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

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