# nLab inverse semigroup

## Definition

An inverse semigroup is a semigroup $S$ (a set with an associative binary operation) such that for every element $s\in S$, there exists a unique “inverse” $s^*\in S$ such that $s s^* s = s$ and $s^* s s^* = s^*$. It is evident from this that $s^{\ast\ast} = s$.

## Examples

Needless to say, a group is an inverse semigroup. More to the point however:

• The fundamental example is the following: for any set $X$, let $I(X)$ be the set of all partial bijections on $X$, i.e. bijections between subsets of $X$. The composite of partial bijections is their composite as relations (or as partial functions).

This inverse semigroup plays a role in the theory similar to that of permutation groups in the theory of groups. It is also paradigmatic of the general philosophy that

Groups describe global symmetries, while inverse semigroups describe local symmetries.

Other examples include:

• If $X$ is a topological space, let $\Gamma(X)\subseteq I(X)$ consist of the homeomorphisms between open subsets of $X$. Then $\Gamma(X)$ is a pseudogroup of transformations on $X$ (a general pseudogroup of transformations is a sub-inverse-semigroup of $\Gamma(X)$).

• If $L$ is a meet-semilattice, then $L$ is an inverse semigroup under the meet operation.

## Properties

Lots to say here: the meet-semilattice of idempotents, the connection with ordered groupoid?s, various representation theorems.

###### Proposition

For any $x$ in an inverse semigroup, $x x^\ast$ and $x^\ast x$ are idempotent. If $e$ is idempotent, then $e^\ast = e$.

The proof is trivial.

###### Lemma

In an inverse semigroup, the product of any two idempotents $e, f$ is idempotent, and any two idempotents commute.

###### Proof

One easily checks that $(e f)^\ast = f(e f)^\ast e$, and that $f(e f)^\ast e$ is an idempotent. So $(e f)^\ast$ is idempotent; as a result, $(e f)^\ast = e f$. Thus $e f$ and similarly $f e$ are idempotent. We then have

$e f (f e) e f = e f^2 e^2 f = e f e f = e f, \qquad f e (e f) f e = f e^2 f^2 e = f e f e = f e$

since $e, f, e f, f e$ are all idempotent, and so $f e = (e f)^\ast = e f$, completing the proof.

Thus the idempotents in an inverse semigroup form a subsemigroup which is commutative and idempotent. Such a structure is the same as a meet-semilattice except for the fact that there might not have an empty meet or top element; that is, we define an order $\leq$ on idempotents by $e \leq f$ if and only if $e = e f$, whence multiplication of idempotents becomes the binary meet.

###### Lemma

For any two elements $x, y$ in an inverse semigroup, $(x y)^\ast = y^\ast x^\ast$.

###### Proof

Since the idempotents $x^\ast x, y y^\ast$ commute, we have

$x y (y^\ast x^\ast) x y = x (y y^\ast)(x^\ast x) y = x x^\ast x y y^\ast y = x y$

and similarly $y^\ast x^\ast (x y)y^\ast x^\ast = y^\ast y y^\ast x^\ast x x^\ast = y^\ast x^\ast$, which is all we need.

###### Proposition

For elements $x, y$ in an inverse semigroup, the following are equivalent:

1. There exists an idempotent $e$ such that $x = e y$,
2. $x = x x^\ast y$,
3. There exists an idempotent $f$ such that $x = y f$,
4. $x = y x^\ast x$.
###### Proof

We show $3. \Rightarrow 2.$; a similar proof shows $1. \Rightarrow 4.$ Clearly then we have $3. \Rightarrow 2. \Rightarrow 1. \Rightarrow 4. \Rightarrow 3.$

Given an idempotent $f$ such that $x = y f$, we have

$\array{ x & = & y f & \\ & = & y y^\ast y f & \\ & = & y f y^\ast y & since \; idempotents \; commute \\ & = & y f f y^\ast y & \\ & = & y f f^\ast y^\ast y & \text{Proposition 1} \\ & = & y f (y f)^\ast y & \text{Lemma 2} \\ & = & x x^\ast y & }$

which gives $3. \Rightarrow 2.$

A partial order $\leq$ is defined on an inverse semigroup by saying $x \leq y$ if any of the four conditions of Proposition 2 is satisfied. When restricted to idempotents, this order coincides with the meet-semilattice order.

###### Proposition

If $a \leq b$ and $x \leq y$ in an inverse semigroup, then $a x \leq b y$ and $x^\ast \leq y^\ast$.

###### Proof

We observe from Proposition 2 that for any elements $c, x$ we have $c x^\ast x = x x^\ast c$. From the hypotheses $a = a a^\ast b$ and $x = y x^\ast x$ we have $a x = a a^\ast b y x^\ast x = (x x^\ast) a a^\ast b y$ by our observation. But $e = (x x^\ast)(a a^\ast)$ is idempotent by Lemma 1. This gives $a x \leq b y$. If $x = e y$ for an idempotent $e$, then $x^\ast = (e y)^\ast = y^\ast e^\ast = y^\ast e$; this gives $x^\ast \leq y^\ast$,

## References

• Mark V. Lawson, tutorial lectures on semigroups in Ottawa, notes available here.
• Mark V. Lawson, Constructing ordered groupoids, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123–138, URL stable
• Mark V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, 1998.
• Mark V. Lawson, G. Kurdyavtseva, The classifying space of an inverse semigroup, Period. Math. Hungar., to appear, as preprint: ArXiv 1210.4421
• Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics 170, Birkhäuser 1999, MR 1724106
• Alcides Buss, Ruy Exel, Ralf Meyer, Inverse semigroup actions as groupoid actions, Semigroup Forum 85 (2012), 227–243, arxiv/1104.0811
• Ruy Exel, Inverse semigroups and combinatorial $C^\ast$-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313, doi MR 2419901
• Alcides Buss, Ruy Exel, Fell bundles over inverse semigroups and twisted étale groupoids, J. Oper. Theory 67, No. 1, 153-205 (2012) MR2821242 Zbl 1249.46053 arxiv/0903.3388journal; Twisted actions and regular Fell bundles over inverse semigroups, arxiv/1003.0613
• Pedro Resende, Lectures on étale groupoids, inverse semigroups and quantales, Lecture Notes for the GAMAP IP Meeting, Antwerp, 4-18 Sep 2006, 115 pp. pdf; Étale groupoids and their quantales, Adv. Math. 208 (2007) 147-209; also published electronically: doi math/0412478; A note on infinitely distributive inverse semigroups, Semigroup Forum 73 (2006) 156-158; doi math/0506454
• B. Steinberg, Strong Morita equivalence of inverse semigroups, Houston J. Math. 37 (2011) 895-927
Revised on November 28, 2015 21:09:50 by Todd Trimble (67.81.95.215)