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.