Generalized dimension of an ordered set and its MacNeille completion

Formal Concept Analysis

“…In the complete lattice (2l(T), <;;;) the supremum is equivalent to the topological closure of the union, i.e., (4) convex sets: For ~n the set <t(~n) of all convex subsets is a closure system, i.e., (<t(~n), s;) is a complete lattice and in this lattice the supremum is the convex closure of the union. For many mathematical structures, the system of substructures is a closure system.…”

“…4) follows directly from the definition. A cross in row g and column m means that the object g has the attribute m. Proof.…”

“…Figure 1.20 A nested line diagram of the free distributive lattice FCD ( 4). From x E F and y 2: x it follows that y E F, 2. for any two elements x, y E F there is an u E F with u ~ x and u ~ y.…”

“…Let Q be a complete lattice and, furthermore, let Vq be a complete lattice for every element q E Q. 4. is an order filter of Vq and an order ideal of v;.…”

“…They were formally introduced in (114] and thoroughly examined by Stephan [161], [160]. However, we have introduced a little change in the notation compared with the literature we quote and now write U(a, b)V, where in earlier publications appeared U(b, a)V. 4.4 Tensor products of complete lattices have been introduced in many articles. However, we have introduced a little change in the notation compared with the literature we quote and now write U(a, b)V, where in earlier publications appeared U(b, a)V. 4.4 Tensor products of complete lattices have been introduced in many articles.…”