Lawson Topology in Continuous Lattices

Let S, T be semilattices. Let us assume that if S is upper-bounded, then T is upper-bounded. A map from S into T is said to be a semilattice morphism from S into T if: (Def. 1) For every finite subset X of S holds it preserves inf of X. Let S, T be semilattices. One can check that every map from S into T which is meet-preserving is also monotone. Let S be a semilattice and let T be an upper-bounded semilattice. One can check that every semilattice morphism from S into T is meet-preserving. Next we state a number of propositions: (1) For all upper-bounded semilattices S, T and for every semilattice morphism f from S into T holds f(⊤S) = ⊤T . (2) Let S, T be semilattices and f be a map from S into T . Suppose f is meet-preserving. Let X be a finite non empty subset of S. Then f preserves inf of X. Partially supported by NATO Grant CRG 951368, NSERC OGP 9207 grant and KBN grant 8 T11C 018 12.