On nodal and conodal ideals in residuated lattices

Abstract

In this article, we put forward the concepts of nodal and conodal ideals in a residuated lattice and study some properties. We state some examples and theorems. We investigate the inverse image of a nodal (conodal) ideal under a homomorphism. In addition, we pay attention to the relationships with the other types of ideals and special sets in varieties of residuated lattices. At the same time, we give a characterization of nodal ideals in terms of congruences and we show that if L is an MTL-algebra and I is a non-principal nodal ideal, then L/I is a chain. We propose a characterization for Boolean residuated lattices (L is a Boolean residuated lattice if and only if L is an involution semi-G-agebra) and we discuss briefly the applications of our results in varieties of residuated lattices. Finally, we introduce the concept of a fuzzy (nodal) ideal of a residuated lattice, and give some related results. After that we define the concept of fuzzy ideal of a residuated lattice with respect to a t-conorm briefly, S-fuzzy ideals and we prove Representation Theorem in residuated lattices.