Interval valued $$(\in,\in\!\vee\,q)$$ -fuzzy filters of pseudo BL-algebras

Abstract

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued (∈, ∈ ∨ q)-fuzzy filters of pseudo B L-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo B L-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo B L-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.

On open problems based on fuzzy filters of pseudo BCK-algebras

“…Wang and Xin solved an open problem in pseudo BLalgebras between fuzzy normal and fuzzy Boolean filter [20]. Literatures [10,26,29] studied interval valuedfuzzy filters in pseudo BL-algebras and MTL-algebras. Therefore, fuzzy filters are a useful tool to obtain results on algebraic structures of logic algebras.…”

Solutions to Open Problems on Fuzzy Filters ofBL-algebras

“…Wang and Xin solved an open problem in pseudo BL-algebras by discussing the relation between fuzzy normal and fuzzy Boolean filter in [ 16 ]. In [ 17,18,19 ], interval valued (∈, ∈ ∨q)-fuzzy filters of residuated lattice and MT L-algebras were studied. Fuzzy filters are not only an extension of classical filter, but also a useful tool to infer some important results on classical filters due to the investigation from different point of views.…”

($\bar{\alpha}$, $\bar{\beta}$)-fuzzy Congruence Relation on Lattice Implication Algebras

“…The idea of fuzzy point and 'belongingness' and 'quasi-coincidence' with a fuzzy set were given by Pu and Liu (Pu & Liu, 1980). The idea of 'belongingness' and 'quasi-coincidence' with a fuzzy set have been applied some important algebraic system (Naraganan & Manikantan, 2005;Davvaz, 2006;Dudek et al, 2009;. We introduced the notion of fuzzy congruence relation on a lattice implication algebra and investigated the properties of it.…”