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.
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) Zbl 1249.46053
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: doimath/0412478; A note on infinitely distributive inverse semigroups, Semigroup Forum 73 (2006) 156-158; doimath/0506454
B. Steinberg, Strong Morita equivalence of inverse semigroups, Houston J. Math. 37 (2011) 895-927
Revised on March 31, 2014 09:27:58
by Tim Porter
(2.26.41.29)