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ℒ-fuzzy Annihilators in Residuated Lattices
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ABSTRACT
In this paper, we provide a new characterization of
ℒ
-fuzzy
ideals of residuated lattices, which allows us to describe
ℒ
-fuzzy
ideals generated by
ℒ
-fuzzy
sets. Thanks to the latter, we endow the lattice of
ℒ
-fuzzy
ideals of a residuated lattice with suitable operations. Moreover, we introduce the notion of
ℒ
-fuzzy
annihilator of an
ℒ
-fuzzy
subset of a residuated lattice with respect to an
ℒ
-fuzzy
ideal and investigate some of its properties. To this extent, we show that the set of all
ℒ
-fuzzy
ideals of a residuated lattice is a complete Heyting algebra. Furthermore, we define some types of
ℒ
-fuzzy
ideals of residuated lattices, namely stable
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
set, and involutory
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
ideal. Finally, we prove that the set of all stable
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
set is also a complete Heyting algebra, and that the set of involutory
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
ideal is a complete Boolean algebra.
Walter de Gruyter GmbH
Title: ℒ-fuzzy
Annihilators in Residuated Lattices
Description:
ABSTRACT
In this paper, we provide a new characterization of
ℒ
-fuzzy
ideals of residuated lattices, which allows us to describe
ℒ
-fuzzy
ideals generated by
ℒ
-fuzzy
sets.
Thanks to the latter, we endow the lattice of
ℒ
-fuzzy
ideals of a residuated lattice with suitable operations.
Moreover, we introduce the notion of
ℒ
-fuzzy
annihilator of an
ℒ
-fuzzy
subset of a residuated lattice with respect to an
ℒ
-fuzzy
ideal and investigate some of its properties.
To this extent, we show that the set of all
ℒ
-fuzzy
ideals of a residuated lattice is a complete Heyting algebra.
Furthermore, we define some types of
ℒ
-fuzzy
ideals of residuated lattices, namely stable
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
set, and involutory
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
ideal.
Finally, we prove that the set of all stable
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
set is also a complete Heyting algebra, and that the set of involutory
ℒ
-fuzzy
ideals relative to an
ℒ
-fuzzy
ideal is a complete Boolean algebra.
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