The semigroup efficiency of groups and monoids (Q2732520)

A semigroup presentation is an ordered pair \(\langle A\mid R\rangle\), where \(A\) is an alphabet and \(R\subseteq A^+\times A^+\), where \(A^+\) is the free semigroup over \(A\). (The set \(R\) is the set of defining relations.) Analogously a monoid presentation and a group presentation are defined. The deficiency of a finite semigroup presentation \(\sigma=\langle A\mid R\rangle\) is \(\text{def}(\sigma)=|R|-|A|\). The minimum of \(\text{def}(\sigma)\) over all finite semigroup presentations \(\sigma\) of a semigroup \(S\) is the deficiency \(\text{def}_S(S)\) of \(S\). If \(\text{def}_S(S)=\text{rank }H_2(S)\), where \(H_2S\) is the second integral homology of \(S\), then \(S\) is called efficient as a semigroup. Analogously a monoid efficient as a monoid and a group efficient as a group is defined. Some theorems concerning monoids and groups which are efficient as semigroups are proved.