Meet Continuous Lattices Revisited 1 Artur

The articles [25], [20], [8], [9], [1], [23], [18], [24], [19], [26], [22], [6], [3], [7], [14], [4], [17], [15], [16], [2], [11], [12], [13], [21], and [5] provide the terminology and notation for this paper. The following two propositions are true: (1) For every set x and for every non empty set D holds x∩ ⋃ D = ⋃ {x∩d : d ranges over elements of D}. (2) Let R be a non empty reflexive transitive relational structure and D be a non empty directed subset of 〈Ids(R),⊆〉. Then ⋃ D is an ideal of R. Let R be a non empty reflexive transitive relational structure. Observe that 〈Ids(R),⊆〉 is up-complete. We now state two propositions: (3) Let R be a non empty reflexive transitive relational structure and D be a non empty directed subset of 〈Ids(R),⊆〉. Then supD = ⋃ D.