On the efficiency of finite simple semigroups (Q2703128)

The deficiency \(\text{def}(S)\) of a finitely presented semigroup \(S\) is defined to be the minimum of the differences \(|R|-|A|\) of the finite presentations \(\langle A\mid R\rangle\) of \(S\). By an unpublished result due to J. Pride, \(\text{def}(S)\geq\text{rank}(H_2(S))\), where \(H_2(S)\) denotes the second integral homology of \(S^1\) (and \(S^1\) equals \(S\) with an identity element adjoined). The authors consider finite simple semigroups. Such a semigroup \(S\) is isomorphic to a finite Rees semigroup \({\mathcal M}[G;I,\Lambda;P]\) over a group \(G\). It is first shown that the second homology of \(S\) is \(H_2(S)=H_2(G)\times\mathbb{Z}^{(|I|-1)(|\Lambda|-1)}\). By constructing specific presentations, the authors show that \(\text{def}(S)\leq\text{def}(G)+(|I|-1)(|\Lambda|-1)+1\), and if \(G\) is a finite Abelian group or a dihedral group with even degree, then \(S\) is efficient, that is, \(\text{def}(S)=\text{rank}(H_2(S))\).