Efficiency of finite simple semigroups

ABSTRACTMSc THESISON THE EFFİCİENCY OF FİNİTE SİMPLE SEMİGROUPSSadi İNCEOĞLUDEPARTMANT OF MATHEMATICSINSTITUTE OF NATURAL AND APPLIED SCIENCESUNIVERSITY OF ÇUKUROVASupervisor: Prof. Dr. Bilal VATANSEVERYear: 2006, Pages: 58Jury: Prof. Dr. Bilal VATANSEVERAssis.Prof. Dr. Fikret KUYUCUAssis.Prof.. Dr. Ersin KIRALLet S be a finite simple semigroup, given as a Rees matrix semigroupµ [G; I , Λ; P] over a group G .In this thesis we review progress on the efficiency of simple semigroups.Second homology of S is H 2 (S ) = H 2 (G ) × Z ( Ι −1)×( Λ −1) .It is known that for any finite presentation A R of S we haveR − A ≥ rankH 2 ( S ) ; we say that S is efficient if equality is attained for somepresentation. Given a presentation A1 R1 for G , there is a presentation A R forS such that R − A = R1 − A1 + ( I − 1)( Λ − 1) + 1 . Further if R1 contains a relationof a special form, it is shown that R − A can be reduced by one. This result hasbeen used to prove that S is efficient whenever G is abelian or dihedral of evendegree.Key Words: Semigroups, Efficiency, Rees Matrix Semigroup, Second HomologyGroupII