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A note on the atomicity of arithmeticity

Michael Hoefnagel, Pierre-Alain Jacqmin

Abstract

The main aim of this note is to show that, in the regular context, every matrix property in the sense of Z. Janelidze either implies the Mal'tsev property, or is implied by the majority property. When the regular category is arithmetical, i.e., both Mal'tsev and a majority category, then we show that satisfies every non-trivial matrix property.

A note on the atomicity of arithmeticity

Abstract

The main aim of this note is to show that, in the regular context, every matrix property in the sense of Z. Janelidze either implies the Mal'tsev property, or is implied by the majority property. When the regular category is arithmetical, i.e., both Mal'tsev and a majority category, then we show that satisfies every non-trivial matrix property.
Paper Structure (3 sections, 9 theorems, 14 equations, 1 figure)

This paper contains 3 sections, 9 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A strong majority property
  3. The regular context

Key Result

Proposition 2.1

Let $n\geqslant 3$ be an integer. A finitely complete category $\mathbb{C}$ satisfies $(M_n)$ if and only if $\mathbb{C}$ has $\mathsf{M}_n$-closed relations, i.e., $\mathbb{C}$ satisfies the matrix property corresponding to $\mathsf{M}_n$.

Figures (1)

  • Figure 1: The poset of non-degenerate elements of $\mathsf{Mclex}[3,7,2]$. The matrix furthest to the left represents the matrix class $\mathsf{mclex}\{\mathsf{Ari}\}$ and the only matrix with two rows represents $\mathsf{mclex}\{\mathsf{Mal}\}$. The matrix which is just above this latter matrix represents the matrix class $\mathsf{mclex}\{\mathsf{Maj}\}$.

Theorems & Definitions (17)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • proof
  • ...and 7 more