Let A be a fuzzy subset of a set X and t ∈ [ 0, 1 ].
Let A be a fuzzy subset of a group G. Then A is a -anti-fuzzy subGroup of G if and only if A x-1y) ∧ μ ≤ (A x-1y A(y)) ∨ λ, ∀x, y ∈ G. Proof.
Theorem 3 Let A be a -fuzzy left ideal, and let B be a fuzzy subset of R. Then A ⊙ B is a -fuzzy left ideal of R. Proof For all z 1, z 2 ∈ R, we have.
Theorem 13 Let A be a fuzzy subset of R. If for all t ∈ ( λ, μ ], A t is a subring (ideal) of R or A t = ∅, then A is a -fuzzy subring (fuzzy ideal) of R. Proof The proof can be obtained from Theorem 6.
Assume R is not right noetherian, then there is a strictly ascending chain I 1 ⊆̷ I 2 ⊆̷ I 3 ⋯ of right ideals of R. Let f be a fuzzy subset of R × R constructed by f ( x, y ) = { 1, x ∈ I 1 and y = 0, 1 k, k ≥ 2, x ∈ I k, x ∉ I k − 1 and y = 0, 0, others.
Let μ, λ be two fuzzy subsets of X, we say that μ is contained in λ if μ ( x ) ≤ λ ( x ), ∀ x ∈ X. Definition 2.2 Let μ be a fuzzy subset of a set X, and let t ∈ [ 0, 1 ].
