Artinian and Noetherian vector lattices

TL;DR

This paper investigates Artinian and Noetherian properties in vector lattices, providing representations, characterizations of prime ideals, and exploring the stationary behavior of prime ideals in various lattice spaces.

Contribution

It offers a concrete representation of Artinian and Noetherian vector lattices and characterizes prime ideals in specific classes, introducing the concept of prime Artinian property.

Findings

Characterization of prime ideals in vector lattices of continuous root functions and piecewise polynomials

Identification of conditions for stationary decreasing sequences of prime ideals

Demonstration that stationary decreasing sequences do not imply stationary increasing sequences

Abstract

In this paper, we study Artinian and Noetherian properties in vector lattices and provide a concrete representation of these spaces. Furthermore, we describe for which Archimedean uniformly complete vector lattices every decreasing sequence of prime ideals is stationary (a property that we refer to as prime Artinian). We also completely characterize the prime ideals in vector lattices of continuous root functions and piecewise polynomials. This is a useful space for studying how having decreasing stationary sequences of prime ideals does not imply having increasing stationary sequences of prime ideals, and vice versa.