Minimal presentations and efficiency of semigroups (Q1971716)

Let \(A\) be an alphabet and let \(A^+\) be the free semigroup generated by \(A\). A semigroup presentation is an ordered pair \({\mathcal P}=\langle A|R\rangle\), where \(R\subset A^+\times A^+\). The deficiency \(\text{def}({\mathcal P})\) of \(\mathcal P\) is the number \(|R|-|A|\). The deficiency \(\text{def}(S)\) of a finitely presented semigroup \(S\) is defined to be \(\min\{\text{def}({\mathcal P})\mid{\mathcal P}\) is a finite presentation of \(S\}\). In general, \(\text{def}(S)\geq\text{rank}(H_2(S))\) for any finite semigroup \(S\), where \(H_2(S)=\text{Tor}^{\mathbb{Z} S}_2(\mathbb{Z},\mathbb{Z})\) is the second homology group of \(S\). A finite semigroup \(S\) is efficient if \(\text{def}(S)=\text{rank}(H_2(S))\). The authors show that finite Abelian groups, the dihedral groups \(D_{2n}\) with \(n\) even, and finite rectangular bands are efficient. On the other hand, finite zero semigroups and finite semilattices, though they are Abelian, are not efficient.