| In recent years,order/lattice theory has been widely used in Topology,Combinatorial Mathematics,Fuzzy Mathematics,Rough Set Theory and Theoretical Computer Science,there are very close relationships among them.Based on the theory of posets and lattices,we study partial ordered structures in topological spaces and formal contexts as well as their concept lattices.The main contents of this paper are as follows:Content One: Preorder-typed sum of topological spaces and their specialization order.We take preordered sets as index sets,and define preorder-typed sum of topological spaces.It mainly shows that,the specialization order of preorder-typed sum of topological spaces is the same as the lexicographic order(w.r.t.the index set)of the specialization order of those topological spaces.Content Two: Partially ordered formal contexts and their concept lattices.On the basis of the classical formal context,the partially ordered formal context is defined.The main conclusions of the formal contexts and their concept lattices remain true in the partially ordered formal contexts and their concept lattices.Content Three: Double Dedekind-MacNeille completion.Double Dedekind-MacNeille completion is defined,it is proved that under some necessary conditions,the ordered ideal and the ordered filter are stable under the of the composition of the intent and the extent operators,the mapping from the object set(resp.,the attribute set)to the corresponding complete lattice is join-dense,which preserve inf(resp.,sup)respectively. |
