Comparing Semigroup and Monoid Presentations for Finite Monoids

It is known that for any finite group G given by a finite group presentation (A\R) there exists a finite semigroup presentation <B\Q> for G of the same deficiency, i.e. satisfying \Q\ - \B\ = \R\ - \A\. It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency.