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| In mathematics, an '''inverse semigroup''' ''S'' is a [[semigroup]] in which every element ''x'' in ''S'' has a unique '''inverse'''
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| ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy''. Inverse semigroups appear in a range of contexts; for
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| example, they can be employed in the study of [[partial symmetries]].<ref>Lawson 1998.</ref>
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| (The convention followed in this article will be that of writing a function on the right of its argument, and
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| composing functions from left to right — a convention often observed in semigroup theory.)
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| == Origins ==
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| Inverse semigroups were introduced independently by [[Viktor Vladimirovich Wagner]]<ref>Since his father was German, Wagner preferred the German transliteration of his name (with a "W", rather than a "V") from Cyrillic - see Schein 1981.</ref> in the [[Soviet Union]] in 1952,<ref>First a short announcement in Wagner 1952, then a much more comprehensive exposition in Wagner 1953.</ref> and by [[Gordon Preston]] in [[Great Britain]] in 1954.<ref>Preston 1954a,b,c.</ref> Both authors arrived at inverse semigroups via the study of partial one-one transformations of a [[Set (mathematics)|set]]: a [[partial function|partial transformation]] α of a set ''X'' is a [[Function (mathematics)|function]] from ''A'' to
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| ''B'', where ''A'' and ''B'' are subsets of ''X''. Let α and β be partial transformations of a set
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| ''X''; α and β can be composed (from left to right) on the largest [[Domain (mathematics)|domain]] upon
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| which it "makes sense" to compose them:
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| :dom αβ = [im α <math>\cap</math> dom β]α<sup>−1</sup> | |
| where α<sup>−1</sup> denotes the [[preimage]] under α. Partial transformations had already been studied
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| in the context of [[pseudogroup]]s.<ref>See, for example, Golab 1939.</ref> It was Wagner, however, who
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| was the first to observe that the composition of partial transformations is a special case of the multiplication of
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| [[binary relations]].<ref>Schein 2002 : 152.</ref> He recognised also that the domain of composition of two partial
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| transformations may be the [[empty set]], so he introduced an ''empty transformation'' to take account of this.
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| With the addition of this empty transformation, the composition of partial transformations of a set becomes an
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| everywhere-defined [[associative]] [[binary operation]]. Under this composition, the collection
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| <math>\mathcal{I}_X</math> of all partial one-one transformations of a set ''X'' forms an inverse semigroup, called
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| the ''symmetric inverse semigroup'' (or monoid) on ''X''.<ref>Howie 1995 : 149.</ref> This is the "archetypal"
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| inverse semigroup, in the same way that a [[symmetric group]] is the archetypal [[group (mathematics)|group]]. For
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| example, just as every [[group (mathematics)|group]] can be embedded in a [[symmetric group]], every inverse
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| semigroup can be embedded in a symmetric inverse semigroup (see below).
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| == The basics ==
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| The inverse of an element ''x'' of an inverse semigroup ''S'' is usually written ''x''<sup>−1</sup>. Inverses in an
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| inverse semigroup have many of the same properties as inverses in a [[group (mathematics)|group]], for example,
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| (''ab'')<sup>−1</sup> = ''b''<sup>−1</sup>''a''<sup>−1</sup>. In an inverse [[monoid]], ''xx''<sup>−1</sup> and
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| ''x''<sup>−1</sup>''x'' are not (necessarily) equal to the identity, but they are both [[idempotent]].<ref>Howie
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| 1995 : Proposition 5.1.2(1).</ref> An inverse monoid ''S'' in which ''xx''<sup>−1</sup> = 1 =
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| ''x''<sup>−1</sup>''x'', for all ''x'' in ''S'' (a ''unipotent'' inverse monoid), is, of course, a [[group (mathematics)|group]].
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| There are a number of equivalent characterisations of an inverse semigroup ''S'':<ref>Howie 1995 : Theorem
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| 5.1.1.</ref>
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| * Every element of ''S'' has a unique inverse, in the above sense.
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| * Every element of ''S'' has at least one inverse (''S'' is a [[regular semigroup]]) and [[idempotent]]s commute (that is, the [[idempotent]]s of ''S'' form a [[semilattice]]).
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| * Every <math>\mathcal{L}</math>-class and every <math>\mathcal{R}</math>-class contains precisely one [[idempotent]], where <math>\mathcal{L}</math> and <math>\mathcal{R}</math> are two of [[Green's relations]].
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| The [[idempotent]] in the <math>\mathcal{L}</math>-class of ''s'' is ''s''<sup>−1</sup>''s'', whilst the
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| [[idempotent]] in the <math>\mathcal{R}</math>-class of ''s'' is ''ss''<sup>−1</sup>. There is therefore a simple
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| characterisation of [[Green's relations]] in an inverse semigroup:<ref>Howie
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| 1995 : Proposition 5.1.2(1).</ref>
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| :<math>a\,\mathcal{L}\,b\Longleftrightarrow a^{-1}a=b^{-1}b,\quad a\,\mathcal{R}\,b\Longleftrightarrow
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| aa^{-1}=bb^{-1}</math>
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| Examples of inverse semigroups:
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| *Every [[group (mathematics)|group]] is an inverse semigroup.
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| *The [[bicyclic semigroup]] is inverse, with (''a'',''b'')<sup>−1</sup> = (''b'',''a'').
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| *Every [[semilattice]] is inverse.
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| *The [[Brandt semigroup]] is inverse.
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| *The [[Munn semigroup]] is inverse.
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| Unless stated otherwise, ''E(S)'' will denote the semilattice of idempotents of an inverse semigroup ''S''.
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| == The natural partial order ==
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| An inverse semigroup ''S'' possesses a ''natural [[partial order]]'' relation ≤ (sometimes denoted by ω)
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| which is defined by the following:<ref>Wagner 1952.</ref>
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| :<math>a \leq b \Longleftrightarrow a=eb,</math>
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| for some [[idempotent]] ''e'' in ''S''. Equivalently,
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| :<math>a \leq b \Longleftrightarrow a=bf,</math>
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| for some (in general, different) [[idempotent]] ''f'' in ''S''. In fact, ''e'' can be taken to be
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| ''aa''<sup>−1</sup> and ''f'' to be ''a''<sup>−1</sup>''a''.<ref>Howie 1995 : Proposition 5.2.1.</ref>
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| The natural [[partial order]] is compatible with both multiplication and inversion, that is,<ref>Howie 1995 :
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| 152–3</ref>
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| :<math>a \leq b, c \leq d \Longrightarrow ac \leq bd</math>
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| and
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| :<math>a \leq b \Longrightarrow a^{-1} \leq b^{-1}.</math>
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| In a [[group (mathematics)|group]], this [[partial order]] simply reduces to equality, since the identity is the
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| only [[idempotent]]. In a symmetric inverse semigroup, the [[partial order]] reduces to restriction of mappings,
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| i.e., α ≤ β if, and only if, the domain of α is contained in the domain of β and
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| ''x''α = ''x''β, for all ''x'' in the domain of α.<ref>Howie 1995 : 153.</ref>
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| The natural partial order on an inverse semigroup interacts with [[Green's relations]] as follows: if ''s'' ≤
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| ''t'' and ''s''<math>\,\mathcal{L}\,</math>''t'', then ''s'' = ''t''. Similarly, if
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| ''s''<math>\,\mathcal{R}\,</math>''t''.<ref>Lawson 1998 : Proposition 3.2.3.</ref>
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| On ''E(S)'', the natural [[partial order]] becomes:
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| :<math>e \leq f \Longleftrightarrow e = ef,</math>
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| so the product of any two [[idempotent]]s in ''S'' is equal to the lesser of the two, with respect to
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| ≤. If ''E(S)'' forms a [[chain (order theory)|chain]] (i.e., ''E(S)'' is [[totally ordered]] by ≤), then
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| ''S'' is a [[union (set theory)|union]] of [[group (mathematics)|groups]].<ref>Clifford & Preston 1967 : Theorem
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| 7.5</ref>
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| == Homomorphisms and representations of inverse semigroups ==
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| A [[homomorphism]] (or ''morphism'') of inverse semigroups is defined in exactly the same way as for any other
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| semigroup: for inverse semigroups ''S'' and ''T'', a [[function (mathematics)|function]] θ from ''S'' to ''T''
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| is a morphism if (''s''θ)(''t''θ) = (''st'')θ, for all ''s'',''t'' in ''S''. The definition of a
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| morphism of inverse semigroups could be augmented by including the condition (''s''θ)<sup>−1</sup> =
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| ''s''<sup>−1</sup>θ, however, there is no need to do so, since this property follows from the above
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| definition, via the following theorem:
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| '''Theorem.''' The homomorphic [[image (mathematics)|image]] of an inverse semigroup is an inverse semigroup; the
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| inverse of an element is always mapped to the inverse of the [[image (mathematics)|image]] of that
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| element.<ref>Clifford & Preston 1967 : Theorem 7.36.</ref>
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| One of the earliest results proved about inverse semigroups was the ''Wagner-Preston Theorem'', which is an analogue
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| of [[Cayley's Theorem]] for [[group (mathematics)|groups]]:
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| '''Wagner-Preston Theorem.''' If ''S'' is an inverse semigroup, then the [[function (mathematics)|function]] φ
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| from ''S'' to <math>\mathcal{I}_S</math>, given by
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| :dom (''a''φ) = ''Sa''<sup>−1</sup> and ''x''(''a''φ) = ''xa''
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| is a [[faithful representation]] of ''S''.<ref>Howie 1995 : Theorem 5.1.7. Originally, Wagner 1952 and,
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| independently, Preston 1954c.</ref>
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| Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup.
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| == Congruences on inverse semigroups ==
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| [[Congruences]] are defined on inverse semigroups in exactly the same way as for any other semigroup: a
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| ''congruence'' ρ is an [[equivalence relation]] which is compatible with semigroup multiplication, i.e.,
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| :<math>a\,\rho\,b,\quad c\,\rho\,d\Longrightarrow ac\,\rho\,bd.</math><ref>Howie 1995 : 22</ref>
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| Of particular interest is the relation <math>\sigma</math>, defined on an inverse semigroup ''S'' by
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| :<math>a\,\sigma\,b\Longleftrightarrow</math> there exists a <math>c\in S</math> with <math>c\leq
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| a,b.</math><ref>Lawson 1998 : 62</ref>
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| It can be shown that σ is a congruence and, in fact, it is a [[group congruence]], meaning that the factor semigroup ''S''/σ is a group. In the set of all group congruences on a semigroup ''S'', the minimal element (for the partial order defined by inclusion of sets) need not be the smallest element. In the specific case in which ''S'' is an inverse semigroup σ is the ''smallest'' congruence on ''S'' such that ''S''/σ is a group, that is, if τ is any
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| other congruence on ''S'' with ''S''/τ a group, then σ is contained in τ. The congruence σ is
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| called the ''minimum group congruence'' on ''S''.<ref>Lawson 1998 : Theorem 2.4.1.</ref> The minimum group
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| congruence can be used to give a characterisation of ''E''-unitary inverse semigroups (see below).
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| A congruence ρ on an inverse semigroup ''S'' is called ''idempotent pure'' if
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| :<math>a\in S, e\in E(S), a\,\rho\,e\Longrightarrow a\in E(S).</math><ref>Lawson 1998 : 65</ref>
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| == ''E''-unitary inverse semigroups ==
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| One class of inverse semigroups which has been studied extensively over the years is the class of ''E-unitary
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| inverse semigroups'': an inverse semigroup ''S'' (with [[semilattice]] ''E'' of [[idempotent]]s) is
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| ''E-unitary'' if, for all ''e'' in ''E'' and all ''s'' in ''S'',
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| :<math>es \in E \Longrightarrow s \in E.</math>
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| Equivalently,
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| :<math>se \in E \Rightarrow s \in E.</math><ref>Howie 1995 : 192.</ref>
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| One further characterisation of an ''E''-unitary inverse semigroup ''S'' is the following: if ''e'' is in ''E'' and
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| ''e'' ≤ ''s'', for some ''s'' in ''S'', then ''s'' is in ''E''.<ref>Lawson 1998 : Proposition 2.4.3.</ref>
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| '''Theorem.''' Let ''S'' be an inverse semigroup with [[semilattice]] ''E'' of idempotents, and minimum group
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| congruence σ. Then the following are equivalent:<ref>Lawson 1998 : Theorem 2.4.6.</ref>
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| *''S'' is ''E''-unitary;
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| *σ is idempotent pure;
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| *<math>\sim</math> = σ,
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| where <math>\sim</math> is the ''compatibility relation'' on ''S'', defined by
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| :<math>a\sim b\Longleftrightarrow ab^{-1},a^{-1}b</math> are idempotent. | |
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| '''McAlister's Covering Theorem.''' Every inverse semigroup S has a E-unitary cover; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S.<ref>Grillet p. 248</ref>
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| Central to the study of ''E''-unitary inverse semigroups is the following construction.<ref>Howie 1995 : 193–4</ref> Let <math>\mathcal{X}</math> be a [[partially ordered set]], with ordering ≤, and let <math>\mathcal{Y}</math> be a [[subset]] of <math>\mathcal{X}</math> with the properties that
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| *<math>\mathcal{Y}</math> is a [[semilattice|lower semilattice]], that is, every pair of elements ''A'', ''B'' in <math>\mathcal{Y}</math> has a [[greatest lower bound]] ''A'' <math>\wedge</math> ''B'' in <math>\mathcal{Y}</math> (with respect to ≤);
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| *<math>\mathcal{Y}</math> is an [[order ideal]] of <math>\mathcal{X}</math>, that is, for ''A'', ''B'' in <math>\mathcal{X}</math>, if ''A'' is in <math>\mathcal{Y}</math> and ''B'' ≤ ''A'', then ''B'' is in <math>\mathcal{Y}</math>.
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| Now let ''G'' be a [[group (mathematics)|group]] which [[group action|acts]] on <math>\mathcal{X}</math> (on the left), such that
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| *for all ''g'' in ''G'' and all ''A'', ''B'' in <math>\mathcal{X}</math>, ''gA'' = ''gB'' if, and only if, ''A'' = ''B'';
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| *for each ''g'' in ''G'' and each ''B'' in <math>\mathcal{X}</math>, there exists an ''A'' in <math>\mathcal{X}</math> such that ''gA'' = ''B'';
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| *for all ''A'', ''B'' in <math>\mathcal{X}</math>, ''A'' ≤ ''B'' if, and only if, ''gA'' ≤ ''gB'';
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| *for all ''g'', ''h'' in ''G'' and all ''A'' in <math>\mathcal{X}</math>, ''g''(''hA'') = (''gh'')''A''.
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| The triple <math>(G, \mathcal{X}, \mathcal{Y})</math> is also assumed to have the following properties:
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| *for every ''X'' in <math>\mathcal{X}</math>, there exists a ''g'' in ''G'' and an ''A'' in <math>\mathcal{Y}</math> such that ''gA'' = ''X'';
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| *for all ''g'' in ''G'', ''g''<math>\mathcal{Y}</math> and <math>\mathcal{Y}</math> have nonempty intersection.
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| Such a triple <math>(G, \mathcal{X}, \mathcal{Y})</math> is called a ''McAlister triple''. A McAlister triple is
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| used to define the following:
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| :<math>P(G, \mathcal{X}, \mathcal{Y}) = \{ (A,g) \in \mathcal{Y}\times G: g^{-1}A \in \mathcal{Y} \}</math>
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| together with multiplication
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| :<math>(A,g)(B,h)=(A \wedge gB, gh)</math>.
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| Then <math>P(G, \mathcal{X}, \mathcal{Y})</math> is an inverse semigroup under this multiplication, with
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| (''A'',''g'')<sup>−1</sup> = (''g''<sup>−1</sup>''A'', ''g''<sup>−1</sup>). One of the main results in the study of
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| ''E''-unitary inverse semigroups is ''McAlister's P-Theorem'':
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| '''McAlister's P-Theorem.''' Let <math>(G, \mathcal{X}, \mathcal{Y})</math> be a McAlister triple. Then <math>P(G,
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| \mathcal{X}, \mathcal{Y})</math> is an ''E''-unitary inverse semigroup. Conversely, every ''E''-unitary inverse
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| semigroup is [[isomorphic]] to one of this type.<ref>Howie 1995 : Theorem 5.9.2. Originally, McAlister
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| 1974a,b.</ref>
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| === ''F''-inverse semigroups ===
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| An inverse semigroup is said to be ''F''-inverse if every element has a ''unique'' maximal element above it in the natural partial order, i.e. ever σ-class has a maximal element. Every ''F''-inverse semigroup is an ''E''-unitary monoid. McAlister's covering theorem has been refined by [[M.V. Lawson]] to:
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| '''Theorem.''' Every inverse semigroup has an ''F''-inverse cover.<ref>Lawson 1998 p. 230</ref>
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| McAlister's ''P''-theorem has been used to characterize ''F''-inverse semigroups as well. A McAlister triple <math>(G, \mathcal{X}, \mathcal{Y})</math> is an ''F''-inverse semigroups if and only if <math>\mathcal{Y}</math> is a principal ideal of <math>\mathcal{X}</math> and <math>\mathcal{X}</math> is a semilattice.
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| == Free inverse semigroups ==
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| A construction similar to a [[free group]] is possible for inverse semigroups. A [[presentation]] of the free inverse semigroup on a set ''X'' may be obtained by considering the [[free semigroup with involution]], where involution is the taking of the inverse, and then [[Semigroup#Homomorphisms and congruences|taking the quotient]] by the '''Vagner congruence'''
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| :<math>\{ (xx^{-1}x, x),\; (xx^{-1}yy^{-1}, yy^{-1}xx^{-1})\;|\;x,y \in (X\cup X^{-1})^+ \}. </math>
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| The [[Word problem for groups|word problem]] for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to [[W. D. Munn]] who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees. Multiplication in the free inverse semigroup has a correspondent on [[Munn tree]]s, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details) <!-- can't say much more without getting into way more details-->
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| Any free inverse semigroup is ''F''-inverse.<ref>Lawson 1998, p.230</ref>
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| == Connections with category theory ==
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| The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is
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| another way of composing partial transformations, which is more restrictive than that used above: two partial
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| transformations α and β are composed if, and only if, the image of α is equal to the domain of
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| β; otherwise, the composition αβ is undefined. Under this alternative composition, the collection
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| of all partial one-one transformations of a set forms not an inverse semigroup but an [[inductive groupoid]], in the
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| sense of [[category theory]]. This close connection between inverse semigroups and inductive groupoids is
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| embodied in the ''Ehresmann-Schein-Nambooripad Theorem'', which states that an inductive groupoid can always be
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| constructed from an inverse semigroup, and conversely.<ref>Lawson 1998 : 4.1.8.</ref> More precisely, an inverse semigroup is precisely a groupoid in the category of posets which is an [[étale groupoid]] with respect to its (dual) [[Alexandrov topology]] and whose poset of objects is a meet-semilattice.
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| == Generalisations of inverse semigroups ==
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| As noted above, an inverse semigroup ''S'' can be defined by the conditions (1) ''S'' is a [[regular semigroup]],
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| and (2) the [[idempotent]]s in ''S'' commute; this has led to two distinct classes of generalisations of
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| an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.
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| Examples of regular generalisations of an inverse semigroup are:<ref>Howie 1995 : Section 2.4 & Chapter 6.</ref>
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| *''[[Regular semigroup]]s'': a [[semigroup]] ''S'' is ''regular'' if every element has at least one inverse; equivalently, for each ''a'' in ''S'', there is an ''x'' in ''S'' such that ''axa'' = ''a''.
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| *''Locally inverse semigroups'': a [[regular semigroup]] ''S'' is ''locally inverse'' if ''eSe'' is an inverse semigroup, for each [[idempotent]] ''e''.
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| *''Orthodox semigroups'': a [[regular semigroup]] ''S'' is ''orthodox'' if its subset of [[idempotent]]s forms a subsemigroup.
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| *''Generalised inverse semigroups'': a [[regular semigroup]] ''S'' is called a ''generalised inverse semigroup'' if its [[idempotent]]s form a normal band, i.e., ''xyzx'' = ''xzyx'', for all [[idempotent]]s ''x'', ''y'', ''z''.
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| The [[class (set theory)|class]] of generalised inverse semigroups is the [[intersection (set theory)|intersection]] of the class of
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| locally inverse semigroups and the class of orthodox semigroups.<ref>Howie 1995 : 222.</ref>
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| Amongst the non-regular generalisations of an inverse semigroup are:<ref>Fountain
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| 1979.</ref>[http://www-users.york.ac.uk/~varg1/finitela.ps]
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| *(Left, right, two-sided) adequate semigroups.
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| *(Left, right, two-sided) ample semigroups.
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| *(Left, right, two-sided) semiadequate semigroups.
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| *Weakly (left, right, two-sided) ample semigroups.
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| == See also ==
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| *[[Biordered set]]
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| *[[Pseudogroup]]
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| *[[Partial symmetries]]
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| *[[Regular semigroup]]
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| *[[Semilattice]]
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| *[[Green's relations]]
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| *[[Category theory]]
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| *[[Special classes of semigroups]]
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| *[[Weak inverse]]
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| *[[Nambooripad order]]
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| == Notes ==
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| {{reflist|2}}
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| == References ==
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| *A. H. Clifford and G. B. Preston, ''The Algebraic Theory of Semigroups'', Volume 2, Mathematical Surveys of the American Mathematical Society, No. 7, Providence, R.I., 1967.
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| *{{cite journal | author=J. B. Fountain | title=Adequate semigroups | journal=Proceedings of the Edinburgh Mathematical Society | year=1979 | volume=22 | pages=113–125 | doi=10.1017/S0013091500016230 | issue=02}}
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| *{{cite journal | author=St. Golab | title=Über den Begriff der "Pseudogruppe von Transformationen" | journal=Mathematische Annalen | year=1939 | volume=116 | pages=768–780 | doi=10.1007/BF01597390}}
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| *{{cite journal | author=R. Exel | title=Partial actions of groups and actions of inverse semigroups | journal=Proceedings of the American Mathematical Society | year=1998 | volume=126 | pages=3481–3494 | issue=12 | doi=10.1090/S0002-9939-98-04575-4 }}
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| *V. Gould, [http://www-users.york.ac.uk/~varg1/finitela.ps "(Weakly) left E-ample semigroups"]
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| *J. M. Howie, ''Fundamentals of Semigroup Theory'', Clarendon Press, Oxford, 1995.
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| *M. V. Lawson, ''Inverse Semigroups: The Theory of Partial Symmetries'', World Scientific, 1998.
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| *{{cite journal | author=D. B. McAlister | title=Groups, semilattices and inverse semigroups | journal=Transactions of the American Mathematical Society | year=1974a | volume=192 | pages=227–244 | doi=10.2307/1996831 | publisher=American Mathematical Society | jstor=1996831 }}
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| *{{cite journal | author=D. B. McAlister | title=Groups, semilattices and inverse semigroups II | journal=Transactions of the American Mathematical Society | year=1974b | volume=196 | pages=351–370 | doi=10.2307/1997032 | publisher=American Mathematical Society | jstor=1997032 }}
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| *M. Petrich, ''Inverse semigroups'', Wiley, New York, 1984.
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| *{{cite journal | author=G. B. Preston | title=Inverse semi-groups | journal=Journal of the London Mathematical Society | year=1954a | volume=29 | pages=396–403 | doi=10.1112/jlms/s1-29.4.396 | issue=4 }}
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| *{{cite journal | author=G. B. Preston | title=Inverse semi-groups with minimal right ideals | journal=Journal of the London Mathematical Society | year=1954b | volume=29 | pages=404–411 | doi=10.1112/jlms/s1-29.4.404 | issue=4 }}
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| *{{cite journal | author=G. B. Preston | title=Representations of inverse semi-groups | journal=Journal of the London Mathematical Society | year=1954c | volume=29 | pages=411–419 | doi=10.1112/jlms/s1-29.4.411 | issue=4 }}
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| *{{cite journal | author=B. M. Schein | authorlink=Boris Schein | title=Obituary: Viktor Vladimirovich Vagner (1908–1981) | journal=Semigroup Forum | year=1981 | volume=28 | pages=189–200 | doi=10.1007/BF02676643 }}
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| *{{cite journal | author=B. M. Schein | authorlink=Boris Schein | title=Book Review: "Inverse Semigroups: The Theory of Partial Symmetries" by Mark V. Lawson | journal=Semigroup Forum | year=2002 | volume=65 | pages=149–158 | doi=10.1007/s002330010132 }}
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| *{{cite journal | author=V. V. Wagner | title=Generalised groups | journal=[[Proceedings of the USSR Academy of Sciences]] | year=1952 | volume=84 | pages=1119–1122 }} {{ru icon}} English translation: [http://www.webcitation.org/query?url=http://uk.geocities.com/cdhollings/vagner1.pdf&date=2009-10-25+04:13:20 ]
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| *{{cite journal | author=V. V. Wagner | title=The theory of generalised heaps and generalised groups | journal=Matematicheskii Sbornik|series=Novaya Seriya | year=1953 | volume=32 | pages=545–632 | issue=74}} {{ru icon}}
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| == Further reading ==
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| *For a brief introduction to inverse semigroups, see either Clifford & Preston 1967 : Chapter 7 or Howie 1995 : Chapter 5.
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| *More comprehensive introductions can be found in Petrich 1984 and Lawson 1998.
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| [[Category:Algebraic structures]]
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| [[Category:Semigroup theory]]
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