An inverse semigroup is a semigroup (a set with an associative binary operation) such that for every element , there exists a unique “inverse” such that and .
The fundamental example is the following: for any set , let be the set of all partial bijections on , i.e. bijections between subsets of . 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 is a topological space, let consist of the homeomorphisms between open subsets of . Then is a pseudogroup of transformations on (a general pseudogroup of transformations is a sub-inverse-semigroup of ).
Lots to say here: the meet-semilattice of idempotents, the connection with ordered groupoid?s, various representation theorems.
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