A STRUCTURAL INVARIANT ON CERTAIN TWO-DIMENSIONAL NOETHERIAN PARTIALLY ORDERED SETS

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 $(\text{Str\hspace{0.17em}}X,{\le }_{\text{Str\hspace{0.17em}}X})$ that completely determines $X$ up to isomorphism. The order relation ${\le }_{\text{Str\hspace{0.17em}}X}$ imposed on $\text{Str\hspace{0.17em}}X$ 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 $(\text{Spec\hspace{0.17em}}\mathbb{Z}[x],\subseteq )$.