•  
  •  
 

Turkish Journal of Mathematics

DOI

-

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.

First Page

129

Last Page

146

Download

Included in

Mathematics Commons

Share

COinS