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# Inverse semigroup

Practice
An inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy. Inverse semigroups appear in a range of contexts; forexample, they can be employed in the study partial symmetries.Lawson 1998.
(The convention followed in this article will be that of writing a function on the right of its argument, andcomposing functions from left to right - a convention often observed in semigroup theory.)

## Origins

Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in1952First a short announcement in Wagner 1952, then a much more comprehensive exposition in Wagner1953., and by Gordon Preston in Great Britain in 1954Preston 1954a,b,c.. Both authorsarrived at inverse semigroups via the study of partial one-one transformations of a set: a partialfunction|partial transformation α of a set X is a function from A toB, where A and B are subsets of X. Let α and β be partial transformations of a setX; α and β can be composed (from left to right) on the largest domain uponwhich it "makes sense" to compose them:
dom αβ = α $cap$ dom βα-1
where α-1 denotes the preimage under α. Partial transformations had already been studiedin the context of pseudogroups.See, for example, Golab 1939. It was Wagner, however, whowas the first to observe that the composition of partial transformations is a special case of the multiplication ofbinary relations.Schein 2002 : 152. He recognised also that the domain of composition of two partialtransformations may be the empty set, so he introduced an empty transformation to take account of this.With the addition of this empty transformation, the composition of partial transformations of a set becomes aneverywhere-defined associative binary operation. Under this composition, the collection$mathcal\left\{I\right\}_X$ of all partial one-one transformations of a set X forms an inverse semigroup, calledthe symmetric inverse semigroup (or monoid) on X.Howie 1995 : 149. This is the "archetypal"inverse semigroup, in the same way that a symmetric group is the archetypal group. Forexample, just as every group can be embedded in a symmetric group, every inversesemigroup can be embedded in a symmetric inverse semigroup (see below).

## The basics

The inverse of an element x of an inverse semigroup S is usually written x-1. Inverses in aninverse semigroup have many of the same properties as inverses in a group, for example,(ab)-1 = b-1a-1. In an inverse monoid, xx-1 andx-1x are not (necessarily) equal to the identity, but they are both idempotent.Howie1995 : Proposition 5.1.2(1). An inverse monoid S in which xx-1 = 1 =x-1x, for all x in S (a unipotent inverse monoid), is, of course, a group.
There are a number of equivalent characterisations of an inverse semigroup S:Howie 1995 : Theorem5.1.1.
• Every element of S has a unique inverse, in the above sense.
• Every element of S has at least one inverse (S is a regular semigroup) and idempotents commute (that is, the idempotents of S form a semilattice).
• Every $mathcal\left\{L\right\}$-class and every $mathcal\left\{R\right\}$-class contains precisely one idempotent, where $mathcal\left\{L\right\}$ and $mathcal\left\{R\right\}$ are two of Green's relations.
The idempotent in the $mathcal\left\{L\right\}$-class of s is s-1s, whilst theidempotent in the $mathcal\left\{R\right\}$-class of s is ss-1. There is therefore a simplecharacterisation of Green's relations in an inverse semigroup:Howie 1995 : Proposition 5.1.2(1).
$a,mathcal\left\{L\right\},bLongleftrightarrow a^\left\{-1\right\}a=b^\left\{-1\right\}b,quad a,mathcal\left\{R\right\},bLongleftrightarrow$
aa^{-1}=bb^{-1}
Examples of inverse semigroups:
• Every group is an inverse semigroup.
• The bicyclic semigroup is inverse, with (a,b)-1 = (b,a).
• Every semilattice is inverse.
• The Brandt semigroup is inverse.
• The Munn semigroup is inverse.

Unless stated otherwise, E(S) will denote the semilattice of idempotents of an inverse semigroup S.

## The natural partial order

An inverse semigroup S possesses a natural partial order relation ≤ (sometimes denoted by ω)which is defined by the following:Wagner 1952.
$a leq b Longleftrightarrow a=eb,$
for some idempotent e in S. Equivalently,
$a leq b Longleftrightarrow a=bf,$
for some (in general, different) idempotent f in S. In fact, e can be taken to beaa-1 and f to be a-1a.Howie 1995 : Proposition 5.2.1.
The natural partial order is compatible with both multiplication and inversion, that is,Howie 1995 :152-3
$a leq b, c leq d Longrightarrow ac leq bd$
and
$a leq b Longrightarrow a^\left\{-1\right\} leq b^\left\{-1\right\}.$

In a group, this partial order simply reduces to equality, since the identity is theonly idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings,i.e., α ≤ β if, and only if, the domain of α is contained in the domain of β andxα = xβ, for all x in the domain of α.Howie 1995 : 153.
The natural partial order on an inverse semigroup interacts with Green's relations as follows: if st and s$,mathcal\left\{L\right\},$t, then s = t. Similarly, ifs$,mathcal\left\{R\right\},$t.Lawson 1998 : Proposition 3.2.3.
On E(S), the natural partial order becomes:
$e leq f Longleftrightarrow e = ef,$
so the product of any two idempotents in S is equal to the lesser of the two, with respect to≤. If E(S) forms a chain (i.e., E(S) is totally ordered by ≤), thenS is a union of groups.Clifford & Preston 1967 : Theorem7.5

## Homomorphisms and representations of inverse semigroups

A homomorphism (or morphism) of inverse semigroups is defined in exactly the same way as for any othersemigroup: for inverse semigroups S and T, a function θ from S to Tis a morphism if (sθ)(tθ) = (st)θ, for all s,t in S. The definition of amorphism of inverse semigroups could be augmented by including the condition (sθ)-1 =s-1θ, however, there is no need to do so, since this property follows from the abovedefinition, via the following theorem:
Theorem. The homomorphic image of an inverse semigroup is an inverse semigroup; theinverse of an element is always mapped to the inverse of the image of thatelement.Clifford & Preston 1967 : Theorem 7.36.
One of the earliest results proved about inverse semigroups was the Wagner-Preston Theorem, which is an analogueof Cayley's Theorem for groups:
Wagner-Preston Theorem. If S is an inverse semigroup, then the function φfrom S to $mathcal\left\{I\right\}_S$, given by
dom φ = Sa-1 and x(aφ) = xa
is a faithful representation of S.Howie 1995 : Theorem 5.1.7. Originally, Wagner 1952 and,independently, Preston 1954c.
Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup.

## Congruences on inverse semigroups

Congruences are defined on inverse semigroups in exactly the same way as for any other semigroup: acongruence ρ is an equivalence relation which is compatible with semigroup multiplication, i.e.,
$a,rho,b,quad c,rho,dLongrightarrow ac,rho,bd.$Howie 1995   22

Of particular interest is the relation $sigma$, defined on an inverse semigroup S by
$a,sigma,bLongleftrightarrow$ there exists a $cin S$ with $cleq$
a,b.Lawson 1998 : 62It can be shown that σ is a congruence and that the factor semigroup S/σ is, in fact, a group.Indeed, σ is the smallest congruence on S such that S/σ is a group, that is, if τ is anyother congruence on S with S/τ a group, then σ is contained in τ. The congruence σ iscalled the minimum group congruence on S.Lawson 1998 : Theorem 2.4.1. The minimum groupcongruence can be used to give a characterisation of E-unitary inverse semigroups (see below).
A congruence ρ on an inverse semigroup S is called idempotent pure if
$ain S, ein E\left(S\right), a,rho,eLongrightarrow ain E\left(S\right).$Lawson 1998   65

## E-unitary inverse semigroups

One class of inverse semigroups which has been studied extensively over the years is the class of E-unitaryinverse semigroups: an inverse semigroup S (with semilattice E of idempotents) isE-unitary if, for all e in E and all s in S,
$es in E Longrightarrow s in E.$
Equivalently,
$se in E Rightarrow s in E.$Howie 1995   192.

One further characterisation of an E-unitary inverse semigroup S is the following: if e is in E andes, for some s in S, then s is in E.Lawson 1998 : Proposition 2.4.3.
Theorem. Let S be an inverse semigroup with semilattice E of idempotents, and minimum groupcongruence σ. Then the following are equivalent:Lawson 1998 : Theorem 2.4.6.
• S is E-unitary;
• σ is idempotent pure;
• $sim$ = σ,
where $sim$ is the compatibility relation on S, defined by
$asim bLongleftrightarrow ab^\left\{-1\right\},a^\left\{-1\right\}b$ are idempotent.

Central to the study of E-unitary inverse semigroups is the following construction.Howie 1995 : 193-4 Let $mathcal\left\{X\right\}$ be a partially ordered set, with ordering ≤, and let $mathcal\left\{Y\right\}$ be a subset of $mathcal\left\{X\right\}$ with the properties that
• $mathcal\left\{Y\right\}$ is a lower semilattice, that is, every pair of elements A, B in $mathcal\left\{Y\right\}$ has a greatest lower bound A $wedge$ B in $mathcal\left\{Y\right\}$ (with respect to ≤);
• $mathcal\left\{Y\right\}$ is an order ideal of $mathcal\left\{X\right\}$, that is, for A, B in $mathcal\left\{X\right\}$, if A is in $mathcal\left\{Y\right\}$ and BA, then B is in $mathcal\left\{Y\right\}$.

Now let G be a group which acts on $mathcal\left\{X\right\}$ (on the left), such that
• for all g in G and all A, B in $mathcal\left\{X\right\}$, gA = gB if, and only if, A = B;
• for each g in G and each B in $mathcal\left\{X\right\}$, there exists an A in $mathcal\left\{X\right\}$ such that gA = B;
• for all A, B in $mathcal\left\{X\right\}$, AB if, and only if, gAgB;
• for all g, h in G and all A in $mathcal\left\{X\right\}$, g(hA) = (gh)A.

The triple $\left(G, mathcal\left\{X\right\}, mathcal\left\{Y\right\}\right)$ is also assumed to have the following properties:
• for every X in $mathcal\left\{X\right\}$, there exists a g in G and an A in $mathcal\left\{Y\right\}$ such that gA = X;
• for all g in G, g$mathcal\left\{Y\right\}$ and $mathcal\left\{Y\right\}$ have nonempty intersection.

Such a triple $\left(G, mathcal\left\{X\right\}, mathcal\left\{Y\right\}\right)$ is called a McAlister triple. A McAlister triple isused to define the following:
together with multiplication
$\left(A,g\right)\left(B,h\right)=\left(A wedge gB, gh\right)$.
Then $P\left(G, mathcal\left\{X\right\}, mathcal\left\{Y\right\}\right)$ is an inverse semigroup under this multiplication, with(A,g)-1 = (g-1A, g-1). One of the main results in the study ofE-unitary inverse semigroups is McAlister's P-Theorem:
McAlister's P-Theorem. Let $\left(G, mathcal\left\{X\right\}, mathcal\left\{Y\right\}\right)$ be a McAlister triple. Then $P\left(G,mathcal\left\{X\right\}, mathcal\left\{Y\right\}\right)$ is an E-unitary inverse semigroup. Conversely, every E-unitary inversesemigroup is isomorphic to one of this type.Howie 1995 : Theorem 5.9.2. Originally, McAlister1974a,b.

## Connections with category theory

The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is ananother way of composing partial transformations, which is more restrictive than that used above: two partialtransformations α and β are composed if, and only if, the image of α is equal to the domain ofβ; otherwise, the composition αβ is undefined. Under this alternative composition, the collectionof all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in thesense of category theory. This close connection between inverse semigroups and inductive groupoids isembodied in the Ehresmann-Schein-Nambooripad Theorem, which states that an inductive groupoid can always beconstructed from an inverse semigroup, and conversely.Lawson 1998 : 4.1.8.

## Generalisations of inverse semigroups

As noted above, an inverse semigroup S can be defined by the conditions (1) S is a regular semigroup,and (2) the idempotents in S commute; this has led to two distinct classes of generalisations ofan inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.
Examples of regular generalisations of an inverse semigroup are:Howie 1995 : Section 2.4 & Chapter 6.
• Regular semigroups: 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.
• Locally inverse semigroups: a regular semigroup S is locally inverse if eSe is an inverse semigroup, for each idempotent e.
• Orthodox semigroups: a regular semigroup S is orthodox if its subset of idempotents forms a subsemigroup.
• Generalised inverse semigroups: a regular semigroup S is called a generalised inverse semigroup if its idempotents form a normal band, i.e., xyzx = xzyx, for all idempotents x, y, z.

The class of generalised inverse semigroups is the intersection of the class oflocally inverse semigroups and the class of orthodox semigroups.Howie 1995 : 222.
Amongst the non-regular generalisations of an inverse semigroup are:Fountain1979.[1]
• (Left, right, two-sided) adequate semigroups.
• (Left, right, two-sided) ample semigroups.
• (Left, right, two-sided) semiadequate semigroups.
• Weakly (left, right, two-sided) ample semigroups.

For a brief introduction to inverse semigroups, see either Clifford & Preston 1967 : Chapter 7 or Howie 1995 :Chapter 5. More comprehensive introductions can be found in Petrich 1984 and Lawson 1998.

## References

• 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.
• V. Gould, "(Weakly) left E-ample semigroups"
• J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.
• M. V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998.
• M. Petrich, Inverse semigroups, Wiley, New York, 1984.