Ordinal measures of the set of finite multisets
Isa Vialard
TL;DR
The paper studies ordinal invariants of two natural finite multiset orderings on a wpo X, namely the multiset embedding M diamond X and the multiset ordering M r X. It derives compositional formulas for the width of the embedding and the height of the ordering, and demonstrates that the width of the ordering is not determined by the standard invariants alone, motivating a new invariant called the maximal safe order type underline o. The work establishes o(M diamond X) = ω^{ o(X) hat} and w(M diamond X) = ω^{ o(X) hat minus 1}, as well as o(M r X) = ω^{o(X)} and h(M r X) = ω^{h(X)}, with w(M r X) captured by the new underline o via w(M r X) = ω^{underline o(X)}; a key technique involves quasi-incomparable subsets and a safe-subset construction to obtain functional width expressions. Collectively these results advance compositionally computable invariants for multisets and illuminate the limits and extensions needed for width analysis in wpo-based data structures.
Abstract
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.