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A new class of hyperstructures. (English) Zbl 0874.20052

The paper deals with two large new classes of hyperstructures larger than the classical ones, introduced by the author in 1990 [in AHA, Proc. 4th Int. Congr., Xanthi/Greece 1990, 203-211 (1991; Zbl 0763.16018)]. In these hyperstructures, named \(H_v\)-structures, the usual axioms are replaced by the weak ones, i.e. the non-empty intersection replaces the equality of sets. In this paper it is proved that in a \(H_v\)-group \(G\) the fundamental relation \(\beta^*\) is the smallest equivalence relation such that the quotient \(G/\beta^*\) is a group. Moreover, it is proved that using the uniting elements method to obtain stricter structures one obtains the same fundamental structure, i.e. associativity and commutativity are valid. The single elements are introduced and their properties are studied. Finally, an ordering on \(H_v\)-structures defined in the same set, is introduced, and using this ordering the set of all \(H_v\)-groups, up to isomorphism, is obtained.

MSC:

20N20 Hypergroups

Citations:

Zbl 0763.16018
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