Rishi Banerjee of University of Illinois Chicago : Infinitary positive existential normal forms for modules

Posted by Rishi Banerjee , part of the Logic Seminar.

At

Aug. 25, 2026, 3 p.m.

In

636 SEO

Abstract

We will prove an infinitary logic analogue of the classical first order quantifier elimination theorem for modules. Fix a ring $R$ and a regular cardinal $\theta$. For every left $R$-module $M$, there is an increasing sequence of parameter sets $I_\alpha \subseteq M^{<\theta}, |I_\alpha| \leq \beth_\alpha(|R| + \theta)$, such that every parameter-free $L_{\infty,\theta}$ formula of quantifier rank $\alpha$ is equivalent in $M$ to an infinitary Boolean combination of formulas of the form $\psi(\overline{x} - \overline{a})$, where $\overline{a} \in I_\alpha$ and $\psi$ belongs to level $\alpha$ of Shelah's hierarchy of infinitary positive existential formulas. This hierarchy is the natural generalization of positive primitive formulas to infinitary logic. The main ingredient in the proof is a combinatorial lemma giving a small test set for deciding whether cosets of a fixed family of subgroups cover an abelian group. We will also show that the model-dependent parameters are necessary in general, which is an important difference from the first-order case.