Sharp Vaught's Conjecture for Some Classes of Partial Orders

TL;DR

This paper extends the sharp Vaught's conjecture, originally proven for linear orders, to broader classes of partial orders and trees, demonstrating the conjecture's validity in these more complex structures.

Contribution

It proves the sharp Vaught's conjecture for certain classes of partial orders and trees, expanding its applicability beyond linear orders.

Findings

The conjecture holds for partial orders built from linear orders via finite products and disjoint unions.

It also applies to trees that are infinite disjoint unions of linear orders.

The results confirm the conjecture's validity in new classes of structures.

Abstract

Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, $I({\mathcal T},\omega )\in \{ 0,1,{\mathfrak{c}}\}$, holds for each complete theory of linear order ${\mathcal T}$. We show that the same is true for each complete theory of partial order having a model in the the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension of the class of rooted trees admitting a finite monomorphic decomposition, obtained in the same way. The sharp version of Vaught's conjecture also holds for the theories of trees which are infinite disjoint unions of linear orders.