On Some Questions Of Quasi-semicontinuous Lattices And Some Category Properties Of Strongly Algebraic Lattices

In this paper, as a generalization of semi-continuous lattices, the concepts of quasi-semicontinuous lattices and quasi-semialgebraic lattices are introduced and their some basic properties are investigated. The topological representation theorems of strongly algebraic lattices by the category of sober spaces with a basis of super-compact open set and strongly arithmetic lattices by the category of sober spaces with a basis of super-compact open set closed under finite intersections are obtained. Finally, we prove that the category of Z-algebraic posets is dually equivalent to a full subcategory of the category of strongly algebraic lattices.