Lattice ordered semigroup, a kind of important ordered semigroup with interesting properties, has always been an important direction in the field of ordered semigroup. Some special types of lattice ordered semigroups such as Brouwierian semigroup, Artinian l-semigroup, Dedekindian l-semigroup, divisibility semigroups, semi-dually residuated lattice ordered semigroups, lattice ordered periodic semigroups, etc. were proposed and their algebraic properties have been investigated (see references). For the general development of lattice ordered semigroups, and ordered semigroups, the ideal theory and the fuzzy ideal theory play an important role. A fuzzy subset f of a given set S is described as an arbitrary function  where  is the usual closed interval of real numbers, which laid the foundations of fuzzy set theory, the literature on fuzzy set theory and its applications such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others has been growing rapidly amounting by now to several papers. In this paper, we introduce the concept of an -fuzzy quasi-ideal in ordered semigroups, which is a generalization of -fuzzy quasi-ideals in ordered semigroups, where  and  By using this concept, we define semiprime -fuzzy quasi-ideals and some basic results of an ordered semigroup are discussed. The upper/lower parts of -fuzzy quasi-ideals are introduced and some characterizations of regular ordered semigroups are given. Using implication operators and the notion of implication-based   fuzzy quasi-ideals, characterizations of a fuzzy quasi-ideal and an -fuzzy quasi-ideal are considered.

fuzzy quasi-ideals, -fuzzy quasi-ideals, semiprime -fuzzy quasi-ideals, implication-based fuzzy quasi-ideals.