Statures and Sobrification Ranks of Noetherian Spaces

TL;DR

This paper extends the theory of maximal order types from well-partial-orders to a broader class of Noetherian spaces by defining and calculating the 'stature' of these spaces, including various constructions and their sobrification ranks.

Contribution

It introduces the notion of stature for Noetherian spaces, provides formulas for their statures and sobrification ranks, and extends existing theories beyond well-partial-orders.

Findings

Formulas for statures of sums, products, and spaces of words on Noetherian spaces.

Extension of known formulas from wpos to Noetherian spaces.

Explicit characterizations of sobrifications and their ranks.

Abstract

There is a rich theory of maximal order types of well-partial-orders (wpos), pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is Noetherian in its Alexandroff topology, and there are more; this prompts us to investigate an analogue of that theory in the wider context of Noetherian spaces. The notion of maximal order type does not seem to have a direct analogue in Noetherian spaces per se, but the equivalent notion of stature, investigated by Blass and Gurevich (2008) does: we define the stature $||X||$ of a Noetherian space $X$ as the ordinal rank of its poset of proper closed subsets. We obtain formulas for statures of sums, of products, of the space of words on a space $X$, of the space of finite multisets on $X$, in particular. They confirm previously known formulas on wpos, and extend them to Noetherian spaces. The proofs are, by necessity, rather different from their wpo counterparts, and rely on explicit characterizations of the sobrifications of the corresponding spaces, as obtained by Finkel and the first author (2020). We also give formulas for the statures of some natural Noetherian spaces that do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces, powersets, and spaces of words on $X$ with the so-called prefix topology. Finally, because our proofs require it, and also because of its independent interest, we give formulas for the ordinal ranks of the sobrifications of each of those spaces, which we call their sobrification ranks.