Structures naturelles des demi-groupes et des anneaux réguliers ou involutés
Mathématiques informatique et sciences humaines, Tome 128 (1994), pp. 15-39.
corrigé par Errata

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.

@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 informatique et sciences humaines},
     pages = {15--39},
     publisher = {Ecole des hautes-\'etudes en sciences sociales},
     volume = {128},
     year = {1994},
     mrnumber = {1329054},
     zbl = {0828.20058},
     language = {fr},
     url = {http://www.numdam.org/item/MSH_1994__128__15_0/}
}
TY  - JOUR
AU  - Calmes, Jean
TI  - Structures naturelles des demi-groupes et des anneaux réguliers ou involutés
JO  - Mathématiques informatique et sciences humaines
PY  - 1994
SP  - 15
EP  - 39
VL  - 128
PB  - Ecole des hautes-études en sciences sociales
UR  - http://www.numdam.org/item/MSH_1994__128__15_0/
LA  - fr
ID  - MSH_1994__128__15_0
ER  - 
%0 Journal Article
%A Calmes, Jean
%T Structures naturelles des demi-groupes et des anneaux réguliers ou involutés
%J Mathématiques informatique et sciences humaines
%D 1994
%P 15-39
%V 128
%I Ecole des hautes-études en sciences sociales
%U http://www.numdam.org/item/MSH_1994__128__15_0/
%G fr
%F MSH_1994__128__15_0
Calmes, Jean. Structures naturelles des demi-groupes et des anneaux réguliers ou involutés. Mathématiques informatique et sciences humaines, Tome 128 (1994), pp. 15-39. http://www.numdam.org/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 | Zbl

[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

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

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

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

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

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

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

[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 | Zbl

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

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

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

[ 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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl