A Structural Invariant On Certain Two-Dimensional Noetherian Partially Ordered Sets

TL;DR

This paper introduces a new structural invariant for certain two-dimensional Noetherian partially ordered sets, enabling complete classification up to isomorphism based on a poset modeled after a known spectral condition.

Contribution

It defines a novel poset invariant that fully characterizes these partial orders, extending spectral theory insights to a broader class of Noetherian domains.

Findings

The invariant uniquely determines the poset up to isomorphism.

It generalizes the P5 condition for spectral sets.

Provides a new tool for classifying two-dimensional Noetherian spectra.

Abstract

If $(X, \le_X)$ is a partially ordered set satisfying certain necessary conditions for $X$ to be order-isomorphic to the spectrum of a Noetherian domain of dimension two, we describe a new poset $(\text{str } X, \le_{\text{str } X})$ that completely determines $X$ up to isomorphism. The order relation $\le_{\text{str } X}$ imposed on $\text{str } X$ is modeled after R. Wiegand's well-known "P5" condition that can be used to determine when a given partially ordered set $(U, \le_U)$ of a certain type is order-isomorphic to $(\text{Spec } \mathbb Z[x], \subseteq).$