一句话：证明薄余代数可由初始代数归纳刻画，并给出其语法与单方程公理化。
本文研究解析函子的余代数，这类结构统一建模状态后继可具有对称性的图状系统（如有序元组、循环列表、多重集），其动机来自基于自动机的验证和薄树理论。作者引入薄余代数，即每个状态仅有可数多条无限路径的余代数。核心结果是通过初始代数对薄性给出归纳刻画；为达此目的，他们为薄行为建立了语法，并证明可用单个方程精确刻画两个项表示同一薄行为的条件。最后，作为多项式函子的特例，他们从该语法中重新导出了薄树的 Cantor-Bendixson 秩概念。
Coalgebras for analytic functors uniformly model graph-like systems where the successors of a state may admit certain symmetries. Examples of successor structure include ordered tuples, cyclic lists and multisets. Motivated by goals in automata-based verification and results on thin trees, we introduce thin coalgebras as those coalgebras with only countably many infinite paths from each state. Our main result is an inductive characterisation of thinness via an initial algebra. To this end, we develop a syntax for thin behaviours and capture with a single equation when two terms represent the same thin behaviour. Finally, for the special case of polynomial functors, we retrieve from our syntax the notion of Cantor-Bendixson rank of a thin tree.