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

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.