Well-partial orders, and the ordinal invariants used to measure them, are relevant in set theory, program verification, proof theory and many other areas of computer science and mathematics. In this article we focus on one of the most common data structure in programming, the finite multiset of some wpo. There are two natural orders one can define on the set of finite multisets $M(X)$ of a partial order $X$: the multiset embedding and the multiset ordering, for which $M(X)$ remains a wpo when $X$ is. Though the maximal order type of these orders is already known, the other ordinal invariants remain mostly unknown. Our main contributions are expressions to compute compositionally the width of the multiset embedding and the height of the multiset ordering. Furthermore, we provide a new ordinal invariant useful for characterizing the width of the multiset ordering.
翻译:局部订单和用于测量这些订单的正方位变量与设定理论、程序核查、证据理论和计算机科学和数学的许多其他领域相关。在本条中,我们侧重于编程中最常见的数据结构之一,即某些 wpo 的有限多组。在部分顺序的限定多组集($M(X)$X) 上有两个自然顺序:多组嵌入和多组定($X) $X$ ),对于多组定($X$ ) 仍然为 wpo) 。虽然这些订单的最大订单类型已经为已知,但其他的正方位变量仍然大多未知。我们的主要贡献是从构成上计算多组嵌入的宽度和多组定的高度。此外,我们提供了一个用于描述多组定宽度的新的或方位变量。