Minimal presentations and efficiency of semigroups

A finite semigroup S is said to be efficient if it can be defined by a presentation [A \ R] with \R\ - \A\ = rank(H-2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D-2n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.