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${}^{♯}$-decomposition} iff ${\mathbb{X}}_i$ satisfies Vaught's conjecture (VC) (resp. VC${}^{♯}$: $I({\mathbb{X}}_i)\in \{1,\mathfrak{c}\}$), 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 ${\tau }_i\in \mathrm{Th}({\mathbb{X}}_i)$, $i\in I$, 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${}^{♯}$ is true for $\mathcal{T}$, if there is a VC${}^{♯}$-decomposition of a model of $\mathcal{T}$. VC is true for the partial orders from the closure $⟨{\mathcal{C}}_0^{\mathrm{reticle}}\cup {\mathcal{C}}^{\mathrm{ba}}{⟩}_{\Sigma }$, where $⟨\mathcal{C}{⟩}_{\Sigma }$ denotes the closure of a class $\mathcal{C}$ under finite lexicographic sums. VC${}^{♯}$ 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.

13 pages