Semi-continuous Grid And Compatibility Of Continuous Posets

This article is mainly about semi-continuous lattices, consistently continuous posets and exact posets.Firstly, semi-continuous lattices--generalizations of continuous lattices are concerned. This paper explores deeply the properties of semi-continuous lattices and the semi-Scott topology of them. It is proved that in semi-continuous lattices, directed sups of semi-prime ideals are also semi-prime ideals and that each principle ideal is semi-Scott closed. It is also proved that if a subset could be expressed as the complement of a principle ideal, then the subset is a prime element of the semi-Scott topology. We also give a sufficient condition for a complete lattice to be a semi-continuous lattice. These results enrich the theory of semi-continuous lattices. Secondly, concepts of semi-bases and local-semi bases on complete lattices are introduced.Some properties and equivalent characterizations of semi-bases and local semi-bases are obtained. It is proved that a complete lattice is semi-continuous if and only if it has a semi-base or a local semi-base.Based on these results, we give the concepts of weights and characters of semi-continuous lattices. We discuss relations between weights, characters of a semi-continuous lattice and that of related topological spaces in the semi-Scott topology or the semi-Lawson topology. And we give a negative answer to an open problem posed by professor Bin Zhao, etc.Finally, we explore consistently continuous posets and exact posets. Consistently continuous posets are minor generalizations of continuous domains. The concept of exact posets is a generalizations of the concept of continuous posets, similar formally but different essentially. For consistently continuous posets, we deeply study their directed completions. The following results are obtained: (1) for a continuous domain P and a subset A?max(P), P\A is a consistently continuous poset whenever P\A is not empty; (2) for a continuous domain P and a subset A?max(P), if the Scott interior of A is empty, then the directed completion of P\A is isomorphic to P. Two examples are given to show the condition that the Scott interior of A is empty is sufficient but not necessary. For exact posets, it is proved that every continuous poset is an exact poset and that exact domains are hereditary to Scott-open sets. It is also proved that every domain is a weak domain, and that a weak domain is a domain if and only if the weak way-below upper set of every its element is an upper set.