Duality Based on Galois Connection. Part I

In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [15, p. 179–183; 1.1–1.12]. The duality is based on the concept of the Galois connection. Let S, T be complete lattices. Note that there exists a connection between S and T which is Galois. One can prove the following proposition (1) Let S, T , S , T be non empty relational structures. Suppose that (i) the relational structure of S = the relational structure of S , and (ii) the relational structure of T = the relational structure of T. Let c be a connection between S and T and c be a connection between S and T. If c = c , then if c is Galois, then c is Galois. Let S, T be lattices and let g be a map from S into T. Let us assume that S is complete and T is complete and g is infs-preserving. The lower adjoint of g is a map from T into S and is defined by: (Def. 1) g, the lower adjoint of g is Galois. Let S, T be lattices and let d be a map from T into S. Let us assume that S is complete and T is complete and d is sups-preserving. The upper adjoint of d is a map from S into T and is defined as follows: (Def. 2) the upper adjoint of d, d is Galois. Let S, T be complete lattices and let g be an infs-preserving map from S into T. Note that the lower adjoint of g is lower adjoint. Let S, T be complete lattices and let d be a sups-preserving map from T into S. One can verify that the upper adjoint of d is upper adjoint. One can prove the following propositions: (2) Let S, T be complete lattices, g be an infs-preserving map from S into T , and t be an element of T. Then (the lower adjoint of g)(t) = inf(g −1 (↑t)).