Structures naturelles des demi-groupes et des anneaux réguliers ou involutés

Calmes, Jean

Mathématiques informatique et sciences humaines, Tome 128 (1994), pp. 15-39.

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.

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