Structures naturelles des demi-groupes et des anneaux réguliers ou involutés
Calmes, Jean
Mathématiques et Sciences humaines, Tome 128 (1994), p. 15-39 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Certaines relations binaires sont définies sur les demi-groupes et les demi-groupes à involution. On examine comment elles peuvent en ordonner les éléments: notamment les idempotents, les éléments réguliers au sens de von Neumann, ceux qui possédent un inverse ponctuel ou de Moore-Penrose ; et en fonction aussi de conditions sur l'involution. Ces relations peuvent alors coïncider avec les ordres naturels des idempotents et des demi-groupes inverses, avec les ordres de Drazin et de Hartwig : elles en sont des extensions. On s'attache à définir sur les demi-groupes et les anneaux, mais aussi sur les modules de matrices, les conditions de ces ordres, celles de leur compatibilité et de leur égalité. Le sujet est inscrit dans le thème des ensembles ordonnés et ses applications aux sciences sociales : l'accent est mis sur la possibilité de disposer en Analyse de tableaux numériques d'un riche réseau de relations binaires, compatibles avec l'ordre semi-défini positif et présentant une affinité particulière avec les différentes formes de projections.

Some binary relations are defined on semigroups and semigroups with involutions. We show how they may order their elements : especially idempotents, regular elements in von Neumann's sense, elements that possess Moore-Penrose or pointwise inverses ; according to the nature of involutions as well. In these cases, the relations may coincide with the natural orders on the set of idempotents and on inverse semigroups, with the orders of Drazin and Hartwig; and so they extend them. We look for conditions of these ordering relations and for conditions of their compatibility and equality, on semigroups and rings but also on modules of matrices. The subject is included in the general theme of order sets and its applications in social sciences : we place emphasis on the possibility in data analysis to dispose of a rich network of binary relations, which are compatible with the positive-semi-definite order and possess close links with projections of different kinds.

Publié le : 1994-01-01
@article{MSH_1994__128__15_0,
     author = {Calmes, Jean},
     title = {Structures naturelles des demi-groupes et des anneaux r\'eguliers ou involut\'es},
     journal = {Math\'ematiques et Sciences humaines},
     volume = {128},
     year = {1994},
     pages = {15-39},
     mrnumber = {1329054},
     zbl = {0828.20058},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/MSH_1994__128__15_0}
}

Calmes, Jean. Structures naturelles des demi-groupes et des anneaux réguliers ou involutés. Mathématiques et Sciences humaines, Tome 128 (1994) pp. 15-39. http://gdmltest.u-ga.fr/item/MSH_1994__128__15_0/

[1] Adkins W.A., Weintraub S.H., Algebra : an approach via module theory, New York, Springer Verlag,1992. | MR 1181420 | Zbl 0768.00003

[2] Barbut M., "Ensembles ordonnés", Rev. franç. Rech. Opération., 20 (1961), 175-198.

[3] Barbut M., Monjardet B., Ordre et classification, Algèbre et combinatoire, Paris, Hachette,1970. | Zbl 0267.06001

[4] Berberian S.K., Baer *-rings, Berlin, Springer Verlag,1972. | MR 429975 | Zbl 0242.16008

[5] Berberian S.K., "The regular ring of a finite Baer *-ring ", J. Alg., 23 (1972), 35-65. | MR 308179 | Zbl 0243.16008

[6] Birkhoff G., Lattice theory, Providence, Amer. Math. Soc.,1967. | MR 227053 | Zbl 0153.02501

[7] Birkhoff G., Von Neumann J., "The logic of quantum mechanics", Ann. of Math., 37 (1936), 823-842. | JFM 62.1061.04 | Zbl 0015.14603

[8] Boyd J.P., "Structural similarity, semi groups and idempotents", Social networks, 5 (1983), 157-182. | MR 718090

[9] Clifford A., Preston G., The algebraic theory of semigroups, Providence, Amer. Math. Soc. , 1961 (vol. I),1967 (vol. II). | Zbl 0178.01203

[10] Degenne A., "Un domaine d'interaction entre les mathématiques et les sciences sociales: les réseaux sociaux", Math. Inf. Sci. hum., 104 (1988), 5-18. | Numdam | MR 989253 | Zbl 0666.92018

[11] Deheuvels R., Formes quadratiques et groupes classiques, Paris, Presses Universitaires de France,1981. | MR 657579 | Zbl 0467.15015

[12] Drazin M.P., "Natural structures on semigroups with involution", Bul. Amer. Math. Soc., 84 (1978), 139-141. | MR 486234 | Zbl 0395.20044

[13] Foulis D.J., "Relative inverses in Baer *-semigroups", Michigan Math. J.,10 (1963), 65-85. | MR 154939 | Zbl 0116.25404

[ 14] Gaffke N., Krafft O., "Matrix inequalities in the LÖwner ordering", in KORTE B, Modern applied mathematics : optimization and operations research, Amsterdam, North Holland, 1982, pp. 595-622. | MR 663205 | Zbl 0495.15012

[15] Guénoche A., Monjardet B., "Méthodes ordinales et combinatoires en analyse des données", Math. Sci. hum., 100 (1987), 5-47. | Numdam | MR 941908 | Zbl 0641.68031

[16] Hartwig R.E., "How to partially order regular elements", Math. Japonica, 25 (1980), 1-13. | MR 571255 | Zbl 0442.06006

[17] Hartwig R.E., Styan G.P.H., "On some characterizations of the star partial ordering for matrices and rank subtractivity", Linear Alg. Appl., 82 (1986),145-161. | MR 858968 | Zbl 0603.15001

[18] Kaplansky L., Rings of operators, New-York, W.A. Benjamin Inc.,1968. | MR 244778 | Zbl 0174.18503

[19] Lesieur L., Tenam R., Lefebvre J., Compléments d'algèbre linéaire, Paris, Armand Colin,1978. | Zbl 0398.15002

[20] Mac Coy N.E., The theory of rings, New York, Mc Millan,1966.

[21] Mallol C., Olivier J.-P., Serrato D., "Groupoïds, idempotents and pointwise inverses in relational categories", J. Pure Appl. Alg., 36 (1985), 23-51. | MR 782638 | Zbl 0555.18002

[22] Marshall A., Olkin I., Inequalities : Theory of majorization and its applications, New York, Academic Press, 1979. | MR 552278 | Zbl 0437.26007

[23] Mitra S.K., "Generalized inverse of matrices and applications to linear models ", in KRISHNAIAH P.R., Handbook of Statistics, vol. 1, Amsterdam, North Holland, 1980, pp. 471-512. | Zbl 0465.62055

[24] Mitsch H., "Inverse semigroups and their natural order", Bull. Austral. Math. Soc., 19 (1978), 59-65. | MR 522180 | Zbl 0386.20044

[25] Nashed M., Votruba G., "An unified approach to generalized inverses of linear operators : I Algebraic, topological and projectional properties ", Bull. Amer. Math. Soc., 80 (1974), 825-830. | MR 365190 | Zbl 0289.47011

[26] Nashed M., Votruba G., "An unified approach to generalized inverses of linear operators : II External and proximal properties", Bull. Amer. Math. Soc., 80 (1974), 831-835. | MR 365191 | Zbl 0289.47011

[27] Nordstrom K., "Some further aspects of the Lowner-ordering antitonicity of the Moore-Penrose inverse", Comm. Statist. Theory Meth., 18, 12 (1989), 4471-4489. | MR 1046720 | Zbl 0707.62096

[28] Petit J.-L., Terouanne E., "Balayage et cumul", in Séminaire Math. & Info. appl., Montpellier, Université Paul Valéry,1992, pp. 24-56.

[29] Petrich M., Inverse semigroups, New York, Wiley,1984. | MR 752899 | Zbl 0546.20053

[30] Pollock D.S., The algebra of econometrics, New York, Wiley,1979. | MR 558960

[31] Prijatelj N., Vidav I., "On special *-regular rings", Michigan Math. J., 18 (1971), 213-221. | MR 283024 | Zbl 0216.33402

[32] Rao C.R., Mitra S.K., Generalized inverse of matrices and its applications, New York, Wiley, 1971. | MR 338013 | Zbl 0236.15004

[33] Rao C.R., Yanai H., "General definition and decomposition of projectors and some applications to statistical problems", J. Statist. Plan. Inference, 3 (1979),1-17. | MR 529869 | Zbl 0427.62046

[34] Schein B.M., "Regular elements of the semigroup of all binary relations", Semigroup forum,13 (1976), 95-102. | MR 432790 | Zbl 0355.20058

[35] Timm N.H., Multivariate analysis with applications in education and psychology, Monterey (Californie), Brooke-Cole, 1975. | MR 443216

[36] Von Neumann J., "On regular rings", Proc. of the National Acad. of Sci. U.S.A., 22 (1936), 296-300. | JFM 62.1103.03 | Zbl 0015.38802

[37] Yanai H., "Some generalized forms of least squares g-inverses, minimum norm g-inverse, and Moore-Penrose inverse matrices", Comput. Statist. & Data Anal.,10 (1990), 251-260. | MR 1086039 | Zbl 0825.62550