Hyperstructures are

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...

s equipped with at least one

multi-valued In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...

operation, called a ''hyperoperation''. The largest classes of the hyperstructures are the ones called

Hv

– structures. A hyperoperation

(\star)

on a

nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

H

is a mapping from

H \times H

to the nonempty

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

P^\!(H)

, meaning the set of all nonempty subsets of

H

, i.e. :

\star: H \times H \to P^\!(H)

\quad\ (x,y) \mapsto x \star y \subseteq H.

For

A,B \subseteq H

we define :

A \star B = \bigcup_ a \star b

and

A \star x = A \star \,\,

x \star B = \ \star B.

(H, \star )

is a ''semihypergroup'' if

(\star)

is an

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...

hyperoperation, i.e.

x \star (y \star z) = (x \star y)\star z

for all

x, y, z \in H.

Furthermore, a hypergroup is a semihypergroup

(H, \star )

, where the

reproduction axiom Reproduction (or procreation or breeding) is the biological process by which new individual organisms – "offspring" – are produced from their "parent" or parents. Reproduction is a fundamental feature of all known life; each individual or ...

is valid, i.e.

a \star H = H \star a = H

for all

a \in H.

References

*AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece
aha.eled.duth.grApplications of Hyperstructure Theory
Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, ,
Functional Equations on Hypergroups
László, Székelyhidi, World Scientific Publishing, 2012, Abstract algebra {{abstract-algebra-stub