Filters of residuated lattices and triangle algebras

Abstract

An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters.In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections. Triangle algebras characterize interval-valued residuated lattices.

A new approach for classification of filters in residuated lattices

“…A residuated lattice L in which x 2 = x for all x ∈ L (or, equivalently, in which x y = x ∧ y for all x, y ∈ L) is also called a Heyting algebra or pseudo-Boolean algebra [36,40]. We denote by H the class of Heyting algebras.…”

“…In other works such as [5,10,27,37,40,49,50] a theory of filters is studied in the more general context of a residuated lattice, thus generalizing the results from BL-algebras.…”

Characterizations of intuitionistic fuzzy ideals and filters based on lattice operations

“…These notions are mainly used to translate connections between properties on algebraic structures and to define congruence relations and quotient algebras [32,25]. They are played a central role in Stone representation theorem for Boolean lattice [28] and in the extensive theory of representation of distributive lattice [13,15,27].…”

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

“…For example, based on filters and prime filters in BL-algebras, Hájek proved the completeness of Basic Logic BL [18]. Literatures [1,4,5,8,12,17,24,30], further studied filters of BL-algebras, lattice implication algebras, pseudo BLalgebras, pseudo effect-algebras, residuated lattices, triangle algebras and the corresponding algebraic structures.…”