Many valued lattices and their representations

This paper presents an investigation of many valued lattices from the point of view of enriched category theory. For a bounded partially ordered set P, the conditions for P to become a lattice can be postulated as existence of certain adjunctions. Reformulating these adjunctions, by aid of enriched category theory, in many valued setting, two kinds of many valued lattices, weak Ω-lattices and Ω-lattices, are introduced. It is shown that the notion of Ω-lattices coincides with that of lattice fuzzy orders of Bělohlávek; and the notion of weak Ω-lattices coincides with that of vague lattices of Demirci.