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Minh Chieu Tran

Publications and source records attributed to Minh Chieu Tran.

5 recordsLinked to original sources
arXiv2022

Interpolative fusions II: Preservation results

We study interpolative fusion, a method of combining theories $T_1$ and $T_2$ in distinct languages in a "generic" way over a common reduct $T_\cap$, to obtain a theory $T_\cup^*$. When each $T_i$ is model-complete, $T_\cup^*$ is the model companion of the union $T_1\cup T_2$. Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of $T_1$ and $T_2$ are inherited by $T_\cup^*$. We first prove preservation results for quantifier elimination, model-completeness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of $T_\cap$, the property $\mathrm{NSOP}_1$ is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in $T_1$ and $T_2$. This generalizes many previous results; for example, simplicity of $\mathrm{ACFA}$ and the random $n$-hypergraph are both non-obvious corollaries. We also address preservation of stability, $\mathrm{NIP}$, and $\aleph_0$-categoricity, and we describe examples which witness that these results are sharp.

math.LO↗
arXiv2021

Interpolative Fusions I

We define the interpolative fusion $T^*_\cup$ of a family $(T_i)_{i \in I}$ of first-order theories over a common reduct $T_\cap$, a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each $T_i$ is model-complete, $T^*_\cup$ coincides with the model companion of $T_\cup = \bigcup_{i \in I} T_i$. By obtaining sufficient conditions for the existence of $T^*_\cup$, we develop new tools to show that theories of interest have model companions.

math.LO↗
arXiv2021

The additive groups of $\mathbb{Z}$ and $\mathbb{Q}$ with predicates for being square-free

We consider the four structures $(\mathbb{Z}; \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Z}; <, \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Q}; \mathrm{Sqf}^\mathbb{Q})$, and $(\mathbb{Q}; <, \mathrm{Sqf}^\mathbb{Q})$ where $\mathbb{Z}$ is the additive group of integers, $\mathrm{Sqf}^\mathbb{Z}$ is the set of $a \in \mathbb{Z}$ such that $v_{p}(a) < 2$ for every prime $p$ and corresponding $p$-adic valuation $v_{p}$, $\mathbb{Q}$ and $\mathrm{Sqf}^\mathbb{Q}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

math.LO↗
arXiv2017

Algebraically Closed Fields with a Generic Multiplicative Character

We study the model theory of the $2$-sorted structure $(\mathbb{F}, \mathbb{C};χ)$, where $\mathbb{F}$ is an algebraic closure of a finite field of characteristic $p$, $\mathbb{C}$ is the field of complex numbers and $χ: \mathbb{F} \to \mathbb{C}$ is an injective, multiplication preserving map. We obtain an axiomatization $\mathrm{ACFC}_p$ of $\mathrm{Th}(\mathbb{F},\mathbb{C};χ)$ in a suitable language $L$, classify the models of $\mathrm{ACFC}_p$ up to isomorphism, prove a modified model companion result, give various descriptions of definable sets inside a model of $\mathrm{ACFC}_p$, and deduce that $\mathrm{ACFC}_p$ is $ω$-stable and has definability of Morley rank in families.

math.LO↗