A study of generalized fuzzy ideals in ordered semigroups

In mathematics, an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders, and error-correcting codes. A theory of fuzzy generalized sets on ordered semigroups can be developed. Using the notion of “belongingness (∈)” and “quasi-coincidence (q)” of fuzzy points with a fuzzy set, we introduce the concept of an (α, β)-fuzzy left (resp. right) ideal of an ordered semigroup S, where α, β  ∈ {∈, q,  ∈ ∨ q,  ∈ ∧ q} with α ≠  ∈ ∧ q. Since the concept of (∈, ∈∨ q)-fuzzy left (resp. right) ideals is an important and useful generalization of ordinary fuzzy left (resp. right) ideal, we discuss some fundamental aspects of (∈, ∈ ∨ q)-fuzzy left (resp. right) ideals and (\(\overline{\in},\overline{\in}\vee \overline{q}\))-fuzzy left (resp. right) ideals. A fuzzy subset μ of an ordered semigroup S is an (∈, ∈ ∨ q)-fuzzy left (resp. right) ideal if and only if μ _t , the level cut of μ is a left (resp. right) ideal of S, for all 0 < t ≤ 0.5 and μ is an (\(\overline{\in},\overline{\in}\vee \overline{q}\))-fuzzy left (resp. right) ideal if and only if μ _t is a left (resp. right) ideal of S, for all 0.5 < t ≤ 1. This means that an (∈, ∈ ∨ q)-fuzzy left (resp. right) ideal and (\(\overline{\in},\overline{\in}\vee \overline{q}\))-fuzzy left (resp. right) ideal are generalizations of the existing concept of fuzzy left (resp. right) ideals. Finally, we characterize regular ordered semigroups in terms of (∈, ∈ ∨ q)-fuzzy left (resp. right) ideals.