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 , together with a multivalued function called the multiplication on .
We write , or simply , instead of . Furthermore, if , then we use the abbreviation .
Given a hypergroupoid , the multiplication induces a binary operation (also written ) on , the powerset of , given by
As a result, we have an induced groupoid . Instead of writing , , and , we write , and instead. From now on, when we write , we mean “first, take the product of and via the multivalued binary operation on , then take the product of the set with the element , under the induced binary operation on ”. Given a hypergroupoid , there are two types of associativity we may define:
- Type 1.
- Type 2.
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 such that for all .
Then is a hypergroup with multiplication .
If the multiplication in a hypergroup is single-valued, then is a -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 , as well as a designated inverse for every element with respect . Actually identities and inverses may be defined more generally for hypergroupoids:
Let be a hypergroupoid. Identity elements are defined via the following three sets:
(set of left identities): ,
(set of right identities): , and
(set of identities): .
is called an absolute identity if for all . If are both absolute identities, then , so can have at most one absolute identity.
Suppose and . An element is said to be a left inverse of with respect to if . Right inverses of are defined similarly. If is both a left and a right inverse of with respect to , then is called an inverse of with respect to .
Thus, one may say that a multigroup is a hypergroup with an identity , and a function such that is an inverse of with respect to .
In the example above, is a multigroup in the sense given in the remark above. The designated identity is (in fact, this is the only identity in ), and for every , its designated inverse is provided by (of course, this may not be its only inverse, as any such that for some will do).
- 1 R. H. Bruck, A Survey on Binary Systems, Springer-Verlag, New York, 1966.
- 2 M. Dresher, O. Ore, Theory of Multigroups, Amer. J. Math. vol. 60, pp. 705-733, 1938.
- 3 J.E. Eaton, O. Ore, Remarks on Multigroups, Amer. J. Math. vol. 62, pp. 67-71, 1940.
- 4 L. W. Griffiths, On Hypergroups, Multigroups, and Product Systems, Amer. J. Math. vol. 60, pp. 345-354, 1938.
- 5 A. P. Dičman, On Multigroups whose Elements are Subsets of a Group, Moskov. Gos. Ped. Inst. Uč. Zap. vol. 71, pp. 71-79, 1953
|Date of creation||2013-03-22 18:38:22|
|Last modified on||2013-03-22 18:38:22|
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