We prove that every poset with bounded cliquewidth and with sufficiently large dimension contains the standard example of dimension $k$ as a subposet. This applies in particular to posets whose cover graphs have bounded treewidth, as the cliquewidth of a poset is bounded in terms of the treewidth of the cover graph. For the latter posets, we prove a stronger statement: every such poset with sufficiently large dimension contains the Kelly example of dimension $k$ as a subposet. Using this result, we obtain a full characterization of the minor-closed graph classes $\mathcal{C}$ such that posets with cover graphs in $\mathcal{C}$ have bounded dimension: they are exactly the classes excluding the cover graph of some Kelly example. Finally, we consider a variant of poset dimension called Boolean dimension, and we prove that posets with bounded cliquewidth have bounded Boolean dimension. The proofs rely on Colcombet's deterministic version of Simon's factorization theorem, which is a fundamental tool in formal language and automata theory, and which we believe deserves a wider recognition in structural and algorithmic graph theory.

翻译：我们证明：每个具有有界团宽且维数足够大的偏序集，必包含维数为$k$的标准例子作为子偏序集。这一结论特别适用于覆盖图具有有界树宽的偏序集，因为偏序集的团宽可由其覆盖图的树宽界定。对于后一类偏序集，我们证明了更强的结论：每个维数足够大的此类偏序集必包含维数为$k$的Kelly例子作为子偏序集。利用这一结果，我们完整刻画了使得覆盖图属于$\mathcal{C}$的偏序集具有有界维数的极小图类$\mathcal{C}$：恰为排除所有Kelly例子的覆盖图的图类。最后，我们考虑偏序集维数的一种变体——布尔维数，并证明具有有界团宽的偏序集具有有界布尔维数。证明依赖于Colcombet对西蒙因子分解定理的确定性版本，该定理是形式语言与自动机理论中的基础工具，我们相信它在结构图论与算法图论中应获得更广泛的认知。