On Galois Connections between External Functions and Relational Constraints: Arity Restrictions and Operator Decompositions

We study the basic Galois connection induced by the ”satisfaction” relation between external functions An → B defined on a set A and valued in a possibly different set B, and ordered pairs (R, S) of relations R ⊆ Am and S ⊆ Bm, called relational constraints. We represent the induced Galois operators as compositions of closure operators associated with necessary and sufficient conditions describing the corresponding Galois closed sets of functions and constraints. We consider further Galois correspondences by restricting the sets of primal and dual objects to fixed arities, and present factorizations of the restricted Galois operators by means of parametrized analogues of the closures mentioned above.