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).

Title	hypergroup
Canonical name	Hypergroup
Date of creation	2013-03-22 18:38:22
Last modified on	2013-03-22 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