Some Properties Of Quasi-C Poset And Generalized Completely Distributive Poset

Since the pioneer work of D. S. Scott, the structure theory of domains hasbeen a focal point for the research in denotational semantics for computer functionallanguages. Seen from the mathematical angle, the research objects in domain theorymain consists of posets satisfying certain conditions and morphisms between them.Continuous dcpo (i.e., domain) is one of the main research objects in domain theory.One of the important aspects of domain theory is to carry as much as possibleof the theory of continuous domain to as general order structure as possible. In 2008,basing on discussing continuous posets'ordered structure, Menon had brought inthe concept of C-posets. It is turned out that C-posets have many properties thatare analogous to continuous posets. It is well known that Quasicontinuous domain isone of the most successfully generalization of domain. Its basic idea is to generalizeway below relation between"point"and"point"to the one between"set"and"set". Following Menon's idea, we generalize quasicontinuous posets to quasi-Cposets and discuss some properties of quasi-C posets. As a kind of speci?c quasi-continuous posets, generalized completely distributive posets were put forward whenMenon gave the description of a poset that is a compact pospace with respect toits interval topology. We will further discuss properties of generalized completelydistributive posets. C-space is an important kind of space that put forward by Ern′ewhen he discussed the relations between topology and order. We prove that theproduct space of two C-space is a C-space, and C-space is hereditary with respectto relatively closed sets, etc.