Vaught's Conjecture and Theories of Partial Order Admitting a Finite Lexicographic Decomposition

TL;DR

This paper investigates Vaught's conjecture within theories of partial orders that admit finite lexicographic decompositions, establishing conditions under which the conjecture holds for these structures and related classes.

Contribution

It introduces the concept of VC-decompositions for partial orders and proves the conjecture for certain classes of theories and models based on their decompositions.

Findings

VC is true for large theories or those with atomic models having VC decompositions.

VC is confirmed for all actually Vaught's FLD$_1$ theories.

VC$^ ext{sharp}$ holds for models with VC$^ ext{sharp}$-decompositions.

Abstract

A complete theory ${\mathcal T}$ of partial order is an FLD$_1$-theory iff some (equivalently, any) of its models ${\mathbb X}$ admits a finite lexicographic decomposition ${\mathbb X} =\sum _{{\mathbb I}}{\mathbb X} _i$, where ${\mathbb I}$ is a finite partial order and ${\mathbb X} _i$-s are partial orders with a largest element. Then we write $\sum _{{\mathbb I}}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ and call $\sum _{{\mathbb I}}{\mathbb X}_i$ a VC-decomposition (resp. a VC$^\sharp$-decomposition} iff ${\mathbb X} _i$ satisfies Vaught's conjecture (VC) (resp. VC$^\sharp$: $I({\mathbb X} _i)\in \{ 1,{\mathfrak{c}}\}$), for each $i\in I$. ${\mathcal T}$ is called actually Vaught's iff for some $\sum _{{\mathbb I}}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ there are sentences $\tau _i\in \mathop{\rm Th}\nolimits ({\mathbb X} _i)$, $i\in I$, providing VC. We prove that: (1) VC is true for ${\mathcal T}$ iff ${\mathcal T}$ is large or its atomic model has a VC decomposition; (2) VC is true for each actually Vaught's FLD$_1$ theory; (3) VC$^\sharp$ is true for ${\mathcal T}$, if there is a VC$^\sharp$-decomposition of a model of ${\mathcal T}$. VC is true for the partial orders from the closure $\langle {\mathcal C} ^{\rm reticle}_0\cup {\mathcal C} ^{\rm ba}\rangle _{\Sigma}$, where $\langle {\mathcal C}\rangle _{\Sigma}$ denotes the closure of a class ${\mathcal C}$ under finite lexicographic sums. VC$^{\sharp}$ is true for a large class of partial orders of the form $\sum _{{\mathbb I}}(\dot{\bigcup}_{j<n_i}\prod _{k<m_i^j}{\mathbb X} _i^{j,k})_r$, where ${\mathbb X} _i^{j,k}$-s can be linear orders, or Boolean algebras, or belong to a wide class of trees.