More General Form of Interval-Valued Fuzzy Ideals of BCK/BCI-Algebras

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.

Retracted: More General Form of Interval-Valued Fuzzy Ideals of BCK/BCI-Algebras

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