A simplification of D'Alarcao's idempotent separating extensions of inverse semigroups

In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.