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.
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) MR2821242Zbl 1249.46053arxiv/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: 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 January 5, 2015 05:32:50
by David Roberts
(127.0.0.1)