On polynomial completeness properties of finite Mal'cev algebras

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.