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We study the basic Galois connection induced by the "satisfaction" relationbetween external operations $A^n\rightarrow B$ defined on a set $A$ and valuedin a possibly different set $B$ on the one hand, and ordered pairs $(R,S)$ ofrelations $R\subseteq A^m$ and $S\subseteq B^m$, called relational constraints,on the other hand. We decompose the closure maps associated with this Galoisconnection, in terms of closure operators corresponding to simple closureconditions describing the corresponding Galois closed sets of functions andconstraints. We consider further Galois correspondences by restricting the setsof primal and dual objects to fixed arities. We describe the restricted Galoisclosure systems by means of parametrized analogues of the simpler closureconditions, and present factorizations of the corresponding Galois closuremaps, similar to those provided in the unrestricted case.

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