Abstract
The notion of bipolar fuzzy implicative ideals of a BCK-algebra is introduced, and several properties are investigated. The relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal is studied. Characterizations of a bipolar fuzzy implicative ideal are given. Conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal are provided. Extension property for a bipolar fuzzy implicative ideal is stated.
Highlights
Fuzzy sets are characterized by a membership function which associates elements with real numbers in the interval [0, 1] that represents its membership degree to the fuzzy set
We apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain further results in this manner
We apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain more related results
Summary
Fuzzy sets are characterized by a membership function which associates elements with real numbers in the interval [0, 1] that represents its membership degree to the fuzzy set. E bipolar-valued fuzzy set notion [1] was introduced to treat imprecision as in traditional fuzzy sets, where the degree of membership belongs to the interval [0, 1], and we cannot tell apart unrelated elements from the opposite elements. The bipolar fuzzy BCI-implicative ideals of BCI-algebras were studied in [32]. We apply the notion of a bipolar-valued fuzzy set to implicative ideals of BCK-algebras and obtain further results in this manner. We say I is an ideal if (c1) 0 ∈ I, (c2)(∀κ ∈ Ł)(∀l ∈ I)(κ ∗ l ∈ I⇒κ ∈ I) A nonempty subset I of a BCK-algebra Ł is called an implicative ideal of Ł if it satisfies (c1) and (c3) (∀κ, l, υ ∈ Ł)((κ ∗ l) ∗ υ ∈ I, l ∗ υ ∈ I⇒κ ∗ υ ∈ I)
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