Z-continuous Poset

Z-continuous posets were introduced by Wright Wagner and Thatcher as a generalization of continuous lattices. Z-continuous posets and generalized Z-continuous posets were a successful generalization of continuous lattice. In recent twenty years, many people devoted on it. The algebra properties of Z-continuous posets have been studied by several authors, e.g [6],[7],[8],[9], but the topological properties of Z-continuous posets have seldom studied.In this paper, I will study detail the topological properties of Z-continuous posets and generalized Z-continuous posets and give some categorical properties of Z-complete posets and Z-continuous functions.In chapter 1, we will give some definitions about Z-continuous posets and some kinds of Z-topology, this definitions are due to the papers of [4], [5], [9].In chapter 2, we introduce Z-continuous function and Z-minimal set. We prove that if / : P - Q and its inverse f-1 : Q - P are two Z-continuous functions then / preserves Z-minimal set, we also prove that if P and Q are two Z-complete posets, then / : P - Q is Z-continuous if and only if / : (P, z(P)) - (Q,z(Q)) is continuous.In chapter 3, we discuss some topological properties of Generalized Z-continuous posets. In [2], the authors introduced a kind of Generalized Z-continuous posets and proved that if P is a strong GZCP, then z(P) is a T2-topology, I will prove that if P is a Z-meet continuous strong GZCP, then z(P) is a T3-topology.In chapter 4, we will discuss some categorical properties of Z-complete posets and Z-continuous functions.