A generalization of the Curtis–Hedlund theorem

Let A be a set and let G be a group, and equip AG with its prodiscrete uniform structure. Let τ:AG→AG be a map. We prove that τ is a cellular automaton if and only if τ is uniformly continuous and G-equivariant. We also give an example showing that a continuous and G-equivariant map τ:AG→AG may fail to be a cellular automaton when the alphabet set A is infinite.