The semigroup efficiency of direct powers of groups (Q2709033)

A semigroup \(S\) is said to be defined by the presentation \(\langle A\mid R\rangle\), with \(A\) an alphabet and \(R\subseteq A^+\times A^+\) (\(A^+\) the free semigroup on \(A\)), if \(S\) is isomorphic with the quotient semigroup \(A^+/\rho\) where \(\rho\) is the congruence on \(A^+\) generated by \(R\). A finite semigroup \(S\) is called efficient if it can be defined by a presentation \(\langle A\mid R\rangle\) with \(|R|-|A|=\text{rank }H_2(S^1)\), the rank of the second integral homology of the monoid \(S^1\). In this paper the direct product of groups is considered with respect to this kind of semigroup presentation. It is shown that the direct products of the dihedral groups \(D_{2n}\) and the alternating group \(A_4\) are efficient in the sense above.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00028].