Polynomial interpolation of partial functions in finite algebras with a Mal'cev term

TL;DR

This paper establishes polynomial completeness results for finite algebras with a Mal'cev term in congruence permutable varieties, focusing on polynomially rich and strictly polynomially rich algebras.

Contribution

It characterizes strictly polynomially rich finite algebras in congruence permutable varieties, extending the concept of polynomial richness introduced in 2001.

Findings

Characterization of strictly polynomially rich algebras

Extension of polynomial completeness results to finite algebras with a Mal'cev term

Clarification of polynomially rich properties in congruence permutable varieties

Abstract

We provide polynomial completeness results for finite algebras in congruence permutable varieties. In 2001, Idziak and S{\l}omczy{\'n}ska introduced the completeness concept of being \emph{polynomially rich}: a finite algebra is polynomially rich if every function preserving congruences and the Tame Congruence Theory labelling of prime quotients in the congruence lattice is a polynomial function of the algebra. We call a finite algebra \emph{strictly polynomially rich} if every partial congruence and type preserving function is a polynomial function, and we describe strictly polynomially rich algebras in congruence permutable varieties.