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 non-empty 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 $ab=\lbrace c\rbrace$ , 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 $$A\cdot B:=\bigcup \lbrace a\cdot b\mid a\in A\mbox{ and } b\in B\rbrace.$$ 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:

Type 1.

$(ab)c\subseteq a(bc)$ , and

Type 2.

$a(bc)\subseteq (ab)c$ .

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 $$aH\cdot bH := \lbrace cH\mid c=ahb\mbox{, }h\in H\rbrace.$$ 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 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): $I_L(G):=\lbrace e\in G\mid a\in ea\mbox{ for all }a\in G\rbrace$ ,
2. (set of right identities): $I_R(G):=\lbrace e\in G\mid a\in ae\mbox{ for all }a\in G\rbrace$ , and
3. (set of identities): $I(G)=I_L(G)\cap I_R(G)$ .

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).

Bibliography

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M. Dresher, O. Ore, Theory of Multigroups, Amer. J. Math. vol. 60, pp. 705-733, 1938.

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L. W. Griffiths, On Hypergroups, Multigroups, and Product Systems, Amer. J. Math. vol. 60, pp. 345-354, 1938.

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A. P. Dicman, On Multigroups whose Elements are Subsets of a Group, Moskov. Gos. Ped. Inst. Uc. Zap. vol. 71, pp. 71-79, 1953