Ordinal Measures of the Set of Finite Multisets

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 a common data structure in programming, finite multisets of some well partial order. There are two natural orders one can define on the set of finite multisets of a partial order: the multiset embedding and the multiset ordering. Though the maximal order type of these orders is already known, 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

On the width of the cartesian product of ordinals

The ordinal invariants, i.e., maximal order type, height, and width, are measures of a well quasi-ordering (wqo) based on the ordinal rank of the trees of its bad sequences, strictly decreasing sequences, and antichain sequences, respectively. Complex wqos are often built from simpler wqos through basic constructions such as disjoint sum, direct sum, cartesian product, and higher-order constructions like powerset or sequences. One main challenge is to compute the ordinal invariants of such wqos compositionally. This article focuses on the width of the cartesian product of wqos, for which no general formula is known. The particular case of the cartesian product of two ordinals has already been solved (Abraham, Order 4, 1987). We generalize this study and compute the width of the cartesian product of finitely many ordinals. To this end, we develop new tools for proving lower bounds on the width of wqos. Finally, we leverage our main result to compute the width of a generic family of elementary wqos that is closed under cartesian product

Ordinal measures of the set of finite multisets

International audienceWell-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 a common data structure in programming, finite multisets of some well partial order. There are two natural orders one can define on the set of finite multisets of a partial order: the multiset embedding and the multiset ordering. Though the maximal order type of these orders is already known, 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