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

Erhard Aichinger, Mario Kapl, Bernardo Rossi

Abstract

We provide polynomial completeness results for finite algebras in congruence permutable varieties. In 2001, Idziak and Słomczy{ń}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.

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

Abstract

We provide polynomial completeness results for finite algebras in congruence permutable varieties. In 2001, Idziak and Słomczy{ń}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.

Paper Structure

This paper contains 13 sections, 58 theorems, 121 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Completeness and near unanimity functions
  3. Characterization of strictly polynomially rich Mal'cev algebras
  4. Preliminaries on modules over simple rings
  5. Strictly polynomially rich modules over simple rings
  6. Preliminaries on Mal'cev algebras
  7. Sufficient conditions for (SC1)
  8. Homogeneous series in Mal'cev algebras with (SC1)
  9. Remarks on a well known strategy for interpolating partial functions with polynomials
  10. Partial type-preserving functions that are constant modulo an homogeneous abelian congruence with (AB$p$)
  11. Partial type-preserving functions that are constant modulo an homogeneous abelian atom
  12. Interpolation of partial functions that preserve types
  13. Proofs of the results from Section \ref{['sec:content']}

Key Result

Lemma 2.1

Let $A$ be a set, let $k \ge 3$, let $n \in \mathbb{N}$ be such that $n < k$, and let $B \subseteq A^n$. Then the partial function $u_k$ preserves the relation $B$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (104)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 3.1: cf. IS:PRAAI:PIIEAM:TOPC
  • Definition 3.2: cf. Ros24
  • Definition 3.3
  • Definition 3.4
  • ...and 94 more