Left restriction semigroups - embeddability into W-products

Left restriction (called also weakly left E-ample) semigroups have appeared in the literature many times, under different names, since the 1960's. They are considered non-regular generalizations of inverse semigroups. The fundamental structure theorems, due to D. McAlister and L. O'Carroll, giving inverse semigroups by means of semilattices and groups via E-unitary inverse semigroups were generalized for left restriction semigroups by J. Fountain, G. Gomes and V. Gould. However, a generalization of the approach via almost factorizable inverse semigroups, due to D. McAlister and M. Lawson, is still missing. A question which might be important in developing such a generalization is to ask which left restriction semigroups are embeddable into W-products of semilattices by monoids. We present characterizations of these left restriction semigroups obtained recently in a joint work with V. Gould.