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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'' × ''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'' = ''H''*''a'' = ''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 × H → P*(H): (x, y) → x*y ⊆ H.
If Α, Β ⊆ Η then we define
(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
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- 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