The notion of BCI-algebras was introduced by Iséki [3] which is a generalization of BCKalgebras [2]. This notion is originated from two different ways: one of the motivations is based on set theory, another motivation is from classical and nonclassical propositional calculi. Zadeh [8] introduced the notion of fuzzy sets. Many researchers have applied this concept to mathematical branches, such ...

The study of symmetry is one the most important and beautiful themes uniting various areas contemporary arithmetic. Algebraic structures are useful in pure mathematics for learning a geometrical object’s symmetries. In this paper, we introduce new concepts an algebraic structure called BCI-algebra, where present bipolar fuzzy (closed) BCI-positive implicative ideals BCI-commutative BCI-algebras...

In this paper, we introduce and study the concepts of prime left, semiprime left and irreducible left hyperideals in ternary semihyper- groups and investigate some basic properties of them. We introduce the concepts of hyper lter and hypersemilattice congruence of ternary semi- hypergroups. We give some characterizations of hyper lters in ternary semihypergroups. Some relationships between hype...

The notion of a (closed) double-framed soft ideal (briefly, a (closed) DFS-ideal) in BCK/BCI-algebras is introduced, and related properties are investigated. Several examples are provided. The relation between a DFS-algebra and a DFS-ideal is considered, and characterizations of a (closed) DFS-ideal are established. A new DFS-ideal from old one is constructed, and we show that the int-uni DFS-s...

In 1982, Pawlak introduced the concept of a rough set (see [6]). This concept is fundamental for the examination of granularity in knowledge. It is a concept which has many applications in data analysis (see [7]). Rough set theory is applied to semigroups and groups (see [4, 5]), and BCK-algebras (see [3]). In this paper, we apply the rough set theory to BCC-algebras, and we discuss the notion ...

We consider the fuzzification of the notion of a positive implicative ordered filter in implicative semigroups. We show that every fuzzy positive implicative ordered filter is both a fuzzy ordered filter and a fuzzy implicative ordered filter. We give examples that a fuzzy (implicative) ordered filter may not be a fuzzy positive implicative ordered filter. We also state equivalent conditions of...