Transitive Filters

Abstract

It is a general observation in a BE-algebra that the BE-ordering $\le$ is reflexive but neither antisymmetric nor transitive. If the BE-algebra X is commutative, then the pair $(X, \le )$ is a partially ordered set. If X is transitive, then $\le$ satisfies only the transitive property. This is the exact reason to concentrate on the transitive property of the filters and to introduce the filters called transitive filters. It is also observed that every filter of a BE-algebra satisfies the transitive property whenever the BE-algebra is transitive. Along with the transitive property of filters, the distributive property is also studied in BE-algebras and introduced the notion of distributive and strong distributive filters in BE-algebras. It is also observed that every filter of a BE-algebra satisfies the distributive property whenever the BE-algebra is self-distributive.