Weakly Smooth Posets And Z-weakly Continuous Posets

In this paper,it is proved that the product of exact posets(reps.weak domains) with the least element is still an exact posets(reps.weak domains),and the images of exact domain under a projection operator which preserves directed sup or has a lower adjoint or an onto mapping which has a lower adjoint and preserves directed sups.Also in the paper,the basis of an exact poset is investigated.Second,it is introduce lower below relation,weakly smooth posets and discuss some basic property.It is proved that if a sup semilattice is weakly smooth and continuous then equivalence completely distributive poset,weakly smooth lattices equivalence F-distributive lattices,a complete semilattice is weakly smooth if it has a injective adjoint which preserves closed set of the upper topology.The importance result is complete semilattices isomorphic when their upper topology are isomorphic,the closed set of the upper topology for a a complete semilat-tice(complete lattice) is isomorphic to a weakly lower stable(lower stable) and strongly algebra smooth complete lattice.Third,we introduce Z-weakly continuous posets and discuss some basic property of it.It is proved that the images of Z-weakly continuous posets is still a Z-weakly continuous posets under a projection operator which preserves Z-sup or has a idempotent upper adjoint or an onto mapping which has a upper adjoint and preserves Z-sups.Also in the paper,the basis of an Z-weakly continuous posets and Z-weakly algebra posets is investigated.