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\begin{document}

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\section*{hypermagma}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak


[[!redirects hypergroup]] [[!redirects hypermagmas]] [[!redirects hypergroups]] [[!redirects comagma]]

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{hypermagma} $X$ relates to a [[magma]] like a [[hypergraph]] relates to an ordinary [[graph]] i. e. the binary operation on $X$ becomes multi-valued by taking value in $2^X$ instead of $X$.

By imposing further axioms one obtains the concept of a \textbf{hypergroup} corresponding to the generalization of the [[group|group concept]] in this context.

With the commutative variant of a [[canonical hypergroup]] this ``\emph{multi-valued hyperalgebra}'' has recently gained prominence in [[number theory]] and algebraic geometry over $\mathbb{F}_1$ (cf. [[hyperring]]).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A set $X$ together with a function $X\times X\to 2^X$ is called a \emph{hypermagma}. The function is normally denoted by $\cdot$ and called the \emph{hyperlaw} (, \emph{hyperoperation} or \emph{hyperproduct}) of $X$. A \emph{morphism of hypermagmas} from $X$ to $Y$ is a function $f:X\to Y$ such that $f(x\cdot y)\subset f(x)\cdot f(y)$ for all $x,y\in X$. The morphism is called \emph{good} if $f(x\cdot y) = f(x)\cdot f(y)$.

A hypermagma $X$ that satisfies furthermore:

\begin{itemize}%
\item $(x\cdot y)\cdot z = x\cdot (y\cdot z)$ for all $x,y,z\in X$ ( \emph{associativity} ) and


\item $x\cdot X= X = X\cdot x$ for all $x\in X$ ( \emph{reproduction} )



\end{itemize}
is called a \emph{hypergroup}.

\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

Given a hypermagma $X$ the hyperlaw is extended to $A,B\in 2^X$ by $A \cdot B=\cup_{a\in A,b\in B} a\cdot b$. Hence, $x\cdot Y$ is understood as $\{x\}\cdot Y$ etc.

Suppose $X$ is an associative hypermagma and $x\cdot y=\emptyset$ then $x\cdot y \cdot X=\emptyset$ and, accordingly, $y\cdot X\neq\X$ or $x\cdot X\neq X$ whence $X$ can't be a hypergroup. So we see that the hyperlaws of hypergroups are in fact valued in \emph{non-empty} subsets - hypermagmas with this property are sometimes called \emph{hypergroupoids}.

But note that $\emptyset$ together with the empty map is nevertheless a hypergroup, in fact, the initial hypergroup in the obvious \emph{category of hypergroups} $\mathbf{HypGrp}$.

By imposing commutativity one arrives at the notion of a [[canonical hypergroup]] that enters into the definition of a [[hyperring]].

Hypergroups whose hyperlaw is valued in singleton subsets correspond to groups.

\hypertarget{example}{}\subsection*{{Example}}\label{example}

Let $G$ be a compact [[Lie group]] and $\hat{G}$ the set of its irreducible representations. Given $a,b\in\hat{G}$ define $a\cdot b$ as the set of irreducible representations $\mu_1,\dots,\mu_k$ occurring in the decomposition $a\otimes b=\Sum m_i \mu_i$. This endows $\hat{G}$ with the structure of a hypergroup. (For further details and the connection to operator algebra see \hyperlink{Lit11}{Litvinov (2011)}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[magma]]


\item [[hypermonoid]]


\item [[canonical hypergroup]]


\item [[hyperring]]


\item [[hypergraph]]


\item [[historical notes on quasigroups]]



\end{itemize}
\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

\begin{itemize}%
\item S. D. Comer, \emph{Polygroups derived from cogroups} , J. Algebra \textbf{89} no.2 (1984) pp.397-405.


\item P. Corsini, V. Leoreanu-Fotea, \emph{Applications of Hyperstructure Theory} , Kluwer Dordrecht 2003.


\item L. Haddad, Y. Sureau, \emph{Les cogroupes et la construction de Utumi} , Pacific J. Math. \textbf{145} no.1 (1990) pp.17-58. (\href{http://projecteuclid.org/euclid.pjm/1102645606}{abstract})


\item L. Haddad, Y. Sureau, \emph{Les groupes, les hypergroupes et l'\'e{}nigme des Murngin} , JPAA \textbf{87} (1993) pp.221-235.


\item J. Jantosciak, \emph{Transposition Hypergroups: Noncommutative Join Spaces} , J. Algebra \textbf{187} (1997) pp.97-119.


\item G. L. Litvinov, \emph{Hypergroups and Hypergroup Algebras} , arXiv:1109.6596 (2011). (\href{http://arxiv.org/abs/1109.6596}{abstract})


\item F. Marty, \emph{Sur une g\'e{}n\'e{}ralization de la notion de groupe} , IV Congr\`e{}s des Math\'e{}maticiens Scandinaves, Stockholm 1934.



\end{itemize}


\end{document}