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Vector spaces with a union of independent subspaces

Alessandro Berarducci, Marcello Mamino, Rosario Mennuni

TL;DR

The paper analyzes the theory $T_K$ of $K$-vector spaces equipped with a predicate $X$ for the union of an infinite family of independent subspaces. It proves quantifier elimination and completeness for infinite $K$ (in the language augmented by $X^n$ predicates), and shows total transcendence and superstability, while treating the finite-field case via a natural near-model-complete completion. It develops a detailed combinatorial and geometric framework based on supports, axes, and weights to achieve quantifier elimination in the infinite-field setting and to characterize models. For finite $K$, it demonstrates incompleteness and near-model-completeness, and it connects the model theory to locally definable groups by showing that a definably generated group arising in this context cannot contain a discrete $Z$-subgroup, informing questions about the structure of locally definable groups in o-minimal-like settings.

Abstract

Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and admits quantifier elimination in the language of $K$-vector spaces with predicates for the $n$-fold sums of $X$ with itself. If $K$ is finite this is no longer true, but we still have that a natural completion is near-model-complete.

Vector spaces with a union of independent subspaces

TL;DR

The paper analyzes the theory of -vector spaces equipped with a predicate for the union of an infinite family of independent subspaces. It proves quantifier elimination and completeness for infinite (in the language augmented by predicates), and shows total transcendence and superstability, while treating the finite-field case via a natural near-model-complete completion. It develops a detailed combinatorial and geometric framework based on supports, axes, and weights to achieve quantifier elimination in the infinite-field setting and to characterize models. For finite , it demonstrates incompleteness and near-model-completeness, and it connects the model theory to locally definable groups by showing that a definably generated group arising in this context cannot contain a discrete -subgroup, informing questions about the structure of locally definable groups in o-minimal-like settings.

Abstract

Motivated by the theory of locally definable groups, we study the theory of -vector spaces with a predicate for the union of an infinite family of independent subspaces. We show that if is infinite then the theory is complete and admits quantifier elimination in the language of -vector spaces with predicates for the -fold sums of with itself. If is finite this is no longer true, but we still have that a natural completion is near-model-complete.
Paper Structure (5 sections, 11 theorems, 5 equations)

This paper contains 5 sections, 11 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. The theory
  3. Infinite fields
  4. The finite field case
  5. Locally definable groups

Key Result

Theorem 2.5

For every ordinal $\alpha$ such that $\lvert K\rvert\le \aleph_\alpha$, there are at most $2^{\aleph_0+\lvert\alpha\rvert}$ models of $T_K$ of size $\aleph_\alpha$. In particular, (every completion of) $T_K$ is superstable.

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.2
  • proof
  • Example 2.3
  • Theorem 2.5
  • proof
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 22 more