Factorisability in certain classes over inverse semigroups

In the structure theory of inverse semigroups, there are two approaches, basically from the 1970’s, to build up inverse semigroups from semilattices and groups via their semidirect products. These approaches are dual to each other in the sense that one produces any inverse semigroup from a semidirect product of a semilattice by a group by taking an idempotent separating homomorphic image of an inverse subsemigroup, and the other one by taking an inverse subsemigroup of an idempotent separating homomorphic image. A crucial role is played by E-unitary inverse semigroups and by almost factorisable inverse semigroups since they turn out to be just the inverse subsemigroups and the (idempotent separating) homomorphic images, respectively, of semidirect products of semilattices by groups. Since then a number of structure theorems have been obtained generalising the first approach. Generalisations go in two main directions. In one of them the regularity of the semigroups considered is retained and the condition that the idempotents commute is weakened, and in the other one the other way around. The main question in these investigations is to determine which semigroups are embeddable in a semidirect product of a special kind. The second approach seems to be more difficult to generalise, and till now very little is known for wider classes. Here the crucial problem is to determine the (idempotent separating) homomorphic images of semidirect products of a given type. We present results obtained in this direction for the class of orthodox semigroups (partly by M. Hartmann) and for the class of weakly ample semigroups (in a joint work with G. Gomes).