arXiv subjects
Publications and source records attributed to Angus Matthews.
We show that large rosy fields are bounded, and substantially simplify the proofs that large stable fields are separably closed and that large simple fields are bounded. Our proofs go through in a general topological setting. We obtain instability, non-simplicity, and non-rosiness explicitly by building the appropriate trees of definable sets. We also show that orders on large rosy fields have several properties of orders on pseudo real closed fields.
In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical. Combining the first equivalence with an earlier result of Hrushovski, we deduce that algebraically bounded stable fields are precisely expansions of algebraically closed fields by constants. We then consider algebraically bounded and o-minimal expansions of fields with generic derivations. We show that if $\mathbb{M}$ is a simple algebraically bounded structure and $\Delta$ is a generic tuple of derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is supersimple if and only if the derivations commute. Similarly, if $\mathbb{M}$ is an o-minimal structure and $\Delta$ is a generic tuple of $T$-derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is superrosy if and only if the derivations commute. We obtain explicit bounds on ranks using the Kolchin polynomial.
We present a Myhill-Nerode theorem for hypergraphs. The theorem involves an operation which takes two input structures and produces a hypergraph as output. Using this operation, we define a Myhill-Nerode-type equivalence relation and show that if a class of hypergraphs is definable in the counting monadic second-order logic of hypergraphs, then the equivalence relation has finite index. We apply this tool to classes of gain-graphic matroids, and show that if the group $\Gamma$ is not uniformly locally finite, then the class of $\Gamma$\dash gain-graphic matroids is not monadically definable. (A group is uniformly locally finite if, for every $k$, there is a maximum size amongst subgroups generated by at most $k$ elements.) In addition, we define the conviviality graph of a group, and show that if the group $\Gamma$ has an infinite conviviality graph, then the class of $\Gamma$\dash gain-graphic matroids is not monadically definable. This will be useful in future constructions.
Let $T$ be an o-minimal theory expanding $\mathrm{RCF}$ and $T_\mathrm{convex}$ be the common theory of its models expanded by predicate for a non-trivial $T$-convex valuation ring. We call an elementary extension $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_*, \mathcal{O}_*) \models T_{\mathrm{convex}}$ $\textit{res-constructible}$ if there is a tuple $\overline{s}$ in $\mathcal{O}_*$ such that $\mathbb{E}_* = \mathrm{dcl}(\mathbb{E},\overline{s})$, and the projection $\mathbf{res}(\overline{s})$ of $\overline{s}$ in the residue field sort is $\mathrm{dcl}$-independent over the residue field $\mathbf{res}(\mathbb{E}, \mathcal{O})$ of $(\mathbb{E}, \mathcal{O})$. We study factorization properties of res-constructible extensions. Our main result is that a res-constructible extension $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_*, \mathcal{O}_*)$ has the property that all $(\mathbb{E}_1, \mathcal{O}_1)$ with $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_1, \mathcal{O}_1) \prec (\mathbb{E}_*, \mathcal{O}_*)$ are res-constructible over $(\mathbb{E}, \mathcal{O})$, if and only if $\mathbb{E}_*$ has countable $\mathrm{dcl}$-dimension over $\mathbb{E}$ or the value group $\mathbf{val}(\mathbb{E}_*, \mathcal{O}_*)$ is $\textit{short}$ (i.e. contains no uncountable well-ordered subset). This analysis entails complete answers to [11, Problem 5.12].