Finite nilpotent semigroups, Ash's Theorem and one dimensional tilings

In one of his beautiful theorems, C.J. Ash proved that the pseudo variety generated by finite inverse semigroups is precisely the pseudo variety of finite semigroups with commuting idempotents. It follows that every finite nilpotent semigroup divides a finite inverse semigroup. In this talk we consider a specific division of this type and show how it is related to free inverse semigroups and to the one dimensional tiling semigroups of Mark Lawson.