1. Introduction
In the algebraic hypercompositional structures theory, the most natural link with the classical algebraic structures theory is assured by certain equivalences, that work as a bridge between both theories. More explicitly, the quotient structure modulo the equivalence
defined on a hypergroup is always a group [
1], the quotient structure modulo the equivalence
defined on a hyperring is a ring [
2], while a commutative semigroup can be obtained factorizing a semihypergroup by the equivalence
[
3]. A more completed list of such equivalences is very clearly presented in [
4]. All the equivalences having this property, i.e., they are the smallest equivalence relations defined on a hypercompositional structure such that the corresponding quotient (modulo that relation) is a classical structure with the same behaviour, are called
fundamental relations, while the associated quotients are called
fundamental structures. The study of the fundamental relations represents an important topic of research in Hypercompositional Algebra also nowadays [
5,
6,
7]. But this is not the unique case when the name “fundamental” is given to an equivalence defined on a hyperstructure. Indeed, there exist three other equivalences, called
fundamental by Jantosciak [
8], who observed that, unlike what happens in classical algebraic structures, two elements can play interchangeable roles with respect to a hyperoperation. In other words, the hyperoperation does not distinguish between the action of the given elements. These particular roles have been translated by the help of the following three equivalences. We say that two elements
x and
y in a hypergroup
are:
operationally equivalent, if their hyperproducts with all elements in H are the same;
inseparable, if x belongs to the same hyperproducts as y belongs to;
essentially indistinguishable, if they are both operationally equivalent and inseparable.
Then a hypergroup is called
reduced [
8] if the equivalence class of any element with respect to the essentially indistinguishable relation is a singleton. Besides the associated quotient structure (with respect to the same fundamental relation) is a reduced hypergroup, called the
reduced form of the original hypergroup. Jantosciak explained the role of these fundamental relations by a very simple and well known result. Define on the set
, where
is the set of integers and
the equivalence∼that assigns equivalent fractions in the same class:
if and only if
for
Endow
H with a hypercompositional structure, considering the hyperproduct
where by
we mean the equivalence class of the element
x with respect to the equivalence ∼. It follows that the equivalence class of the element
with respect to all three fundamental relations defined above is equal to the equivalence class of
with respect to the equivalence
Therefore
H is not a reduced hypergroup, while its reduced form is isomorphic with
the set of rationals.
Motivated by this example, in the same article [
8], Jantosciak proposed a method to obtain all hypergroups having a given reduced hypergroup as their reduced form. This aspect of reducibility of hypergroups has been later on considered by Cristea et al. [
9,
10,
11], obtaining necessary and sufficient condition for a hypergroup associated with a binary or
ary relation to be reduced. The same author studied the regularity aspect of these fundamental relations and started the study of fuzzyfication of the concept of reducibility [
12]. This study can be considered in two different directions, corresponding to the two different approaches of the theory of hypergroups associated with fuzzy sets. An overview of this theory is covered by the monograph [
13] written by Davvaz and Cristea, the only one on this topic published till now. The fuzzy aspect of reducibility could be investigated in two ways: by studying the indistinguishability between the elements of a fuzzy hypergroup (i.e., a structure endowed with a fuzzy hyperoperation) or by studying the indistinguishability between the images of the elements of a crisp hypergroup through a fuzzy set. By a crisp hypergroup we mean a hypergroup, but the attribute “crisp” is used to emphasize that the structure is not fuzzy. The study conducted in this article follows the second direction, while the first direction will be developed in our future works.
The aim of this note is to study the concept of
fuzzy reduced hypergroup as a crisp hypergroup which is fuzzy reduced with respect to the associated fuzzy set. One of the most known fuzzy sets associated with a hypergroup is the grade fuzzy set
introduced by Corsini [
14]. It was studied by Corsini and Cristea [
15] in order to define the
fuzzy grade of a hypergroup as the length of the sequence of join spaces and fuzzy sets associated with the given hypergroup. For any element
x in a hypergroup
the value
is defined as the average value of the reciprocals of the sizes of all hyperproducts containing
x. The properties of this particular fuzzy set, in particular those related to the fuzzy grade, have been investigated for several classes of finite hypergroups, as: complete hypergroups, non-complete 1-hypergroups or i.p.s. hypergroups (i.e., hypergroups with partial scalar identities). Inspired by all these studies, first we introduce the definition of fuzzy reduced hypergroups and present some combinatorial aspects related to them. Then we focus on the fuzzy reducibility of i.p.s. hypergroups, complete hypergroups and non-complete 1-hypergroups with respect to the grade fuzzy set
Theorem 3 states that any proper complete hypergroup is not reduced, either fuzzy reduced with respect to
. Regarding the i.p.s. hypergroups, we show that they are reduced, but not fuzzy reduced with respect to
(see Theorems 4 and 5). Finally, we present a general method to construct a non-complete 1-hypergroup, that is not reduced either fuzzy reduced with respect to
. Some conclusions and new research ideas concerning this study are gathered in the last section.
2. Review of Reduced Hypergroups
In this section we briefly recall the basic definitions which will be used in the following, as well as the main properties of reduced hypergroups. We start with the three fundamental relations defined by Jantosciak [
8] on an arbitrary hypergroup. Throughout this note, by a hypergroup
, we mean a non-empty set
H endowed with a hyperoperation, usually denoted as
satisfying the associativity, i.e., for all
there is
and the reproduction axiom, i.e., for all
there is
where
denotes the set of all non-empty subsets of
Definition 1 ([
8])
. Two elements in a hypergroup are called:- 1.
operationally equivalent or by short o-equivalent, and write , if , and , for any ;
- 2.
inseparable or by short i-equivalent, and write , if, for all , ;
- 3.
essentially indistinguishable or by short e-equivalent, and write , if they are operationally equivalent and inseparable.
Definition 2 ([
8])
. A reduced hypergroup has the equivalence class of any element with respect to the essentially indistinguishable relation a singleton, i.e., for any , there is . Example 1. Let be a hypergroup, where the hyperoperation “∘” is defined by the following table.∘ | a | b | c | d |
a | a | a | | |
b | a | a | | |
c | | | | |
d | | | | |
One notices that because the lines (and columns) corresponding to a and b are exactly the same, thereby: while and But, on the other side, the equivalence class of any element in H with respect to the relation is a singleton, as well as with respect to the relation by consequence is reduced.
In [
11] Cristea et al. discussed about the regularity of these fundamental relations, proving that in general none of them is strongly regular. This means that the corresponding quotients modulo these equivalences are not classical structures, but hypergroups. Moreover, Jantosciak [
8] established the following result.
Theorem 1 ([
8])
. For any hypergroup H, the associated quotient hypergroup is a reduced hypergroup, called the reduced form of Motivated by this property, Jantosciak concluded that the study of the hypergroups can be divided into two parts: the study of the reduced hypergroups and the study of all hypergroups having the same reduced form [
8].
3. Fuzzy Reduced Hypergroups
As already mentioned in the introductory part of this article, the extension of the concept of reducibility to the fuzzy case can be performed on a crisp hypergroup endowed with a fuzzy set, by defining, similarly to the classical case, three equivalences as follows.
Definition 3. Let be a crisp hypergroup endowed with a fuzzy set μ. For two elements , we say that
- 1.
x and y are fuzzy operationally equivalent and write if, for any , and ;
- 2.
x and y are fuzzy inseparable and write if , for ;
- 3.
x and y are fuzzy essentially indistinguishable and write , if they are fuzzy operationally equivalent and fuzzy inseparable.
Definition 4. The crisp hypergroup is called fuzzy reduced if the equivalence class of any element in H with respect to the fuzzy essentially indistinguishable relation is a singleton, i.e., Notice that the notion of fuzzy reducibility of a hypergroup is strictly connected with the definition of the involved fuzzy set.
Remark 1. It is easy to see that, for each hypergroup H endowed with an arbitrary fuzzy set the following implication holds: for any , Remark 2. First of all, let us better explain the meaning of for any and any arbitrary fuzzy set μ defined on Generally, is a subset of so is the direct image of this subset through the fuzzy set i.e., .
From here, two things need to be stressed on. Firstly, if is a singleton, i.e., , then is a set containing the real number . Therefore we can write , but it is not correct writing , because the first member is a real number, while the second one is a set containing the real number .
Secondly, if then, clearly , but not also viceversa because it could happen that for .
Generally, as illustrated in the next example. Indeed, if then if and only if . But it can happen that with , so also . And if , then , thus .
Finally, it is simple to see the implication
The following example illustrates all the issues in the above mentioned remark.
Example 2 ([
16])
. Let be a hypergroup represented by the following commutative Cayley table:∘ | e | | | |
e | e | | | |
| | | e | e |
| | | | |
| | | | |
One notices immediately that while - (a)
Define now on H the fuzzy set μ as follows: Since it follows that Moreover, since we havewhile it is clear that so - (b)
If we define on H the fuzzy set μ by taking it follows that and
Once again, it is evident that the three equivalences , and are strictly related with the definition of the fuzzy set considered on the hypergroup.
Now we will present an example of an infinite hypergroup and study its fuzzy reducibility.
Example 3. Consider the partially ordered group with the usual addition and orderings of integers. Define on the hyperoperation . Then is a hypergroup [17]. Define now on the fuzzy set μ as follows: and , for any . We obtain , for any , therefore is fuzzy reduced with respect to μ. Indeed, for two arbitrary elements x and y in , we have if and only if , for any , where and similarly, . Since a is an arbitrary integer, for any x and y we always find a suitable integer a such that and . This means that the sets and contain descending sequences of positive integers, so they are equal only when . Therefore . In the following, we will study the fuzzy reducibility of some particular types of finite hypergroups, with respect to the
grade fuzzy set defined by Corsini [
14]. We recall here its definition. With any crisp hypergroupoid
(not necessarily a hypergroup) we may associate the fuzzy set
considering, for any
where
and
. By convention, we take
anytime when
In other words, the value
is the average value of reciprocals of the sizes of all hyperproducts
containing the element
u in
In addition, sometimes when we will refer to formula (
1), we will denote its numerator by
, while the denominator is already denoted by
.
Remark 3. As already explained in Remark 2 (ii), generally, for an arbitrary fuzzy set, , while the implication holds if we consider the grade fuzzy set . Indeed, if , then if and only if and therefore , implying that , and moreover . This leads to the equality . By consequence, based on Remark 2 (iii), it holds , with respect to .
Example 4. Let us consider now a total finite hypergroup H, i.e., , for all . It is easy to see that for any meaning that for any Thus, a total hypergroup is not reduced. What can we say about the fuzzy reducibility with respect to the grade fuzzy set
Since, for any , it follows that , for any Then, it is clear that , for all , implying that , for all Concluding, it follows that any total finite hypergroup is neither reduced, nor fuzzy reduced.
Based now on Remarks 1 and 3, the following assertion is clear.
Corollary 1. If is a not reduced hypergoup, then it is also not fuzzy reduced with respect to the grade fuzzy set .
3.1. Fuzzy Reducibility in Complete Hypergroups
The complete hypergroups form a particular class of hypergroups, strictly related with the join spaces and the regular hypergroups. Their definition is based on the notion of
complete part, introduced in Koskas [
1], with the main role to characterize the equivalence class of an element under the relation
More exactly, a nonempty set
A of a semihypergroup
is called a complete part of
if for any natural number
n and any elements
in
the following implication holds:
We may say, as it was mentioned in the review written by Antampoufis et al. [
18], that a complete part
A absorbs all hyperproducts of the elements of
H having non-empty intersection with
The intersection of all complete parts of
H containing the subset
A is called the
complete closure of
A in
H and denoted by
Definition 5. A hypergroup is called a complete hypergroup if, for any there is
As already explained in the fundamental book on hypergroups theory [
19] and the other manuscripts related with complete hypergroups [
16,
20], in practice, it is more useful to use the following characterization of the complete hypergroups.
Theorem 2 ([
19])
. Any complete hypergroup may be constructed as the union of its subsets, where- (1)
is a group.
- (2)
The family is a partition of i.e., for any , , there is
- (3)
If , then
Example 5. The hypergroup presented in Example 2 is complete, where the group and the partition set contains , , and . It is clear that all conditions in Theorem 2 are fulfilled.
Let
be a proper complete hypergroup (i.e.,
H is not a group). Define now on
H the equivalence “∼” by:
Proposition 1. On a proper complete hypergroup the equivalence∼in (2) is a representation of the essentially indistinguishability equivalence Proof. By Theorem 2, one notices that, for any element there exists a unique such that . In the following, we will denote this element by . First, suppose that i.e., there exists such that For any arbitrary element we can say that with and by the definition of the hyperproduct in the complete hypergroup there is (and similarly, ) implying that (i.e., x and y are operationally equivalent). Secondly, for any it follows that but so Thereby, if and only if meaning that (i.e., x and y are inseparable). We have proved that
Conversely, let us suppose that Since x and y are inseparable, i.e., if and only if we may write Therefore there exists such that so
Example 6. If we continue with Example 2, we notice that the equivalence classes of the elements of H with respect to the equivalence∼defined in (2) are: , , . Regarding now the essentially indistinguishability equivalence , we have the same equivalences classes: , , . This is clear from the Cayley table of the hypergroup: the lines corresponding to and are the same and every time and are in the same hyperproduct, so they are equivalent. Since these properties are not satisfied for and e, their equivalence classes are singletons. Proposition 2. Let be a proper complete hypergroup and consider on H the grade fuzzy set . Then (with respect to the fuzzy set ).
Proof. By the definition of the grade fuzzy set
one obtains that
Take now such that There exists such that thereby and by Remark 2 (ii) we have Moreover, by Proposition 1 there is and by Remark 1 we get that Concluding, we have proved that with respect to
Theorem 3. Any proper complete hypergroup is not reduced, either fuzzy reduced with respect to the grade fuzzy set.
Proof. Since is a proper complete hypergroup, there exists at least one element g in G such that i.e., there exist two distinct elements a and b in H with the property that Then and meaning that is not reduced, either fuzzy reduced with respect to the grade fuzzy set
3.2. Fuzzy Reducibility in i.p.s. Hypergroups
An i.p.s. hypergroup is a canonical hypergroup with partial scalar identities. The name, given by Corsini [
21], originally comes from Italian, the abbreviation “i.p.s.” standing for “identità parziale scalare” (in English, partial scalar identity). First we dwell on the terminology connected with the notion of identity in a hypergroup
. An element
is called a
scalar, if
, for any
An element
is called
partial identity of
H if it is a
left identity (i.e., there exists
such that
) or a
right identity (i.e., there exists
such that
). We denote by
the set of all partial identities of
Besides, for a given element
, a
partial identity of x is an element
such that
. The element
is a
partial scalar identity of x whenever from
it follows
and from
it follows that
. We denote by
the set of all partial identities of
x, by
the set of all partial scalar identities of
x, and by
the set of all scalars of
It is obvious that
Remark 4. The term “partial” here must not be confused with the “left or right” (identity), but it must be connected with the fact that the element u has a partial behaviour of identity with respect to the element So, u is not a left/right (i.e., partial) identity for the hypergroup Besides, an i.p.s. hypergroup is a commutative hypergroup, so the concept of partial intended as left/right element satisfying a property (i.e., left/right unit) has no sense. It is probably better to understand an element u having the property of being partial identity for x as an element having a similar behaviour as an identity but only with respect to so a partial role of being identity.
Let us recall now the definition of an i.p.s. hypergroup. All finite i.p.s. hypergroups of order less than 9 have been determined by Corsini [
21,
22,
23].
Definition 6. A hypergroup is called i.p.s. hypergroup, if it satisfies the following conditions.
- 1.
It is canonical, i.e.
it is commutative;
it has a scalar identity 0 such that , for any ;
every element has a unique inverse , that is ;
it is reversible, so , for any .
- 2.
It satisfies the relation: for any , if , then .
The most useful properties of i.p.s. hypergroups are gathered in the following result.
Proposition 3 ([
21])
. Let be an i.p.s. hypergroup.- 1.
For any element x in H, the set is a subhypergroup of H.
- 2.
For any element x in H, different from , we have: or x is a scalar of H, or there exists such that . Moreover .
- 3.
If x is a scalar of H, then the set of all partial scalar identities of x contains just 0.
If x is not a scalar of H, then and therefore .
Proposition 4. Let be an i.p.s. hypergroup. For any scalar and for any element , there exists a unique such that .
Proof. The existence part immediately follows from the reproducibility property of H. For proving the unicity, assume that there exist such that . Then, by reversibility, it follows that Since u is a scalar element, we get and then .
Example 7 ([
21])
. Let us consider the following i.p.s. hypergroup .H | 0 | 1 | 2 | 3 |
0 | 0 | 1 | 2 | 3 |
1 | 1 | 2 | | 1 |
2 | 2 | | 1 | 2 |
3 | 3 | 1 | 2 | 0 |
Here we can notice that 0 is the only one identity of In addition, and only for so Also, only for there is thus (In general, if then , according with Proposition 3). Similarly, one gets and since it follows that
Note that, in an i.p.s. hypergroup, the Jantosciak fundamental relations have a particular meaning, in the sense that, for any two elements there is
By consequence, one obtains the following result.
Theorem 4. Any i.p.s. hypergroup is reduced.
In the following we will discuss the fuzzy reducibility of an i.p.s. hypergroup with respect to the grade fuzzy set
Theorem 5. Any i.p.s. hypergroup is not fuzzy reduced with respect to the fuzzy set
Proof. Since any i.p.s. hypergroup contains at least one non-zero scalar, take arbitrary such a . We will prove that and therefore meaning that H is not fuzzy reduced.
First we will prove that, for any , there is , equivalently with For doing this, based on the fact that , for all , we show that and
Let us start with the computation of and If it follows that that is and then Thereby On the other hand, by Proposition 4, we have (since for any there exists a unique such that )
Let us calculate now
By formula (
1), we get that
Since, for
such that
we similarly get that
and it is clear that
so
Therefore
It remains to prove the second part of the theorem, that is equivalently with
If , then and by the first part of the theorem, there is
If then since for any , it follows that so and then Now the proof is complete.
3.3. Fuzzy Reducibility in Non-Complete 1-Hypergroups
In this subsection we will study the reducibility and fuzzy reducibility of some particular finite non-complete 1-hypergroups defined and investigated by Corsini, Cristea [
24] with respect to their fuzzy grade. A hypergroup
H is called
1-hypergroup if the cardinality of its heart
is 1.
The general construction of this particular hypergroup is the following one. Consider the set , where and with and , such that and . Define on H the hyperoperation “∘” by the following rule:
for all
for all such that , set
for all , set
for all , there is
for all , set
for all and
is an 1-hypergroup which is not complete.
We will discuss the (fuzzy) reducibility of this hypergroup for different cardinalities of the sets A and B.
Let us suppose
, where
,
,
with
,
Thus the Cayley table of
is the following one
H | e | | | | | |
e | e | A | A | B | B | B |
| | | B | e | e | e |
| | | | e | e | e |
| | | | A | A | A |
| | | | | A | A |
| | | | | | A |
From the table, we notice immediately that the elements and are essentially indistinguishable, while the equivalence class with respect to the relation of all other elements is a singleton. Thereby H is not reduced.
Calculating now the values of the grade fuzzy set , one obtains Since , it follows that . But , while , so , meaning that that is .
On the other side, we have because they are also operationally equivalent, and because . This is equivalent with and therefore H is not reduced with respect to .
Consider now the most general case. The Cayley table of the hypergroup
H is the following one:
H | e | | | ⋯ | | | | ⋯ | |
e | e | A | A | ⋯ | A | B | B | ⋯ | B |
| | | B | ⋯ | B | e | e | ⋯ | e |
| | | | ⋯ | B | e | e | ⋯ | e |
⋮ | | | | ⋱ | | | | | ⋮ |
| | | | | | e | e | ⋯ | e |
| | | | | | A | A | ⋯ | A |
| | | | | | | A | ⋯ | A |
⋮ | | | | | | | | ⋱ | ⋮ |
| | | | | | | | | A |
As already calculated in [
24], there is
,
, for any
, while
, and
, for any
. As in the previous case, we see that any two elements in
are operational equivalent, by consequence also fuzzy operational equivalent, and indistinguishable and fuzzy indistinguishable (because their values under the grade fuzzy set
are the same). Concluding, this non complete 1-hypergroup is always not reduced, either not fuzzy reduced with respect to
.