Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.13111 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911007824150528 |
|---|---|
| author | Abd-Aldaim, A. Conant, G. Terry, C. |
| author_facet | Abd-Aldaim, A. Conant, G. Terry, C. |
| contents | The $k$-dimensional functional order property ($\text{FOP}_k$) is a combinatorial property of a $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf, which identified $\text{NFOP}_2$ as a ternary analogue of stability in the context of two finitary combinatorial problems related to hypergraph regularity and arithmetic regularity. In this paper we show $\text{NFOP}_k$ has equally strong implications in model-theoretic classification theory, where its behavior as a $(k+1)$-ary version of stability is in close analogy to the behavior of $k$-dependence as a $(k+1)$-ary version of $\text{NIP}$. Our results include several new characterizations of $\text{NFOP}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when $k=2$. As a corollary of our collapsing theorem, we show $\text{NFOP}_k$ is closed under Boolean combinations, and that $\text{FOP}_k$ can always be witnessed by a formula where all but one variable have length $1$. When $k=2$, we prove a composition lemma analogous to that of Chernikov and Hempel from the setting of $2$-dependence. Using this, we provide a new class of algebraic examples of $\text{NFOP}_2$ theories. Specifically, we show that if $T$ is the theory of an infinite dimensional vector space over a field $K$, equipped with a bilinear form satisfying certain properties, then $T$ is $\text{NFOP}_2$ if and only if $K$ is stable. Along the way we provide a corrected and reorganized proof of Granger's quantifier elimination and completeness results for these theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13111 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher arity stability and the functional order property Abd-Aldaim, A. Conant, G. Terry, C. Logic The $k$-dimensional functional order property ($\text{FOP}_k$) is a combinatorial property of a $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf, which identified $\text{NFOP}_2$ as a ternary analogue of stability in the context of two finitary combinatorial problems related to hypergraph regularity and arithmetic regularity. In this paper we show $\text{NFOP}_k$ has equally strong implications in model-theoretic classification theory, where its behavior as a $(k+1)$-ary version of stability is in close analogy to the behavior of $k$-dependence as a $(k+1)$-ary version of $\text{NIP}$. Our results include several new characterizations of $\text{NFOP}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when $k=2$. As a corollary of our collapsing theorem, we show $\text{NFOP}_k$ is closed under Boolean combinations, and that $\text{FOP}_k$ can always be witnessed by a formula where all but one variable have length $1$. When $k=2$, we prove a composition lemma analogous to that of Chernikov and Hempel from the setting of $2$-dependence. Using this, we provide a new class of algebraic examples of $\text{NFOP}_2$ theories. Specifically, we show that if $T$ is the theory of an infinite dimensional vector space over a field $K$, equipped with a bilinear form satisfying certain properties, then $T$ is $\text{NFOP}_2$ if and only if $K$ is stable. Along the way we provide a corrected and reorganized proof of Granger's quantifier elimination and completeness results for these theories. |
| title | Higher arity stability and the functional order property |
| topic | Logic |
| url | https://arxiv.org/abs/2305.13111 |