Prime Filters on Residuated Lattices

Fuzzyn-Fold Filters of Pseudoresiduated Lattices

“…In this paper, we introduced the notions of n-fold implicative filters, n-fold positive implicative filters, n-fold boolean filters, n-fold fantastic filters, n-fold normal filters and n-fold obstinate filters in residuated lattices and study the relations among them. This generalized the similar existing results in BL-algebra with the connection of the work of Kerre and all in [14], Kondo and all in [7], [11] and Motamed and all in [9]. At the end of this paper, we draw two diagrams; the first one describe the relations between some type of n-fold filters in residuated lattices and the second one describe the relations between some type of n-fold residuated lattices.…”

“…In [2], [3], [9] and [13] the authors defined the notion of n-fold implicative filters, n-fold positive implicative filters, n-fold boolean filters, n-fold fantastic filters, n-fold obstinate filters, n-fold normal filters in BL-algebras and studied the relation among many type of n-fold filters in BL-algebra. The aim of this paper is to extend this research to residuated lattices with the connection of the results obtaining in [14], [11], [7].…”

Filters on Some Classes of Quantum B-Algebras

“…In Section 5 we extend the result of Kondo and Turunen (see [21]) to the setting of noncommutative residuated ∨-semilattices that, if prime filters and ∨-prime filters of a residuated ∨-semilattice A coincide, then A must be a pseudo MTL-algebra.…”

“…In the paper [15] Van Gasse et al asked whether, for any commutative residuated lattice L, if prime filters and ∨-prime filters coincide, then L must be an MTLalgebra. The affirmative answer was given in [21] by Kondo and Turunen. We extend their result to the setting of noncommutative residuated semilattices.…”

Folding Theory Applied to Residuated Lattices

“…A close analysis of the situation reveals that the main drive in all the previously mentioned works resides in the existence of an adjoint pair of operations. Just as the foldness theory for filters in BL-algebras generalizes filters introduced by Hájek, our foldness theory for filters in residuated lattices builds on recently published works on filters in residuated lattices by Haveshki et al [6], Van Gasse et al [7], Kondo and Dudek [8], Kondo and Turunen [9], Borumand Saeid and Pourkhatoun in [10], and Zahiri and Farahani in [11].…”