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Spectral Real Semigroups
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 2, pp. 359-412.

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.

Dans [8] nous avons introduit la notion de semigroupe réel dans le but de donner un cadre général pour l’étude des formes quadratiques diagonales sur des anneaux commutatifs, unitaires, semi-réels. Dans cet article nous étudions une classe de semigroupes réels avec des propriétés remarquables  : les semigroupes réels spectraux (SRS). Nos résultats principaux sont  : (i) l’existence d’une dualité fonctorielle naturelle entre la catégorie des SRS et celle des espaces spectraux héréditairement normaux  ; (ii) la caractérisation des SRS comme étant les semigroupes réels dont l’ordre de représentation est un treillis distributif  ; (iii) la détermination des quotients des SRS  ; (iv) le caractére spectral des semigroupes réels associés aux anneaux réticulés.

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Accepted:
Published online:
DOI: https://doi.org/10.5802/afst.1338
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
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M. Dickmann; A. Petrovich. Spectral Real Semigroups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. 2, pp. 359-412. doi : 10.5802/afst.1338. https://afst.centre-mersenne.org/articles/10.5802/afst.1338/

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