Structures paragraduées (groupes, anneaux, modules). I. (Paragraded structures (groups, rings, modules). I)
As a generalization of graded structures [cf. \textit{M. Krasner}, Rev. Roum. Math. Pures Appl. 27, 443-472 (1982; Zbl 0507.01008), Int. J. Math. Math. Sci. 6, 307-311 (1983; Zbl 0516.16030), Colloq. d'algèbre, Univ. Rennes I 1980, 209-308 (1980; Zbl 0521.16001)] the authors introduce paragraded structures. Given a group (G,.) and a left complete semi- lattice \((\Delta,<)\) which is right inductive; let Sg(G) be the set of all subgroups of G. A mapping \(\pi\) : \(x\in \Delta \to G_ x\in Sg(G)\) is called a postgraduation [paragraduation] of G with grades in \(\Delta\) if a list of 5 [6] axioms is satisfied. If VI is replaced by VI' one gets an extragraduation of G. The union of values of \(\pi\) is called the homogeneous part of G with respect to \(\pi\) ; for each member x of this union one defines a grade \(\delta\) x as inf\(\{\) \(y\in \Delta\), \(x\in G_ y\}\). Parallelly to such a ''non homogeneous'' approach, a semi-homogeneous approach is described as well: 3 axioms are listed in order that a given subset H of G be a homogeneous part for some \(\pi\). Then (G,H) is quoted as a paragraded group.
scientific article; zbMATH DE number 3997872