On polynomial completeness properties of finite Mal'cev algebras

Abstract: 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.