群逆伪BCI-代数与群逆滤子(英文)

Anti-grouped Pseudo-BCI Algebras and Anti-grouped Filters

伪BCK-代数是非可换模糊逻辑(蕴涵片段)的基本代数框架,伪BCI-代数是伪BCK-代数的推广,本文研究伪BCI-代数的结构。首先,借助BZ-代数(又称弱BCC-代数)给出伪BCI-代数的一个特征性质;其次,通过引入群逆伪BCI-代数的概念,研究了伪BCI-代数与(非可换)群之间的关系;接着,引入群逆滤子、优滤子和正规滤子的概念,并通过它们给出伪BCI-代数成为群逆伪BCI-代数(以及滤子成为p-滤子)的充要条件;最后,证明了如下结论:(1)平均伪BCI-代数等价于p-半单BCI-代数;(2)伪BCI-代数的每一个滤子是p-滤子,当且仅当它是群逆的且其伴随群的每一个子群是正规子群。

The notion of Pseudo-BCK algebras can be regarded as an algebraic framework （implicational fragment） of non-commutative fuzzy logics. This paper investigates the structures of pseudo-BCI algebras which are generalization pseudo BCK algebras. First, a characterization of pseudo BCI algebra is constructed by the notion of BZ-algebra （or weak BCC-algebra）. Second, the relationship between pseudo-BCI algebras and （non-commutative） groups are investigated by introducing the notion of anti-grouped pseudo-BCl algebra. Third, the notions of anti-grouped （well, normal） filter are introduced, and by these notions the necessary and sufficient conditions for pseudo-BCI algebra to be anti grouped （and filter to be p-filter） are given. Finally, the following results are proved： （1） medial pseudo-BCI algebra is equivalent to p-semisimple BCI atgebra （2） every filter of a pseudo-BCI algebra X is p-filter, if and only if, X is anti-grouped and every subgroup of adjoint group of X is normal subgroup.