Abstract
AbstractThis paper focuses on the investigation of fuzzy filters of BL-algebras, an important and popular generic logical algebra. By studying the equivalent conditions of fuzzy fantastic filter and fuzzy normal filter of BL-algebras, the relation between these two filters are revealed. Two open problems of BL-algebras, i.e.,”Under what suitable condition a normal filter becomes a fantastic filter? ” and ” Under what suitable condition extension property for normal filter holds? (Extension property for a normal filter)”, are solved accordingly.
Highlights
Logic algebras and the concept of filters play a vital role in reasoning mechanism in logic-based reasoning systems
In[6], there are two open problems, i.e., 1) ”Under what suitable condition a normal filter becomes a fantastic filter? ” and 2) ”(Extension property for a normal filter) Under what suitable condition extension property for normal filter holds? ” To solve the open problems, it is very useful to discuss the relation between fuzzy fantastic filter and fuzzy normal filter of BL-algebras
After giving the equivalent conditions of these two fuzzy filters, we present the relation between fuzzy fantastic filter and fuzzy normal filter of BL-algebras and solve the two open problems
Summary
Logic algebras and the concept of filters play a vital role in reasoning mechanism in logic-based reasoning systems. In [3,5,6], the notions of prime filter, Boolean filter, implicative filters, normal filters, fantastic filters and positive implicative filters in BL-algebras were proposed, along with their properties investigated. Xu and Qin [12] proposed the notions of fuzzy positive implicative filters in lattice implication algebras. Jun et al derived several characterizations of fuzzy positive implicative filters of lattice implication algebras [13,14]. In [17,18,19], interval valued (∈, ∈ ∨q)-fuzzy filters of residuated lattice and MT L-algebras were studied.
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