Abstract
The notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are introduced, and related properties are investigated. Characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are given, and several properties are discussed. Finally, we prove that if T is an implicative BCK-algebra, then a fuzzy subset µ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T.
Highlights
BCK-algebras and BCI-algebras are two classes of non-classical logic algebras which were introduced by Y
Lee [16, 17] studied the bipolar fuzzy sets in which a negative membership degree is given for each element in the fuzzy set that ranges over the interval [−1, 0]
We introduce the notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras and investigate related properties
Summary
BCK-algebras and BCI-algebras are two classes of non-classical logic algebras which were introduced by Y. After the introduction of fuzzy sets by Zadeh, fuzzy set theory has become an active area of research in various fields. Positive) membership degree is given for each element in the fuzzy set that ranges over the interval [−1, 0] A number of research papers have been devoted to the study of bipolar fuzzy set theory in several algebraic structures We introduce the notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras and investigate related properties. We give characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras and several properties are discussed. We prove that if T is an implicative BCK-algebra, a fuzzy subset μ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T
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