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\(n\)-valued groups: theory and applications. (English) Zbl 1129.20045
The author gives a survey of his most important results in algebraic and topological theory of \(n\)-valued groups. The basic definitions are richly exemplified by \(n\)-valued groups having applications in various fields of research, e.g., a class of \(n\)-valued groups as deformations of a finite group. Algebraic representations of \(n\)-valued groups and their representations on graphs are also given. Using the method of generalized shift operators developed in his previous papers, some formal two-valued groups over the field of complex numbers are defined. A new definition of integrability of multivalued dynamics with discrete time taking action of \(n\)-valued groups for the base is also given. An \(n\)-valued group defined on a compact Hausdorff space \(X\) such that the ring of functions \(C[X]\) is endowed with an \(n\)-Hopf structure is described.

MSC:
20N15 \(n\)-ary systems \((n\ge 3)\)
17A42 Other \(n\)-ary compositions \((n \ge 3)\)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S34 Group rings
05E30 Association schemes, strongly regular graphs
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