Generalized fuzzy filters in ordered semigroups

Fuzzy semigroup theory concentrates on theoretical aspects, but also includes applications in the areas of fuzzy coding theory, fuzzy finite state machines, and fuzzy languages. In this paper, we introduce the concept of an (??, ??)-fuzzy filter of an ordered semigroup ??, where ??, ?? ?? ????, ??, ???? ??, ???? ???? with ?? ?????? ??. Since the concept of (??, ???? ??)-fuzzy filter is an important and useful generalization of ordinary fuzzy filter, we discuss some fundamental aspects of (??, ???? ??)-fuzzy filters. An (??, ???? ??)-fuzzy filter is a generalization of the existing concept of a fuzzy filter. The concept of (??, ???? ??)-fuzzy filters is also introduced and some related properties are investigated. The relationships among ordinary fuzzy filters, (??, ???? ??)-fuzzy filters and (??, ???? ??)-fuzzy filters are discussed. We discuss the concept of (??, ???? ??)-fuzzy left (right and bi)-filters and provide some characterization theorems. Finally, we extend the concept of a fuzzy subgroup with thresholds to the concept of a fuzzy left (right and bi)-filter with thresholds of ??.