兰州大学机构库 >数学与统计学院
关于半群和半群代数的维数的若干研究
Alternative TitleSome studies on some dimensions of semigroups and semigroup algebras
崔冉冉
Thesis Advisor罗彦锋
2012-12-01
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword本原富足半群 本原可分解富足半群 Rees 矩阵半群 逆半群 Gelfand-Kirillov 维数 同调维数 极大可消子幺半群 *-理想
Abstract本文主要研究某些本原富足半群和 Ress 矩阵半群及其半群代数的Gelfand-Kirillov 维数和同调维数. 共分六章. 第一章为本文的引言和预备知识. 第二章研究有限生成的本原正则半群的增长和 Gelfand-Kirillov 维数.同时给出线性正则半群的增长和线性增长的性质刻画. 第三章研究一类本原富足半群的 Gelfand-Kirillov 维数. 对于某类本原富足半群 S, S 及其半群代数 K[S]有多项式增长当且仅当它的所有的可消子幺半群 T 及其半群代数 K[T] 有多项式增长. 作为应用,给出有限生成的有置换性质的本原逆幺半群的关于GK-维数的一个结论定理. 第四章研究一类 Rees 矩阵半群的 Gelfand-Kirillov 维数. 对于 Rees 矩阵半群 S, S 有多项式增长当且仅当它的所有的子幺半群 M 有多项式增长. 第五章首先给出本原富足半群的一个结构定理. 如果 S 是幺半群,有主 *-理想链 $S=S_1\supseteq\cdots \supseteq S_n$, 且满足每个 $S_i/S_{i+1}$ 是本原可分解富足半群, 那么我们得到 K[S]的同调维数的一个上界. 作为应用, 考虑有限完全 $\mathscr{J}^*$-单半群和有限逆幺半群的同调维数. 第六章研究 Rees 矩阵半群代数 K[S] 的同调维数.首先, 我们给出 Rees 矩阵半群代数的一些特征刻画. 之后给出 K[S] 的同调维数的一个上界. 最后考虑有限型 A 半群的代数的同调维数和它的 PA 对角块 Rees矩阵的同调维数的关系.
Other AbstractThis 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 Chapter 2, 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 Chapter 3, Gelfand-Kirillov dimensions of some primitive abundant semigroups 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 Chapter 4, 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 Chapter 5, first a structure theorem of primitively abundant semigroup is given. If S is a monoid with a principal *-ideal chain $S=S_1\supseteq\cdots \supseteq S_n$ such that each $S_i/S_{i+1}$ is a primitively decomposable abundant semigroup, we obtain an upper bound for the homological dimension of S. As applications, the homological dimensions of finite completely $\mathscr{J}^*$-simple semigroup and inverse monoid are given. In Chapter 6, we consider the homological dimension of finite Rees matrix semigroup algebra K[S]. First we give some characterizations of Rees matrix semigroup 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.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225364
Collection数学与统计学院
Recommended Citation
GB/T 7714
崔冉冉. 关于半群和半群代数的维数的若干研究[D]. 兰州. 兰州大学,2012.
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