Fuzzy (Ordered) Filters of Ordered BCI-Algebras

Fuzzy points are concepts used in mathematics, particularly in fuzzy set theory and related fields, where the membership of a given set is described using a membership function that allocates values in the unit interval [0, 1] rather than assigning binary values (true or false). This idea of a fuzzy point is used to consider the fuzzification of (ordered) filters in ordered BCI-algebras. Consequently, the concept of fuzzy (ordered) filters is introduced, and several related properties are investigated. The relationships between fuzzy filter, fuzzy (ordered) filter, and fuzzy (ordered) subalgebra are discussed, and characteristics of the fuzzy ordered filter are considered. This paper provides the conditions for a fuzzy set to become a fuzzy-ordered filter. We also investigate when a positive subset forms an ordered filter and examine whether the level and q-sets satisfy the same properties.