The Bicyclic Monoid! Also, semilattices are inverse (they are regular because of the idempotents (every band is regular) and as they are commutative all their idempotents commute. Thus, they are inverse).

Bicyclic Monoid: The bicyclic monoid is the monoid given by the presentation .

Semilattice: A semilattice is a commutative semigroup of idempotents. For example, take a set, . Then the set of all finite subsets of forms a semigroup under the operation of union. This operation is commutative, and all elements are idempotents. I believe that this is the free join-semilattice. Note that there is a formal definition using the `meet' operation.

Howie's rather fine book, `Fundamentals of Semigroup Theory', contains a chapter on inverse semigroups, and Mark Lawson has a book in this area (it's called something like `inverse semigroups: the theory of partial symmetry'). If you are doing a thesis about semigroups then I presume someone in your department studies them, so Howie's book will be in your uni's library. Lawson's may or may not be (apparently it is in mine!)