Buchstaber, V. M.; Evdokimov, S. A.; Ponomarenko, I. N.; Vershik, A. M. Combinatorial algebras and multivalued involutive groups. (English. Russian original) Zbl 0878.05082 Funct. Anal. Appl. 30, No. 3, 158-162 (1996); translation from Funkts. Anal. Prilozh. 30, No. 3, 12-18 (1996). The main result of the paper says that the category of multivalued involutive groups is equivalent to the category of combinatorial algebras. For a group \(G\) and a finite subgroup \(H\) of the group of automorphisms of \(G\) one defines the product of two elements \(g,h\) of \(G\) as the multiset of all elements in the \(A\) orbit of \(gh\) (i.e., elements counted with multiplicity). Thus we obtain an operation that maps \(G \times G\) to the set of \(n\)-element, \(n = |A|\), multisets of elements of \(G\). This gives an example of a multivalued group. In particular, it satisfies suitably chosen axioms of existence of identity and inverse element and associativity. If there is an involution on a multivalued group that fulfills some axioms assuring compatibility with the structure of the multivalued group then the multivalued group is called involutive. The authors demonstrate that if one defines the product of two elements of an \(n\)-valued group as one \(n\)th of the sum of the elements of their group product then one can define on the complex linear span of the group elements the structure of a combinatorial algebra. This concept—with slightly stronger axioms—appears in the context of association schemes [see E. Bannai and T. Ito, Algebraic Combinatorics. I: Association schemes (1984; Zbl 0555.05019)]. Roughly speaking it is an algebra given combinatorially by a set of structure constants. This map from multivalued groups to combinatorial algebras is shown to induce an equivalence of categories. Finally, relations to Schur rings and other algebraic structures are discussed. Reviewer: V.Welker (Essen) Cited in 2 ReviewsCited in 8 Documents MSC: 05E30 Association schemes, strongly regular graphs 20N20 Hypergroups Keywords:multivalued group; combinatorial algebra; Schur ring Citations:Zbl 0555.05019 PDFBibTeX XMLCite \textit{V. M. Buchstaber} et al., Funct. Anal. Appl. 30, No. 3, 158--162 (1996; Zbl 0878.05082); translation from Funkts. Anal. Prilozh. 30, No. 3, 12--18 (1996) Full Text: DOI References: [1] E. Bannai and T. Ito, Algebraic Combinatorics, I. Association Schemes, Benjamin/Comings, London-Amsterdam-Don Mills, Ontario-Sydney-Tokyo, 1984. · Zbl 0555.05019 [2] V. M. Buchshtaber and E. G. Rees, Multivalued Groups, their Representation and Hopf Algebras, Preprint of Edinburgh Univ., 1995. [3] A. M. Vershik, ”A geometric theory of states von Neumann boundary, and duality ofC *-algebras,” Zap. Nauchn. Sem. LOMI,29, 147–154 (1972). [4] S. V. Kerov, ”Duality of finite-dimensionalC *-algebras,” Vestn. Leningr. Univ., Mat., Mekh. Astronom.,7, 23–29 (1974). · Zbl 0291.46041 [5] H. Wielandt, Finite Permutation Groups, Academic Press, New York-London, 1964. · Zbl 0138.02501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.