Ideal And Finite Co-cover Property In L-precotopological Spaces

An L—precotopological space differs from an L—cotopological space (the latter is a particular example of L-precotopological space), and its conception is more extensive and it also has good nature. With the deep developing of L-topology, L-precotopological space will play a more important role in the study. Based on re-ferrence [2], ideal and finite co-cover property in L-precotopological space are mainly studied in this paper .The following are the construction and main contents of this paper:In chapter 1, we give the basic concepts and conclusions in lattice theory for convenience of the following discussion.In chapter 2, the fact that L—precotopological space is a generalization of L—cotopological space is showed and ideal in L—precotopological space is studied. Firstly, we show that an L—precotopological space is necessarily an L—cotopological space, while an L—cotopological space may not be an L—precotopological space. Secondly, we introduce remote neighborhood method based on closed element, which is often used when studying L—precotopological space. In the following, we give the concepts of accumulation point, cluster point and concerning basic conclusions in L-precotopological space. Finally, we give the stronger and weaker relation between two ideals and some concerning conclusions in L—precotopological space. Besides, the properties of maximal ideal are studied specially.In chapter 3, convergence for net and ideal in L—precotopological space is studied and order homomorphism is introduced as basic mapping between two L—precotopological spaces. Firstly, we introduce the concepts of convergence, limit point and cluster point for net in L—precotopological space and discuss their properties, respectively. Secondly, we introduce the concepts of convergence, limit point and cluster point for ideal in L—precotopological space and discuss their properties, respectively. Thirdly, we study the relation between convergence for molecular net and ideal in L—precotopological space. Finally, we introduce order homomorphism between two L—precotopological spaces, and give the concepts of continuous order homomorphism, open and closed order homomorphism as well as some equivalent characterations of these order homomorphisms. Also the images of molecular net and ideal under order homomorphism are studied and some...