Almost factorisable straight locally inverse semigroups

We introduce the notion of a factorisable and an almost factorisable straight locally inverse semigroup, and prove that every straight locally inverse semigroup can be obtained as a subsemigroup of an idempotent separating homomorphic image of a Pastijn product of a semilattice by a completely simple semigroup. Moreover, we give an alternative proof of the fact that each straight locally inverse semigroup has a weakly E-unitary cover, implicitely due to Pastijn.