Z-exact Posets And Quasi Z-exact Posets

In this paper, for a general subset Z, we define the concept of Z-minimalsets, Z-exact posets, quasi Z-exact posets, Z-meet exact posets and quasi Z-algebraic exact posets. And the equivalent characterizations between the mapping preserving the Z-minimal sets and the mapping preserving <<_Z-ω and the Z-exact posets are discussed and we also study some properities of Z-exact posets. The corresponding extension theorem is proved. For the general function space of Z-exact posets, we discuss its some properties. The characterizations of quasi Z-exact posets, Z-meet exact posets and quasi Z-algebraic exact posets are given. And their some characterization are discussed. We prove that Z is the property M of Rudin subset system and P is Z- exact posets, then P is quasi Z-exact posets. We prove that Z is finite join Rudin subset system and P is quasi Z-exact posets, then (P,λ_Z(P)) is Hausdroff space. We prove that P is Z-meet exact posets andλ_Z(P) is Huasdorff, then P is weakly Z-exact posets. Also we prove that P is quasi Z-algebraic exact posets, then P is quasi Z-exact posets.