Prime ideals and Noetherian properties in vector lattices

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 fin...

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Bibliographic Details
Published in:Positivity : an international journal devoted to the theory and applications of positivity in analysis 2022-02, Vol.26 (1), Article 13
Main Authors: Kandić, Marko, Roelands, Mark
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Chains
Econometrics
Fourier Analysis
Lattices (mathematics)
Mathematics
Mathematics and Statistics
Operator Theory
Polynomials
Potential Theory
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Summary: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.
ISSN:1385-1292
1572-9281
DOI:10.1007/s11117-022-00887-0