On Some Questions Of Z-connected Continuous Posets And Z-connected Smooth Posets

For a Z—connected set system, we study some mapping properties of (weakly) Z—connected continuous posets which are characterized by Z—connected basis, Z—connected embedded basis, local Z—connected basis, and discuss a function space of Z—connected continuous lattices.Generalize an important lattice—Smooth lattice to the Z—connected set system is another important work in this paper, the concepts of Z—connected smooth posets and Z—connected smooth algebraic lattices are introduced. It is proved that a complete lattice M is isomorphic to the Z—connected closed set lattice of a complete lattice P if and only if M are Z—connected stable and Z—connected smooth algebraic. So these work broaden the research field of Z—connected set system.Finally, as another way of a generalization of Smooth lattices, we introduce the concepts of semi-Smooth lattices and semi-Smooth algebraic lattices on the base of semicontinuous lattices theory. Some basic properties of them and the relation with semicontinuous lattices are investigated. It turns out that a complete lattice L is completely distributive lattice if and only if it is both semi-Smooth lattice and semicontinuous lattice.