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Epimorphisms between finitely generated algebras

Luca Carai, Miriam Kurtzhals, Tommaso Moraschini

TL;DR

The paper investigates when epimorphisms between finitely generated algebras in a quasivariety are surjective, formalizing the weak ES property and its logical connection to finite Beth definability. It develops a characterization and reduction tools for detecting weak ES, notably in contexts with a near-unanimity term and in congruence-permutable varieties, and shows that weak ES often forces arithmeticity. Key contributions include a near-unanimity based reduction to subdirect products of fewer factors, a CP-compatible obstruction theory for fully epic subalgebras, and a consequence that filtral varieties with weak ES are discriminator varieties. The results illuminate a deep link between definability phenomena and algebraic structure, providing practical criteria for verifying weak ES and revealing when such algebras exhibit arithmetical or discriminator behavior.

Abstract

A quasivariety has the weak ES property when the epimorphisms between its finitely generated members are surjective. A characterization of quasivarieties with the weak ES property is obtained and a method for detecting failures of this property in quasivarieties with a near unanimity term and in congruence permutable varieties is given. It is also shown that under reasonable assumptions the weak ES property implies arithmeticity. In particular, every filtral variety with the weak ES property is a discriminator variety.

Epimorphisms between finitely generated algebras

TL;DR

The paper investigates when epimorphisms between finitely generated algebras in a quasivariety are surjective, formalizing the weak ES property and its logical connection to finite Beth definability. It develops a characterization and reduction tools for detecting weak ES, notably in contexts with a near-unanimity term and in congruence-permutable varieties, and shows that weak ES often forces arithmeticity. Key contributions include a near-unanimity based reduction to subdirect products of fewer factors, a CP-compatible obstruction theory for fully epic subalgebras, and a consequence that filtral varieties with weak ES are discriminator varieties. The results illuminate a deep link between definability phenomena and algebraic structure, providing practical criteria for verifying weak ES and revealing when such algebras exhibit arithmetical or discriminator behavior.

Abstract

A quasivariety has the weak ES property when the epimorphisms between its finitely generated members are surjective. A characterization of quasivarieties with the weak ES property is obtained and a method for detecting failures of this property in quasivarieties with a near unanimity term and in congruence permutable varieties is given. It is also shown that under reasonable assumptions the weak ES property implies arithmeticity. In particular, every filtral variety with the weak ES property is a discriminator variety.
Paper Structure (7 sections, 28 theorems, 49 equations)

This paper contains 7 sections, 28 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Varieties and quasivarieties
  3. Epimorphism surjectivity
  4. Quasivarieties with a near unanimity term
  5. Congruence permutable varieties
  6. Arithmeticity is often necessary
  7. Acknowledgements.

Key Result

Proposition 2.2

Let $\A$ be an algebra and $X \subseteq \mathsf{Con}(\A)$. Then the map defined by the rule $f(a / \bigcap X) \coloneqq \langle a / \theta : \theta \in X \rangle$ is a subdirect embedding.

Theorems & Definitions (55)

  • Proposition 2.2
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['Prop : epic']}.
  • Definition 3.4
  • ...and 45 more