n-Fold Boolean, Implicative and Integral Ideals on Bounded Commutative Residuated Lattices

Abstract

In this paper, we introduce the concepts of [Formula: see text]-fold boolean ideals, [Formula: see text]-fold implicative ideals and [Formula: see text]-fold integral ideals in residuated lattices and we state and prove their properties. Several characterizations of these notions are derived and the relations between those notions are investigated. Also, we construct the correspondence between the notions of [Formula: see text]-fold ideal and [Formula: see text]-fold filter in residuated lattices.

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

“…By the fact that ideals and filters are not dual notions in residuated lattices, it appears natural to built up the foldness theory of ideals in the framework of residuated lattices. In [19], authors introduce and investigate the concepts of n-fold boolean ideals, n-fold implicative ideals and n-fold integral ideals in residuated lattices; but they have not studied the relations between these different n-folds. In this paper, we introduce the concepts of n-fold obstinate ideals, n-fold normal ideals, n-fold fantastic ideals and n-fold involutive ideals in residuated lattices.…”

“…In [19], we introduced a new operation and gave another description of ideals in residuated lattice.…”

“…It is easy to see that x ≤ y ⊘ z if and only if x ⇀ y ≤ z. Proposition 5 [19]. Let L be a residuated lattice, for any x, y, z ∈ L and n, m ∈ N * , we have:…”