Laminar set systems consist of non-crossing subsets of a universe with set inclusion essentially corresponding to the descendant relationship of a tree, the so-called laminar tree. Laminar set systems lie at the core of many graph decompositions such as modular decompositions, split decompositions, and bi-join decompositions. We show that from a laminar set system we can obtain the corresponding laminar tree by means of a monadic second order logic (MSO) transduction. This resolves an open question originally asked by Courcelle and is a satisfying resolution as MSO is the natural logic for set systems and is sufficient to define the property ``laminar''. Using results from Campbell et al. [STACS 2025], we can now obtain transductions for obtaining modular decompositions, co-trees, split decompositions and bi-join decompositions using MSO instead of CMSO. We further gain some insight into the expressive power of counting quantifiers and provide some results towards determining when counting quantifiers can be simulated in MSO in laminar set systems and when they cannot.

翻译：层状集合系统由宇宙集合中互不交叉的子集构成，其包含关系本质上对应树的子孙关系，即所谓的层状树。层状集合系统是许多图分解（如模分解、分裂分解和双联结分解）的核心基础。我们证明，通过一元二阶逻辑（MSO）转导，可以从层状集合系统获得相应的层状树。这解决了Courcelle最初提出的一个开放问题，并且是一个令人满意的解答，因为MSO是集合系统的自然逻辑，足以定义"层状"这一性质。利用Campbell等人[STACS 2025]的研究成果，我们现在可以获得使用MSO而非CMSO的转导方法，用于获取模分解、余树、分裂分解和双联结分解。我们进一步深入理解了计数量词的表达能力，并提供了关于在层状集合系统中何时可以在MSO中模拟计数量词、何时不能模拟的相关结论。