Vaught's Conjecture for Sums of Products of Rooted Trees

Abstract

We confirm Vaught&#39;s conjecture for each partial order ${\mathbb X}=\sum _{\mathbb I}\prod _{j<m_i}{\mathbb X}_{i,j}$ from the closure $\langle \langle {\mathcal C}^{\rm rt}\rangle _Π\rangle _Σ$ of the class ${\mathcal C}^{\rm rt}$ of rooted trees under finite direct products and lexicographic sums. In addition, (a) ${\mathcal T}:=\mathop{\rm Th}\nolimits ({\mathbb X})$ is $ω$-categorical iff all the theories ${\mathcal T}_{i,j}:=\mathop{\rm Th}\nolimits ({\mathbb X}_{i,j})$ are $ω$-categorical; (b) ${\mathcal T}$ satisfies VC$^\sharp$ (that is, $I({\mathcal T})\in \{1,{\mathfrak{c}}\}$), if VC$^\sharp$ holds for all ${\mathcal T}_{i,j}$. If ${\mathbb X}\in \langle \langle {\mathcal C}^{\rm rt}_{\rm fa}\rangle _Π\rangle _Σ$, where ${\mathcal C}^{\rm rt}_{\rm fa}$ is the class of finitely axiomatizable rooted trees, then (c) ${\mathcal T}$ is finitely axiomatizable and (d) ${\mathbb Y}\in \mathop{\rm Mod}\nolimits ({\mathcal T})$ iff ${\mathbb Y} \cong \sum _{\mathbb I}\prod _{j<m_i}{\mathbb Y}_{i,j}$, where ${\mathbb Y}_{i,j}\in \mathop{\rm Mod}\nolimits ({\mathcal T}_{i,j})$, for all indices. As a by-product we prove that (e) For each $n$ the class ${\mathcal C} _n$ of partial orders isomorphic to a direct product of $n$ rooted trees of size $>1$ is first-order definable; (f) Each $ω$-categorical partial order from the class $\langle \langle {\mathcal C}^{\rm rt}_{\rm fb}\rangle _Π\rangle _Σ$, where ${\mathcal C}^{\rm rt}_{\rm fb}$ is the class of finite-branching rooted trees, is finitely axiomatizable. Statement (f) is related to the results of Rosenstein (for the class of linear orders) and Schmerl (for the class of partial orders of finite width).