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On the hypergroups with four proper pairs and without scalars. (English) Zbl 0869.20049

This paper forms part of a series of three papers [the two others will appear in Rend. Circ. Mat. Palermo, II. Ser. and An. Stiinţ. Univ. Iaşi] in which the authors study and classify the hypergroups having four proper hyperproducts. This problem is difficult but using elegant techniques of calculation the authors solve it here for hypergroups without scalars (a hyperproduct \(ab\) of a hypergroup is proper if and only if \(\text{Card }ab>1\); an element \(a\) of a hypergroup is a scalar if and only if \(\text{Card}(ax)=1=\text{Card}(xa)\), for every element \(x\) of the hypergroup). Earlier the hypergroups with one, two or three proper hyperproducts have been classified respectively by C. Gutan and M. Gutan [Atti. Semin. Mat. Fis. Univ. Modena, 33, 155-159 (1984; Zbl 0591.20068)], M. Gutan [Rend. Circ. Mat. Palermo, II. Ser. 43, No. 1, 107-118 (1994; Zbl 0832.20085)] and D. Freni, M. Gutan and Y. Sureau [New frontiers in hyperstructures, Hadronic Press, 103-125 (1996)]. The results of this paper are an important progress in order to classify all the finite (with small cardinality) hypergroups. The paper is very well written, contains many interesting examples and the presentation of the results is very clear.

MSC:

20N20 Hypergroups
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References:

[1] Bruck, R.H. : A survey of binary systems, Springer-Verlag (1966). · Zbl 0141.01401
[2] Corsini, P. : Prolegomena of hypergroup theory, Aviani Editore, Udine, (1993). · Zbl 0785.20032
[3] Freni, D., Gutan, M., Sureau, Y. : Sur le groupe des scalaires d’un hypergroupe, submitted. · Zbl 0893.20050
[4] Gutan, C., Gutan, M. : Hypergroupes commutatifs infinis, Atti Sem.Mat.Fis.Univ. Modena, 33 (1984), 155-159. · Zbl 0591.20068
[5] Gutan, M. : Hypergroupes infinis ayant un nombre fini d’hyperproduits propres, Rendiconti del Circolo Matematico di Palermo, 43 (1994). · Zbl 0832.20085
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