Weights of Continuous Lattices

The papers [25], [20], [1], [9], [12], [10], [22], [3], [15], [2], [23], [19], [26], [24], [27], [21], [8], [18], [5], [11], [6], [17], [16], [4], [14], and [7] provide the terminology and notation for this paper. In this article we present several logical schemes. The scheme UparrowUnion deals with a relational structure A and a unary predicate P, and states that: Let S be a family of subsets of the carrier of A. If S = {X; X ranges over subsets of A : P[X]}, then ↑ ⋃ S = ⋃ {↑X;X ranges over subsets of A : P[X]} for all values of the parameters. The scheme DownarrowUnion deals with a relational structure A and a unary predicate P, and states that: Let S be a family of subsets of the carrier of A. If S = {X; X ranges over subsets of A : P[X]}, then ↓ ⋃ S = ⋃ {↓X;X ranges over subsets of A : P[X]} for all values of the parameters. Let L1 be a lower-bounded continuous sup-semilattice and let B1 be a CLbasis of L1 with bottom. One can verify that 〈Ids(sub(B1)),⊆〉 is algebraic. Let L1 be a continuous sup-semilattice. The functor CLweightL1 yields a cardinal number and is defined as follows: (Def. 1) CLweightL1 = ⋂ {B1 : B1 ranges over CLbasis of L1 with bottom}. We now state a number of propositions: