On the Efficiency of Finite Simple Semigroups

Authors: H. AYIK, C. M. CAMPBELL, J. J. O'CONNOR, N. RUSKUC

Abstract: Let $S$ be a finite simple semigroup, given as a Rees matrix semigroup $\mathcal{M}[G;I,\Lambda ;P]$ over a group $G$. We prove that the second homology of $S$ is $H_{2}(S)=H_{2}(G)\times {\mathbb Z}^{(|I|-1)(|\Lambda |-1)}$. It is known that for any finite presentation $\langle \: A\: |\: R\: \rangle$ of $S$ we have $|R|-|A|\geq \mbox{rank}(H_{2}(S))$; we say that $S$ is efficient if equality is attained for some presentation. Given a presentation $\langle \: A_{1}\: |\: R_{1}\: \rangle$ for $G$, we find a presentation $\langle \: A\: |\: R\: \rangle$ for $S$ such that $|R|-|A|=|R_{1}|-|A_{1}|+(|I|-1)(|\Lambda |-1)+1$. Further, if $R_{1}$ contains a relation of a special form, we show that $|R|-|A|$ can be reduced by one. We use this result to prove that $S$ is efficient whenever $G$ is finite abelian or dihedral of even degree.

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