Some Weakly Auxiliary Relation And Weak Continuity On Partially Ordered Structure

The auxiliary relation on complete lattice has important effect on the research to continuous lattice. Firstly, the auxiliary relation is generalized to directed-complete partial order based on complete lattice in this paper. Some weakly auxiliary relations are introduced in directed-complete partial order in the following. Then C-continuous poset, D-continuous poset, E-continuous poset and F-continuous poset are defined on this foundation, some concrete examples are given for every kind of posets. And some constitutive properties and category nature of these posets are researched. Furthermore, the relation between these posets and the relation between them and traditional continuous poset are discussed and studied. Next, the weakly auxiliary relations above are generalized to general lattices, so C-continuous lattice, D-continuous lattice, E-continuous lattices and F-continuous lattice are defined, some concrete examples are given for every kind of lattices. Then some constitutive properties and category nature of these lattices are discussed simply. And the relation between these lattices and relation between them and traditional continuous lattice are studied.