The Properties Of Categories CL And PairCL

Closed-set lattice (or be called moleculely generated lattice) differs from the completely distributive lattice, which is a particular example of closed-set lattice, and its conception is more extensive and it also has good nature. With the deep developing of fuzzy topology , closed-set lattice plays a more important role in the study of lattice theory and topology. In this paper, we mainly study the categorical properties of categories CL and PairCL (that is a generalization of the notions of topological molecular lattice and quesi subspace). We firstly discuss minimal sets characterizations of closed-set lattice , then systematically study the categorical properties of categories CL and PairCL by the means of the join-semilattice's closed-set-latticefies, and the constructions of monomorphism, epimor-phism, extremal monomorphism, extremal epimorphism, section, retraction, subob-ject, quotient object, product, co-product, equalizer, coequalizer, inverse limits and direct limits in the categories CL and PairCL are given. The main content of this paper is as follows:1. Minimal sets characterizations of closed-set lattice. Following the means of minimal sets characterizations of the completely distributive lattice, we define F minimal sets in the closed-set lattice and draw some conclusions which are similar to the completely distributive lattice's: (1) Complete lattice L becomes closed-set lattice if and only if every elment of L has T minimal sets ; (2) T minimal sets of closed-set lattice are all lower set. (3) F minimal-mapping preserves T join.2. The properties of category CL. We systematically study the categorical properties of category CL by the means of the join-semilattice's closed-set-latticefies, and the constructions of monomorphism, epimorphism, extremal monomorphism, extremal epimorphism, section, retraction, subobject, quotient object, product, co-product, equalizer, coequalizer, inverse limits and direct limits in the category CL are given.3. The properties of category PairCL. We discuss the continuous mapping's properties of category PairCL, and the equivalent characterizations of continuous...