Galois-type Connections and Closure Operations on Preordered Sets

For a function f of one preordered set X to another Y , we shall establish several consequences of the following two definitions: (a) f is increasingly φ–regular, for some function φ of X to itself, if for any x1 , x2 ∈ X we have x1 ≤ φ (x2) if and only if f (x1) ≤ f (x2); (b) f is increasingly g–normal, for some function g of Y to X , if for any x ∈ X and y ∈ Y we have f (x) ≤ y if and only if x ≤ g (y). These definitions have been mainly suggested to us by a recent theory of relators (families of relations) worked out by Á. Száz and G. Pataki and the extensive literature on Galois connections and residuated mappings.