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

TL;DR

This paper explores how the Eklof-Mekler-Shelah Construction Principle influences the superstability of free objects in algebraic varieties, revealing that strong forms of this principle lead to unsuperstability in many cases.

Contribution

It establishes a connection between the Construction Principle and superstability of free objects, extending the analysis to AEC-coverings and applications in modules and groups.

Findings

Strong Construction Principle implies most AEC-coverings are unsuperstable.

Applications demonstrate the principle's impact on modules and group varieties.

Results generalize to first-order logic and broader algebraic contexts.

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.