Fuzzy Commutative Algebra

Theoretical approaches on certain characterization of L‐fuzzy soft modules

“…Then f is called increasing if a ≤ b implies f ( a ) ≤ f ( b ) for all a , b ∈ L 1 . Definition [37] Let ( L , ≤) be a complete lattice. Then L is called infinite ∨‐distributive lattice iff a ∧ (∨ i ∈ Λ b i ) = ∨ i ∈ Λ ( a ∧ b i ) for all a , b i ∈ L and i ∈ Λ. Definition [1, 38] A L ‐fuzzy subset μ of M is defined as a map from M to L . The set of all L‐fuzzy subsets of M is denoted by ℱ L ( M ).…”

“…Let μ ∈ ℱ L ( M ). Then the set μ α = { x | x ∈ M , μ ( x ) ≥ α } is said to be α ‐level subset of μ . Theorem [38] Let μ , ν ∈ ℱ L ( M ) and L be a lattice. If μ ≤ ν and α ∈ L , then μ α ⊆ ν α . Definition [38] Let { X i | i ∈ Λ} be a family of non‐empty sets and { μ i | i ∈ Λ} be a family of L‐fuzzy subsets of X i for all i ∈ Λ.…”

“…Then the set μ α = { x | x ∈ M , μ ( x ) ≥ α } is said to be α ‐level subset of μ . Theorem [38] Let μ , ν ∈ ℱ L ( M ) and L be a lattice. If μ ≤ ν and α ∈ L , then μ α ⊆ ν α . Definition [38] Let { X i | i ∈ Λ} be a family of non‐empty sets and { μ i | i ∈ Λ} be a family of L‐fuzzy subsets of X i for all i ∈ Λ. Then the mapping μ : ∏ i ∈ Λ μ i → L given by μ (( x i )) = ∧ i ∈ Λ μ i ( x i ) is called direct product of the family of { μ i | i ∈ Λ} and is denoted by ∏ i ∈ Λ μ i . Definition [38] Let ( G , +) be a group and { μ i | i ∈ Λ} be a family of L‐fuzzy subsets of G .…”

Vague ideals of implication groupoids

“…al introduced the concept of fuzzy ideal, fuzzy deductive systems in Hilbert algebras and discuss the relation between fuzzy ideals and fuzzy deductive systems. It is noticed that fuzzy algebra is now a well established branch of study [10]. Recently W.L.…”

Quotient of Ideals of an Intuitionistic Fuzzy Lattice

“…The concept of ideal of a fuzzy subring was introduced by Mordeson and Malik in [9]. After that N Ajmal and A.S Prajapathi introduced the concept of residual of ideals of an L-Ring in [10].…”