Duality Based on the Galois Connection. Part I

Let S, T be complete lattices. One can check that there exists a connection between S and T which is Galois. Next we state the 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 as follows: (Def. 1) 〈g, the lower adjoint of g〉 is Galois.