导　　师： 方捷

学科专业： G0101

授予学位： 博士

作　　者： ;

机构地区： 汕头大学

摘　　要： 在这篇博士论文中,我们首先将布尔代数的互补性推广,提出了GBω代数. GBω代数/(L;f/)是Ockham代数的一类子代数,在这类代数中,对于任意的x∈L,存在n≥0,使得fn/(x/)与fn+1/(x/)互补.我们研究了此类代数的基本性质,刻画了它的同余关系的结构.特别的我们得出/(L;f/)的同余格ConL是布尔格当且仅当/(L;f/)是有限的布尔代数.我们还给出了次直不可约的GBω代数/(L;f/)是链的充要条件. 其次,我们研究了S1代数的主同余性质. S1代数是GBω代数的子代数,它包含布尔代数和Stone代数.我们得出一个S1代数具有主同余性质当且仅当它是有限的,且它的确定同余是主同余,它的每一个确定同余类都具有格的主同余性质.之后我们刻画了具有主同余性质的GBn链. 然后,我们研究了bpO代数上的同余交换性质. bpO代数是Ockham代数和伪补代数的一种结合代数,我们运用Priestley拓扑对偶空间的方法刻画了具有同余交换性质的bpO代数. 最后,我们研究了具有正则图的有限格,称之为正则图格.我们得出,一个有限格是分配的正则图格当且仅当它是布尔格.而且我们找出了所有的1阶和2阶正则图格,还证明了8-元素的布尔格是最小的3阶正则图格. In this thesis we ?rstly study GBωalgebras. A GBωalgebra /(L; f/) is anOckham algebra, where ?x∈L, there exists n≥0 such that fn/(x/) and fn+1/(x/)are complementary. We characterized the structure of the lattice of congruenceson such an algebra /(L; f/). In particular we show that the lattice of congruenceson L is Boolean if and only if L is a ?nite Boolean algebra. We also give asu?cient and necessary condition for the subdirectly irreducible GBωchains. Then we study the principal congruences on S1-algebras. The class of S1-algebras is a subclass of GBωalgebras, and it includes Boolean algebras andStone algebras. We show that an S1 algebra has the principal congruenceproperty if and only if it is ?nite, the determination congruence is principal,and every determination class has the same property. We also ?nd all GBnchains which have the principal congruence property. And then we look for those bpO algebras on which all congruences are per-mutable. bpO algebras are the mixture of Ockham algebras and pseudocom-plemented algebras. We give a characterization of the permutable congruenceson bpO algebras via Priestley duality. Finally, we study the lattices whose Hass diagrams are regular graphs, namedregular graph lattices. We show that a ?nite distributive lattice is a regulargraph lattice if and only if it is a Boolean lattice. We also ?nd out all the1-degree graph lattices and 2-degree graph lattices. In particular, we show thatthe 8-element Boolean lattice is the smallest 3-degree regular graph lattice.

关 键 词： 代数 代数 主同余 同余交换性质 正则图格

分 类 号： [TM]

领　　域： [电气工程]