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

CEMAT - Center for Computational and Stochastic Mathematics