The Theory Of Uniform Semicontinuous Lattices

In the early1970s, D.Scott due to the semantic problems of The-oretical Computer puts forward the concept of continuous lattices. It marked theemergence of the classical theory of Domain which caused wide attention at the sametime. In1989, Ray.Y firstly proposed the semiprime ideal on lattices. In1997, basedon semiprime ideal, zhao D gave a new relationship between point and point oncomplete lattices, and then introduced the concept of semicontinuous lattices. In2007, Zeng Lihua promoted this new relationship to set and set, further more de-fined Quasi-semicontinuous lattices.In this thesis, we give the definition of uniformsemiprime subset by way of semiprime ideal. On this basis, a new relationship whichacting on point and point is obtained, and then we give definition of uniform semi-continuous lattices. In addition, some properties of uniform semicontinuous latticesare further studied. We disuss it mainly from three parts:In the first part, the concept of uniform semicontinuous lattice is introducedand examined. Using the method of uniform semiprime minimal subsets, we explaina few relationships which are among the uniform semicontinuous of the mapping,preserving and the uniform semiprime minimal subsets, In addition, we obtainthat any retract of a uniform semicontinuous lattice is still uniform semicontinuous.In the second part, we introduce a concept of quasi-uniform semicontinuouslattice and examine its properties. We obtain that the Cartesian product of finitequasi-uniform semicontinuous lattices is a quasi-uniform semicontinuous lattice. Inaddition,we give a sufficient condition for the inverse image of a quasi-uniform semi-continuous lattice is a quasi-uniform semicontinuous lattice.In the third part, the uniform semi-Scott topology on uniform complete lattice isintroduced, And some basic properties of the uniform semi-Scott topology on uniformsemicontinuous lattices is investigated.