It is known that for any finite group G given by a finite group presentation \(\) there exists a finite semigroup presentation \(\) for G of the same deficiency, i.e. satisfying \(\). 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.
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