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 H v {\displaystyle Hv} – structures.

A hyperoperation ( ⋆ ) {\displaystyle (\star )} on a nonempty set H {\displaystyle H} is a mapping from H × H {\displaystyle H\times H} to the nonempty power set P ∗ ( H ) {\displaystyle P^{*}\!(H)} , meaning the set of all nonempty subsets of H {\displaystyle H} , i.e.

⋆ : H × H → P ∗ ( H ) {\displaystyle \star :H\times H\to P^{*}\!(H)}

  ( x , y ) ↦ x ⋆ y ⊆ H . {\displaystyle \quad \ (x,y)\mapsto x\star y\subseteq H.}

For A , B ⊆ H {\displaystyle A,B\subseteq H} we define

A ⋆ B = ⋃ a ∈ A , b ∈ B a ⋆ b {\displaystyle A\star B=\bigcup _{a\in A,\,b\in B}a\star b} and A ⋆ x = A ⋆ { x } , {\displaystyle A\star x=A\star \{x\},\,} x ⋆ B = { x } ⋆ B . {\displaystyle x\star B=\{x\}\star B.}

( H , ⋆ ) {\displaystyle (H,\star )} is a semihypergroup if ( ⋆ ) {\displaystyle (\star )} is an associative hyperoperation, i.e. x ⋆ ( y ⋆ z ) = ( x ⋆ y ) ⋆ z {\displaystyle x\star (y\star z)=(x\star y)\star z} for all x , y , z ∈ H . {\displaystyle x,y,z\in H.}

Furthermore, a hypergroup is a semihypergroup ( H , ⋆ ) {\displaystyle (H,\star )} , where the reproduction axiom is valid, i.e. a ⋆ H = H ⋆ a = H {\displaystyle a\star H=H\star a=H} for all a ∈ H . {\displaystyle a\in H.}

• 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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