Affiliations: Department of Systems Engineering-Automatic and Informatics Applied, Technical University of Civil Engineering and International Theoretical High School of Informatics Bucharest, Bucharest, Romania
Correspondence:
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Corresponding author. Liviu-Constantin Holdon, Department of Systems Engineering-Automatic and Informatics Applied, Technical University of Civil Engineering and International Theoretical High School of Informatics Bucharest, 648, Colentina st., 021187 Bucharest, Romania. Tel.: +40 212 403035, E-mails: holdon liviu@yahoo.com and holdon.liviu@ichb.ro.
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.
