Spectral Real Semigroups

M. Dickmann[1]; A. Petrovich[2]

  • [1] Projets Logique Mathématique et Topologie et Géométrie Algébriques, Institut de Mathématiques de Jussieu, Universités Paris 6 et 7, Paris, France
  • [2] Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina

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

  • Volume: 21, Issue: 2, page 359-412
  • ISSN: 0240-2963

Abstract

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The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the real semigroups whose representation partial order is a distributive lattice; (iii) Determination of all quotients of SRSs, and (iv) Spectrality of the real semigroup associated to any lattice-ordered ring.

How to cite

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Dickmann, M., and Petrovich, A.. "Spectral Real Semigroups." Annales de la faculté des sciences de Toulouse Mathématiques 21.2 (2012): 359-412. <http://eudml.org/doc/250995>.

@article{Dickmann2012,
abstract = {The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the real semigroups whose representation partial order is a distributive lattice; (iii) Determination of all quotients of SRSs, and (iv) Spectrality of the real semigroup associated to any lattice-ordered ring.},
affiliation = {Projets Logique Mathématique et Topologie et Géométrie Algébriques, Institut de Mathématiques de Jussieu, Universités Paris 6 et 7, Paris, France; Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina},
author = {Dickmann, M., Petrovich, A.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {spectral real semigroups; abstract real spectra; real reduced multirings; quadratic forms over semi-real rings},
language = {eng},
month = {4},
number = {2},
pages = {359-412},
publisher = {Université Paul Sabatier, Toulouse},
title = {Spectral Real Semigroups},
url = {http://eudml.org/doc/250995},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Dickmann, M.
AU - Petrovich, A.
TI - Spectral Real Semigroups
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/4//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - 2
SP - 359
EP - 412
AB - The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the real semigroups whose representation partial order is a distributive lattice; (iii) Determination of all quotients of SRSs, and (iv) Spectrality of the real semigroup associated to any lattice-ordered ring.
LA - eng
KW - spectral real semigroups; abstract real spectra; real reduced multirings; quadratic forms over semi-real rings
UR - http://eudml.org/doc/250995
ER -

References

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  10. Dickmann (M.), Petrovich (A.).— Real Semigroups, Real Spectra and Quadratic Forms over Rings, in preparation, approx. 250 pp. Preliminary version available online at http://www.maths.manchester.ac.uk/raag/index.php?preprint=0339 MR2058670
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