Infinitary positive existential normal forms for modules
Abstract
Let θ be a regular cardinal, R a ring, and M a left R-module. We prove that for every ordinal α there is a set Iα⊆ M<θ of size at most bethα(|R| + θ) such that every parameter-free L∞,θ formula of rank at most α is equivalent in M to an infinitary Boolean combination of cosets a̅ + ϕ(M), where a̅ ∈ Iα and ϕ is an infinitary positive existential formula of rank at most α. The main ingredient in the proof is a combinatorial lemma which says that given κ subgroups of an abelian group, there is a set of at most 2κ points which tests whether any family in which each member is either empty or a coset of the corresponding subgroup covers the whole group. The proof proceeds by applying this lemma fiberwise to show that the relevant complete Boolean algebras of positive-existentially definable cosets are closed under projections.
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BibTeX
@misc{banerjee2026infinitary,
title = {{Infinitary positive existential normal forms for modules}},
author = {Rishi Banerjee},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.16692},
eprint = {2608.16692},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16692},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinitary positive existential normal forms for modules
AU - Rishi Banerjee
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.16692
UR - https://arxiv.org/abs/2608.16692
AB - Let θ be a regular cardinal, R a ring, and M a left R-module. We prove that for every ordinal α there is a set I_α⊆ M^<θ of size at most beth_α(|R| + θ) such that every parameter-free L_∞,θ formula of rank at most α is equivalent in M to an infinitary Boolean combination of cosets a̅ + ϕ(M), where a̅ ∈ I_α and ϕ is an infinitary positive existential formula of rank at most α. The main ingredient in the proof is a combinatorial lemma which says that given κ subgroups of an abelian group, there is a set of at most 2^κ points which tests whether any family in which each member is either empty or a coset of the corresponding subgroup covers the whole group. The proof proceeds by applying this lemma fiberwise to show that the relevant complete Boolean algebras of positive-existentially definable cosets are closed under projections.
ER -