Vaught's Conjecture for Unions of Products of Rooted Trees

Abstract

Let ${\mathcal{C}}^{\mathrm{rt}}$ be the class of rooted trees and $⟨{\mathcal{C}}^{\mathrm{rt}}{⟩}_{\dot{\cup }\Pi }$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to $\mathbb{X}={\dot{\bigcup }}_{i<n}{\prod }_{j<m_i}{\mathbb{X}}_i^j$, where ${\mathbb{X}}_i^j$ are rooted trees. Defining $\mathcal{T}=\mathrm{Th}(\mathbb{X})$, ${\mathcal{T}}_i^j=\mathrm{Th}({\mathbb{X}}_i^j)$, for $i<n$ and $j<m_i$, and $\kappa ={\prod }_{i<n}{\prod }_{j<m_i}I({\mathcal{T}}_i^j)$, we have (a) Vaught's conjecture is true for $\mathcal{T}$: $I(\mathcal{T})=\kappa$, if $\kappa \in \{1,\omega ,\mathfrak{c}\}$, and, otherwise, $I(\mathcal{T})\in [3,\omega )$; (b) $\mathbb{Y}\equiv \mathbb{X}$ iff $\mathbb{Y}\cong {\dot{\bigcup }}_{i<n}{\prod }_{j<m_i}{\mathbb{Y}}_i^j$, where ${\mathbb{Y}}_i^j\equiv {\mathbb{X}}_i^j$, for $i<n$ and $j<m_i$; (c) $\mathbb{E}\preccurlyeq \mathbb{X}$ iff $\mathbb{E}={\dot{\bigcup }}_{i<n}{\prod }_{j<m_i}{\mathbb{E}}_i^j$, where ${\mathbb{E}}_i^j\preccurlyeq {\mathbb{X}}_i^j$, for $i<n$ and $j<m_i$; (d) $\mathcal{T}$ is atomic iff ${\mathcal{T}}_i^j$, for $i<n$ and $j<m_i$, are atomic; then ${\dot{\bigcup }}_{i<n}{\prod }_{j<m_i}{\mathbb{A}}_i^j$ is a countable atomic model of $\mathcal{T}$, where ${\mathbb{A}}_i^j$ is a countable atomic model of ${\mathcal{T}}_i^j$, for $i<n$ and $j<m_i$; (e) $\mathcal{T}$ is small iff ${\mathcal{T}}_i^j$, for $i<n$ and $j<m_i$, are small; then ${\dot{\bigcup }}_{i<n}{\prod }_{j<m_i}{\mathbb{S}}_i^j$ is a countably saturated model of $\mathcal{T}$, where ${\mathbb{S}}_i^j$ is a countably saturated model of ${\mathcal{T}}_i^j$, for $i<n$ and $j<m_i$.

16 pages