Ordinal measures of the set of finite multisets

TL;DR

This paper investigates ordinal invariants of finite multisets over well-partial orders, providing formulas for width and height of specific multiset orders, and introduces a new invariant for characterizing multiset ordering width.

Contribution

It offers compositional formulas for the width and height of multiset embedding and ordering, and introduces a new invariant for multiset ordering width characterization.

Findings

Formulas for the width of multiset embedding.

Formulas for the height of multiset ordering.

Introduction of a new ordinal invariant for multiset ordering width.

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.