Rishi Banerjee
University of Illinois Chicago (UIC)
Title
Infinitary positive existential normal forms for modules
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.
