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 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 $ab=\{c\}$, 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 single-valued, 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).