Of Fuzzy Implication Algebra And Its Topology

Non-classical logic is a theoretical basis of fuzzy reasoning and fuzzy control. With non-classical logic gradually coming to maturation and perfection, many scholars have posed various logical algebras based on different implication operators, such as MV-algebras, BCK-algebras, BL-algebras, lattice implicative algebras and R₀-algebras and so on. The purpose of this paper is to study various logic algebras and their topologies with algebraic tools, topological methods and techniques coming from domain theory. We maimly explore characterizations and relations of various logical algebras, instilling energy into the development of fuzzy logic and the interaction between fuzzy logic and other subjects.In chapter one, we explore further properties of fuzzy implicative algebras (in short, FI-algebras). In chapter two, we systematically study FI-algebras satisfying the condition (IC): (x→y)→y=(y→x)→x, which is called commutative FI-algebras (in short, CFI-algebras). Some important properties of this kind of algebras are obtained. Relations between CFI-algebras and other logical algebras, such as HFI-algebras, lattice implicative algebras, R₀-algebras, Heyting algebras and residual lattices are investigated. A necessary and sufficient condition for CFI algebras to be HFI algebras is given. It is proved that CFI algebras are weak R₀-algebras and that CFI algebras can be viewed as lattice implicative algebras and vice versa. It is also proved that CFI algebras in the induced order form distributive residual lattices. In addition, some properties of the subalgebras and product properties of CFI-algebras are given. In chapter three, further characterizations for MP filters of FI-algebras are given. A representation theorem of MP filter which is created by a nonempty set of FI-algebra is obtained. It is proved that the set of all MP filters on FI-algebras in set-inclusion order forms a distributive algebraic lattice, especially a frame. We also introduce the notions of fuzzy (prime) MP-filters. Three characterizations theorems of fuzzy MP filters and a representation theorem of fuzzy MP filter which is generated by a fuzzy set and an extension theorem of fuzzy prime MP filter are given. Relations between them and (prime) MP-filters are discussed. Finally in chapter four, we construct uniformities useing congruence relations and collections closed under finite intersection of MP filters on an FI-algebra, which induce topology spaces. Properties of these uniformity and relevant topologies are discussed. It is proved that this kind of spaces is rarely connected. A sufficiented condition for this topological space to be compact is obtained.