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.