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| Title of object: |
hypergroup |
| Canonical Name: |
Hypergroup |
| Type: |
Definition |
| Created on: |
2008-12-24 10:45:25 |
| Modified on: |
2008-12-24 15:35:52 |
| Classification: |
msc:20N20 |
| Defines: |
hypergroupoid, hypersemigroup, left identity, right identity, identity, absolute identity, left inverse, right inverse, inverse |
| Synonyms: |
hypergroup=multigroupoid
hypergroup=multisemigroup
hypergroup=multigroup |
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Content:
\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:
\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 $G$.
We write $a\cdot b$, or simply $ab$, instead of $\cdot(a,b)$. Furthermore, if $ab=\lbrace c\rbrace$, then we use the abbreviation $ab=c$.
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:
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
``first, 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}
\item[Type 1.] $(ab)c\subseteq a(bc)$, and
\item[Type 2.] $a(bc)\subseteq (ab)c$.
\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.
\textbf{Definition}. A \emph{hypergroup} is a hypersemigroup $G$ such that $aG=Ga=G$ for all $a\in G$.
For example, let $G$ be a group and $H$ a subgroup of $G$. Let $M$ be the collection of all left cosets of $H$ in $G$. For $aH,bH\in M$, set $$aH\cdot bH := \lbrace cH\mid c=ahb\mbox{, }h\in H\rbrace.$$ Then $M$ is a hypergroup with multiplication $\cdot$.
If the multiplication in a hypergroup $G$ is single-valued, then $G$ is a $2$-group, and therefore a group (see \PMlinkname{proof here}{PolyadicSemigroup}).
\textbf{Remark}. A hypergroup is also known as a \emph{multigroup}, although some call a multigroup as a hypergroup with a designated identity element $e$, as well as a designated inverse for every element with respect $e$. Actually identities and inverses may be defined more generally for hypergroupoids:
Let $G$ be a hypergroupoid. Identity elements are defined via the following three sets:
\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{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}): $I(G)=I_L(G)\cap I_R(G)$.
\end{enumerate}
$e \in L(G)$ 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.
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 $e$.
Thus, one may say that a multigroup is a hypergroup $G$ with an identity $e\in G$, and a function $^{-1}:G\to G$ such that $a^{-1}:=^{-1}(a)$ is an inverse of $a$ with respect to $e$.
In the example above, $M$ is a multigroup in the sense given in the remark above. The designated identity is $H$, and for every $aH \in M$, its designated inverse is provided by $a^{-1}H$ (of course, this may not be its only inverse, as any $bH$ such that $ahb=e$ for some $h\in H$ will do).
More to come... |
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