hypergroup
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
$$A\cdot B:=\bigcup \{a\cdot b\mid a\in A\text{and}b\in B\}.$$ 
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
$$aH\cdot bH:=\{cH\mid c=ahb\text{,}h\in H\}.$$ 
Then $M$ is a hypergroup with multiplication $\cdot $.
If the multiplication in a hypergroup $G$ is singlevalued, then $G$ is a $2$group (http://planetmath.org/PolyadicSemigroup), and therefore a group (see proof here (http://planetmath.org/PolyadicSemigroup)).
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): ${I}_{L}(G):=\{e\in G\mid a\in ea\text{for all}a\in G\}$,

2.
(set of right identities): ${I}_{R}(G):=\{e\in G\mid a\in ae\text{for all}a\in G\}$, 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
Title  hypergroup 
Canonical name  Hypergroup 
Date of creation  20130322 18:38:22 
Last modified on  20130322 18:38:22 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 20N20 
Synonym  multigroupoid 
Synonym  multisemigroup 
Synonym  multigroup 
Related topic  group 
Defines  hypergroupoid 
Defines  hypersemigroup 
Defines  left identity 
Defines  right identity 
Defines  identity 
Defines  absolute identity 
Defines  left inverse 
Defines  right inverse 
Defines  inverse 
Defines  absolute identity 