Hyperstructure
Hyperstructures are algebraic structures equipped with at least one multi-valued operation, called a hyperoperation. The largest classes of the hyperstructures are the ones called [math]\displaystyle{ Hv }[/math] – structures.
A hyperoperation [math]\displaystyle{ (\star) }[/math] on a nonempty set [math]\displaystyle{ H }[/math] is a mapping from [math]\displaystyle{ H \times H }[/math] to the nonempty power set [math]\displaystyle{ P^{*}\!(H) }[/math], meaning the set of all nonempty subsets of [math]\displaystyle{ H }[/math], i.e.
- [math]\displaystyle{ \star: H \times H \to P^{*}\!(H) }[/math]
- [math]\displaystyle{ \quad\ (x,y) \mapsto x \star y \subseteq H. }[/math]
For [math]\displaystyle{ A,B \subseteq H }[/math] we define
- [math]\displaystyle{ A \star B = \bigcup_{a \in A,\, b \in B} a \star b }[/math] and [math]\displaystyle{ A \star x = A \star \{ x \},\, }[/math] [math]\displaystyle{ x \star B = \{x\} \star B. }[/math]
[math]\displaystyle{ (H, \star ) }[/math] is a semihypergroup if [math]\displaystyle{ (\star) }[/math] is an associative hyperoperation, i.e. [math]\displaystyle{ x \star (y \star z) = (x \star y)\star z }[/math] for all [math]\displaystyle{ x, y, z \in H. }[/math]
Furthermore, a hypergroup is a semihypergroup [math]\displaystyle{ (H, \star ) }[/math], where the reproduction axiom is valid, i.e. [math]\displaystyle{ a \star H = H \star a = H }[/math] for all [math]\displaystyle{ a \in H. }[/math]
References
- AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece. aha.eled.duth.gr
- Applications of Hyperstructure Theory, Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, ISBN:1-4020-1222-5, ISBN:978-1-4020-1222-8
- Functional Equations on Hypergroups, László, Székelyhidi, World Scientific Publishing, 2012, ISBN:978-981-4407-00-7
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