Hyperstructure

This article is about a mathematical concept. For the architectural concept, see arcology.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Hv} – structures.

A hyperoperation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \star ) } , where the reproduction axiom is valid, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \star H = H \star a = H} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in H.}

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