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 x 1 , x 2 Î X we have x 1 £ φ( x 2 ) if and only if f ( x 1 ) £ f ( x 2 ); (b) f is increasingly g -normal, for some function g of Y to X , if for any x Î X and y Î 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.