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:
- 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 ).
If is a meet-semilattice, then is an inverse semigroup under the meet operation.
Lots to say here: the meet-semilattice of idempotents, the connection with ordered groupoid?s, various representation theorems.
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. We then have
since are all idempotent, and so , completing 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:
- There exists an idempotent such that ,
- There exists an idempotent such that ,
We show ; a similar proof shows Clearly then we have
Given an idempotent such that , we have
A partial order is defined on an inverse semigroup by saying if any of the four conditions of Proposition 2 is satisfied. When restricted to idempotents, this order coincides with the meet-semilattice order.
If and in an inverse semigroup, then and .
We observe from Proposition 2 that for any elements we have . From the hypotheses and we have by our observation. But is idempotent by Lemma 1. This gives . If for an idempotent , then ; this gives ,
- Mark V. Lawson, tutorial lectures on semigroups in Ottawa, notes available here.
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