Closed classes of functions, generalized constraints and clusters

TL;DR

This paper extends Galois theory to infinite domains, characterizing classes of functions and operations through generalized constraints and clusters, respectively, and establishing a Galois connection between them.

Contribution

It introduces a framework for characterizing function classes and operation classes on infinite domains using generalized constraints and clusters, extending existing Galois theory.

Findings

Characterization of classes of functions via generalized constraints.

Characterization of classes of operations via clusters.

Establishment of a Galois connection between operations and clusters.

Abstract

Classes of functions of several variables on arbitrary non-empty domains that are closed under permutation of variables and addition of dummy variables are characterized in terms of generalized constraints, and hereby Hellerstein's Galois theory of functions and generalized constraints is extended to infinite domains. Furthermore, classes of operations on arbitrary non-empty domains that are closed under permutation of variables, addition of dummy variables and composition are characterized in terms of clusters, and a Galois connection is established between operations and clusters.