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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:
Definition. A hypergroupoid, or multigroupoid, is a nonempty set $G$, together with a multivalued function $\cdot:G\times G\Rightarrow G$ called the multiplication on $G$.
We write $a\cdot b$, or simply $ab$, instead of $\cdot(a,b)$. Furthermore, if $ab=\{c\}$, then we use the abbreviation $ab=c$.
A hypergroupoid is said to be 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
$A\cdot B:=\bigcup\{a\cdot b\mid a\in A\mbox{ and }b\in B\}.$ 
As a result, we have an induced groupoid $P(G)$. Instead of writing $\{a\}B$, $A\{b\}$, and $\{a\}\{b\}$, 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:
 Type 1.

$(ab)c\subseteq a(bc)$, and
 Type 2.

$a(bc)\subseteq(ab)c$.
$G$ is said to be associative if it satisfies both types of associativity laws. An associative hypergroupoid is called a hypersemigroup. We are now ready to formally define a hypergroup.
Definition. A 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
$aH\cdot bH:=\{cH\mid c=ahb\mbox{, }h\in H\}.$ 
Then $M$ is a hypergroup with multiplication $\cdot$.
If the multiplication in a hypergroup $G$ is singlevalued, then $G$ is a $2$group, and therefore a group (see proof here).
Remark. A hypergroup is also known as a 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:
1. (set of left identities): $I_{L}(G):=\{e\in G\mid a\in ea\mbox{ for all }a\in G\}$,
2. (set of right identities): $I_{R}(G):=\{e\in G\mid a\in ae\mbox{ for all }a\in G\}$, and
3. (set of identities): $I(G)=I_{L}(G)\cap I_{R}(G)$.
$e\in L(G)$ is called an 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 left inverse of $a$ with respect to $e$ if $e\in ba$. 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 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 $H$ (in fact, this is the only identity in $M$), 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).
References
 1 R. H. Bruck, A Survey on Binary Systems, SpringerVerlag, New York, 1966.
 2 M. Dresher, O. Ore, Theory of Multigroups, Amer. J. Math. vol. 60, pp. 705733, 1938.
 3 J.E. Eaton, O. Ore, Remarks on Multigroups, Amer. J. Math. vol. 62, pp. 6771, 1940.
 4 L. W. Griffiths, On Hypergroups, Multigroups, and Product Systems, Amer. J. Math. vol. 60, pp. 345354, 1938.
 5 A. P. Dičman, On Multigroups whose Elements are Subsets of a Group, Moskov. Gos. Ped. Inst. Uč. Zap. vol. 71, pp. 7179, 1953
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