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Complementary hypergroups. (English) Zbl 1065.20082

Starting from a given hypergroupoid \((H,\circ)\) and using the notion of the complete closure (of the non-empty hyperproduct of two arbitrary elements of \((H,\circ)\)), a new hyperproduct is defined on \(H\) and the concept of the complementary hypergroupoid is introduced.
The authors study the properties of the previous complementary hyperstructures, in connection with the starting hyperstructures. Many examples are given and a class of the starting hypergroups is presented so that the complementary hypergroupoids be quasi-hypergroups.
The paper concludes with two different complementary hypergroupoids constructed from the Cartesian product \(H\times H\), of the starting hypergroupoid. Some basic results concerning the previous hyperstructures are obtained through the identity map which is a weak homomorphism.

MSC:

20N20 Hypergroups
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