RATIO MATHEMATICA 25(2013), 59–66 ISSN:1592-7415

The LV-hyperstructures

N. Lygeros*, T. Vougiouklis**
*Lgpc, University of Lyon, Lyon, France,

**Democritus University of Thrace, School of Education

w@lygeros.org, tvougiou@eled.duth.gr

Abstract
The largest class of hyperstructures is the one which satisfy the

weak properties and they are called Hv-structures introduced in 1990.
The Hv-structures have a partial order (poset) on which gradations
can be defined. We introduce the LV-construction based on the Levels
Variable.

Key words: hyperstructures, Hv-structures, hopes, weak hopes.

MSC2010: 20N20.

1 Fundamental Definitions

In a set H is called hyperoperation (abbreviation hyperoperation=hope)
in a set H, is called any map · : H ×H →P(H) −{∅}.
Definition 1.1 (Marty 1934). A hyperstructure (H, ·) is a hypergroup if (·)
is an associative hyperoperation for which the reproduction axiom: hH =
Hh = H,∀x ∈ H, is valid.
Definition 1.2 (Vougiouklis 1990). In a set H with a hope we abbreviate by
WASS the weak associativity : (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ H and by COW
the weak commutativity : xy∩yx 6= ∅,∀x,y ∈ H. The hyperstructure (H, ·) is
called Hv-semigroup if it is WASS, it is called Hv-group if it is reproductive
Hv-semigroup, i.e. xH = Hx = H,∀x ∈ H. The hyperstructure (R, +, ·) is
called Hv-ring if both (+) and (·) are WASS, the reproduction axiom is valid
for (+) and (·) is weak distributive with respect to

(+) : x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅,∀x,y,z ∈ R

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N. Lygeros, T. Vougiouklis

Definition 1.3 (Santilly-Vougiouklis). A hyperstructure (H, ·) which con-
tain a unique scalar unit e, is called e-hyperstructure. A hyperstructure
(F, +, ·), where (+) is an operation and (·) is a hyperoperation, is called
e-hyperfield if the following axioms are valid:

1. (F, +) is an abelian group with the additive unit 0,

2. (·) is WASS,

3. (·) is weak distributive with respect to (+),

4. 0 is absorbing element: 0 ·x = x · 0 = 0,∀x ∈ F ,

5. there exists a multiplicative scalar unit 1, i.e. 1 ·x = x ·1 = x,∀x ∈ F ,

6. for every x ∈ F there exists a unique inverse x−1, such that

1 ∈ x ·x−1 ∩x−1 ·x.

The elements of an e-hyperfield are called e-hypernumbers. In the case
that the relation: 1 = x ·x−1 = x−1 ·x, is valid, then we say that we have a
strong e-hyperfield.

Construction 1.4. The Main e-Construction. Given a group (G, ·), where
e is the unit, then we define in G, a large number of hyperoperations (⊗) as
follows:

x⊗y = {xy,g1,g2, . . .},∀x,y ∈ G−{e}, and g1,g2, . . . ∈ G−{e}

g1,g2, . . . are not necessarily the same for each pair (x,y). Then (G,⊗)
becomes an Hv-group, in fact is Hb-group which contains the (G, ·). The
Hv-group (G,⊗) is an e-hypergroup. Moreover, if for each x,y such that
xy = e, so we have x⊗y = xy, then (G,⊗) becomes a strong e-hypergroup.

For more definitions and applications on Hv-structures, see the books and
papers [1-20].

The main tool to study hyperstructures are the fundamental relations β∗,
γ∗ and ε∗, which are defined, in Hv-groups, Hv-rings and Hv-vector spaces,
resp., as the smallest equivalences so that the quotient would be group, ring
and vector space, resp. Fundamental relations are used for general defini-
tions. Thus, an Hv-ring (R, +, ·) is called Hv-field if R/γ∗ is a field.

Definition 1.5. Let (H, ·), (H,∗) be Hv-semigroups defined on the same set
H. Then (·) is called smaller than (∗), and (∗) greater than (·), iff there
exists an f ∈ Aut(H,∗) such that xy ⊂ f(x∗ y),∀x,y ∈ H. Then we write
· ≤ ∗ and we say that (H,∗) contains (H, ·). If (H, ·) is a structure then it
is called basic structure and (H,∗) is called Hb-structure.

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The LV-hyperstructures

Theorem 1.6 (The Little Theorem). Greater hopes than the ones which are
WASS or COW, are also WASS or COW, respectively.

This Theorem leads to a partial order on Hv-structures, thus we have
posets. The determination of all Hv-groups and Hv-rings is very interesting.
To compare classes we can see the small sets. The problem of enumeration of
classes of Hv-structures was started very early but recently we have results
by using computers. The partial order in Hv-structures restricts the problem
in finding the minimals.

2 Enumeration Theorems

Theorem 2.1 (Chung-Choi). There exists up to isomorphism, 13 minimal
Hv-groups of order 3 with scalar unit, i.e. minimal e-hyperstructures of or-
der 3.

Theorem 2.2 (Bayon-Lygeros).

• There exist, up to isomorphism, 20 Hv-groups of order 2.

• There exist, up to isomorphism, 292 Hv-groups of order 3 with scalar
unit, i.e. e-hyperstructures of order 3.

• There exist, up to isomorphism, 6494 minimal Hv-groups of order 3.

• There exist, up to isomorphism, 1026462 Hv-groups of order 3.

Theorem 2.3 (Bayon-Lygeros).

• There exist, up to isomorphism, 631609 Hv-groups of order 4 with scalar
unit, i.e. e-hyperstructures of order 4.

• There exist, up to isomorphism, 8.028.299.905 abelian Hv-groups of
order 4.

Theorem 2.4 (Bayon-Lygeros).

• The number of abelian Hv-groups of order 4 with scalar unit (i.e. abelian
e-hyperstructures) in respect with their automorphism group are the fol-
lowing

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N. Lygeros, T. Vougiouklis

|Aut (Hv)| 1 2 3 4 6 8 12 24
— — — 32 — 46 5510 626021

• There are 63 isomorphism classes of hyperrings of order 2.

• There are 875 isomorphism classes of Hv-rings of order 2.

• There are 33277642 isomorphism classes of hyperrings of order 3.

In all the above results we construct the poset of hyperstructures of order
2 and 3 in the sense of inclusion for hyperproducts. We compute the Betti
numbers of the poset of Hv-groups of order 2 and we have the following re-
sults: (1, 5), (2, 4), (3, 6), (4, 4), (5, 1). We also compute the Betti numbers
of the poset of hypergroups of order 3 and we have the following results:
(1, 59), (2, 168), (3, 294), (4, 438), (5, 568), (6, 585), (7, 536), (8, 480), (9, 358),
(10, 245), (11, 160), (12, 66), (13, 29), (14, 10), (15, 2), (16, 1).

We explicitly compute the Cayley subtables of the minimal e-hyperstruc-
tures with H = {e,a,b} and we have for the products (aa, ab, ba, bb) the
following results: (b; e; e; a), (eb; a; a; e), (e; ab; ab; e), (a; eb; eb; a), (ab;
ea; ea; e), (H; eb; a; ea), (H; a; eb; ea), (a; H; H; e), (b; H; H; e), (a; H; H;
b), (H; b; a; H), (H; a; b; H), (H; e; ab; H).

3 Construction Theorems

There are several ways to organize such posets using hyperstructure the-
ory. We present now a new construction on posets and we name this LV-
construction since it is based on gradations where the Levels are used as
Variable. Thus LV means Level Variable.

Theorem 3.1. The LV-Construction I
Consider the set Pn of all Hv-groups defined on a set of n elements. Take

the following gradation on Pn based on posets:
Level 0 (or grade 0), denoted by g0, is the set of all minimals of Pn. Level

(grade) 1, denoted by g1, is the set of all Hv-groups obtained from minimals
by adding one only element to anyone of the results of the products of two
elements on the minimals of Pn, i.e. of g0. Level 2 (or grade 2), denoted
by g2, is the set of all Hv-groups obtained from minimals by adding only
two elements to anyone of the results of the products of two elements of the
minimals g0. Then inductively the Level k is defined, denoted by gk. In the

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The LV-hyperstructures

case that an Hv-group is obtained by adding k1 elements of one minimal and
by adding k2 elements of another minimal then we consider that it belongs to
the Level min(k1,k2).

Denote by r the cardinality of the minimals, |g0| = r, and by s the number
of levels. Take any Hv-group with r elements corresponding to the r elements
of g0, so we have an Hv-group (g0,∗). Then we define a hope on

Pn = g0 ∪ g1∪, . . . ,∪gs−1,

as follows

x⊗y =

{
x∗y, ∀x,y ∈ g0
gκ+λ, ∀x ∈ gκ,y ∈ gλ, where (κ,λ) 6= (0, 0)

Then the hyperstructure (Pn,⊗) is an Hv-group where its fundamental
group is isomorphic to Zs, thus we have

Pn/β
∗ ≈ Zs.

Proof. Let us correspond, numbered, the levels with the elements of Zs :
gi → i, i = 0, . . . ,s− 1.

From the definition of (⊗) any hyperproduct of elements from several
levels, apart of g0, equals to only one special set of Hv-groups that constitute
one level. Moreover we have

x⊗y = g0,∀x ∈ gκ,y ∈ g−κ, for any κ 6= 0.

That means that the elements of g0are β*-equivalent. Therefore all elements
of each level are β∗-equivalent and there are no β∗-equivalent elements from
different levels. That proves that

Pn/β
∗ ≈ Zs.

The above is a construction similar to the one from the book [15, p.27]
A generalization of the above construction is the following:

Theorem 3.2. The LV-Construction II
Consider a graded finite poset with n elements: Pn = g0∪g1∪, . . . ,∪gs−1,

with s levels (grades) g0, g1, . . . , gs−1, such that

s−1∑
i=0

|gi| = n.

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N. Lygeros, T. Vougiouklis

Denoting |g0| = r, we consider two Hv-groups (E, ·) and (S,∗) such that
|E| = r, |S| = s and moreover S has a unit single element e. Then we
take 1:1 maps from E onto g0 and from S onto {g0, g1, . . . , gs−1}, so we
obtain two Hv-groups: (g0, ·) and

(
G = {g0, g1, . . . , gs−1},∗

)
where E = g0

corresponds to the single element e. We define a hope on Pn as follows:

x⊗y =

{
x ·y, ∀x,y ∈ g0
gκ ∗ gλ, ∀gκ, gλ ∈ G, where (κ,λ) 6= (0, 0)

Then the hyperstructure (Pn,⊗) is an Hv-group where its fundamental group
is isomorphic to the fundamental group of (S,∗), therefore we have

(Pn,⊗)/β
∗ ≈ (S,∗)/β∗.

Proof. From the reproductivity of (G,∗), for each gκ,κ 6= 0, there exists a
gλ such that g0 ∈ gκ ∗ gλ. But g0 is a single element of (S,∗), therefore we
have g0 = gκ ∗ gλ. Then, by the definition, for any x ∈ gκ, y ∈ gλ we have,
x⊗y = g0. Therefore, all the elements of g0 are β∗-equivalent. On the other
side, from the definition, all elements of each level are β*-equivalent and
they are β∗-equivalent elements with different levels if and only if they are
β∗-equivalent in (G,∗). In other wards they follow exactly the β∗-equivalence
of (G,∗).

That proves that
(Pn,⊗)/β

∗ ≈ (S,∗)/β∗.

With this LV-construction we can define the poset for Hv-groups of order
2. So we get a non-connected poset with Betti numbers for the two subposets
(1,4), (2,4), (3,1) and (1,1), (2, 4), (3,6).

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