We define representations for downward-closed subsets of a rich family of well-quasi-orders, and more generally for closed subsets of an even richer family of Noetherian topological spaces. This includes the cases of finite words, of multisets, of finite trees, notably. Those representations are given as finite unions of ideals, or more generally of irreducible closed subsets. All the representations we explore are computable, in the sense that we exhibit algorithms that decide inclusion, and compute finite unions and finite intersections. The origin of this work lies in the need for computing finite representations of sets of successors of the downward closure of one state, or more generally of a downward-closed set of states, in a well-structured transition system, and this is where we start: we define adequate notions of completions of well-quasi-orders, and more generally, of Noetherian spaces. For verification purposes, we argue that the required completions must be ideal completions, or more generally sobrifications, that is, spaces of irreducible closed subsets.

中文翻译：

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WSTS 前向分析，第一部分：完井

我们定义了一个丰富的准有序家族的向下闭合子集的表示，并且更一般地定义了一个更丰富的诺特拓扑空间家族的闭合子集。这尤其包括有限词、多集、有限树的情况。这些表示被给出为理想的有限联合，或更一般地是不可约的封闭子集。我们探索的所有表示都是可计算的，因为我们展示了决定包含的算法，并计算有限联合和有限交集。这项工作的起源在于需要在结构良好的转换系统中计算一个状态的向下闭合的后继集的有限表示，或者更一般地说，计算向下闭合的一组状态的有限表示，这就是我们的地方开始：我们定义了完全准好阶的完备概念，更一般地，定义了诺特空间的完备概念。出于验证目的，我们认为所需的补全必须是理想的补全，或者更一般地称为 sobrification，即不可约封闭子集的空间。