- Source: Hyperstructure
Hyperstructures are algebraic structures equipped with at least one multi-valued operation, called a hyperoperation. The largest classes of the hyperstructures are the ones called
H
v
{\displaystyle Hv}
– structures.
A hyperoperation
(
⋆
)
{\displaystyle (\star )}
on a nonempty set
H
{\displaystyle H}
is a mapping from
H
×
H
{\displaystyle H\times H}
to the nonempty power set
P
∗
(
H
)
{\displaystyle P^{*}\!(H)}
, meaning the set of all nonempty subsets of
H
{\displaystyle H}
, i.e.
⋆
:
H
×
H
→
P
∗
(
H
)
{\displaystyle \star :H\times H\to P^{*}\!(H)}
(
x
,
y
)
↦
x
⋆
y
⊆
H
.
{\displaystyle \quad \ (x,y)\mapsto x\star y\subseteq H.}
For
A
,
B
⊆
H
{\displaystyle A,B\subseteq H}
we define
A
⋆
B
=
⋃
a
∈
A
,
b
∈
B
a
⋆
b
{\displaystyle A\star B=\bigcup _{a\in A,\,b\in B}a\star b}
and
A
⋆
x
=
A
⋆
{
x
}
,
{\displaystyle A\star x=A\star \{x\},\,}
x
⋆
B
=
{
x
}
⋆
B
.
{\displaystyle x\star B=\{x\}\star B.}
(
H
,
⋆
)
{\displaystyle (H,\star )}
is a semihypergroup if
(
⋆
)
{\displaystyle (\star )}
is an associative hyperoperation, i.e.
x
⋆
(
y
⋆
z
)
=
(
x
⋆
y
)
⋆
z
{\displaystyle x\star (y\star z)=(x\star y)\star z}
for all
x
,
y
,
z
∈
H
.
{\displaystyle x,y,z\in H.}
Furthermore, a hypergroup is a semihypergroup
(
H
,
⋆
)
{\displaystyle (H,\star )}
, where the reproduction axiom is valid, i.e.
a
⋆
H
=
H
⋆
a
=
H
{\displaystyle a\star H=H\star a=H}
for all
a
∈
H
.
{\displaystyle a\in H.}
References
AHA (Algebraic Hyperstructures & Applications). A scientific group at Democritus University of Thrace, School of Education, Greece. aha.eled.duth.gr
Applications of Hyperstructure Theory, Piergiulio Corsini, Violeta Leoreanu, Springer, 2003, ISBN 1-4020-1222-5, ISBN 978-1-4020-1222-8
Functional Equations on Hypergroups, László, Székelyhidi, World Scientific Publishing, 2012, ISBN 978-981-4407-00-7