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ℒ -Fuzzy Cosets of ℒ -Fuzzy Filters of Residuated Multilattices
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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.
Title: ℒ
-Fuzzy Cosets of
ℒ
-Fuzzy Filters of Residuated Multilattices
Description:
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.
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