发表机构
Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院数学与信息系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文确认了Vaught猜想对于由有根树经有限直积和字典序和构成的偏序类成立，并证明了相关理论的可公理化性质。
AI 中文摘要
我们确认了Vaught猜想对于每个偏序${\mathbb X}=\sum _{\mathbb I}\prod _{j<m_i}{\mathbb X}_{i,j}$成立，其中${\mathbb X}$来自类${\mathcal C}^{\rm rt}$（有根树）在有限直积和字典序和下的闭包$\langle \langle {\mathcal C}^{\rm rt}\rangle _\Pi\rangle _\Sigma$。此外，(a) ${\mathcal T}:=\mathop{\rm Th}\nolimits ({\mathbb X})$是$\omega $-范畴的当且仅当所有理论${\mathcal T}_{i,j}:=\mathop{\rm Th}\nolimits ({\mathbb X}_{i,j})$都是$\omega $-范畴的；(b) 如果VC$^\sharp$对所有${\mathcal T}_{i,j}$成立，则${\mathcal T}$满足VC$^\sharp$（即$I({\mathcal T})\in \{1,{\mathfrak{c}}\}$）。如果${\mathbb X}\in \langle \langle {\mathcal C}^{\rm rt}_{\rm fa}\rangle _\Pi\rangle _\Sigma$，其中${\mathcal C}^{\rm rt}_{\rm fa}$是有限可公理化有根树的类，那么(c) ${\mathcal T}$是有限可公理化的，且(d) ${\mathbb Y}\in \mathop{\rm Mod}\nolimits ({\mathcal T})$当且仅当${\mathbb Y} \cong \sum _{\mathbb I}\prod _{j<m_i}{\mathbb Y}_{i,j}$，其中${\mathbb Y}_{i,j}\in \mathop{\rm Mod}\nolimits ({\mathcal T}_{i,j})$，对所有指标成立。作为副产品，我们证明了(e) 对于每个$n$，与$n$个大小$>1$的有根树的直积同构的偏序类${\mathcal C} _n$是一阶可定义的；(f) 来自类$\langle \langle {\mathcal C}^{\rm rt}_{\rm fb}\rangle _\Pi\rangle _\Sigma$的每个$\omega $-范畴偏序，其中${\mathcal C}^{\rm rt}_{\rm fb}$是有限分支有根树的类，都是有限可公理化的。陈述(f)与Rosenstein（对于线性序类）和Schmerl（对于有限宽度偏序类）的结果相关。
英文摘要
We confirm Vaught'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).
Comments19 pages
