In this paper we study certain types of positive fuzzy quasi-orders on semigroups and their links with completely semiprime fuzzy ideals, completely prime fuzzy ideals, and fuzzy filters. We characterize various properties of a positive fuzzy quasi-order Q in terms of the properties of its left and right eigen spaces I Q and C Q , the solution sets of eigen fuzzy set equations corresponding to Q. We also demonstrate certain applications of the obtained results to semilattice decompositions of semigroups. The results of this paper shed new light on the known links between positive (crisp) quasi-orders, completely semiprime ideals, completely prime ideals, and filters of semigroups, and make these links much clearer.