BL代数的(∈,∈∨q)-模糊滤子理论

Theory of(∈,∈∨q)-fuzzy Filters in BL-algebras

对BL代数的(∈,∈∨q)-模糊滤子理论作系统研究。首先,在BL代数中引入(∈,∈∨q)-模糊对合滤子和(∈,∈∨q)-模糊结合滤子两类新概念,获得了这两类(∈,∈∨q)-模糊滤子的几个等价刻画。其次,详细讨论了BL代数中各类(∈,∈∨q)-模糊滤子间的关系,证明了一个模糊集为(∈,∈∨q)-模糊布尔(关联)滤子当且仅当它既是(∈,∈∨q)-模糊正关联滤子又是(∈,∈∨q)-模糊对合滤子。最后,以直观图示的方式对BL代数中各类(∈,∈∨q)-模糊滤子间关系进行了总结。

In this paper, the theory of （∈,∈∨q）-fuzzy filters in BL-algebras was studied systematically. Firstly, two notions of （∈,∈∨q）-fuzzy involution filters and （∈,∈∨q）-fuzzy associative filters are introduced and some characterizations of them are obtained. Secondly, relations among all kinds of （∈,∈∨q）-fuzzy filters are discussed. It is proved that a fuzzy set is a （∈,∈∨q）-fuzzy Boolean （implicative） filter if and only if it is both a （∈,∈∨q）-fuzzy positive implicative and a （∈,∈∨q）-fuzzy involution filter. Finally, we summarized the relations among all kinds of （∈,∈∨q）-fuzzy filters by an illustrative figure.