## Infinite hypergroups having a finite number of proper hyperproducts. (Hypergroupes infinis ayant un nombre fini d’hyperproduits propres.)(French)Zbl 0832.20085

For non-commutative hypergroups a theorem of P. Corsini (1980) is proved: If $$(H, \cdot)$$ is an infinite hypergroup with finite set of hyperproducts of cardinality greater than 1, then there exists $$(a,b) \in H^2$$ with $$\text{card} (a\cdot b) = \text{card }H$$. Moreover, all hypergroups having at most two proper hyperproducts are determined. Therefore, these hypergroups include the very thin ones.

### MSC:

 20N20 Hypergroups
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### References:

 [1] Corsini P.,Contributo alla teoria degli ipergruppi, Atti Soc. Pelor. Sc. Mat. Fis. Nat. (1980). [2] Corsini P.,Ipergruppi semiregolari e regolari, Rend. Sem. Mat. Univ. Politecn. Torino40 (1982), 35–46. [3] Gutan C., Gutan M.,Hypergroupes commutatifs infinis, Atti Sem. Mat. Fis. Univ. Modena,33 (1984), 155–159. · Zbl 0591.20068 [4] Howie J. M.,An Introduction to Semigroup Theory, Academic Press (1976). · Zbl 0355.20056 [5] Scott W. R.,Group Theory, Prentice Hall (1964). [6] Sureau Y.,Contribution à la théorie des hypergroupes et hypergroupes opérant transitivement sur un ensemble, Thèse de doctorat, Université de Clermont-Ferrand II (1980). [7] Vougiouklis T.,The very thin hypergroups and the S-construction, Combinatorics ’88, vol. 2 (1991), 471–477. · Zbl 0945.20524
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