An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable pure set and the countable dense linear order without endpoints have this property. We generalise these results to (a) any structure approximated by finite substructures with few orbits, provided the field is of characteristic zero, and (b) any Fraïssé limit with free amalgamation in a finite vocabulary consisting of unary and binary relations, possibly expanded with a generic total order. As a special case, we deduce the finite length property of the Rado graph using both methods. We also describe some connections with function spaces, weighted register automata, and orbit-finite systems of linear equations.

翻译：摘要：一个无限结构具有（在给定域上的）有限长度性质，如果其每个有限幂在相应自由向量空间中的等变子空间链的长度有界。先前研究表明，可数纯集和可数无端点稠密线性序具有该性质。我们将这些结果推广至：(a) 在特征为零的域上，任何可由具有少量轨道的有限子结构逼近的结构；(b) 在仅包含一元和二元关系的有限词汇表中具有自由融合的Fraïssé极限，可能扩展有泛型全序。作为特例，我们通过两种方法推导出Rado图的有限长度性质。本文还描述了该性质与函数空间、加权寄存器自动机以及轨道有限线性方程系统之间的联系。