Further Research On Fuzzy W-eo Algebras And Z-semicontinuous Lattices

In 1965, Zadeh put forward to the concept of fuzzy set which marked the birth of fuzzy mathematics. In 1973, Zadeh put the thoughts and meth-ods of fuzzy mathematics into fuzzy reasoning, and it achieved a great success. In recent years, ideal and filter theory applied to the research on the algebraic structure became one of the hotspots in the field of logic. On the basis of these theories, this paper mainly studies weak extended-order algebra and its filter theory.D. Scott put forward to the concept of continuous lattice because of the need of computer problems. It is closely related to algebra, analysis, topology, and has achieved fruitful results. Affected by it, some generalization structures of continuous lattice, for example,Z-continuous posets, hyper continuous posets, semi-continuous posets, etc., also got certain research. In particular, on the basis of the subset system of semiprime ideals, Zhao dong-sheng replaced the<< relation with<= relation, and introduced the concept of semi-continuous lattice. In recent years, many scholars studied the topology of semi-continuous lattice and its properties. On the basis of these theories, this paper puts forward to the concepts of Z-semialgebraic lattice and strongly Z-algebraic lattice, and makes a systematic study on the properties of Z-semialgebraic lattice and Z-semi Scott topology on Z-semicontinuous lattices.The arrangement of this thesis is as follows:Chapter One:Preliminaries. In this chapter, we introduce the basic concepts and results of logic algebra theory, Domain theory and lattice theory which will be used throughout this thesis.Chapter Two:Fuzzy weak extended-order algebra. Firstly, in this chapter, we give the concepts of subalgebra and filter about weak extended-order algebra. It is proved that the product of weak extended-order algebras (extended-order algebras) is also weak extended-order algebras(extended-order algebras). Secondly, we introduce the concepts of fuzzy weak extended-order filter and fuzzy weak extended-order subalgebra and discuss the relationship between weak extended-order filters and fuzzy weak extended-order filters. It is proved that the set of fuzzy weak extended-order filters is a complete weak extended-order algebra, and the product of fuzzy weak extended-order filters is also fuzzy weak extended-order filters. Finally, we use fuzzy points to describe fuzzy weak extended-order filters, and define the concept of weak filter, and obtain the equivalence relation of strong weak extended-order filters, fuzzy weak extended-order filters and weak filters.Chapter Three:Z-Semialgebraic Lattices and Z-Semi Scott Topology. The Z-compact elements are defined on the base of Z-way below relation, and the concepts of Z-semialgebraic lattices and strongly Z-algebraic lattices are introduced. It is proved that under certain conditions the image of a closure operator on the Z-semialgebraic lattice is also a Z-semialgebraic lattice, and a strongly Z-algebraic lattice is isomorphic to the set of all Z-ideals of its Z-compact elements. Finally, some properties of Z-semi Scott topology on Z-semicontinuous lattices are studied, and it is obtained that the retract of a Z-semicontinuous lattice is still a Z-semicontinuous lattice.