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 the carrier of L such that X ⊆ Y holds finsups(X) ⊆ finsups(Y ). (3) Let L be a non empty transitive relational structure, S be a supsinheriting non empty full relational substructure of L, X be a subset of the carrier of L, and Y be a subset of the carrier 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 the carrier of L, and Y be a subset of the carrier of S. If X = Y, then finsups(X) = finsups(Y ). This work has been supported by KBN Grant 8 T11C 018 12.