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.
@article{AFST_2012_6_21_2_359_0,
author = {M. Dickmann and A. Petrovich},
title = {Spectral {Real} {Semigroups}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {359--412},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {2},
year = {2012},
doi = {10.5802/afst.1338},
mrnumber = {2978099},
zbl = {1271.11047},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1338/}
}
TY - JOUR AU - M. Dickmann AU - A. Petrovich TI - Spectral Real Semigroups JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 359 EP - 412 VL - 21 IS - 2 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1338/ DO - 10.5802/afst.1338 LA - en ID - AFST_2012_6_21_2_359_0 ER -
%0 Journal Article %A M. Dickmann %A A. Petrovich %T Spectral Real Semigroups %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 359-412 %V 21 %N 2 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1338/ %R 10.5802/afst.1338 %G en %F AFST_2012_6_21_2_359_0
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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