Polynomially rich algebras

An algebra A is said to be polynomially rich if every mapping f:As→A that preserves congruences of A and their labeling (in the sense of Tame Congruence Theory) is already a polynomial of A. We show that a locally finite variety in which all finite algebras are polynomially rich must be congruence modular. Moreover, we give a full and relatively simple characterization of polynomially rich congruence permutable varieties.