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The Construction Principle and superstability of free objects in varieties of algebras

Tapani Hyttinen, Gianluca Paolini, Davide Emilio Quadrellaro

Abstract

We investigate the relationship between the Eklof-Mekler-Shelah Construction Principle for a variety of algebras $\mathbf{V}$ and the question of superstability of the free objects in $\mathbf{V}$, denoted as $\mathcal{F}_\mathbf{V}$. We consider this question in the general setting of AEC-coverings of $\mathcal{F}_\mathbf{V}$, with applications to first-order logic and beyond. Our main result is that if a strong form of the Construction Principle is satisfied, then almost all AEC-covering of $\mathcal{F}_\mathbf{V}$ are unsuperstable. Concrete applications to $R$-modules and varieties of groups are also considered.

The Construction Principle and superstability of free objects in varieties of algebras

Abstract

We investigate the relationship between the Eklof-Mekler-Shelah Construction Principle for a variety of algebras and the question of superstability of the free objects in , denoted as . We consider this question in the general setting of AEC-coverings of , with applications to first-order logic and beyond. Our main result is that if a strong form of the Construction Principle is satisfied, then almost all AEC-covering of are unsuperstable. Concrete applications to -modules and varieties of groups are also considered.
Paper Structure (11 sections, 14 theorems, 12 equations)

This paper contains 11 sections, 14 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. The first venue
  3. $R$-modules
  4. Groups
  5. The second venue
  6. Preliminaries
  7. The first venue
  8. Applications to $R$-modules
  9. Applications to groups
  10. The second venue
  11. Applications

Key Result

Theorem 1.7

Let $\mathbf{V}$ be a countable variety of algebras which satisfies $\mathrm{RCP}$ (cf. RCP_intro) and let $(\mathcal{K}, \preccurlyeq)$ be a strong enough (cf. def_enough_strong) $\mathrm{AEC}$-covering of $(\mathcal{F}_\mathbf{V}, \leqslant_{\ast}^{\mathrm{ff}})$. Then $(\mathcal{K}, \preccurlyeq)

Theorems & Definitions (44)

  • Definition 1.2
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.9
  • Corollary 1.10
  • Corollary 1.11
  • Theorem 1.12
  • Definition 1.13
  • Corollary 1.14
  • Definition 2.1
  • ...and 34 more