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

TL;DR

This paper explores the Galois connection between external operations and relational constraints, focusing on arity restrictions and operator decompositions to understand their closure properties.

Contribution

It introduces a decomposition of Galois closure maps and analyzes the effects of arity restrictions on these correspondences.

Findings

Decomposition of Galois closure maps into simpler operators

Characterization of closure systems under arity restrictions

Factorization results for restricted Galois closures

Abstract

We study the basic Galois connection induced by the "satisfaction" relation between external operations $A^n\rightarrow B$ defined on a set $A$ and valued in a possibly different set $B$ on the one hand, and ordered pairs $(R,S)$ of relations $R\subseteq A^m$ and $S\subseteq B^m$, called relational constraints, on the other hand. We decompose the closure maps associated with this Galois connection, in terms of closure operators corresponding to simple closure 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. We describe the restricted Galois closure systems by means of parametrized analogues of the simpler closure conditions, and present factorizations of the corresponding Galois closure maps, similar to those provided in the unrestricted case.