On Some Questions Of Hypercontinuous Lattices And The Biscott Topologies

In the paper, for a complete lattice, the hyper-approximating element is obtained. We get a category property on hypercontinuous lattice, and prove that the complete lattice L is a hypercontinuous lattice iff the relation"<"satisfies the interpolation property and (?)x, y∈L, x≠y <=> i(x)≠i(y). The definition of base B(L), weight W(L) and character x(L) on a hypercontinuous domain are given, and some characteristic theorems are obtained. The relation between the weight and character on a hypercontinuous domain and that of the related topological space with v(L) topology are discussed, and we prove that W(L, v(L))≤W(L), x(L)≤W(L), x(L)=x(L, v(L)) on the quasi-hypercomtinuous domain. We also discuss the topology on the quasi-hypercomtinuous domain, get a condition that let the quasi-hypercomtinuous domain to be a quasi-hyperalgebraic domain, and prove that (L, v(L)) is locally compact space on the quasi-hypercomtinuous domain. At last, the definition of the biscott topology on a double directed complete poset is introduced, and gets some result about the biscott topology compactness.