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The purpose of this paper is to introduce the notions of ∈, ∈ ∨ q   k  -fuzzy ideals of a fuzzy ordered semigroup with the ordering being a fuzzy relation. Several characterizations of ∈, ∈ ∨ q   k  -fuzzy left (resp. right) ideals and ∈, ∈ ∨ q   k  -fuzzy interior ideals are derived. The lattice structures of all ∈, ∈ ∨ q   k  -fuzzy (interior) ideals on such fuzzy ordered semigroup are studied and some methods are given to construct an ∈, ∈ ∨ q   k  -fuzzy (interior) ideals from an arbitrary fuzzy subset. Finally, the characterizations of generalized semisimple fuzzy ordered semigroups in terms of ∈, ∈ ∨ q   k  -fuzzy ideals (resp. ∈, ∈ ∨ q   k  -fuzzy interior ideals) are developed.

Fuzzy ideals of ordered semigroups with fuzzy orderings