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The '''hyperstructures''' are [[algebraic structure]]s 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 (*) on a non-empty set ''H'' is a mapping from ''H''&nbsp;×&nbsp;''H'' to [[power set]] ''P''*(''H'') (the set of all non-empty sets of ''H''), i.e.
 
(*): ''H'' × ''H'' → ''P''*(''H''): (''x'', ''y'') → ''x''*''y'' ⊆ ''H''.
 
If ''Α'', ''Β'' ⊆ ''Η'' then we define
 
: ''A''*''B'' = <math>\bigcup_{a \in A, b\in B} (a \star b)</math> and ''A''*''x'' = ''A''*{''x''}, ''x''*''B'' = {''x''}* ''B''.
 
(''Η'',*) is a ''semihypergroup'' if (*) is an [[associative]] hyperoperation, i.e. ''x''*(''y''*''z'') = (''x''*''y'')*''z'', for all ''x'',''y'',''z'' of ''H''.
Furthermore, a ''hypergroup'' is a semihypergroup (''H'', *), where the [[reproduction axiom]] is valid, i.e. ''a''*''H''&nbsp;=&nbsp;''H''*''a''&nbsp;=&nbsp;''H'', for all ''a'' of ''H''.
 
==References==
{{reflist}}
*AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus  University of Thrace, School of Education, Greece. [http://aha.eled.duth.gr aha.eled.duth.gr]
 
*[http://books.google.co.uk/books?id=uvCrZ3iGur4C Applications of Hyperstructure Theory], Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, ISBN 1-4020-1222-5, ISBN 978-1-4020-1222-8
 
*[http://www.worldscientific.com/worldscibooks/10.1142/8481 Functional Equations on Hypergroups], László, Székelyhidi, World Scientific Publishing, 2012, ISBN 978-981-4407-00-7
 
 
[[Category:Abstract algebra]]

Revision as of 18:58, 30 December 2013

Template:Cleanup 29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. The 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 (*) on a non-empty set H is a mapping from H × H to power set P*(H) (the set of all non-empty sets of H), i.e.

(*): H × HP*(H): (x, y) → x*yH.

If Α, ΒΗ then we define

A*B = aA,bB(ab) and A*x = A*{x}, x*B = {x}* B.

(Η,*) is a semihypergroup if (*) is an associative hyperoperation, i.e. x*(y*z) = (x*y)*z, for all x,y,z of H. Furthermore, a hypergroup is a semihypergroup (H, *), where the reproduction axiom is valid, i.e. a*H = H*a = H, for all a of H.

References

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  • AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece. aha.eled.duth.gr