Many-valued Lattices And Their Representations

In this thesis, we study basic properties and representations of many-valued lattices from the view of category theory.LetΩ= (Ω, *, I) be a commutative unital quantale. From the view of category theory,Ωis a symmetric monoidal closed category. The categories enriched overΩare calledΩ-categories for short. Preordered sets and fuzzy preordered sets are both special cases ofΩ-categories, andΩ-categories can be interpreted as many-valued preordered sets. Therefore,Ω-categories are usually calledΩ-preordered sets.Ω-category has both categorical meanings and order theoretical meanings and is the foundation of researching many-valued order structures. For this reason, we can use categorical tools to study many-valued lattices.Since the classical notion of lattice can be described as existence of certain adjunctions, many-valued adjunction (calledΩ-adjunction) is important when we consider many-valued lattice. We consider two kinds of many-valued lattices,Ω-lattice and weakΩ-lattice. Both of them can be described as existence of certain many-valued adjunctions. We compare them with similar notions in the literature, and prove that the notion ofΩ-lattice is equivalent to that of lattice fuzzy order defined by R.BÄ›lohlávek and that the notion of weakΩ-lattice is equivalent to that of vague lattice defined by Demirci. We also study the representations of many-valued lattices. Join(meet)Ω-semilattices areΩ-modules in the category of semilattices while anΩ-lattice A can be represented as the underlying poset A₀ (a classical lattice) with a function fromΩto Adj(A₀→A₀) which satisfies certain conditions. Adj(A₀→A₀) is the join semilattice consisting of all the adjunctions from A₀ to itself. In the last section we show that the poser BX of formal balls in a metric space X is the underlying poset of the tensor completion of X considered as a ([0, oo]^OP, +, 0)-enriched category. Thus, a categorical description of BX is obtained.