On polynomial completeness properties of finite Mal'cev algebras

TL;DR

This paper extends the characterization of polynomial completeness to finite congruence regular Mal'cev algebras, focusing on functions that preserve congruences and can be interpolated by polynomials.

Contribution

It generalizes existing results from expanded groups to a broader class of finite Mal'cev algebras, advancing the understanding of polynomial functions in algebraic structures.

Findings

Extended the characterization of strictly 1-affine complete algebras

Applied the results to finite congruence regular Mal'cev algebras

Connected polynomial completeness with congruence preservation

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.