RATIO MATHEMATICA 23 (2012), 21–38 ISSN: 1592-7415

(Intuitionistic) Fuzzy Grade of a
Hypergroupoid:

A Survey of Some Recent Researches

Irina Cristea
Centre for Systems and Information Technologies

University of Nova Gorica, Slovenia

irinacri@yahoo.co.uk

Abstract

This paper aims to present a short survey on two numerical func-
tions determined by a hypergroupoid, called the fuzzy grade and the
intuitionistic fuzzy grade of a hypergroupoid. It starts with the main
construction of the sequences of join spaces and (intuitionistic) fuzzy
sets associated with a hypergroupoid. After some computations of
the above grades, we discuss some similarities and differences between
the two grades for the complete hypergroups and for the i.p.s. hyper-
groups. We conclude with some open problems.

Key words: (Intuitionistic) Fuzzy Grade, Join Space, I.p.s. Hy-
pergroup, Complete Hypergroup.

MSC2012: 20N20; 03E72.

1 Introduction

The study of hyperstructures connected with fuzzy sets represents a grow-
ing and new line of research spanning over the last twenty years. Till now
one can distinguish three principal approaches of this theme: the study of
new crisp hyperoperations obtained by means of fuzzy sets; the study of fuzzy
subhyperstructures (i.e. fuzzy sets those level sets are crisp hyperstructures);
the study of structures endowed with fuzzy hyperoperations and called fuzzy
hyperstructures.



Irina Cristea

We concentrate here on the first line of reserach. Its origin is due to
Corsini [12] in 1993, when he defined a join space from a nonempty set en-
dowed with a fuzzy set. Another definition of a join space induced by a
fuzzy set was given in 1997 by Ameri and Zahedi [1]. The Corsini’s direc-
tion was then investigated by Corsini and Leoreanu [16]. Ten years later,
the same Corsini [13] considered the inverse problem: to define a fuzzy set
from a hypergroupoid. Iterating both constructions he obtained a sequence
of join spaces and fuzzy sets that stops when there exist two consecutive
non-isomorphic join spaces. The length of the sequence, i.e. the number
of the non-isomorphic join spaces in it, was called by Corsini and Cristea
[14, 15] the fuzzy grade of a hypergroupoid. At the beginning the sequence
was studied for some particular hypergroups: the complete hypergroups and
the i.p.s. hypergroups. Recently this grade has been investigated with a gen-
eral and innovative method by Cristea [22, 23], Ştefănescu and Cristea [30],
Angheluţă and Cristea [2, 3] making use of an ordered n-tuple determined
by an equivalence relation. We will see the details in the next section.

One of the principal concerns of several researchers is that to obtain new
hyperstructures using modern tools for investigation of uncertain problems,
like intuitionistic fuzzy sets, rough sets, soft sets, interval-valued fuzzy sets.
Following Corsini’s idea, Cristea and Davvaz [24] associated with any hy-
pergroupoid a sequence of join spaces and intuitionistic fuzzy sets with the
length called the intuitionistic fuzzy grade. Although the fuzzy grade and
the intuitionistic fuzzy grade have similar definitions, the corresponding as-
sociated sequences of join spaces have different properties. In order to high-
light the differences between them, Davvaz, Hassani-Sadrabadi and Cristea
[25, 26, 27] have investigated this problem for the complete hypergroups and
i.p.s. hypergroups of order less than 8. In this survey our main goal is to
realize a comparison between the two sequences.

The second section of the paper gives a survey of the construction of the
sequences of join spaces associated with a hypergroupoid or with a non-empty
set endowed with an (intuitionistic) fuzzy set. Section 3 is dedicated to some
computations of the two grades: the fuzzy and the intuitionistic fuzzy grade
of a hypergroupoid. In Section 4 we focus our comparison between the two
grades on the cases of the complete hypergroups and of the i.p.s. hypergroups
of order less than 8. In the last section we conclude with the statement of
some open problems.

22



Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

2 Sequences of the join spaces associated with

a hypergroupoid

We briefly recall the construction of the sequences of join spaces and
fuzzy sets/intuitionistic fuzzy sets associated with a hypergroupoid. For
a comprehensive overview of hypergroups theory the reader is refereed to
[11, 17, 31] and for fuzzy and intuitionistic fuzzy sets theory see [32, 5, 6, 7, 8].
We will use the terminology and the notations from [13, 23, 30].

Let X be a nonempty set. A fuzzy set A in X is characterized by a
membership function µA : X −→ [0, 1], where for any x ∈ X, the value µA(x)
represents the grade of membership of x in A. More generally, any function
µ : X −→ [0, 1] is called a fuzzy subset of X. We denote by FS(X) the set
of all fuzzy subsets of X.

An Atanassov’s intuitionistic fuzzy set A in X is a triplet of the form
A = {(x,µA(x),λA(x)) | x ∈ X}, where, for any x ∈ X, the degree of
membership of x (namely µA(x)) and the degree of non-membership of x
(namely λA(x)) verify the relation 0 ≤ µA(x) + λA(x) ≤ 1; for simplicity, we
denote it by A = (µ,λ). We denote by IFS(X) the set of all intuitionistic
fuzzy subsets of X.

For any nonempty set H endowed with a fuzzy subset α ∈ FS(H),
Corsini[12] defined a join space (H,◦α), where the hyperproduct is defined
as it follows:

x◦α y = {z ∈ H | α(x) ∧α(y) ≤ α(z) ≤ α(x) ∨α(y)} (1)

Conversely, from any hypergroupoid (H,◦) Corsini[13] defined a fuzzy
subset µ̃ ∈ FS(H) by the following formula:

µ̃(u) =

∑
(x,y)∈Q(u)

1

|x◦y|

q(u)
(2)

where Q(u) = {(a,b) ∈ H2 | u ∈ a◦ b}, q(u) = |Q(u)|.

If Q(u) = ∅, we set µ̃(u) = 0.
In 2010 Cristea and Davvaz[24] defined in a similar way an intuitionistic

23



Irina Cristea

fuzzy subset associated with a hypergroupoid (H,◦):

µ(u) =

∑
(x,y)∈Q(u)

1

|x◦y|

n2
,

λ(u) =

∑
(x,y)∈Q(u)

1

|x◦y|

n2

(3)

Iterating equations (1) and (2) one obtains a sequence (iH = (H,◦i), µ̃i−1)i≥1
of join spaces and fuzzy sets associated with H. The sequence stops when
there exist two consecutive isomorphic join spaces. The length of this se-
quence, that is the number of the non-isomorphic join spaces in the sequence,
is called the fuzzy grade of the hypergroupoid H. More exactly we have the
following definition.

Definition 2.1. [14] A hypergroupoid H has the fuzzy grade m ∈ N∗, and
we write f.g.(H) = m if, for any 1 ≤ i < m, the join spaces iH and i+1H
associated with H are not isomorphic and, for any s > m, sH is isomorphic
with mH.
We say that the hypergroupoid H has the strong fuzzy grade m, and we write
s.f.g.(H) = m, if f.g.(H) = m and, for all s,s > m, sH = mH.

Similarly, iterating equations (1) and (3) one obtains two sequences (iH =
(H,◦µi∧λi ); Ai = (µi,λi))i≥0 and (

iH = (H,◦µi∨λi ); Ai = (µi,λi))i≥0 of join
spaces and intuitionistic fuzzy sets associated with H. The sequences stop
when there exist two consecutive isomorphic join spaces and their lengths
are called the lower/upper intuitionistic fuzzy grade of the hypergroupoid H.
Let’s see better their definitions.

Definition 2.2. [24] A set H endowed with an intuitionistic fuzzy set A =
(µ,λ) has the lower intuitionistic fuzzy grade m and we write l.i.f.g.(H) = m
if, for any 0 ≤ i < m, the join spaces (H,◦µi∧λi ) and (H,◦µi+1∧λi+1 ) associated
with H are not isomorphic and, for any s ≥ m, sH is isomorphic with m−1H.

Definition 2.3. [24] A set H endowed with an intuitionistic fuzzy set A =
(µ,λ) has the upper intuitionistic fuzzy grade m and we write u.i.f.g.(H) = m
if, for any 0 ≤ i < m, the join spaces (H,◦µi∨λi ) and (H,◦µi+1∨λi+1 ) associated
with H are not isomorphic and, for any s ≥ m, sH is isomorphic with m−1H.

The construction of the above sequences can start in two different ways:

1. from a hypergroupoid (H,◦);

24



Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

2. from a nonempty set H endowed with a fuzzy set or an intuitionistic
fuzzy set.

It is easy to prove that in both cases one obtains the same fuzzy grade.
But what can we say about the other two sequences of join spaces and intu-
itionistic fuzzy sets?

If we start the construction from a hypergroupoid (H,◦), we obtain only
one sequence of join spaces, because the first join spaces 0H = (H,◦µ∧λ) and
0H = (H,◦µ∨λ) are always isomorphic. Indeed, since, for any x ∈ H, the
sum µ(x) + λ(x) is always constant, it follows that µ(x) = µ(y) if and only
if λ(x) = λ(y).
Therefore in this case l.i.f.g.(H) = u.i.f.g.(H) and it is simply called the
intuitionistic fuzzy grade of H (shortly i.f.g.(H)).

On the other side, if we start the construction from a nonempty set H
endowed with an intuitionistic fuzzy set, one may obtain two different se-
quences with different lengths. We only illustrate this case by the following
example.

Example 2.1. [24] On H = {a,b,c,d} we define the Atanassov’s intuition-
istic fuzzy set:

µ(a) = 0.25 µ(b) = 0.25 µ(c) = 0.30 µ(d) = 0.10

λ(a) = 0.40 λ(b) = 0.40 λ(c) = 0.50 λ(d) = 0.90.

To start with, we construct the first sequence of join spaces and we determine
its l.i.f.g.(H).

For the associated join space

◦µ∧λ a b c d
a {a,b} {a,b} {a,b,c} {a,b,d}
b {a,b} {a,b} {a,b,c} {a,b,d}
c {a,b,c} {a,b,c} c H
d {a,b,d} {a,b,d} H d

we calculate the Atanassov’s intuitionistic fuzzy set associated with H as in
(3) and we obtain the following values:

µ1(a) = 31/96 µ1(b) = 31/96 µ1(c) = 17/96 µ1(d) = 17/96

λ1(a) = 12/96 λ1(b) = 12/96 λ1(c) = 26/96 λ1(d) = 26/96,

therefore µ1 ∧ λ1(a) = µ1 ∧ λ1(b) < µ1 ∧ λ1(c) = µ1 ∧ λ1(d) and thereby it

25



Irina Cristea

results the following join space

◦µ1∧λ1 a b c d
a {a,b} {a,b} H H
b {a,b} {a,b} H H
c H H {c,d} {c,d}
d H H {c,d} {c,d}

For any x ∈ H, we compute µ2(x) = 4/16 and λ2(x) = 2/16; thus, for any
x,y ∈ H, x◦µ2∧λ2 y = H. So the first sequence of join spaces associated with
H has 3 elements, that is l.i.f.g.(H) = 3.

Now, in order to determine the u.i.f.g.(H), we start with the join space

◦µ∨λ a b c d
a {a,b} {a,b} {a,b,c} H
b {a,b} {a,b} {a,b,c} H
c {a,b,c} {a,b,c} c {c,d}
d H H {c,d} d

Note that 〈0H,◦µ∧λ〉 and 〈
0H,◦µ∨λ〉 are not isomorphic. Since

µ1(a) = 13/48 µ1(b) = 13/48 µ1(c) = 13/48 µ1(d) = 9/48

λ1(a) = 9/48 λ1(b) = 9/48 λ1(c) = 9/48 λ(d) = 13/48

it follows that µ1 ∨λ1(x) = 13/48, whenever x ∈ H, which means that, for
any x,y ∈ H, x◦µ1∨λ1 y = H. Therefore the second sequence of join spaces
associated with H has 2 elements, that is u.i.f.g.(H) = 2.

Sometimes the computations of the values of the membership functions
µ̃i,µi,λi, i ≥ 0 could take much time, thus it is useful to determine the
(intuitionistic) fuzzy grade of a hypergroupoid without calculating all these
values. In order to realize this we introduce some notations.

With any join space iH in the sequence of join spaces and fuzzy sets
corresponding to H, one may associate an ordered chain (iC1,

iC2, ...,
iCr)

and an ordered r-tuple (ik1,
ik2, ...,

ikr), where

1. ∀j ≥ 1: x,y ∈iCj ⇐⇒ µ̃i−1(x) = µ̃i−1(y),

2. for x ∈iCj and z ∈iCk, if j < k then µ̃i−1(x) < µ̃i−1(z)

3. ikj = |iCj|, for all j.

We have similar notations for the sequence (iH = (H,◦µi∧λi ); Ai = (µi,λi))i≥0.
With any join space iH one may associate an ordered chain (

iC1,
iC2, ...,

iCr)
and an ordered r-tuple (ik1,

ik2, ...,
ikr), where

26



Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

1. ∀j ≥ 1: x,y ∈iCj ⇐⇒ µi ∧λi(x) = µi ∧λi(y),

2. for x ∈iCj and z ∈iCk, if j < k then µi ∧λi(x) < µi ∧λi(z)

3. ikj = |iCj|, for all j.

One of the crucial questions we have to ask is: When two consecutive
join spaces in these sequences are isomorphic?

A first answer was given by Corsini and Leoreanu [16] in 1995.

Theorem 2.1. [16] Let iH and i+1H be two consecutive join spaces associated
with H determined by the membership functions µ̃i−1 and µ̃i, where

iH =
r1⋃
l=1

Cl,
i+1H =

r2⋃
l=1

C′l and (k1,k2, ...,kr1 ) is the r1-tuple associated with
iH,

(k′1,k
′
2, ...,k

′
r2

) is the r2-tuple associated with
i+1H.

The join spaces iH and i+1H are isomorphic if and only if r1 = r2 and
(k1, . . . ,kr1 ) = (k

′
1, . . . ,k

′
r1

) or (k1, . . . ,kr1 ) = (k
′
r1
, . . . ,k′1).

A similar theorem can be formulated also for the join spaces in the se-
quence (iH = (H,◦µi∧λi ); Ai = (µi,λi))i≥0.

In order to use this result we need to determine both r-tuples associated
with iH and i+1H. But Ştefănescu and Cristea [30] have given a sufficient
condition such that two consecutive join spaces are not isomorphic, that is
the sequence doesn’t stop, using only one r-tuple.

Theorem 2.2. [30] Let (k1,k2, ...,kr) be the r-tuple associated with the join
space iH. If (k1, . . . ,kr) = (kr,kr−1, . . . ,k1), then the join spaces

iH and
i+1H are not isomorphic.

A second crucial problem arises: does there exist a hypergroupoid
H such that s.f.g.(H)/i.f.g.(H) = n, whenever n is a natural number?
The anser is yes, and an example is given in the following fundamental result.

Theorem 2.3. [30, 24] Let H = {x1,x2, . . . ,xs}, where s = 2n, n ∈ N∗ \
{1, 2}, be the commutative hypergroupoid defined by the hyperproduct

xi ◦xi = xi, 1 ≤ i ≤ n,
xi ◦xj = {xi,xi+1, . . . ,xj}, 1 ≤ i < j ≤ n.

Then s.f.g.(H) = i.f.g.(H) = n. Moreover, H is a join space.

27



Irina Cristea

3 Computation of the fuzzy grade/intuitionistic

fuzzy grade of a hypergroupoid

Let (H,◦) be a hypergroupoid. We define the following equivalence on H

xReµy ⇐⇒ µ̃(x) = µ̃(y).
We determine f.g.(H) when |H/Reµ| ∈ {2, 3}.

Theorem 3.1. [2] Let (H,◦) be a finite hypergroupoid.

1. If |H/Reµ| = 2, that is (k1,k2) is the pair associated with H, then
s.f.g.(H) = 2 whenever k1 = k2, and f.g.(H) = 1 otherwise.

2. If |H/Reµ| = 3, that is (k1,k2,k3) is the triple associated with H, and
(a) if k1 = k2 = k3, then f.g.(H) = 2

(b) if k1 = k3 6= k2, then f.g.(H) = 2 whenever k2 6= 2k3 and
s.f.g.(H) = 3, otherwise

(c) if k1 < k2 = k3, then f.g.(H) = 1

(d) if k1 = k2 < k3, then f.g.(H) = 1 whenever P = 2k
3
3 − 8k31 −

k21k3 + 5k1k
2
3 > 0 and f.g.(H) = 3, otherwise

(e) if k1 6= k2 6= k3, there is no precise order between µ̃1(x), µ̃1(y), µ̃1(z).

A similar result we obtain for the quotient H/Rµ∧λ.

Theorem 3.2. [3] Let (H,◦) be a finite hypergroupoid.

1. If |H/Rµ∧λ| = 2, that is (k1,k2) is the pair associated with H, then
i.f.g.(H) = 2 whenever k1 = k2, and i.f.g.(H) = 1, otherwise.

2. If |H/Rµ∧λ| = 3, that is (k1,k2,k3) is the triple associated with H, and

(a) if k1 = k2 = k3, then i.f.g.(H) = 2

(b) if k1 = k3 6= k2, then i.f.g.(H) = 2 whenever k2 6= 2k3 and
i.f.g.(H) = 3, otherwise

(c) if k1 < k2 = k3, then i.f.g.(H) = 1 whenever 2k
2
2 > 3k1k2 + 3k

2
1

and i.f.g.(H) = 3, otherwise

(d) if k1 = k2 < k3, then i.f.g.(H) = 1 whenever k3 6= 2k1 and
i.f.g.(H) = 2, otherwise

(e) if k1 6= k2 6= k3, then there is no precise order between µ1(x) ∧
λ1(x),µ1(y) ∧λ1(y),µ1(z) ∧λ1(z).

28



Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

Angheluţă and Cristea [3] have studied the intuitionistic fuzzy grade of a
hypergroupoid in the case of some particular ternaries. We integrate here this
theorem with similar results for the fuzzy grade. In the following the ternary
(k1,k2,k3) associated with H is intended respected with the equivalence Reµ
(when we talk about the fuzzy grade) and Rµ∧λ, respectively, (when we talk
about the intuitionistic fuzzy grade).

Proposition 3.1. Let H be a hypergroupoid of cardinality 2k, k ≥ 3, with
the ternary associated with H of the type (k,k − 1, 1). Then s.f.g.(H) =
i.f.g.(H) = 1.

Generalizing, we obtain the same result for the ternary (pk,p(k−1),p), with
p,k ∈ N \{0, 1}.

Proof. We prove only that s.f.g.(H) = 1. For the proof of the fact that
i.f.g.(H) = 1, see [3].

Let H be a hypergroupoid such that |H| = 2k, k ≥ 3, and (k,k−1, 1) be
the ternary associated with H determined by the equivalence Reµ. Thus H can
be written as the union H = C1∪C2∪C3, with |C1| = k, |C2| = k−1, |C3| = 1.
Using equation (2), for any x ∈ C1,y ∈ C2,z ∈ C3, one obtains

µ̃1(x) =
4k2 −k − 1
3k2(2k − 1)

µ̃1(y) =
8k3 + 2k2 − 12k + 4
2k(2k − 1)(3k2 − 1)

µ̃1(z) =
4k − 2

k(4k − 1)

After some computations that we omit here for lack of space, it results
the following relation

µ̃1(x) ≤ µ̃1(y) ≤ µ̃1(z),

which means that (k,k − 1, 1) is the ternary associated with the join space
1H and thereby s.f.g.(H) = 1.

Proposition 3.2. Let (2k,k, 1), with k ≥ 2, be the ternary associated with
a hypergroupoid H. Then

1. s.f.g.(H) = 1.

2. i.f.g.(H) = 3 and the sequence of join spaces is cyclic, that is 3lH '
3H, 3l+1H ' 1H, 3l+2H ' 2H, for any l ∈ N∗.

29



Irina Cristea

Proof. Considering (2k,k, 1), with k ≥ 2, be the ternary associated with a
hypergroupoid H respected with the equivalence Reµ, we decompose H as the
union H = C1 ∪ C2 ∪ C3, with |C1| = 2k, |C2| = k, |C3| = 1. Using again
equation (2), for any x ∈ C1,y ∈ C2,z ∈ C3, one obtains

µ̃1(x) =
15k + 11

6(2k + 1)(3k + 1)

µ̃1(y) =
21k2 + 58k + 25

3(k + 1)(3k + 1)(5k + 6)

µ̃1(z) =
13k2 + 10k + 1

(k + 1)(3k + 1)(6k + 1)

For any k ≥ 2, it follows that

µ̃1(x) ≤ µ̃1(y) ≤ µ̃1(z),

that is s.f.g.(H) = 1.

For the second part of the theorem see [3].

Proposition 3.3. Let (h,k, 1), with 2 ≤ h ≤ k, be the ternary associated
with a hypergroupoid H.

1. Then s.f.g.(H) = 1.

2. If h = 2 and

(a) k = 2, then i.f.g.(H) = 3;

(b) k = 3, then i.f.g.(H) = 2;

(c) k ≥ 4, then i.f.g.(H) = 3 and the sequence of join spaces associ-
ated with H is cyclic.

3. If h > 2, then i.f.g.(H) = 1, whenever k ≥ h.

Proof. Since (h,k, 1), with 2 ≤ h ≤ k, is the ternary associated with a
hypergroupoid H respected with the equivalence Reµ, we write H as the
union H = C1 ∪C2 ∪C3, with |C1| = h, |C2| = k, |C3| = 1. Calculating the
values of the membership function µ̃1 by using the formula (2), we obtain,

30



Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

for any x ∈ C1,y ∈ C2,z ∈ C3, that

µ̃1(x) =
h2 + 3k2 + 4hk + 3h + 5k

(h + k)(h + k + 1)(h + 2k + 2)

µ̃1(y) =
3h2k2 + 7h2k + 4hk3 + 9hk2 + 7hk + k4 + 4k3 + 3k2 + 2h2

(h + k)(h + k + 1)(k + 1)(2hk + 2h + k2 + 2k)

µ̃1(z) =
5hk + 3h + 3k2 + 4k + 1

(h + k + 1)(k + 1)(2h + 2k + 1)

Similarly as in the previous two lemmas, we prove that

µ̃1(x) ≤ µ̃1(y) ≤ µ̃1(z)

and thus s.f.g.(H) = 1.
For the second part of the theorem see [3].

4 (Intuitionistic) fuzzy grade of special types

of hypergroups

In this section we deal with two classes of hypergroups: the complete
hypergroups and the hypergroups with partial scalar identities, shortly called
i.p.s. hypergroups. We determine the fuzzy and intuitionistic fuzzy grades of
all these hypergroups of order less than 8. This part of the paper is based on
the articles of Corsini and Cristea [14, 15], Cristea [21], Cristea and Davvaz
[24], Angheluţă and Cristea [2, 3], Davvaz, Hassani-Sadrabadi and Cristea
[25, 26, 27].

4.1 The complete hypergroups

We start with the characterization of the complete hypergroups.
A hypergroup (H,◦) is a complete hypergroup if it can be written as the

union of its subsets H = ∪g∈GAg, where

1. (G, ·) is a group;

2. for any (g1,g2) ∈ G2, g1 6= g2, we have Ag1 ∩Ag2 = ∅;

3. if (a,b) ∈ Ag1 ×Ag2 , then a◦ b = Ag1g2.

31



Irina Cristea

Let G = {g1, . . . ,gm} be a finite group. Then with H =
m⋃
i=1

Agi we may

associate an m-tuple [k1, . . . ,km], where ki = |Agi|.
Using these notations and based on the fact that, for any u ∈ H, ∃!gu ∈
G : u ∈ Agu , Corsini [13] has determined the formula for the membership
function µ̃ associated with a complete hypergroup as the following one:

µ̃(u) =
1

|Agu|
.

Similarly, the formula of the membership functions µ,λ for a complete
hypergroup is simpler that those in the general case, as Cristea and Davvaz
proved [24].
Defining on H the following equivalence

u ∼ v ⇐⇒∃g ∈ G : u,v ∈ Ag,

one obtains that:

µ̄(u) =
|Q(u)|
|Agu|

·
1

n2
, λ̄(u) =

(∑
v/∈û

|Q(v)|
|Agv|

)
·

1

n2
.

We notice that, if G1 and G2 are non isomorphic groups of the same
cardinality and H1, H2 are the complete hypergroups obtained by them,
then f.g.(H1) = f.g.(H2), but their i.f.g. may be different.

Cristea [21] determined the fuzzy grade of all complete hypergroups of
order less than 7. Later on, Davvaz together with Hassani-Sadrabadi and
Cristea [27] determined their intuitionistic fuzzy grade. We recall here these
results.

Theorem 4.1. Let H be a complete hypergroup of order n ≤ 6.

1. There are two non isomorphic complete hypergroups of order three that
have s.f.g.(H) = i.f.g.(H) = 1.

2. Among the five non isomorphic complete hypergroups of order four,
three of them have s.f.g.(H) = i.f.g.(H) = 1 and two of them have
s.f.g.(H) = i.f.g.(H) = 2.

3. There are 12 non isomorphic complete hypergroups of order 5. All of
them have s.f.g.(H) = 1. Nine of them have i.f.g.(H) = 1 and the
other three have i.f.g.(H) = 3.

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Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

4. There are 21 non isomorphic complete hypergroups of order 6:
- 17 with s.f.g.(H) = 1 and 4 with s.f.g.(H) = 2.
- 16 with i.f.g.(H) = 1, 3 with i.f.g.(H) = 2 and 2 with i.f.g.(H) = 3.

Angheluţă and Cristea [2] have studied the fuzzy grade of some particular
complete hypergroups. It remains an open problem, and not a very easy one,
to determine their intuitionistic fuzzy grade.

Theorem 4.2. Let H be a complete hypergroup of type

1. [p,p, ...,p︸ ︷︷ ︸
k times

,kp], where n = |H| = 2kp. Then s.f.g.(H) = 2.

2. [p,p, . . . ,p︸ ︷︷ ︸
s times

,k,k, . . . ,k︸ ︷︷ ︸
t times

,ps], 2 ≤ p < k < ps, n = |H| = 2ps + kt.

• for n = 4ps, s.f.g.(H) = 3,
• for kt 6= 2ps, f.g.(H) = 2.

3. [k,k, . . . ,k︸ ︷︷ ︸
l times

,p,p, . . . ,p︸ ︷︷ ︸
s times

,s,s, . . . ,s︸ ︷︷ ︸
p times

, l, l, . . . , l︸ ︷︷ ︸
k times

], 2 ≤ k < p < s < l, n =

|H| = 2(ps + kl).

• if kl = ps, then s.f.g.(H) = 3,
• otherwise f.g.(H) = 2.

4. [1, . . . , 1︸ ︷︷ ︸
k times

, 2, . . . , 2︸ ︷︷ ︸
k
2

times

, 3, . . . , 3︸ ︷︷ ︸
k
3

times

, . . . ,m− 1, . . . ,m− 1︸ ︷︷ ︸
k

m−1 times

,k],

k = m.c.m.(1, 2, . . . ,m− 1), n = |H| = mk = 2s−1k.
Then s.f.g.(H) = s.

4.2 The i.p.s. hypergroups

An i.p.s. hypergroup (i.e. a hypergroup with partial scalar identities)
is a particular type of canonical hypergroup. We recall that a canonical
hypergroup is a commutative hypergroup (H,◦) with a scalar identity, where
every element a ∈ H has a unique inverse a−1 such that it satisfies the
following properties

1. if y ∈ a◦x, then x ∈ a−1 ◦y;

2. x ∈ x◦a =⇒ x = x◦a

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Irina Cristea

The canonical hypergroups were introduced for the first time by Krasner
[28] as the additive structures of the hyperfields and then Mittas [29] studied
them independently from the others operations. Later on Corsini determined
all i.p.s. hypergroups of order less than 9, proving that they are strongly
canonical.

Corsini and Cristea [14, 15] calculated the fuzzy grade for all i.p.s. hy-
pergroups of order less than 8 and their intuitionistic fuzzy grade have been
determined by Davvaz, Hassan-sadrabadi and Cristea [25, 26]. Here we sum-
marize these results.

Theorem 4.3. 1. There exists one i.p.s. hypergroup H of order 3 and
f.g.(H) = i.f.g.(H) = 1.

2. There exist 3 i.p.s. hypergroups of order 4 with f.g.(H1) = i.f.g.(H1) =
1, f.g.(H2) = 1, i.f.g.(H2) = 2, f.g.(H3) = i.f.g.(H3) = 2.

3. There exist 8 i.p.s. hypergroups of order 5: one has f.g.(H) = 2, all
the others have f.g.(H) = 1; four of them have i.f.g.(H) = 1, three
have i.f.g.(H) = 2 and one is with i.f.g.(H) = 3.

4. There exist 19 i.p.s. hypergroups of order 6: 14 of them have f.g.(H) =
1, 4 have f.g.(H) = 2 and one has f.g.(H) = 3;
ten of them have i.f.g.(H) = 1, eight of them have i.f.g.(H) = 2 and
only for one of them we find i.f.g.(H) = 3.

5. There exist 36 i.p.s. hypergroups of order 6: 27 of them with f.g.(H) =
1, 8 with f.g.(H) = 2, 1 with f.g.(H) = 3;
10 of them with i.f.g.(H) = 1, 10 with i.f.g.(H) = 2, 9 with i.f.g.(H) =
3, 5 with i.f.g.(H) = 4, 2 with i.f.g.(H) = 5.
Moreover, for 18 of them we find that the associated sequences of join
spaces and intuitionistic fuzzy sets are cyclic.

5 Future work

Based on the definition of the intuitionistic fuzzy set associated with a hy-
pergroupoid [24], Ashgari-Larimi and Cristea [4] have defined the Atanassov’s
intuitionistic fuzzy index of a hypergroupoid. It would be interesting to ob-
tain some relations between this index and the intuitionistic fuzzy grade.

We think that, finding necessary and sufficient conditions in order that
the sequences of join spaces (iH = (H,◦µi∧λi ); Ai = (µi,λi))i≥0 and (

iH =

(H,◦µi∨λi ); Ai = (µi,λi))i≥0 associated with a nonempty set H endowed with

34



Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

an intuitionistic fuzzy set (like in Section 2) coincide, could be another line
of research on this argument.

We also intend to find in our future work some possible connection be-
tween hypergroups and automata. For example, what can we say about its
(intuitionistic) fuzzy grade? Does there exist an automaton with (intuition-
istic) fuzzy grade of its state hypergroup equal to a given natural number?

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Fuzzy Grade of a Hypergroupoid: A Survey of Some Recent Researches

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