Bases of Continuous Lattices 1 Robert Milewski
نویسنده
چکیده
The article is a Mizar formalization of [13, 168–169]. We show definition and fundamental theorems from theory of basis of continuous lattices. The following proposition is true (1) For every non empty poset L and for every element x of L holds compactbelow(x) = ↓ ↓ x∩the carrier of CompactSublatt(L). Let L be a non empty reflexive transitive relational structure and let X be a subset of Ids(L), ⊆. Then X is a subset of L. Next we state a number of propositions: (2) For every non empty relational structure L and for all subsets X, Y of L such that X ⊆ Y holds finsups(X) ⊆ finsups(Y). (3) Let L be a non empty transitive relational structure, S be a sups-inheriting non empty full relational substructure of L, X be a subset of L, and Y be a subset of S. If X = Y, then finsups(X) ⊆ finsups(Y). (4) Let L be a complete transitive antisymmetric non empty relational structure, S be a sups-inheriting non empty full relational substructure of L, X be a subset of L, and Y be a subset of S. If X = Y, then finsups(X) = finsups(Y). (5) Let L be a complete sup-semilattice and S be a join-inheriting non empty full relational substructure of L. Suppose ⊥ L ∈ the carrier of S. Let X be a subset of L and Y be a subset of S. If X = Y, then finsups(Y) ⊆ finsups(X). (6) For every lower-bounded sup-semilattice L and for every subset X of Ids(L), ⊆ holds sup X = ↓finsups(X). (7) For every reflexive transitive relational structure L and for every subset X of L holds ↓↓X = ↓X.
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Białystok Bases of Continuous Lattices 1 Robert Milewski University of Białystok
The following proposition is true (1) For every non empty poset L and for every element x of L holds compactbelow(x) = ↓x ∩ the carrier of CompactSublatt(L). Let L be a non empty reflexive transitive relational structure and let X be a subset of 〈Ids(L),⊆〉. Then ⋃ X is a subset of L. The following propositions are true: (2) For every non empty relational structure L and for all subsets X, Y of ...
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