ℒ-Fuzzy Cosets of ℒ-Fuzzy Filters of Residuated Multilattices

Abstract

This paper mainly focuses on building the ℒ-fuzzy filter theory of residuated multilattices. Firstly, we introduce the concepts of ℒ-fuzzy filter and ℒ-fuzzy deductive system of residuated multilattices. Then, we highlight their properties and show how they are linked. Secondly, we introduce the concept of prime ℒ-fuzzy filter and propose some illustrative examples. Then, we bring out their properties and show how they are related to the concept of ℒ-fuzzy prime filter. Thirdly, we characterize ℒ-fuzzy maximal filter and maximal ℒ-fuzzy filter by atoms and coatoms. In the case where ℒ is a distributive lattice, we prove that maximal ℒ-fuzzy filters are prime. Finally, we are interested in ℒ-fuzzy cosets of an ℒ-fuzzy filter, and we prove that the set of all ℒ-fuzzy cosets of any ℒ-fuzzy filter of a residuated multilattice is a residuated multilattice.