Generalized fuzzy filters in ordered semigroups

Abstract

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 􀜵.