Characterizations of ternary semigroups by ,qk -fuzzy ideals

Abstract

In this paper, we introduce the concept of (α, β)-fuzzy ideals in ternary semigroups, which is a generalization of fuzzy ideals in ternary semigroups. We investigate the related properties of ternary semigroups. The lower and upper parts of fuzzy subsets of a ternary semigroup are defined. Characterizations of regular ternary semigroups by the properties of the lower part of (∈,∈∨ q)-fuzzy left (right) ideals, (∈, ∈∨ q)-fuzzy quasi ideals and (∈, ∈∨ q)-fuzzy bi-ideals are given. Dudek et al. (19) and Ma and Zhan (20) defined (α,β)-fuzzy ideals in hemirings, and investigated some related properties of hemirings. In (21), Jun and Song initiated the study of (α,β)-fuzzy interior ideals of a semigroup. Kazanci and Yamak (22), studied (∈,∈∨ q)- fuzzy bi-ideals of a semigroup. In (23), Wang and Chen defined (α,β)-fuzzy subalgebra of BCH-algebras. Recently, (∈,∈∨q)-fuzzy K-algebra has been studied by Akram (24). It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems and other algebraic structures. As a first step in this direction, we introduce the concept of (α,β)-fuzzy ternary subsemigroups, (α,β)-fuzzy ideals, (α,β)-fuzzy quasi-ideals, (α,β)-fuzzy generalized bi-ideals, (α,β)- fuzzy bi-ideals in ternary semigroups. We discuss several related properties of (∈,∈∨ q)-fuzzy ideals. We also characterize regular ternary semigroups by the properties of these ideals.