Maximum linearizations of lower sets in $\mathbb{N}^m$ with application to monomial ideals
Harry Altman, Andreas Weiermann
TL;DR
This paper determines the maximal linearization (type) of well partial orders formed by lower sets in $\mathbb{N}^m$, giving exact ordinals for bounded and unbounded lower sets: $o(D(\mathbb{N}^m\times k)) = \omega^{\omega^{m-1}k}$, $o(D(\mathbb{N}^m)) = \omega^{\omega^{m-1}}$, and $o(I(\mathbb{N}^m)) = \omega^{\sum_{k=1}^{m} \omega^{m-k} \binom{m}{k-1}} + 1$. The authors develop inductive and combinatorial techniques, including De Jongh–Parikh theorems, Hessenberg sums/products, and a decomposition into products of lower-set lattices, to obtain tight upper and lower bounds. A key contribution is linking these ordinal bounds to effective lengths of sequences of monomial ideals in two variables over a field via Hardy functions, providing complexity-style insights for algebraic structures. The results clarify how the structure of downward-closed sets drives growth in the corresponding index-ordinals and yield precise bounds useful for algorithmic and algebraic considerations in monomial ideals.
Abstract
We compute the type (maximum linearization) of the well partial order of bounded lower sets in $\mathbb{N}^m$, ordered under inclusion, and find it is $ω^{ω^{m-1}}$. Moreover we compute the type of the set of all lower sets in $\mathbb{N}^m$, a topic studied by Aschenbrenner and Pong, and find that it is equal to \[ ω^{\sum_{k=1}^{m} ω^{m-k}\binom{m}{k-1} }+ 1. \] As a consequence we deduce corresponding bounds on effectively given sequences of monomial ideals in $F[X,Y]$ where $F$ is a field.