Artinian and Noetherian vector lattices

arXiv (Cornell University)

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.

Authors 3

1. Marko Kandić
2. Mark C. P. Roelands Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute , Leiden University , 2300 RA Leiden , The Netherlands

3. Marten Wortel

   University of Pretoria

   Affiliation as printed

   Department of Mathematics and Applied Mathematics , University of Pretoria , Private Bag X20 Hatfield , Pretoria 0028 , South Africa