  Polynomial completeness results aim at characterizing those functions that
are induced by polynomials. Each polynomial function is congruence preserving,
but the opposite need not be true. A finite algebraic structure $\mathbf{A}$ is
called strictly 1-affine complete if every unary partial function from a subset
of $A$ to $A$ that preserves the congruences of $\mathbf{A}$ can be
interpolated by a polynomial function of $\mathbf{A}$. The problem of
characterizing strictly 1-affine complete finite Mal'cev algebras is still
open. In this paper we extend the characterization by E. Aichinger and P.
Idziak of strictly 1-affine complete expanded groups to finite congruence
regular Mal'cev algebras.
