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 . It is evident from this that .
Needless to say, a group is an inverse semigroup. More to the point however:
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 ).
If is a meet-semilattice, then is an inverse semigroup under the meet operation.
For any in an inverse semigroup, and are idempotent. If is idempotent, then .
The proof is trivial.
In an inverse semigroup, the product of any two idempotents is idempotent, and any two idempotents commute.
One easily checks that , and that is an idempotent. So is idempotent; as a result, . Thus and similarly are idempotent. Next we have
since are all idempotent, and so , which completes the proof.
Thus the idempotents in an inverse semigroup form a subsemigroup which is commutative and idempotent. Such a structure is the same as a meet-semilattice except for the fact that there might not have an empty meet or top element; that is, we define an order on idempotents by if and only if , whence multiplication of idempotents becomes the binary meet.
For any two elements in an inverse semigroup, .
Since the idempotents commute, we have
and similarly , which is all we need.
For elements in an inverse semigroup, the following are equivalent:
We show ; a similar proof shows Clearly then we have
Given an idempotent such that , we have
A preorder is defined on an inverse semigroup by saying if any of the four conditions of Proposition 2 is satisfied; transitivity follows by equivalence to 1. and closure of idempotents under multiplication. When restricted to idempotents, this preorder coincides with the meet-semilattice order.
If and in an inverse semigroup, then and .
Writing for some idempotent , we have and so . Similarly , so by transitivity. This gives . If for an idempotent , then ; this gives ,
The preorder on an inverse semigroup is a partial order, i.e., if and , then .
From we derive and , and similarly from we derive . Thus since the preorder on idempotents is a meet-semilattice, which is a partial order. Then from we derive .
In this section, an ordered groupoid means an internal groupoid in the finitely complete category of posets Pos. For any finitely complete category , we observe that the forgetful functor , taking an internal groupoid in to the underlying semicategory (remembering only composition of morphisms, forgetting presence of inverses and identity morphisms), has a right adjoint which takes a semicategory to the core groupoid of the category of idempotents attached to a semicategory (see here for details). (This observation is formulated in finite limit logic, and thus by a Yoneda lemma argument, its validity reduces to that of the observation in the special case .)
In particular, this construction may be applied to an inverse semigroup seen as a semigroup in :
The groupoid attached to an inverse semigroup is the core of the category of idempotents of , which as a semigroup in is viewed as a one-object semicategory in .
In more detail: an arrow in is a triple of elements in , where are idempotent elements and is an element such that . Such an arrow is invertible precisely when and , with inverse . Thus the core consists of such arrows .
A key example to keep in mind is the inverse semigroup of partial bijections on a set, where the arrows of the corresponding groupoid are actual invertible maps between subsets. In general, the object part of the associated groupoid is not just a poset, but a poset with binary meets.
The reason for the notation is that this ordered groupoid is a so-called inductive groupoid, defined as follows:
An inductive groupoid is an internal groupoid in with the following additional properties:
The object part admits binary meets;
Given in and in , there exists a unique in with , called the restriction .
Given in and in , there exists a unique in with , called the corestriction .
In fact conditions 2. and 3. in this definition are equivalent. A morphism of inductive groupoids is an internal functor from to in .
For an inductive groupoid, a tensor product may be defined by the rule
where indicates composition in . It may be shown that is an inverse semigroup , and the two notions are equivalent:
(Ehresmann-Schein-Nampooripad) There are canonical isomorphisms and , providing an equivalent of categories .
With only a subtle change in definition, the result is that one gets only groups:
Let be an inhabited semigroup with the property that for every there exists a unique such that . Then is a group.
Since is inhabited, say by an element , it has an idempotent , for example . We will show that for any ; by a similar argument , so that any idempotent is an identity (the identity ), whence the idempotents and equal for any and is a group.
If for unique , then from it follows and hence is an inverse semigroup. The same observation means it is enough to show , since then also , which by uniqueness implies .
The above results on inverse semigroups apply and we derive
as was to be shown.
Mark V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, 1998.
Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics 170, Birkhäuser 1999, MR 1724106
Ruy Exel, Inverse semigroups and combinatorial -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
Darien DeWolf and Dorette Pronk, The Ehresmann-Schein-Nampooripad Theorem for Inverse Categories, arXiv pdf link.