Abstract
The concepts of interval-valued ∈ , ∈ ∨ κ ∗ ˜ , q κ ˜ -fuzzy subalgebras, interval-valued ∈ , ∈ ∨ κ ∗ ˜ , q κ ˜ -fuzzy ideals, and interval-valued ∈ ∨ κ ∗ ˜ , q κ ˜ , ∈ ∨ κ ∗ ˜ , q κ ˜ -fuzzy ideals are introduced, and related properties are studied. Many examples are given in support of these new notions. Furthermore, interval-valued ∈ , ∈ ∨ κ ∗ ˜ , q κ ˜ -fuzzy commutative ideals are defined, and some important properties are discussed. For a BCK-algebra X , it is proved that every interval-valued ∈ , ∈ ∨ κ ∗ ˜ , q κ ˜ -fuzzy commutative ideal of BCK-algebra X is an interval-valued ∈ , ∈ ∨ κ ∗ ˜ , q κ ˜ -fuzzy ideal of X , but the converse need not be true, in general, and then a counterexample is constructed.
Highlights
As an extension of fuzzy sets, Zadeh defined fuzzy sets with an interval-valued membership function proposing the concept of interval-valued fuzzy sets. is concept has been studied from various points of view in different algebraic structures as BCK-algebras and some of its generalization, groups, and rings
In BCK/BCI-algebras and other related algebraic structures, different kinds of related concepts were investigated in various ways
Zhan et al [35, 36] studied (∈, ∈∨q)-fuzzy ideals of BCI-algebras. e concept of “quasi-coincidence” of an interval-valued fuzzy point together with “belongingness” within an intervalvalued fuzzy set was used in the studies made by Ma et al in [37, 38] where they discussed properties of some types of (∈, ∈∨q)-interval-valued fuzzy ideals of BCI-algebras
Summary
As an extension of fuzzy sets, Zadeh defined fuzzy sets with an interval-valued membership function proposing the concept of interval-valued fuzzy sets. is concept has been studied from various points of view in different algebraic structures as BCK-algebras and some of its generalization (see, for example, [1,2,3,4,5]), groups (see for example, [6,7,8,9,10]), and rings (see, for example, [11,12,13]). E concept of “quasi-coincidence” of an interval-valued fuzzy point together with “belongingness” within an intervalvalued fuzzy set was used in the studies made by Ma et al in [37, 38] where they discussed properties of some types of (∈, ∈∨q)-interval-valued fuzzy ideals of BCI-algebras. It is natural to introduce the general form of the existing interval-valued fuzzy ideals of BCK/BCI-algebras. If X satisfies (1)–(4) and (5) 0 ∗ υ 0, X is a BCKalgebra
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