@MISC{Grabowski_thefollowing, author = {Adam Grabowski}, title = {The following proposition is true 1. PRELIMINARIES}, year = {} }

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Abstract

(1) Let S, T be up-complete Scott top-lattices and M be a subset of SCMaps(S,T). Then SCMaps(S,T) M is a continuous map from S into T. Let S be a non empty relational structure and let T be a non empty reflexive relational structure. One can verify that every map from S into T which is constant is also monotone. Let S be a non empty relational structure, let T be a reflexive non empty relational structure, and let a be an element of T. One can check that S ↦− → a is monotone. One can prove the following propositions: (2) Let S be a non empty relational structure and T be a lower-bounded antisymmetric reflexive non empty relational structure. Then ⊥ MonMaps(S,T) = S ↦− → ⊥T. (3) Let S be a non empty relational structure and T be an upper-bounded antisymmetric reflexive non empty relational structure. Then ⊤ MonMaps(S,T) = S ↦− → ⊤T. (4) Let S, T be complete lattices, f be a monotone map from S into T, and x be an element of S. Then f (x) = sup ( f ◦ ↓x). (5) Let S, T be complete lower-bounded lattices, f be a monotone map from S into T, and x be an element of S. Then f (x) = � T { f (w);w ranges over elements of S: w ≤ x}. (6) Let S be a relational structure, T be a non empty relational structure, and F be a subset of T the carrier of S. Then supF is a map from S into T. 2. ON THE SCOTT CONTINUITY OF MAPS Let X1, X2, Y be non empty relational structures, let f be a map from [:X1, X2:] into Y, and let x be an element of X1. The functor Proj ( f,x) yields a map from X2 into Y and is defined by: (Def. 1) Proj ( f,x) = (curry f)(x).