On regular filters and well filters of pseudo-BCI algebras

Abstract

Pseudo-BCI algebra is a kind of non-classical logic algebra; it is a generalization of pseudo-BCK algebra which is closely related to non-commutative fuzzy logic algebras. In this paper, a new notion of regular filter of pseudo-BCI algebra is proposed, and a characteristic property is given. Moreover, the relationships among regular filters and other filters of pseudo-BCI algebras are presented, and the following important propositions are proved: (1) the notions of well filter and normal filter in pseudo-BCI algebras are coincide; (2) a filter of pseudo-BCI algebra is p-filter iff (that is, if and only if) it is well anti-grouped filter (or normal anti-grouped filter); (3) a filter of pseudo-BCI algebra is regular iff it is closed anti-grouped filter; (4) a filter of pseudo-BCI algebra is an associative filer (or pseudo-a filter) if and only if it is regular T-type filter (or regular pseudo q-filter). Finally, a counterexample is given to show that a regular p-filter (or well regular filter) of pseudo-BCI algebra may be not associative filter (or pseudo-a filter).