Abstract
We inquire further into the properties on fuzzy closed ideals. We give a characterization of a fuzzy closed ideal using its level set, and establish some conditions for a fuzzy set to be a fuzzy closed ideal. We describe the fuzzy closed ideal generated by a fuzzy set, and give a characterization of a finite‐valued fuzzy closed ideal. Using a t‐norm T, we introduce the notion of (imaginable) T‐fuzzy subalgebras and (imaginable) T‐fuzzy closed ideals, and obtain some related results. We give relations between an imaginable T‐fuzzy subalgebra and an imaginable T‐fuzzy closed ideal. We discuss the direct product and T‐product of T‐fuzzy subalgebras. We show that the family of T‐fuzzy closed ideals is a completely distributive lattice.
Highlights
In 1983, Hu et al introduced the notion of a BCH-algebra which is a generalization of a BCK/BCI-algebra
In [8], the first author considered the fuzzification of ideals and filters in BCH-algebras, and described the relation among fuzzy subalgebras, fuzzy closed ideals and fuzzy filters in BCH-algebras
We describe the fuzzy closed ideal generated by a fuzzy set, and give a characterization of a finite-valued fuzzy closed ideal
Summary
In 1983, Hu et al introduced the notion of a BCH-algebra which is a generalization of a BCK/BCI-algebra (see [6, 7]). Using a t-norm T , we introduce the notion of (imaginable) T -fuzzy subalgebras and (imaginable) T fuzzy closed ideals, and obtain some related results. A nonempty subset A of a BCH-algebra X is called a subalgebra of X if x ∗ y ∈ A whenever x, y ∈ A. A nonempty subset A of a BCH-algebra X is called a closed ideal of X if (i) 0 ∗ x ∈ A for all x ∈ A, (ii) x ∗ y ∈ A and y ∈ A imply that x ∈ A. Let X denote a BCH-algebra unless otherwise specified. A fuzzy set μ in X is called a fuzzy subalgebra of X if μ(x ∗ y) ≥ min μ(x), μ(y) , ∀x, y ∈ X. Note that every t-norm T has a useful property: (P4) T (α, β) ≤ min(α, β) for all α, β ∈ [0, 1]
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