Prime ideals and Noetherian properties in vector lattices

TL;DR

This paper investigates the structure of prime ideals in vector lattices, exploring their properties and classifications, including prime Noetherianity, and provides a complete characterization for vector lattices of piecewise polynomials.

Contribution

It introduces the concept of prime Noetherian vector lattices and characterizes prime ideals in piecewise polynomial lattices, advancing the understanding of their algebraic and order-theoretic structure.

Findings

Prime ideals can be characterized in specific classes of vector lattices.

The concept of prime Noetherianity is introduced and analyzed.

Complete characterization of prime ideals in piecewise polynomial lattices.

Abstract

In this paper we study the set of prime ideals in vector lattices and how the properties of the prime ideals structure the vector lattice in question. The different properties that will be considered are firstly, that all or none of the prime ideals are order dense, secondly, that there are only finitely many prime ideals, thirdly, that every prime ideal is principal, and lastly, that every ascending chain of prime ideals is stationary (a property that we refer to as prime Noetherian). We also completely characterize the prime ideals in vector lattices of piecewise polynomials, which turns out to be an interesting class of vector lattices for studying principal prime ideals and ascending chains of prime ideals.