Ratio Mathematica
Vol. 33, 2017, pp. 47-60

ISSN: 1592-7415
eISSN: 2282-8214

A Brief Survey on the two Different
Approaches of Fundamental Equivalence

Relations on Hyperstructures

Nikolaos Antampoufis∗, Šarka Hošková-Mayerová†

‡doi:10.23755/rm.v33i0.388

This paper is dedicated to Prof. Thomas Vougiouklis lifetime work.

Abstract

Fundamental structures are the main tools in the study of hyperstructures.
Fundamental equivalence relations link hyperstructure theory to the theory
of corresponding classical structures. They also introduce new hyperstruc-
ture classes.The present paper is a brief reference to the two different ap-
proaches to the notion of the fundamental relation in hyperstructures. The
first one belongs to Koskas, who introduced the β∗ - relation in hyperstruc-
tures and the second approach to Vougiouklis, who gave the name funda-
mental to the resulting quotient sets. The two approaches, the necessary
definitions and the theorems for the introduction of the fundamental equiva-
lence relation in hyperstructures, are presented.
Keywords: Fundamental equivalence relations, strongly regular relation,
hyperstructures, Hv - structures.
2010 AMS subject classifications: 20N20, 01A99.

∗Lyceum of Avdera, P. Christidou 42, 67132 Xanthi, Greece; antanik@otenet.gr.
†Department of Mathematics and Physics, University of Defence Brno, Czech Republic;

sarka.mayerova@unob.cz
‡ c©Nikolaos Antampoufis and Šarka Hošková-Mayerová. Received: 31-10-2017. Accepted:

26-12-2017. Published: 31-12-2017.

47



Nikolaos Antampoufis and Sarka Hoskova-Mayerova

1 Introduction
Dealing with classical algebraic structures often leads to the study of the be-

haviours of the elements of these sets with respect to the introduced operation(s).
This study focuses, very often, on looking for elements with similar behaviour.
Therefore, the use of the quotient set is intertwined with the search for regularity
and symmetry between elements of algebraic structures and ’similar’ algebraic
structures too.
It is well known that ”... the most powerful tool in order to obtain a stricter struc-
ture from a given one is the quotient out procedure. To use this method in ordi-
nary algebraic domains, one needs special equivalence relations. If one suggests a
method that can applied for every equivalence relation, has to use the hyperstruc-
tures” [6].
In the commutative algebra, many problems of algebraic structures are not always
visible, resulting in a large number of questions and obstacles appearing in the
non-commutative algebra. For example, in classical theory if G is a group and
H ⊆ G is a subgroup, then G/H quotient is a group only when H is a normal
subgroup. This obstacle [21] is overcome by the definition of Fr. Marty (1934)
[17], since

”If G is a group and H ⊆ G is a subgroup of it, then the quotient G/H is a
hypergroup.”

The previous proposition is generalized by the definition [26] of the weak hyper-
structures by Th. Vougiouklis (1990), as follows:

”If G is a group and S is any partition of G, then the quotient G/H is a
Hv-group”.

In these cases, the quotient set functioned as a process that led to ’looser’ struc-
tures than classic algebraic ones, but increased complexity.
The utility of outmost importance of the quotient set in hyperstructures is its use
as a bridge between classical structures and hyperstructures. In 1970, this con-
nection was achieved by M. Koskas [16] using the β - relation and its transitive
closure. Observing the similar behaviour of elements belonging to the same hy-
perproduct leads to the introduction of the β - relation which, clearly, is reflective,
symmetric but not always transitive. The next step is to use the transitive closure
of β to obtain equivalence relation and partition in equivalence classes. Using
the usual definition of operations between classes, we return to classical algebraic
structures. This relation studied mainly by Corsini [5], Vougiouklis [25], Davvaz
[8], Leoreanou-Fotea [7], Freni [12], Migliorato [19] and many others.
The quotient set not only links the hyperstructures with the classical structures

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A Brief Survey on the two Different Approaches of Fundamental Equivalence
Relations on Hyperstructures

as a bridge, but also enhances the view of hyperstructures as a generalization of
the corresponding classical algebraic structures. In this way, as reported in [22],
the algebraic structures are contained in the corresponding hyperstructures as sub-
cases. It seems that Fr. Marty defined the hypergroup replacing the axiom of the
existence of a unitary and inverse element with the axiom of reproduction because
he had ”sensed” this connection and chose the widest possible generalization in
order to create space for the introduction of new types of hypergroups.
In 1988 at a congress in Italy, Th. Vougiouklis presents a paper titled ”How a
hypergroup hides a group” [22], [27] and finds out that
a) Vougiouklis, eighteen years after Koskas worked on the same subject without
having knowledge of his work.
b) Much of the study was completed with the work of Koskas and especially P.
Corsini and his school.
c) Vougiouklis approach was different from that of Koskas and the others.
In the following we will present the two different approaches.

2 Preliminaries
In a set H 6= ∅ equipped with a hyperoperation (·) : H × H → ℘∗(H) we

abbreviate by

WASS the weak associativity: x · (y ·z) ∩ (x ·y) ·z 6= ∅,∀(x,y,z) ∈ H3.
COW the weak commutativity: x ·y ∩y ·x 6= ∅,∀(x,y) ∈ H2.

Definition 2.1. The hyperstructure (H, ·) is called Hv-semigroup if it is WASS
and it is called Hv-group if it is reproductive Hv-semigroup, that is xH = Hx =
H,∀x ∈ H.
Definition 2.2. The hyperstructure (H, ·) is called semihypergroup if x · (y ·z) =
(x ·y) ·z,∀(x,y,z) ∈ H3 and it is called hypergroup if it is reproductive semihy-
pergroup.

Definition 2.3. A Hv-group is called Hb-group if its hyperoperation contains op-
eration which define a group. We define analogously Hb-ring, Hb-vector space.

Definition 2.4. Let (H, ·) be a Hv - structure. An element e ∈ H is called identity
if x ∈ ex∩xe,∀x ∈ H. We define analogously the left (right) identity.
Definition 2.5. Let φ : H → H/β∗ be the fundamental map of a Hv-group then,
the kernel of φ is called core and it is denoted by ωH .

Definition 2.6. A Hv-semigroup or a semihypergroup H is called cyclic if there
exists s ∈ H, called generator, such that: H = s1∪s2∪...∪sn∪...,n ∈ N,n > 0.
For more definitions and applications on Hv-structures, see also the papers [4],
[9], [10], [14], [15], [18], [20], [28].

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Nikolaos Antampoufis and Sarka Hoskova-Mayerova

3 The two approaches of fundamental relations
Searching for the quotient set, the definition of the relation between the ele-

ments of the hyperstructure plays an important role. The observation of a hyperop-
eration leads to the conclusion that elements belonging to the same hyperproduct
act in a similar way with respect to the hyperoperation. This observation is the
basis for defining the relation β. This definition is common to both approaches.
Another common finding of the two approaches is the fact that the relation β is
reflective, symmetric but not always transitive. The need for an equivalence re-
lation that produces a quotient set such that it is a classical algebraic structure,
makes it necessary to consider the β∗ - relation that is the transitive closure of the
β - relation and is, obviously, an equivalence relation. The last common point of
the two approaches is the search for the smallest (with respect to the inclusion)
equivalence relation having as quotient set the corresponding algebraic structure.

3.1 Koskas approach
Koskas, in his approach, introduces the equivalence relation that obtains as

quotient set the corresponding algebraic structure by using the strongly regular
equivalence relation. It is then shown that the transitive closure of the β - relation
is the smallest strongly regular equivalence relation, i.e. β∗ is the targeted relation.
The proof is completed by the obvious finding that β∗ is the desired equivalence
relation such that the H/β∗ is the corresponding algebraic structure. One can say
that Koskas approach is a deductive way of defining fundamental relation on hy-
perstructures, since he starts considering a general definition and then specifying
the β∗-relation as a subcase.

Taking into consideration the approach, the necessary definitions and theorems
are mentioned, having as main sources the books [5], [7], [8].

Definition 3.1. Let (H,◦) be a hypergroupoid, a,b elements of H and ρ be an
equivalence relation on H. Then ρ is strongly regular on the left if the following
implication holds:

aρb ⇒∀u ∈ H,∀x ∈ u◦a,∀y ∈ u◦ b : xρy.

Similarly, the strong regularity on the right can be defined. We call ρ strongly
regular if it is strongly regular on both the left and the right.

Definition 3.2. Let (H, ·) be a semihypergroup and n > 1 be a natural number.
We define the βn relation as follows:

xβny if there exist a1,a2, ...,an elements of H, so subsets {x,y}⊆
n∏

i=1

ai.

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A Brief Survey on the two Different Approaches of Fundamental Equivalence
Relations on Hyperstructures

and let

β =
⋃
n≥1

βn, where β1 = {(x,x)/x ∈ H} is the diagonal relation on H.

Notice that relation β is reflexive and symmetric but, generally, not a transitive
one.

Definition 3.3. We denote β∗ the transitive closure of relation β.

Theorem 3.1. β∗ is the smallest strongly regular equivalence relation on H with
respect to the inclusion.

Theorem 3.2. Let (H, ·) be a semihypergroup (hypergroup), then the transitive
closure of relation β is the smallest equivalence relation such that the quotient
H/β∗is a semigroup (group).

Definition 3.4. β∗ is called the fundamental relation on H and H/β∗ is called the
fundamental semigroup (group).

Notice that [22] the term fundamental, given by Vougiouklis, is subsequent of
Koskas definitions but totally used nowadays.

3.2 Vougiouklis approach
Unlike the previous ones, Vougiouklis approaches the issue in a straightfor-

ward way starting with the acquired question about the appropriate equivalence
relations. He defines the relation that has as quotient set the corresponding al-
gebraic structure. He then defines the relation β in a more general manner than
previously defined. The approach has been completed by proving that the funda-
mental relation is no other than the transitive closure of the relation β. We can
assume that Vougiouklis approach is an inductive way of defining the fundamen-
tal relation in hyperstructures because it starts with the partial and ends in a more
general result.
It is important to note that Vougiouklis definitions were given for Hv-groups which
are a wider class than the one of hypergroups. Also, the proof of theorem about
β∗ relation (see below) follows a remarkable strategy [3].

Taking into consideration the approach, the necessary definitions and theorems
are mentioned, having as main source the book [25] and the papers [24], [26].

Definition 3.5. Let (H, ·) be a Hv-group. The relation β∗ on H is called funda-
mental equivalence relation if it is the smallest equivalence relation on H such
that the quotient set H/β∗ is a group, called fundamental group of H.

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Nikolaos Antampoufis and Sarka Hoskova-Mayerova

Notice that the proof of the fundamental groups existence for any Hv-group is
an obvious result of the following theorem’s proof.

Let us denote by U the set of all finite hyperproducts of elements of H.

Definition 3.6. Let (H, ·) be a Hv-group. We define the relation β on H as fol-
lows:

xβy iff {x,y}⊆ u,u ∈ U.

Theorem 3.3. The fundamental equivalence relation β∗ is the transitive closure
of the relation β on H.

Definition 3.7. Let (R, +, ·) be a Hv-ring. The relation γ∗ on R is called funda-
mental equivalence relation on R if it is the smallest equivalence relation on R
such that the quotient set R/γ∗ is a ring, called fundamental ring of R.

Let us denote by U the set of all finite polynomials of elements of R, over N.

Definition 3.8. Let (R, +, ·) be a Hv-ring. We define the relation γ on H as
follows:

xγy iff {x,y}⊆ u,u ∈ U.

Theorem 3.4. The fundamental equivalence relation γ∗ is the transitive closure
of the relation γ on R.

Definition 3.9. [26] A Hv-ring is called Hv-field if its fundamental ring is a field.

Definition 3.10. Let V be a Hv-vector space over a Hv-field R. The relation ε∗
on V is called fundamental equivalence relation if it is the smallest equivalence
relation on V such that the quotient set V/ε∗ is a vector space over the field R/γ∗,
called fundamental vector space of V over R.

4 Fundamental classes
Searching for the classes of fundamental equivalence relations is a central

question in studying the fundamental structures derived from hyperstructures.
This quest is intertwined with the exploration of the conditions that must be ac-
complished so that the β relation is transitive, that is, β = β∗. It is clear that the
two different approaches to the fundamental equivalence relation in hyperstruc-
tures settle on two different ways of searching or constructing the fundamental
equivalence classes in a hyperstructure. We could also talk, in a similar way with
3.1 and 3.2, about the deductive and inductive way of finding equivalence classes.

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A Brief Survey on the two Different Approaches of Fundamental Equivalence
Relations on Hyperstructures

4.1 Complete Parts
According to the deductive way that Koskas used and Corsini’s school contin-

ued, the equivalence classes, that occur when the equivalence relation is strongly
regular, are used. The notion of complete part of a hyperstructure’s subset plays
a key role in finding the β∗ class of each element. The complete closure C(A) of
the part A is connected with an increasing chain of subsets of the hyperstructure
which, in turn, are related to hyperproducts containing A. It then turns out that
the introduction, in a natural way, of the equivalence relation K is essentially a
consideration of the equivalence β∗. In this way the complete closure coincides to
the fundamental equivalence class.
In particular, the definition of the complete part is used in the case of the singleton
{x}, for each element x of the hyperstructure, so that we find ourselves in the en-
vironment of the fundamental equivalence relation. The increasing chain of sets
created by the set {x} constructs the fundamental class of the arbitrary element x.
It is evident that, as in the introduction of the β∗ - relation [16], a notion is used as
a mediator, which comes in between the questions ”how is the class” and ”what is
the class”. This notion is K relation.
We now present the necessary propositions in order to describe the step by step
approach of the fundamental classes notion. The main references we use are the
books [5], [7], [8] and the papers [13], [19].

Definition 4.1. Let (H, ·) be a semihypergroup and A be a nonempty subset of H.
We say that A is a complete part of H if for any nonzero natural number n and
for all a1,a2, ,an elements of H, the following implication holds:

A∩
n∏

i=1

ai 6= ∅ ⇒
n∏

i=1

ai ⊆ A.

Notice that complete part A absorbs every hyperproduct containing one, at
least, element of A.
According to theorem 3.1, β∗(x) is a complete part of H,∀x ∈ H. (Step 1)

Definition 4.2. Let (H, ·) be a semihypergroup and A be a nonempty subset of H.
The intersection of the complete parts of H which contain A is called the complete
closure of A in H; it will be denoted by C(A).

Denote K1(A) = A and for all n > 0

Kn+1(A) ={
x ∈ H| ∃p ∈ N,∃(h1,h2, ...,hp) ∈ Hp : x ∈

p∏
i=1

(hi), Kn(A) ∩
p∏

i=1

(hi) 6= ∅

}
.

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Nikolaos Antampoufis and Sarka Hoskova-Mayerova

Obviously, Kn(A),n > 0 is an increasing chain of subsets of H as we mentioned.
If x ∈ H, we denote Kn({x}) = Kn(x). This implies that

Kn(A) =
⋃
a∈A

Kn(a)

In particular, if P is the set of all finite hyperproducts of elements of H, and x ∈ H
we have:

K1(x) = {x}, K2(x) =
⋃
u∈P

u : x ∈ u, K3(x) =
⋃
u∈P

u : u∩K2(x) 6= ∅,

...,Kn+1(x) =
⋃
u∈P

u : u∩Kn(x) 6= ∅ and C(A) =
⋃
a∈A

C(a),A ⊆ H.

Notice that K2(x) = {z ∈ H|zβx} = β(x). (Step 2)

Theorem 4.1. Let (H, ·) be a semihypergroup and K a binary relation defined as
follows:

xKy ⇔ x ∈ C(y), (x,y) ∈ H2.

Then,K is an equivalence relation that coincides with β∗. (Step 3)

Thus, the relation K and the chain of sets Kn behave as a mediator, which
comes in between C(x) and β∗, connecting the construction of the class with the
class itself.
Now we present some propositions mainly about the transitivity of β-relation.

Theorem 4.2. [12] Let (H, ·) be a hypergroup then, β = β∗.

Theorem 4.3. Let (H, ·) be a hypergroupoid. Then,

β = β∗ ⇔ C(x) = K2(x),∀x ∈ H.

Theorem 4.4. [13] Let (H, ·) be a Hv-group having, at least, one identity. Then,

β = β∗.

Theorem 4.5. [13] Let (H, ·) be a Hb-group then, β = β∗.

4.2 Constructing Fundamental Classes
According to the inductive way that Vougiouklis introduced, the direct ap-

proach to the fundamental class is used. Vougiouklis, while studying the funda-
mental classes, follows the same philosophy and strategy as he does in his ap-
proach to the introduction of the fundamental relationship and the corresponding

54



A Brief Survey on the two Different Approaches of Fundamental Equivalence
Relations on Hyperstructures

structure. He does not attempt to create an environment of desirable definitions
in which he will then place the fundamental class of each element, as Koskas has
chosen to do.
On the contrary, he considers the fundamental class of each element as a set and
tries to specify its elements through their common properties. He prefers, in short,
a straightforward reference to the common behaviour of these elements in the
generation of hyperproducts. Therefore, he develops propositions that relate each
other the elements which behave in a similar way, thus achieving the accumula-
tion of all the equivalent, with respect to β∗, elements which belong to the same
fundamental class. It is clear that this straightforward dealing with the class study
is an inductive type of answer to the question of the nature and form of the funda-
mental classes.
The fundamental class which is a singleton, whenever it exists, plays an essen-
tial role in the study of Hv-groups and Hvrings. The element of each such class is
called single element. Its value lies in the fact that each hyperproduct having a sin-
gle element as a factor, is an entire fundamental class. Thus, finding the classes is
achieved by multiplying a single element with each element of the hyperstructure.
In fact, it is not necessary to perform the hyperoperation with all the elements.
Moreover, the existence of one, at least, single element is a sufficient condition
such that β = β∗ holds in Hv-groups.
Finally, the above mentioned approach also includes the technique of ”transla-
tion” of hyperproducts which allows us to find the fundamental structure of an
Hv-group and its classes using isomorphism between the quotient sets. We now
present the necessary propositions in order to describe the direct approach of fun-
damental classes. The main references we use are the book [25] and the papers
[24], [26].

Theorem 4.6. Let (H, ·) be a Hvgroup, then

xβ∗y iff there exist A,A
′
⊆ β∗(a), B,B

′
⊆ β∗(b), (a,b) ∈ H2,

such that xA∩B 6= ∅ and yA′ ∩B′ 6= ∅.

Theorem 4.7. Let (H, ·) be a Hv-group, then u ∈ ωH iff there exist A ⊆ β∗(a),
for some a ∈ H, such that uA∩A 6= ∅.

Definition 4.3. Let H be a Hv-structure. An element s ∈ H is called single if
β∗(s) = {s}.

We denote by SH the set of singles elements of H

Theorem 4.8. Let (H, ·) be a Hv-group and s ∈ SH . Let (a,v) ∈ H2 such that
s ∈ av, then

β∗(a) = {h ∈ H : hv = s} and the core of H is ωH = {z ∈ H : zs = s}.

55



Nikolaos Antampoufis and Sarka Hoskova-Mayerova

Theorem 4.9. Let (H, ·) be a Hv-group and SH 6= ∅, then β∗ = β.

Definition 4.4. (Translations) [25] Let (H, ·) be a Hv-group and x ∈ H, then
H/β∗ ∼= (H/lx)/β∗, where lx is the translation equivalence relation.

4.3 Using the fundamental equivalence relations

The fundamental equivalence relations on hyperstructures, on the one hand,
connect the theory of hyperstructures to that of the corresponding classical struc-
tures, and on the other hand, are a tool for the introduction of new hyperstructure
classes. The two approaches to the concept of the fundamental equivalence rela-
tions were initially referred to semihypergroups and hypergroups. However, they
form the driving lever to apply similar definitions to other hyperstructures as hy-
perrrings and hyperfields or to study specific behaviour of some hyperstructures
using the quotient set.
Freni [11] and others (P. Corsini, B. Davazz) introduced and studied equivalence
relations that refer to individual properties or characteristics of elements of a hy-
perstructure. As an example, we mention the equivalence relations ∆h∗ and α∗

which are related to the cyclicity and commutativity of a hyperstructure, respec-
tively. Our main reference is [8].
Relation α was introduced by Freni who used the letter γ instead of α. Thus, there
was a confusion on symbolism since Vougiouklis had already used the letter γ for
the fundamental equivalence relation on hyperrings.

Vougiouklis focused on the study of hyperstructures which have a desirable
quotient.(h/v)-structures are a typical example of generalization using the funda-
mental structures. They are a larger class than that of Hv-structures. Also, the use
of the fundamental classes of equivalence as hyperproducts, led to constructions
of new hyperstructure classes. As additional examples of hyperstructures with
desirable quotient we mention the s1,s2, ...,sn-hyperstructures [3], the complete
(in the sense of Corsini) Hv-groups and the constructions S1 and S2 [25].

Definition 4.5. [23] The Hv-semigroup (H, ·) is called h/v-group if H/β∗ is a
group.

Notice that the reproductivity is not necessarily valid. However, the reproduc-
tivity of classes is valid.

Definition 4.6. [1], [2]. We shall say that a hyperstructure is an sn-hyperstructure
if all its fundamental classes are singleton except for n of them, n ∈ N,n > 0.

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A Brief Survey on the two Different Approaches of Fundamental Equivalence
Relations on Hyperstructures

5 Conclusions - Findings
The aim of this paper was to present the two different approaches to the con-

cept of the fundamental equivalence relation through a comparison of the choices
and methods applied by Koskas and Vougiouklis, but also through the results they
have achieved. Summarizing our conclusions and findings, we can note that:
a) The approach of Koskas is a deductive way based on the introduction of gen-
eral notions willing to draw conclusions to more specific ones. On the contrary,
Vougiouklis makes the reverse. Beginning with the specific notion leads to propo-
sitions generalizing his conclusions.
b) The directness of Vougiouklis approach allows for the ”transfer” of definitions
and corresponding theorems from the Hv-groups to Hv-rings, Hv-modules and the
other weak hyperstructures. In essence, it is a widely applied model that imposed,
among other things, on the terminology of the ”fundamental” relations.
c) The nature of the step-by-step approach of Koskas has led to frequent use of
the mathematical induction method in proving many basic theorems. On the other
hand, due to the direct reference of Vougiouklis to the equivalence classes and
their view as sets of elements, the most frequent proving method used is the proof
by contradiction.
d) The Koskas approach raises the question: What is the quotient set of the hyper-
structure that we are studying? That is why increasing chains of relations are the
main study tool. Vougiouklis approach reverses the question with the following
one: Which hyperstructure produces a particular desired fundamental quotient?
In this inversion, there is a trigger for the introduction of weak and h/v-structures.
Finally, we consider that the mathematical value of fundamental equivalence re-
lations in hyperstructures is important and their asynchronized approaches by
Koskas and Vougiouklis is an interesting moment of the short history of algebraic
hyperstructures from their beginning at 1934 until today.

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Nikolaos Antampoufis and Sarka Hoskova-Mayerova

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