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{{Short description|Semigroup in abstract algebra}}

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.

* [[Uniqueness (mathematics)|Uniqueness]]

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]].

* Double application "cancelling itself out".

* 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.

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>

An example from [[linear algebra]] is the [[Matrix multiplication|multiplicative]]a [[monoid]]set of [[Real number|real]]-valued n-by-n square [[Matrix (mathematics)|matrices]] ofwith order&nbsp;''n''the (calledmatrix-transpose as the [[full linear monoid]])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''^&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 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.

# {{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>{{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>{{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.

== Notions of regularity ==

There are two related, but not identical notions of regularity in *-semigroups. They were introduced nearly simultaneously by Nordahl & Scheiblich (1978) and respectively Drazin (1979).<ref>Crvenkovic and Dolinka</ref>

=== Regular *-semigroups (Nordahl & Scheiblich) ===

As mentioned in the [[#Examples|previous examples]], [[inverse semigroup]]s are a subclass of *-semigroups. It is also textbook knowledge that an inverse semigroup can be characterized as a regular semigroup in which any two idempotents commute. In 1963, [[Boris M. Schein]] hasshowed publishedthat the following two axioms providingprovide an analogous characterization of inverse semigroups as a [[Variety (universal algebra)|subvariety]] of *-semigroups:{{cn|date=March 2025}}

* {{math|''x'' {{=}} ''xx''*''x''}}

* {{math|(''xx''*)(''x''*''x'') {{=}} (''x''*''x'')(''xx''*)}}

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).{{cn|date=March 2025}} 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|exampleExample 7]] is a regular *-semigroup that is not an inverse semigroup.<ref name="Nordahl and Scheiblich"/> It is also easy to verify that in a regular *-semigroup the product of any two projections is an idempotent.<ref>Nordahl and Scheiblich, Theorem 2.5</ref> In the aforementioned rectangular band example, the projections are elements of the form {{math|(''x'', ''x'')}} and [(like all elements of a band]) are idempotent. However, two different projections in this band need not commute, nor is their product necessarily a projection since {{math|(''a'', ''a'')(''b'', ''b'') {{=}} (''a'', ''b'')}}.

Semigroups that satisfy only {{math|''x''** {{=}} ''x'' {{=}} ''xx''*''x''}} (but not necessarily the antidistributivity of * over multiplication) have also been studied under the name of [[I-semigroup]]s.

====P-systems====

The problem of characterizing when a regular semigroup is a regular *-semigroup (in the sense of Nordahl & Scheiblich) wascan be addressed by M. Yamada (1982). He defineddefining a '''P-system'''. F(S) as subset ofFor the idempotents ofsemigroup {{mvar|S}}, denoted as usual bylet {{math|''E''(''S'').}} Usingdenote the usualset notationof idempotents, and let {{math|''V''(''a'')}} fordenote the inverses of ''{{mvar|a}}. A P-system {{math|'', F''(''S'')}} needsis then a subset of {{math|''E''(''S'')}} towhich satisfysatisfies the following axioms:

# For any ''{{mvar|a''}} in {{mvar|S}}, there exists a unique {{math|''a''°}} in {{math|''V''(''a'')}} such that {{math|''aa''°}} and {{math|''a''°''a''}} are in {{math|''F''(''S'')}}

# For any ''{{mvar|a''}} in {{mvar|S}}, and {{mvar|b}} in {{math|''F''(''S'')}}, ''a°ba'' is in F(S), where ° is the well-defined operation from the previous axiom

# For any ''{{mvar|a''}}, ''{{mvar|b''}} in {{math|''F''(''S'')}}, {{math|''ab''}} is in {{math|''E''(''S'')}}; note: not necessarily in {{math|''F''(''S'')}}

A regular semigroup {{mvar|S}} is a *-regular semigroup, as defined by Nordahl & Scheiblich, if and only if it has a p-system {{math|''F''(''S'')}}.<ref>{{harvcoltxt|Yamada|1982}}</ref> In this case {{math|''F''(''S'')}} is the set of projections of {{mvar|S}} with respect to the operation {{math|°}} defined by {{math|''F''(''S'')}}. In an [[inverse semigroup]] the entire semilattice of idempotents is a pP-system. Also, if a regular semigroup {{mvar|S}} has a pP-system that is multiplicatively closed (i.e. subsemigroup), then {{mvar|S}} is an inverse semigroup. Thus, a pP-system may be regarded as a generalization of the semilattice of idempotents of an inverse semigroup.

===*-regular semigroups (Drazin)===

{{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&ndash;PenroseMoore–Penrose inverse of ''x''. This agrees with the classical definition of the [[Moore–Penrose inverse]] of a square matrix.

One motivation for studying these semigroups is that they allow generalizing the Moore–Penrose inverse's properties from {{tmath|\R }} and {{tmath|\C }} to more general sets.

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''.

==Free semigroup with involution ==

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">{{harvcoltxt|Lawson |1998|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> emphasizedemphasizes 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 bookharvcoltxt|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|publisherpages=World Scientific|isbn=97813-981-02-4042-4|pages=13–1414}}</ref> or a ''symmetric alphabet''.<ref name="Sakarovitch">{{cite bookharvcoltxt|title=Elements of Automata TheorySakarovitch|publisher=Cambridge University Press2009|pages=305–306|author=Jacques Sakarovitch305-306}}</ref> Let <math>\theta:X\rightarrow X^\dagger</math> be a bijection; <math>\theta</math> is naturally [[Function (mathematics)#Restrictions and extensions|extended]] to a bijection <math>{ }\dagger: Y \to Y</math> essentially by taking the disjoint union of <math>\theta</math> (as a set) with its [[inverse function|inverse]], or in [[piecewise]] notation:<ref name="Lipscomb1996">{{cite bookharvcoltxt|author=Stephen Lipscomb|title=Symmetric Inverse Semigroups|year=1996|publisher=American Mathematical Soc.|isbn=978-0-8218-0627-2|pagep=86}}</ref>

: <math display=block>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 display=block>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 display=block>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|exampleExample 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"/>

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 it, 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 [[Quotientquotient (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", howeverHowever simplification in the Dyck congruence takes place regardless of order. For example, e.g. 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 bookharvcoltxt|editors=Manfred Droste, Werner Kuich, Heiko VoglerPetre|title=Handbook of Weighted AutomataSalomaa|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|origyearorig-year=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"Daggers, Kernels, Baer *-Semigroups, and Orthomodularity".” ''Journal of Philosophical Logic''. 6 April 6, 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.

Line 116 ⟶ 126:

==See also==

* [[Dagger category]] (aka category with involution) — generalizes the notion*-monoids

* [[*-algebra]]

* [[Special classes of semigroups]]

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==References==

* 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)

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

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

* 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|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}}

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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.

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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), 7-477–47.

* {{cite book|title=Elements of Automata Theory|year=2009|publisher=Cambridge University Press|first=Jacques|last=Sakarovitch}}

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

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

{{DEFAULTSORT:Semigroup With Involution}}