Journals / Turkish Journal of Mathematics / 2000 / Cilt: 24 - Sayı: 2
On the Efficiency of Finite Simple Semigroups
- Pages
- 129–146
- DOI
- —
Abstract
Let S be a finite simple semigroup, given as a Rees matrix semigroup \mathcal{M}[G;I,L ;P] over a group G. We prove that the second homology of S is H2(S)=H2(G)\times {\mathbb Z}(|I|-1)(|L |-1). It is known that for any finite presentation \langle \: A\: |\: R\: \rangle of S we have |R|-|A|\geq \mbox{rank}(H2(S)); we say that S is efficient if equality is attained for some presentation. Given a presentation \langle \: A1\: |\: R1\: \rangle for G, we find a presentation \langle \: A\: |\: R\: \rangle for S such that |R|-|A|=|R1|-|A1|+(|I|-1)(|L |-1)+1. Further, if R1 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.
Özet
Let S be a finite simple semigroup, given as a Rees matrix semigroup \mathcal{M}[G;I,L ;P] over a group G. We prove that the second homology of S is H2(S)=H2(G)\times {\mathbb Z}(|I|-1)(|L |-1). It is known that for any finite presentation \langle \: A\: |\: R\: \rangle of S we have |R|-|A|\geq \mbox{rank}(H2(S)); we say that S is efficient if equality is attained for some presentation. Given a presentation \langle \: A1\: |\: R1\: \rangle for G, we find a presentation \langle \: A\: |\: R\: \rangle for S such that |R|-|A|=|R1|-|A1|+(|I|-1)(|L |-1)+1. Further, if R1 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.