# 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 $s^* s s^* = s^*$.

## Examples

• 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 $\Gamma(X)$).

## Properties

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

## 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.
• M. Lawson, G. Kurdyavtseva, The classifying space of an inverse semigroup, Period. Math. Hungar., to appear, pdf
• 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 April 23, 2014 05:50:21 by Tim Porter (2.26.36.107)