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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 ()-fuzzy left (resp. right) ideals. A fuzzy subset μ of an …