Non-axiomatizability of real spectra in ℒ ∞ λ

Timothy Mellor[1]; Marcus Tressl[2]

  • [1] Universität Regensburg, NWF I - Mathematik, D-93040 Regensburg, Germany
  • [2] The University of Manchester, School of Mathematics, Oxford Road, Manchester M13 9PL, UK

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: 2, page 343-358
  • ISSN: 0240-2963

Abstract

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We show that the property of a spectral space, to be a spectral subspace of the real spectrum of a commutative ring, is not expressible in the infinitary first order language ℒ ∞ λ of its defining lattice. This generalises a result of Delzell and Madden which says that not every completely normal spectral space is a real spectrum.

How to cite

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Mellor, Timothy, and Tressl, Marcus. "Non-axiomatizability of real spectra in $\mathcal{L}_\infty \lambda $." Annales de la faculté des sciences de Toulouse Mathématiques 21.2 (2012): 343-358. <http://eudml.org/doc/251003>.

@article{Mellor2012,
abstract = {We show that the property of a spectral space, to be a spectral subspace of the real spectrum of a commutative ring, is not expressible in the infinitary first order language $\mathcal\{L\}_\infty \lambda $ of its defining lattice. This generalises a result of Delzell and Madden which says that not every completely normal spectral space is a real spectrum.},
affiliation = {Universität Regensburg, NWF I - Mathematik, D-93040 Regensburg, Germany; The University of Manchester, School of Mathematics, Oxford Road, Manchester M13 9PL, UK},
author = {Mellor, Timothy, Tressl, Marcus},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {spectral space; real spectrum; commutative ring; infinitary first-order language},
language = {eng},
month = {4},
number = {2},
pages = {343-358},
publisher = {Université Paul Sabatier, Toulouse},
title = {Non-axiomatizability of real spectra in $\mathcal\{L\}_\infty \lambda $},
url = {http://eudml.org/doc/251003},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Mellor, Timothy
AU - Tressl, Marcus
TI - Non-axiomatizability of real spectra in $\mathcal{L}_\infty \lambda $
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/4//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 2
SP - 343
EP - 358
AB - We show that the property of a spectral space, to be a spectral subspace of the real spectrum of a commutative ring, is not expressible in the infinitary first order language $\mathcal{L}_\infty \lambda $ of its defining lattice. This generalises a result of Delzell and Madden which says that not every completely normal spectral space is a real spectrum.
LA - eng
KW - spectral space; real spectrum; commutative ring; infinitary first-order language
UR - http://eudml.org/doc/251003
ER -

References

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