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!)