We introduce two generalizations of the full inverse symmetric semigroup ISX — the inverse semigroups PIPX and P̃IPX . Both of these semigroups have the same set of elements and contain ISX . The binary operations on PIPX and P̃IPX are two natural generalizations of the multiplication of partial permutations and each of the considered operations coincides with this multiplication when restricted ...

A semilattice decomposition of an inverse semigroup has good internal mapping properties. These are used to give natural proofs of some embedding theorems, which were originally proved in a rather artificial way. The reader is referred to [1] for the basic theory of inverse semigroups. In an earlier paper [3] we proved the following embedding result: (1) An E-unitary inverse semigroup is isomor...

A new construction for free inverse semigroups was obtained by Poliakova and Schein in 2005. Based on their result, we find Gröbner-Shirshov bases for free inverse semigroups with respect to the deg-lex order of words. In particular, we give the (unique and shortest) normal forms in the classes of equivalent words of a free inverse semigroup together with the Gröbner-Shirshov algorithm to trans...