Lesson plan

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 Hv – structures. A hyperoperation (\star) on a nonempty set H is a mapping from H \times H to the nonempty power set P^{*}\!(H) , meaning the set of all nonempty subsets of H , i.e. : \star: H \times H \to P^{*}\!(H) : \quad\ (x,y) \mapsto x \star y \subseteq H. For A,B \subseteq H we define : A \star B = \bigcup_{a \in A,\, b \in B} a \star b and A \star x = A \star \{ x \},\, x \star B = \{x\} \star B. (H, \star ) is a semihypergroup if (\star) is an associative hyperoperation, i.e. x \star (y \star z) = (x \star y)\star z for all x, y, z \in H. Furthermore, a hypergroup is a semihypergroup (H, \star ) , where the reproduction axiom is valid, i.e. a \star H = H \star a = H for all a \in H.

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Article summary

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 Hv – structures. A hyperoperation (\star) on a nonempty set H is a mapping from H \times H to the nonempty power set P^{*}\!(H) , meaning the set of all nonempty subsets of H , i.e. : \star: H \times H \to P^{*}\!(H) : \quad\ (x,y) \mapsto x \star y \subseteq H. For A,B \subseteq H we define : A \star B = \bigcup_{a \in A,\, b \in B} a \star b and A \star x = A \star \{ x \},\, x \star B = \{x\} \star B. (H, \star ) is a semihypergroup if (\star) is an associative hyperoperation, i.e. x \star (y \star z) = (x \star y)\star z for all x, y, z \in H. Furthermore, a hypergroup is a semihypergroup (H, \star ) , where the reproduction axiom is valid, i.e. a \star H = H \star a = H for all a \in H.

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  1. Summarise the article in your own words. Which single claim would you verify first, and why?
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  3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

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