THE SEMIGROUP EFFICIENCY OF DIRECT POWERS OF GROUPS

A finite semigroup S is said to be efficient if it can be defined by a semigroup presentation 〈 A | R 〉 with |R| - |A| = rank(H₂(S¹)) where H₂(S¹)is the second integral homology of the monoid S¹ obtained from S by adjoining an identity. In this paper we show that certain classes of direct powers of finite groups are efficient as semigroups.