Some Studies On Some Dimensions Of Semigroups And Semigroup Algebras

This thesis consists of six chapters. We mainly investigate the Gelfand-Kirillov dimensions and homological dimensions of some primitive abundant semigroups and Rees matrix semigroups and their semigroup algebras.The first chapter is introduction and preliminaries.In Chapter2, we consider the growth of some finitely generated primitive regular semigroups. Then the growth and linear growth of a finitely generated linear primitive regular semigroups are investigated.In Chapter3, Gelfand-Kirillov dimensions of some primitive abundant semi-groups are investigated. It is shown that for certain primitive abundant semigroup S, S as well as the semigroup algebra K[S] has polynomial growth if and only if all of its cancellative submonoids T as well as K[T] have polynomial growth. As an application, it is established a theorem about Gelfand-Kirillov dimension of a finitely generated primitive inverse monoid having the permutational property.In Chapter4, Gelfand-Kirillov dimensions of some Rees matrix semigroups are investigated. It is shown that for Rees matrix semigroup S, S has polynomial growth if and only if all of its submonoids M have polynomial growth.In Chapter5, first a structure theorem of primitively abundant semigroup is given. If S is a monoid with a principal*-ideal chain S=S1(?)…(?)Sn such that each Si/Si+1is a primitively decomposable abundant semigroup, we obtain an upper bound for the homological dimension of S. As applications, the homological dimensions of finite completely (?)*-simple semigroup and inverse monoid are given.In Chapter6, we consider the homological dimension of finite Rees matrix semigroup algebra K[S]. First we give some characterizations of Rees matrix semi-group algebras. Then a bound of homological dimension of K[S] is given. Finally, the relation between the homological dimension of semigroup algebra of a finite type A semigroup and that of semigroup algebra of its PA diagonal block Rees matrix semigroup is given.