Scott - Continuous Functions 1 Adam Grabowski

1. PRELIMINARIES Let S be a non empty set and let a, b be elements of S. The functor a,b, ... yielding a function from N into S is defined by the condition (Def. 1). (Def. 1) Let i be a natural number. Then (i) if there exists a natural number k such that i = 2 · k, then (a,b, ...)(i) = a, and (ii) if it is not true that there exists a natural number k such that i = 2 · k, then (a,b, ...)(i) = b. One can prove the following three propositions: (1) Let S, T be non empty reflexive relational structures, f be a map from S into T , and P be a lower subset of T . If f is monotone, then f−1(P) is lower. (2) Let S, T be non empty reflexive relational structures, f be a map from S into T , and P be an upper subset of T . If f is monotone, then f−1(P) is upper. (3) Let S, T be reflexive antisymmetric non empty relational structures and f be a map from S into T . If f is directed-sups-preserving, then f is monotone. Let S, T be reflexive antisymmetric non empty relational structures. Note that every map from S into T which is directed-sups-preserving is also monotone. Next we state the proposition (4) Let S, T be up-complete Scott top-lattices and f be a map from S into T . If f is continuous, then f is monotone. Let S, T be up-complete Scott top-lattices. One can verify that every map from S into T which is continuous is also monotone. 1This work has been supported by KBN Grant 8 T11C 018 12.