''s'' = ''et'' and ''e'' = ''ett''* for some projection ''e''.<ref name="L117"/> In a *-semigroup, PI(''S'') is an [[ordered groupoid]] with the [[Partial groupoid|partial product]] given by ''s''&sdot;''t'' = ''st'' if ''s''*''s'' = ''tt''*.<ref>{{harvcoltxt|Lawson, |1998|p. =118}}</ref>

=== Examples ===

In terms of examples for these notions, in the *-semigroup of binary relations on a set, the partial isometries are the relations that are [[difunctional]]. The projections in this *-semigroup are the [[partial equivalence relation]]s.<ref>{{harvcoltxt|Lawson |1998|p.=122 and }}</ref><ref>{{harvcoltxt|Lawson|1998|p.=35}}</ref>

The [[partial isometry|partial isometries]] in a C*-algebra are exactly those defined in this section. In the case of ''M''_''n''('''C''') more can be said. If ''E'' and ''F'' are projections, then ''E'' ≤ ''F'' if and only if [[Image (mathematics)|im]]''E'' ⊆ im''F''. For any two projection, if ''E'' ∩ ''F'' = ''V'', then the unique projection ''J'' with image ''V'' and kernel the [[orthogonal complement]] of ''V'' is the meet of ''E'' and ''F''. Since projections form a meet-[[semilattice]], the partial isometries on ''M''_''n''('''C''') form an inverse semigroup with the product <math>A(A^*A\wedge BB^*)B</math>.<ref>{{harvcoltxt|Lawson |1998|p.=120}}</ref>

Another simple example of these notions appears in the next section.

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: <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">{{harvcoltxt|Lawson |1998|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>,

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* {{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.