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| \emph{Hypergroups} are generalizations of groups. Recall that a group is set with a binary operation on it satisfying a number of conditions. If this binary operation is taken to be multivalued, then we arrive at a hypergroup. In order to make this precise, we need some preliminary concepts: |
\emph{Hypergroups} are generalizations of groups. Recall that a group is set with a binary operation on it satisfying a number of conditions. If this binary operation is taken to be multivalued, then we arrive at a hypergroup. In order to make this precise, we need some preliminary concepts: |
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| \textbf{Definition}. A \emph{hypergroupoid}, or \emph{multigroupoid}, is a non-empty set $G$, together with a multivalued function $\cdot: G\times G\Rightarrow G$ called the \emph{multiplication} on $G$. |
\textbf{Definition}. A \emph{hypergroupoid}, or \emph{multigroupoid}, is a non-empty set $G$, together with a multivalued function $\cdot: G\times G\Rightarrow G$ called the \emph{multiplication} on $G$. |
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| We write $a\cdot b$, or simply $ab$, instead of $\cdot(a,b)$. Furthermore, if $ab=\lbrace c\rbrace$, then we use the abbreviation $ab=c$. |
We write $a\cdot b$, or simply $ab$, instead of $\cdot(a,b)$. Furthermore, if $ab=\lbrace c\rbrace$, then we use the abbreviation $ab=c$. |
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| A hypergroupoid is said to be \emph{commutative} if $ab=ba$ for all $a,b\in G$. Defining associativity of $\cdot$ on $G$, however, is trickier: |
A hypergroupoid is said to be \emph{commutative} if $ab=ba$ for all $a,b\in G$. Defining associativity of $\cdot$ on $G$, however, is trickier: |
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| Given a hypergroupoid $G$, the multiplication $\cdot$ induces a binary operation (also written $\cdot$) on $P(G)$, the powerset of $P$, given by $$A\cdot B:=\bigcup \lbrace a\cdot b\mid a\in A\mbox{ and } b\in B\rbrace.$$ |
Given a hypergroupoid $G$, the multiplication $\cdot$ induces a binary operation (also written $\cdot$) on $P(G)$, the powerset of $P$, given by $$A\cdot B:=\bigcup \lbrace a\cdot b\mid a\in A\mbox{ and } b\in B\rbrace.$$ |
| As a result, we have an induced groupoid $P(G)$. Instead of writing $\lbrace a\rbrace B$, $A\lbrace b\rbrace$, and $\lbrace a\rbrace \lbrace b\rbrace$, we write $aB, Ab$, and $ab$ instead. From now on, when we write $(ab)c$, we mean |
As a result, we have an induced groupoid $P(G)$. Instead of writing $\lbrace a\rbrace B$, $A\lbrace b\rbrace$, and $\lbrace a\rbrace \lbrace b\rbrace$, we write $aB, Ab$, and $ab$ instead. From now on, when we write $(ab)c$, we mean |
| ``take the product of $a$ and $b$ via the multivalued binary operation $\cdot$ on $G$, then take the product of the set $ab$ with the element $c$, under the induced binary operation on $P(G)$''. Given a hypergroupoid $G$, there are two types of associativity we may define: |
``take the product of $a$ and $b$ via the multivalued binary operation $\cdot$ on $G$, then take the product of the set $ab$ with the element $c$, under the induced binary operation on $P(G)$''. Given a hypergroupoid $G$, there are two types of associativity we may define: |
| \begin{description} |
\begin{description} |
| \item[Type 1.] $(ab)c\subseteq a(bc)$, and |
\item[Type 1.] $(ab)c\subseteq a(bc)$, and |
| \item[Type 2.] $a(bc)\subseteq (ab)c$. |
\item[Type 2.] $a(bc)\subseteq (ab)c$. |
| \end{description} |
\end{description} |
| $G$ is said to be \emph{associative} if it satisfies both types of associativity laws. An associative hypergroupoid is called a \emph{hypersemigroup}. We are now ready to formally define a hypergroup. |
$G$ is said to be \emph{associative} if it satisfies both types of associativity laws. An associative hypergroupoid is called a \emph{hypersemigroup}. We are now ready to formally define a hypergroup. |
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| \textbf{Definition}. A \emph{hypergroup} is a hypersemigroup $G$ such that $aG=Ga=G$ for all $a\in G$. |
\textbf{Definition}. A \emph{hypergroup} is a hypersemigroup $G$ such that $aG=Ga=G$ for all $a\in G$. |
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| Let $G$ be a hypergroupoid. Identity elements are defined via the following three sets: |
Let $G$ be a hypergroupoid. Identity elements are defined via the following three sets: |
| \begin{enumerate} |
\begin{enumerate} |
| \item (set of \emph{left identities}): $I_L(G):=\lbrace e\in G\mid a\in ea\mbox{ for all }a\in G\rbrace$, |
\item (set of \emph{left identities}): $I_L(G):=\lbrace e\in G\mid a\in ea\mbox{ for all }a\in G\rbrace$, |
| \item (set of \emph{right identities}): $I_R(G):=\lbrace e\in G\mid a\in ae\mbox{ for all }a\in G\rbrace$, and |
\item (set of \emph{right identities}): $I_R(G):=\lbrace e\in G\mid a\in ae\mbox{ for all }a\in G\rbrace$, and |
| \item (set of \emph{identities}): $I(G)=I_L(G)\cap I_R(G)$. |
\item (set of \emph{identities}): $I(G)=I_L(G)\cap I_R(G)$. |
| \end{enumerate} |
\end{enumerate} |
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$e \in L(G)$ is called an \emph{absolute identity} if $ea=ae=a$ for all $a\in G$. If $e,f\in G$ are both absolute identities, then $e = ef = f$, so $G$ can have at most one absolute identity.
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$e \in L(G)$ is called an \emph{absolute identity} if $ea=ae=a$ for all $a\in G$. If $e,f\in G$ are both absolute identities, then $\lbrace e\rbrace = ef = \lbrace f\rbrace$, so $G$ can have at most one absolute identity.
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| Suppose $e\in I_L(G)\cup I_R(G)$ and $a\in G$. An element $b\in G$ is said to be a \emph{left inverse} of $a$ with respect to $e$ if $e\in ba$. \emph{Right inverses} of $a$ are defined similarly. If $b$ is both a left and a right inverse of $a$ with respect to $e$, then $b$ is called an \emph{inverse} of $a$ with respect to $e$. |
Suppose $e\in I_L(G)\cup I_R(G)$ and $a\in G$. An element $b\in G$ is said to be a \emph{left inverse} of $a$ with respect to $e$ if $e\in ba$. \emph{Right inverses} of $a$ are defined similarly. If $b$ is both a left and a right inverse of $a$ with respect to $e$, then $b$ is called an \emph{inverse} of $a$ with respect to $e$. |
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| More to come... |
More to come... |
