Folding Theory Applied to Residuated Lattices

Abstract

Residuated lattices play an important role in the study of fuzzy logic based ont-norms. In this paper, we introduce some notions ofn-fold filters in residuated lattices, study the relations among them, and compare them with prime, maximal and primary, filters. This work generalizes existing results in BL-algebras and residuated lattices, most notably the works of Lele et al., Motamed et al., Haveski et al., Borzooei et al., Van Gasse et al., Kondo et al., Turunen et al., and Borumand Saeid et al., we draw diagrams summarizing the relations between different types ofn-fold filters andn-fold residuated lattices.

Fuzzyn-Fold Filters of Pseudoresiduated Lattices

“…Note. Similar to the case of commutative residuated lattices [24], we establish in [25] that (i) a filter is maximal if and only if, for any ∈ , ∉ implies that there are ∈ and natural numbers , ≥ 1 such that ( ⊗ ) = 0;…”

$n$-fold fantastic and $n$-fold involutive ideals in bounded commutative residuated lattices

“…In the framework of residuated lattices, Kadji et al [8] have studied the notion of n-fold normal filter (n ∈ N * ), and Ahadpanah et al [1] have introduced the notion of normal filter in the same framework. This section is devoted to the study of n-fold normal ideal in residuated lattices.…”

“…Although these structures are ordered by inclusion, the foldness theory for filters has not been studied on these structures in a gradual way. For example, the foldness theory for filters is studied by Haveshki et al [5]- [6] in the BL-algebras in 2006 and 2008, [7] in the RL-monoids in 2010; Motamed et al [14] in the BL-algebras in 2011; Borzooei et al [2] in the BL-algebras in 2013; Zahiri et al [21] in the MTL-algebras in 2014; Kadji et al [8] in the residuated lattices in 2014; Paad et al [15] in the BL-algebras in 2015. In particular, Kadji et al [8] defined the notion of n-fold boolean filters, n-fold implicative filters, n-fold positive implicative filters, n-fold integral filters, n-fold fantastic filters, n-fold obstinate filters, n-fold normal filters and n-fold involutive filters in residuated lattices and studied the relation among many of them (see for example Ref.…”

“…In particular, Kadji et al [8] defined the notion of n-fold boolean filters, n-fold implicative filters, n-fold positive implicative filters, n-fold integral filters, n-fold fantastic filters, n-fold obstinate filters, n-fold normal filters and n-fold involutive filters in residuated lattices and studied the relation among many of them (see for example Ref. [8]).…”

“…Moreover, using the complement elements, we study relationship between n-fold obstinate (respectively normal, fantastic or involutive) ideals and n-fold obstinate (respectively normal, fantastic or involutive) filters in residuated lattices. Moreover, we built the diagram summarizing all the relationships between these types of ideals as in the case of filters [8]. This paper is organized as follows.…”