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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