Atanassov’s intuitionistic fuzzy grade of hypergroups

Information Sciences 180 (2010) 1506–1517

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Atanassov’s intuitionistic fuzzy grade of hypergroups

Irina Cristea a,*, Bijan Davvaz b

a DIEA, University of Udine, Via delle Scienze 208, 33100 Udine, Italyb Department of Mathematics, Yazd University, Yazd, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 October 2008Received in revised form 14 December 2009Accepted 3 January 2010

Keywords:Fuzzy setAtanassov’s intuitionistic fuzzy setHypergroupFuzzy gradeJoin spaceComplete hypergroup

0020-0255/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.ins.2010.01.002

* Corresponding author.E-mail addresses: [email protected] (I. Cristea

This paper deals with connections between hypergroupoids and Atanassov’s intuitionisticfuzzy sets. First a sequence of join spaces is associated with a hypergroupoid H; the lengthof the sequence is called Atanassov’s intuitionistic fuzzy grade of H. Second, a theoremabout the existence of a hypergroup with Atanassov’s intuitionistic fuzzy grade equal ton is proved. Furthermore, some properties of the complete hypergroups in connection withthis argument are presented and discussed.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

The concept of algebraic hyperstructure is a natural generalization of that one of algebraic structure. This theory has beenintroduced by Marty [38] in 1934 at the 8th Congress of Scandinavian Mathematicians, where he defined the hypergroupsand presented some of their applications to non-commutative groups, algebraic functions, rational fractions. Since then var-ious connections between hyperstructures and both theoretical and applied sciences have been established: a book recentlypublished by Corsini and Leoreanu [11] contains numerous applications from the last decades to geometry, hypergraphs,binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, combinatorics, codes, artificial intelligence,and probabilities.

No sooner had Zadeh [45] introduced the fuzzy sets, than the reconsideration of the concept of classical mathematics be-gan. The study of fuzzy hyperstructures started with the notion of fuzzy subgroups defined by Rosenfeld [39] due to theimportance of group theory in mathematics, as well as its many areas of application. There is a considerable amount of workson the connections between fuzzy sets and hyperstructures and they can be classified into three groups. A first group of pa-pers considers crisp hyperoperations defined through fuzzy sets. This field of study was first explored by Corsini [5] and it waslater continued by Corsini, Cristea, Leoreanu (for example see [6–10,13–15,41]). A second group deals with fuzzy hyperoper-ations, with a completely different approach: a fuzzy hyperoperation maps a pair of elements of a set X to a fuzzy subset of X.Corsini and Tofan [12] introduced the fuzzy hypergroups and then Serafimidis and Kehagias [34,40] studied the L-fuzzy Nak-ano hypergroups. Also, Zahedi and Hasankhani [46] introduced the notion of F-polygroups. A third group of papers concernsthe fuzzy hyperalgebras, a direct extension of the well known concept of fuzzy algebras (fuzzy (sub)groups, fuzzy lattices, fuz-zy rings, etc.). For example, given a crisp hypergroup hH; �i and a fuzzy set l, then we say that l is a fuzzy subhypergroup ofhH; �i if every cut of l, say lt , is a (crisp) subhypergroup of hH; �i. In [16], Davvaz applied the concept of fuzzy sets to the

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I. Cristea, B. Davvaz / Information Sciences 180 (2010) 1506–1517 1507

theory of algebraic hyperstructures and defined fuzzy subhypergroup (resp. fuzzy Hv-subgroup) of a hypergroup (resp. Hv-group), with the intent to generalize the concept of Rosenfeld’s fuzzy subgroup of a group. Later on this subject has beenstudied in depth by Davvaz and by others also in connection with rings and modules (see [17–20,24,33,35,36,44]).

Uncertainty is an attribute of information and uncertain data are presented in various domains. The most appropriate the-ory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [45]. Besides several generalizations of fuz-zy sets, the intuitionistic fuzzy sets introduced by Atanassov [1,2] have been found to be highly useful to cope with imperfectand/or imprecise information. Atanassov’s intuitionistic fuzzy sets are an intuitively straightforward extension of Zadeh’sfuzzy sets: while a fuzzy set gives the degree of membership of an element in a given set, an Atanassov’s intuitionistic fuzzyset gives both a degree of membership and a degree of non-membership. Many concepts in fuzzy set theory were also ex-tended to intuitionistic fuzzy set theory, such as intuitionistic fuzzy relations, intuitionistic L-fuzzy sets, intuitionistic fuzzyimplications, intuitionistic fuzzy logics, the degree of similarity between intuitionistic fuzzy sets, intuitionistic fuzzy roughsets (see [28,31,32,37,43,48]). Intuitionistic fuzzy set theory has been successfully applied in decision analysis and patternrecognition, in logic programming and medical diagnosis. Recently, Davvaz et al. [21–23], Dudek et al. [26,27] and Zhan et al.[47] have studied the intuitionistic fuzzification of the concept of algebraic hyperstructures, investigating some of their prop-erties. This paper continues this line of research: the notion of fuzzy grade of a hypergroup is extended through Atanassov’sintuitionistic fuzzy sets.

The paper is organized as follows: in Section 2 we present some fundamental definitions on hypergroups, and in Section 3we recall the construction of the sequence of join spaces and fuzzy sets associated with a hypergroupoid. Section 4 is ded-icated to the concept of Atanassov’s intuitionistic fuzzy grade of a hypergroupoid: we introduce this new notion and we con-struct a hypergroup such that its Atanassov’s intuitionistic fuzzy grade is equal to a given natural number. Section 5 dealswith complete hypergroups in connection with this argument. Furthermore we indicate some conclusions and researchdirections covered in Section 6.

2. Preliminaries

Let us briefly recall some basic notions about hypergroups; for a comprehensive overview of this area, the reader is ref-ereed to [4,11,42].

Let H be a nonempty set and P�ðHÞ the set of all nonempty subsets of H.A set H endowed with a hyperoperation � : H2 ! P�ðHÞ is called a hypergroupoid. If the hyperoperation satisfies the

conditions:

(i) For any ða; b; cÞ 2 H3; ða � bÞ � c ¼ a � ðb � cÞ (the associativity).(ii) For all a 2 H;H � a ¼ a � H ¼ H (the reproduction law),

then the hypergroupoid hH; �i is a hypergroup. A hypergroup hH; �i is called total hypergroup if, for any ðx; yÞ 2 H2; x � y ¼ H.For each pair ða; bÞ 2 H2, we denote:

a=b ¼ fxja 2 x � bg and b n a ¼ fyja 2 b � yg:

A commutative hypergroup hH; �i is called a join space if, for any four elements a; b; c; d 2 H, such that a=b \ c=d – ;, it fol-lows a � d \ b � c – ;. The notion of join space has been introduced and studied for the first time by Prenowitz. Later on, to-gether with Jantosciak, he reconstructed, from the algebraic point of view, several branches of geometry: the projective, thedescriptive and the spherical geometry.

Let hH; �i and hH0; �0i be two hypergroups. A function f : H! H0 is called a homomorphism if it satisfies the condition: forany x; y 2 H,

f ðx � yÞ# f ðxÞ�0f ðyÞ:

f is a good homomorphism if, for any x; y 2 H; f ðx � yÞ ¼ f ðxÞ�0f ðyÞ. We say that the two hypergroups are isomorphic if there is agood homomorphism between them which is also a bijection.

On a hypergroupoid hH; �i one can define several relations: the fundamental relation b, the operational equivalence, theinseparability equivalence, the essential indistinguishable equivalence. The last three relations, introduced by Jantosciak[29], are used to define the notion of reduced hypergroup [30]. We recall here their definitions.

Let hH; �i be a hypergroupoid and x; y arbitrary elements of H.

� x and y are called operationally equivalent if x � a ¼ y � a and a � x ¼ a � y, for any a 2 H;� x and y are called inseparable if for a; b 2 H; x 2 a � b if and only if y 2 a � b;� x and y are called essentially indistinguishable if x and y are operationally equivalent and inseparable.

The three relations, operational, inseparability and essential indistinguishability, denoted by �o;�i and �e respectively,are equivalence relations on H. For any x 2 H, let bxo ; bxi and bxe , respectively denote the equivalence class of x respect to�o;�i and �e.

1508 I. Cristea, B. Davvaz / Information Sciences 180 (2010) 1506–1517

3. Fuzzy grade of hypergroups

In this section we recall the construction and the main property of the sequence of join spaces and fuzzy sets associatedwith a hypergroup H.

For any hypergroup hH; �i, Corsini defined in [6] a fuzzy subset ~l of H in the following way: for any u 2 H, we consider:

~lðuÞ ¼Pðx;yÞ2QðuÞ

1jx�yj

qðuÞ ; ðxÞ

where QðuÞ ¼ fða; bÞ 2 H2ju 2 a � bg; qðuÞ ¼ jQðuÞj. If QðuÞ ¼ ;, we set ~lðuÞ ¼ 0. In other words, ~lðuÞ is the average value ofthe reciprocals of the sizes of x � y for all x � y containing u.

Then, with any hypergroupoid H endowed with a fuzzy set, we can associate a join space h1H; �1i as follows (see [5]):

8ðx; yÞ 2 H2; x�1y ¼ fz 2 Hj~lðxÞ ^ ~lðyÞ 6 ~lðzÞ 6 ~lðxÞ _ ~lðyÞg:

By using the same procedure as in ðxÞ, from 1H we can obtain a membership function ~l1 and the associated join space 2Hand so on. A sequence of fuzzy sets and join spaces ðhrH; �ri; ~lrÞrP1 is determined in this way. We denote ~l0 ¼ ~l; 0H ¼ H. Iftwo consecutive hypergroups of the obtained sequence are isomorphic, then the sequence stops. In Theorem 4.5 [41] it isgiven a sufficient condition such that two consecutive join spaces of the sequence are not isomorphic.

The length of the sequence of join spaces associated with H is called the fuzzy grade of the hypergroupoid H, moreexactly:

Definition 3.1 (see [8]). A hypergroupoid H has the fuzzy grade m;m 2 N�, and we write f :g:ðHÞ ¼ m if, for any i;0 6 i < m,the join spaces iH and iþ1H associated with H are not isomorphic (where 0H ¼ H) and for any s; s > m; sH is isomorphic withmH.

We say that the hypergroupoid H has the strong fuzzy grade m and we write s:f :g:ðHÞ ¼ m if f :g:ðHÞ ¼ m and for alls; s > m; sH ¼ mH.

One of the interesting result on this argument is established by Cristea and S�tefanescu [41]: for any n 2 N, there is alwaysa hypergroup H whose strong fuzzy grade is equal to n.

4. Atanassov’s intuitionistic fuzzy grade of hypergroups

As a generalization of the notion of fuzzy set in a nonempty set X, Atanassov [1] has introduced the concept of intuitionisticfuzzy set in X, as an object having the form A ¼ fðx;lAðxÞ; kAðxÞÞjx 2 Xg, where, for any x 2 X, the degree of membership of x(namely lAðxÞ) and the degree of non-membership of x (namely kAðxÞ) verify the relation 0 6 lAðxÞ þ kAðxÞ 6 1. Throughoutthis paper, we denote an Atanassov’s intuitionistic fuzzy set A ¼ fðx;lAðxÞ; kAðxÞÞjx 2 Xg, by A ¼ ðl; kÞ.

The problem with the name of this set is studied in [25].Let hH; �i be a finite hypergroupoid of cardinality n;n 2 N�. Define on H an Atanassov’s intuitionistic fuzzy set A ¼ ð�l; �kÞ in

the following way: for any u 2 H, denote

QðuÞ ¼ fða; bÞ 2 H2ju 2 a � bgQðuÞ ¼ fða; bÞ 2 H2ju R a � bg

and consider

�lðuÞ ¼Pðx;yÞ2QðuÞ

1jx�yj

n2 ;

�kðuÞ ¼Pðx;yÞ2QðuÞ

1jx�yj

n2 :

ð1Þ

If QðuÞ ¼ ;, then we put �lðuÞ ¼ 0 and similarly, if QðuÞ ¼ ;, then we put �kðuÞ ¼ 0.Moreover, it is clear that, for any u 2 H;0 6 �lðuÞ þ �kðuÞ 6 1.Now, let A ¼ ðl; kÞ be an Atanassov’s intuitionistic fuzzy set on H. We may associate with H two join spaces hH; �l^ki and

hH; �l_ki, where, for any fuzzy set a on H, the hyperproduct ‘‘�a” is defined by

x�ay ¼ fujaðxÞ ^ aðyÞ 6 aðuÞ 6 aðxÞ _ aðyÞg:

Corsini [5] has proved that the associated hypergroup hH; �ai is a join space.In this way we can construct two sequences of join spaces and Atanassov’s intuitionistic fuzzy sets associated with a set H.

More exactly, we consider an arbitrary Atanassov’s intuitionistic fuzzy set A ¼ ðl; kÞ defined on a set H; then we associate thejoin space h0H; �l^ki and we construct the Atanassov’s intuitionistic fuzzy set A1 ¼ ð�l1; �k1Þ as in (1); we associate again thejoin space h1H; ��l1^�k1

i, we determine, like in (1), its Atanassov’s intuitionistic fuzzy set A2 ¼ ð�l2; �k2Þ and we construct the joinspace h2H; ��l2^�k2

i and so on. We obtain the sequence ðiH ¼ hiH; ��li^�kii; Ai ¼ ð�li; �kiÞÞiP0 of join spaces and Atanassov’s intuition-

istic fuzzy sets associated with H. Similarly we may construct the second sequence ðiH ¼ hiH; ��li_�kii; Ai ¼ ð�li; �kiÞÞiP0.


We call the lower, and respectively, the upper Atanassov’s intuitionistic fuzzy grade of H the length of the two corre-sponding sequences associated with H, more exactly:

Definition 4.1. A set H endowed with an Atanassov’s intuitionistic fuzzy set A ¼ ðl; kÞ has the lower Atanassov’s intuitionisticfuzzy grade m;m 2 N�, and we write l:i:f :g:ðHÞ ¼ m if, for any i;0 6 i < m, the join spaces hiH; ��li^�ki

i and hiþ1H; ��liþ1^�kiþ1i

associated with H are not isomorphic (where 0H ¼ hH; ��l^�ki) and, for any s; s P m; sH is isomorphic with m�1H.

Definition 4.2. A set H endowed with an Atanassov’s intuitionistic fuzzy set A ¼ ðl; kÞ has the upper Atanassov’s intuitionisticfuzzy grade m;m 2 N�, and we write u:i:f :g:ðHÞ ¼ m if, for any i;0 6 i < m, the join spaces hiH; ��li_�ki

i and hiþ1H; ��liþ1_�kiþ1i asso-

ciated with H are not isomorphic (where 0H ¼ hH; ��l_�ki) and, for any s; s P m; sH is isomorphic with m�1H.

Note that we may consider a more general situation, that one of a set H endowed with two membership functionsl; k : H! Rþ. But in this paper we only deal with sets endowed with an Atanassov’s intuitionistic fuzzy set A ¼ ðl; kÞ.

For the sake of completeness of our investigations we consider the following example.

Example 4.3. On H ¼ fa; b; c; dg we define the Atanassov’s intuitionistic fuzzy set:

�lðaÞ ¼ 0:25; �lðbÞ ¼ 0:25; �lðcÞ ¼ 0:30; �lðdÞ ¼ 0:10;�kðaÞ ¼ 0:40; �kðbÞ ¼ 0:40; �kðcÞ ¼ 0:50; �kðdÞ ¼ 0:90:

To start with, we construct the first sequence of join spaces and we determine its l:i:f :g:ðHÞ.For the associated join space

we calculate the Atanassov’s intuitionistic fuzzy set associated with H as in (1) and we obtain the following values:

�l1ðaÞ ¼ 31=96; �l1ðbÞ ¼ 31=96; �l1ðcÞ ¼ 17=96; �l1ðdÞ ¼ 17=96;�k1ðaÞ ¼ 12=96; �k1ðbÞ ¼ 12=96; �k1ðcÞ ¼ 26=96; �k1ðdÞ ¼ 26=96;

therefore �l1 ^ �k1ðaÞ ¼ �l1 ^ �k1ðbÞ < �l1 ^ �k1ðcÞ ¼ �l1 ^ �k1ðdÞ and thereby we have the following join space

For any x 2 H, we compute �l2ðxÞ ¼ 4=16 and �k2ðxÞ ¼ 2=16; thus, for any x; y 2 H, x��l2^�k2y ¼ H. So the first sequence of join

spaces associated with H has 3 elements, that is l:i:f :g:ðHÞ ¼ 3.Now, in order to determine the u:i:f :g:ðHÞ, we start with the join space

Note that h0H; ��l^�ki and h0H; ��l_�ki are not isomorphic. Since

�l1ðaÞ ¼ 13=48; �l1ðbÞ ¼ 13=48; �l1ðcÞ ¼ 13=48; �l1ðdÞ ¼ 9=48;�k1ðaÞ ¼ 9=48; �k1ðbÞ ¼ 9=48; �k1ðcÞ ¼ 9=48; �kðdÞ ¼ 13=48

it follows that �l1 _ �k1ðxÞ ¼ 13=48, whenever x 2 H, which means that, for any x; y 2 H; x��l1_�k1y ¼ H. Therefore the second se-

quence of join spaces associated with H has 2 elements, that is u:i:f :g:ðHÞ ¼ 2. .

Note that we may start the construction of the same sequences from a hypergroupoid hH; �i. But, in this case, we obtainonly one sequence of join spaces. Indeed, since, for any x 2 H, the sum �lðxÞ þ �kðxÞ is always constant, it follows that�lðxÞ ¼ �lðyÞ if and only if �kðxÞ ¼ �kðyÞ and then the join spaces h0H; ��l^�ki and h0H; ��l_�ki are always isomorphic. In order to ex-plain this situation, we introduce a new concept.


Definition 4.4. We say that a hypergroupoid H has the Atanassov’s intuitionistic fuzzy grade m;m 2 N�, and we writei:f :g:ðHÞ ¼ m, if l:i:f :g:ðHÞ ¼ m.

Example 4.5. Consider the hypergroupoid H ¼ fa; b; c; dg with the following hyperoperation:

then

�lðaÞ ¼ 14=96; �lðbÞ ¼ 23=96; �lðcÞ ¼ 23=96; �lðdÞ ¼ 36=96�kðaÞ ¼ 66=96; �kðbÞ ¼ 57=96; �kðcÞ ¼ 57=96; �kðdÞ ¼ 44=96

and so we have

then

�l1ðaÞ ¼ 34=192; �l1ðbÞ ¼ 62=192; �l1ðcÞ ¼ 62=192; �l1ðdÞ ¼ 34=192;�k1ðaÞ ¼ 52=192; �k1ðbÞ ¼ 24=192; �k1ðcÞ ¼ 24=192; �k1ðdÞ ¼ 52=192

and so we have

then

�l2ðaÞ ¼ 2=8; �l2ðbÞ ¼ 2=8; �l2ðcÞ ¼ 2=8; �l2ðdÞ ¼ 2=8;�k2ðaÞ ¼ 1=8; �k2ðbÞ ¼ 1=8; �k2ðcÞ ¼ 1=8; �k2ðdÞ ¼ 1=8

At last, we have

Thus, we conclude that i:f :g:ðHÞ ¼ 3.

It is obvious that, for any x 2 H, the numerator of ~lðxÞ is equal to the numerator of �lðxÞ. Moreover, we can give a generalformula to compute �lðxÞ; �kðxÞ as in [6].

Let ðiH ¼ hiH; ��li^�kii; Ai ¼ ð�li; �kiÞÞiP1 be the sequence of join spaces and Atanassov’s intuitionistic fuzzy sets associated with

a hypergroupoid H of cardinality n. Then, for any i, there are r, namely r ¼ ri, and a partition p ¼ fiCjgrj¼1of iH such that, for

any j P 1; x; y 2 iCj () �li�1ðxÞ ¼ �li�1ðyÞ.On the set of classes fiCjgr

j¼1 we define the following ordering relation :iCj <

iCk, if, for elements x 2 iCj and y 2 iCk; �li�1ðxÞ < �li�1ðyÞ.We shall use some notations, for all j; s:

kj ¼ jiCjj; sC ¼[

16j6s

iCj;sC ¼

[s6j6r

iCj; sk ¼ jsCj; sk ¼ jsCj:


With any ordered chain ðiCj1 ;iCj2 ; . . . ; iCjr Þ we may associate an ordered r-tuple ðkj1 ; kj2 ; . . . ; kjr Þ, where kjl ¼ jiCjl j, for all

l;1 6 l 6 r.As in Theorem 1.1 [6] we obtain the following formula.

Theorem 4.6. For any z 2 iCjs; i P 1; s ¼ 1;2; . . . ; r,

�liðzÞ ¼kjs þ 2

Pl6s6ml – m

kjlkjmP

l6t6mkjt

n2 :

Similarly, we find the next result.

Theorem 4.7. For any z2iCs; i P 1; s ¼ 1;2; . . . ; r,

�kiðzÞ ¼

Pl – skjl þ 2

Ps6l<m6r

kjlkjmP

l6t6mkjt

þ 2P

16l<m6skjl

kjmPl6t6m

kjt

n2 :

Proof. Since, for any z 2 iCs; s ¼ 1;2; . . . ; r, the following formula

�liðzÞ þ �kiðzÞ ¼

Prl¼1kjl þ 2

P16l<m6r

kjlkjmP

l6t6mkjt

n2

holds, the statement of this theorem follows immediately from Theorem 4.6. h

Remark 4.8

(i) It is clear that, for any z 2 iH ¼Sr

s¼1iCjs , the sum �liðzÞ þ �kiðzÞ is a constant which does not depend on s, but only on the

r-tuple ðkj1 ; kj2 ; . . . ; kjr Þ associated with the join space iH.(ii) If the r-tuple associated with iH has the form ðkj1 ; kj2 ; . . . ; kjr Þ, with kjl ¼ kjr�lþ1

;1 6 l 6 ½r=2�, it follows that, for anyx 2 iCjl and y 2 iCjr�lþ1

; �liðxÞ ¼ �liðyÞ and now it is clear that �kiðxÞ ¼ �kiðyÞ.

A natural problem arises: does there exist a hypergroupoid H such that i:f :g:ðHÞ ¼ n, whenever n is a natural number? Theanswer is given below.

Lemma 4.9. Let H ¼ fx1; x2; . . . ; xng, where n ¼ 2p; p 2 N�, be the hypergroupoid defined by the hyperproduct

xi � xi ¼ xi; 1 6 i 6 n;

xi � xj ¼ fxi; xiþ1; . . . ; xjg; 1 6 i < j 6 n:

Then, for any s 2 f1;2; . . . ;n=2g, we obtain

�lðxsÞ ¼ �lðxn�sþ1Þ; �kðxsÞ ¼ �kðxn�sþ1Þ:

Proof. The method we use here consists in identifying some blocks (from the table of the hypergroup hH; �i) associated withevery xs 2 H.

The table of the commutative hyperoperation ‘‘�” is the following one:

where we use the notation xi ! xj ¼ fxi; xiþ1; . . . ; xjg, with i < j.


One finds the element xs only in two blocks: one formed with the lines 1st;2nd; . . . ; sth and the columnssth;ðsþ 1Þth; . . . ;nth of the previous table and denoted by AðxsÞ and the other one formed with the linessth;ðsþ 1Þth; . . . ;nth and the columns 1st;2nd; . . . ; sth of the previous table and denoted by A0ðxsÞ.

Take AðxsÞ ¼ ðaljÞ 16l6s16j6n�sþ1

2Ms;n�sþ1ðP�ðHÞÞ, where alj ¼ xl � xsþj�1 ¼ fxl; xlþ1; . . . ; xs; . . . ; xsþj�1g.

With any matrix AðxsÞ; s 2 f1;2; . . . ;n=2g, we associate another matrix

BðxsÞ ¼ ðbljÞ 16l6s16j6n�sþ1

2Ms;n�sþ1ðN�Þ;

where blj ¼ jaljj ¼ jxl � xsþj�1j. The matrix BðxsÞ is the following one:

s sþ 1 sþ 2 � � � 2s� 2 2s� 1 � � � n� 1 n

s� 1 s sþ 1 � � � 2s� 3 2s� 2 � � � n� 2 n� 1s� 2 s� 1 s � � � 2s� 4 2s� 3 � � � n� 3 n� 2

..

. ... ..

. . .. ..

. ...

� � � ... ..

.

2 3 4 � � � s sþ 1 � � � n� sþ 1 n� sþ 21 2 3 � � � s� 1 s � � � n� s n� sþ 1

0BBBBBBBBB@

1CCCCCCCCCA
It can be easily verified that, for every s 2 f1;2; . . . ;n=2g, the matrices BðxsÞ and Bðxn�sþ1Þ have the same elements
distributed in a similar order, more specifically, if BðxsÞ ¼ ðbljÞ 16l6s16j6n�sþ1

and Bðxn�sþ1Þ ¼ ð~bikÞ16i6n�sþ116k6s

, then ~bik ¼ bs�kþ1;n�s�iþ2.Since

QðxsÞ ¼ fðxl; xsþj�1Þ; xsþl0�1; x0j

� �;1 6 l; j0 6 s;1 6 l0; j 6 n� sþ 1g;

it follows that
Xða;bÞ2Qðx1Þ
1ja � bj ¼ 1þ 2

12þ 1

3þ � � � þ 1

n

� �ð2Þ

and, whenever s P 2,

Xða;bÞ2QðxsÞ

1ja � bj ¼ 2

X16l6s

16j6n�sþ1

1blj

0BB@1CCA� 1 ¼ ð3Þ

¼ 1þ 222þ 3

3þ � � � þ s

sþ s

sþ 1þ � � � þ s

n� sþ 1þ s� 1

n� sþ 2þ � � � þ 2

n� 1þ 1

n

� �: ð4Þ

In other words, in order to compute the numerator of �lðxsÞ we add twice all the inverses 1blj

of the elements blj of the matrixBðxsÞ and then subtract 1 (since 1 ¼ jxs � xsj is the unique common element of the matrices BðxsÞ and its symmetric obtainedfrom A0ðxsÞ). The simplest method to compute these operations is to cross the matrix BðxsÞ, starting with the lower left-handcorner, along its ‘‘principal pseudo-diagonals” that contain only the elements 1, or 2, or 3 and so on; thus we obtain the rela-tion (4).

Since the matrices BðxsÞ and Bðxn�sþ1Þ; s 2 f1;2; . . . ;n=2g, contain the same elements, it is clear that

�lðxsÞ ¼ �lðxn�sþ1Þ:

Since, for any x 2 H, the sum �lðxÞ þ �kðxÞ is equal to a certain constant (by Remark 4.8(i)), it follows that �lðxÞ ¼ �lðyÞ if andonly if �kðxÞ ¼ �kðyÞ and thereby

�kðxsÞ ¼ �kðxn�sþ1Þ: �

Lemma 4.10. Let hH; �ibe the hypergroupoid defined in Lemma 4.9. Then, for any s 2 f1;2; . . . ;n=2g, the following formulas hold:

�lðxsÞ < �lðxsþ1Þ; �kðxsÞ > �kðxsþ1Þ:

Proof. Rewriting the relations (2) and (4) in the form

�lðx1Þ ¼1n2 1þ 2

Xn

l¼2

1l

!;

�lðxsÞ ¼1n2 1þ 2ðs� 1Þ þ 2

Xn�2sþ1

l¼1

ssþ l

þ 2Xs�1

l¼1

ln� lþ 1

" #; 2 6 s 6 n=2;

it results that


�lðxsþ1Þ ¼ �lðxsÞ þ2n2

Xn�s

l¼sþ1

1l; for any 1 6 s 6 n=2:

Similarly, in virtue of Theorem 4.7, we find

�kðx1Þ ¼1n2 n� 1þ 2

Xn�1

l¼2

n� ll

!;

�kðxsÞ ¼1n2 n� 1þ 2

Xs

l¼2

n� 2lþ 1l

þ 2Xn�s

l¼sþ1

n� sþ 1� ll

!; 2 6 s 6 n=2

and therefore

�kðxsÞ ¼ �kðxsþ1Þ þ2n2

Xn�s

l¼sþ1

1l; for any 1 6 s 6 n=2:

Then �lðxsÞ < �lðxsþ1Þ and �kðxsÞ > �kðxsþ1Þ, for any s 2 f1;2; . . . ;n=2g.

Lemma 4.11. Let hH; �i be the hypergroupoid defined in Lemma 4.9. Then, for any s 2 f1;2; . . . ;n=2g and n P 6, we obtain

�lðxsÞ < �kðxsÞ:

Proof. It is obvious that, for any n P 6; �lðx1Þ < �kðx1Þ if and only if

1þ 2Xn

l¼2

1l< n� 1þ 2

Xn�1

l¼2

n� ll() n� 2þ 2

n� 32þ n� 4

3þ � � � þ 1

n� 2

� �� 2

n> 0

which is satisfied.Now we prove that, for s 2 f2;3; . . . ; n=2g and n P 6, we obtain �lðxsÞ < �kðxsÞ, that is

sþXn�2sþ1

l¼1

ssþ l

þXs�1

l¼1

ln� lþ 1

<n2þXs

l¼2

n� 2lþ 1l

þXn�s

l¼sþ1

n� sþ 1� ll

: ð5Þ

Since
Xn�2sþ1
l¼1

ssþ l

<sðn� 2sþ 1Þ

sþ 1Xs�1

l¼1

ln� lþ 1

<sðs� 1Þ

2ðn� sþ 2Þ

denoting the first member of the inequality (5) by MI , we find that:

MI < sþ sðn� 2sþ 1Þsþ 1

þ sðs� 1Þ2ðn� sþ 2Þ :

Since
Xs
l¼2

n� 2lþ 1l

>ðs� 2Þðn� sÞ

s� 1Xn�s

l¼sþ1

n� sþ 1� ll

>ðn� 2sþ 1Þðn� 2sþ 2Þ

2ðn� sÞ

denoting the second member of the inequality (5) by MII , we obtain that:

MII >n2þ ðs� 2Þðn� sÞ

s� 1þ ðn� 2sþ 1Þðn� 2sþ 2Þ

2ðn� sÞ :

Then MI < MII if and only if

sþ sðn� 2sþ 1Þsþ 1

þ sðs� 1Þ2ðn� sþ 2Þ <

n2þ ðs� 2Þðn� sÞ

s� 1þ ðn� 2sþ 1Þðn� 2sþ 2Þ

2ðn� sÞ : ð6Þ

For s ¼ 2 the formula (6) becomes

2n2 þ 33n

<2n� 3

2() 2n2 � 9n� 6 > 0


which holds for any n P 6. For s P 3 the formula (6) is equivalent to

f ðnÞ ¼ 2n3s2 � 6n3 � 7n2s3 þ 3n2s2 þ 23n2s� 15n2 þ 8ns4 � 10ns3 � 28ns2 þ 42ns� 8n� 3s5 þ 9s4 þ 5s3 � 25s2

þ 14s� 4 > 0; whenever s 6 n=2:

Calculating the first three derivatives of the function f with respect to n, we obtain that f is a monotonic increasing function,so

f ðnÞ > f ð2sÞ ¼ s5 þ s4 � 7s3 � s2 � 2s� 4 > 0; for any s P 3

and the required result is proved. h

Theorem 4.12. Let H ¼ fx1; x2; . . . ; xng, where n ¼ 2p, p 2 N� n f1;2g, be the hypergroupoid defined by the hyperproduct

xi � xi ¼ xi; 1 6 i 6 n;

xi � xj ¼ fxi; xiþ1; . . . ; xjg; 1 6 i < j 6 n:

Then i:f :g:ðHÞ ¼ p.

Proof. We prove this fact in several steps, inductively by the index i P 0; we show that, for any s 2 f1;2; . . . ;n=2iþ1g, weobtain

� �liðxsÞ ¼ �liðxn�sþ1Þ; �kiðxsÞ ¼ �kiðxn�sþ1Þ;� �liðxsÞ < �liðxsþ1Þ; �kiðxsÞ > �kiðxsþ1Þ;� �liðxsÞ < �kiðxsÞ; therefore ð�li ^ �kiÞðxsÞ < ð�li ^ �kiÞðxsþ1Þ.

Thereby, with any join space hiH; ��li^�kii we may associate the ri-tuple ð2iþ1;2iþ1; . . . ;2iþ1Þ, where ri ¼ n=2iþ1.

Step 1. According to Lemmas 4.9, 4.10 and 4.11 we obtain the previous three relations for i ¼ 0. Thus, with the join spaceh0H; ��l^�ki we associate the r0-tuple ð2;2; . . . ;2Þ; r0 ¼ n=2.

Step 2. Assume that with the join space i�1H we associate the r ¼ ri�1-tuple ð2i;2i; . . . ;2iÞ, with ri�1 ¼ n=2i and we provethat, for any s 2 f1;2; . . . ; n=2iþ1g,

� �liðxsÞ ¼ �liðxn�sþ1Þ; �kiðxsÞ ¼ �kiðxn�sþ1Þ;� �liðxsÞ < �liðxsþ1Þ; �kiðxsÞ > �kiðxsþ1Þ;� �liðxsÞ < �kiðxsÞ; therefore ð�li ^ �kiÞðxsÞ < ð�li ^ �kiÞðxsþ1Þ.

Indeed, by Remark 4.8 ðiÞ it follows the first relation.Then, by Theorem 4.6, we know

�liðx1Þ ¼2i

n2 1þ 2Xr

l¼2

1l

!; ð7Þ

�liðxsÞ ¼2i

n2 1þ 2Xs�1

l¼1

Xr

j¼s

1j� lþ 1

þ 2Xr

j¼sþ1

1j� sþ 1

!ð8Þ

¼ 2i

n2 1þ 2ðs� 1Þ þ 2Xr�2sþ1

l¼1

ssþ l

þ 2Xs�1

l¼1

lr � lþ 1

!; 2 6 s 6 r=2; ð9Þ

and thus, like in Lemma 4.10,

�liðxsþ1Þ � �liðxsÞ ¼2iþ1

n2

Xr�s

l¼sþ1

1l:

Similarly, by Theorem 4.7 we obtain

�kiðx1Þ ¼2i

n2 r � 1þ 2Xr�1

l¼2

r � ll

!; ð10Þ

�kiðxsÞ ¼2i

n2 r � 1þ 2Xs

l¼2

r � 2lþ 1l

þ 2Xr�s

l¼sþ1

r � sþ 1� ll

!; 2 6 s 6 r=2; ð11Þ

consequently,

�kiðxsÞ � �kiðxsþ1Þ ¼2iþ1

n2

Xr�s

l¼sþ1

1l:


So, �liðxsÞ < �liðxsþ1Þ, �kiðxsÞ > �kiðxsþ1Þ, for any s 2 f1;2; . . . ; r=2g. It remains to prove that �liðxsÞ < �kiðxsÞ. Note that the formulasfor �liðxsÞ and �kiðxsÞð1 6 s 6 r=2Þ are similar to that for �lðxsÞ and �kðxsÞð1 6 s 6 n=2Þ and thus, according to Lemma 4.11, we canstate that

�liðxsÞ < �kiðxsÞ; 1 6 s 6 r=2; for any r P 6; ð12Þ

and therefore

ð�li ^ �kiÞðxsÞ < ð�li ^ �kiÞðxsþ1Þ; 1 6 s 6 r=2; r P 6:

Moreover, as we shall see in the next example, the relation ð�li ^ �kiÞðx1Þ < ð�li ^ �kiÞðx2Þ holds too.Step 3. We can now conclude the proof as follows: for any i;1 6 i 6 p, with the join space iH we associate the ri-tuple

ð2iþ1;2iþ1; . . . ;2iþ1Þ, with ri ¼ n=2iþ1. So, with the join space p�1H we associate the 1-tuple ð2pÞ, which leads to the fact thatthe join space p�1H is the total hypergroup and thus i:f :g:ðHÞ ¼ p. h

Example 4.13. Consider r ¼ n=2i ¼ 2p�i ¼ 4; thus, with the join space p�3H, we associate the quadruple ð2p�2;2p�2;2p�2;2p�2Þ.By the relations (7) and (10) for r ¼ 4 we obtain

�lp�2ðx1Þ ¼2p�2

n2 1þ 212þ 1

3þ 1

4

� �� ¼ 2p�2

n2 �196;

�kp�2ðx1Þ ¼2p�2

n2 3þ 222þ 1

3

� �� ¼ 2p�2

n2 �173:

Moreover, by the relations (9) and (11) for r ¼ 4 it follows

�lp�2ðx2Þ ¼2p�2

n2 1þ 2ð2� 1Þ þ 223

� �¼ 2p�2

n2 �296;

�kp�2ðx2Þ ¼2p�2

n2 3þ 212

� �¼ 2p�2

n2 � 4:

Now it is obvious that

ð�lp�2 ^ �kp�2Þðx1Þ < ð�lp�2 ^ �kp�2Þðx2Þ;

therefore with the join space p�2H we associated the pair ð2p�1;2p�1Þ and then with the join space p�1H we associate the 1-tuple ð2pÞ; thus p�1H is the total hypergroup and i:f :g:ðHÞ ¼ p.

5. Atanassov’s intuitionistic fuzzy grade of the complete hypergroups

In this section we determine simpler formulas for the membership functions �l and �k associated with a complete hyper-group. Besides we give some properties of the Atanassov’s intuitionistic fuzzy grade of a complete hypergroup.

Let H ¼S

g2G be a complete hypergroup of cardinality n. We may identify a complete residue system Si ¼ fx1; x2; . . . ; xlg ofH modulo the equivalence �i (the equivalence �i is the relation of inseparability described in Preliminaries): for anyi – j 2 f1;2; . . . ; lg; xi ¿ ixj, where l ¼ jGj. Indeed, it is obvious that, for any u 2 H, there exists an unique gu 2 G such thatu 2 Agu

and therefore u 2 x � y ¼ Agxgy¼ Agu

. Then

�lðuÞ ¼Pðx;yÞ2QðuÞ

1jx�yj

n2 ¼Pðx;yÞ2QðuÞ

1jAgu j

n2 ¼ qðuÞjAguj �

1n2 ;

�kðuÞ ¼Pðx;yÞ2QðuÞ

1jx�yj

n2 ¼

PvRuiv2Si

qðvÞjAgv j

n2 :

ð13Þ

Proposition 5.1. Let H ¼S

g2G be a complete hypergroup of cardinality n. If the group G is isomorphic with the additive group Z2,then i:f :g:ðHÞ ¼ 1.

Proof. If GwZ2, then we write H ¼ Ag1[ Ag2

; g1 – g2 2 G. Therefore, for any u 2 Ag1and v 2 Ag2

, by (4) we get

�lðuÞ ¼ qðuÞjAg1j �

1n2 ;

�lðvÞ ¼ qðvÞjAg2j �

1n2 :

Moreover, for any x 2 Ag and y 2 Ag0 , with g – g0 2 G, we have x¿ iy (otherwise, if x�iy and x 2 a � b ¼ Agagb¼ Agx

¼ Ag ,then y 2 Ag , thus Ag \ Ag0 – ;, false).


Since u 2 Ag1and v 2 Ag2

, it follows that u¿ iv and by the relation (4) we obtain

�kðuÞ ¼ qðvÞjAg2j �

1n2 ;

�kðvÞ ¼ qðuÞjAg1j �

1n2 :

Thereby �lðuÞ ¼ �kðvÞ and �kðuÞ ¼ �lðvÞ. Therefore, for any x 2 H; �lðxÞ ^ kðxÞ ¼ constant; thus the hypergroup hH; ��l^ki is thetotal hypergroup and i:f :g:ðHÞ ¼ 1. h

6. Conclusions and future work

With any hypergroupoid H we have associated, in a certain manner, a sequence of join spaces and Atanassov’s intuition-istic fuzzy sets; this sequence stops when two consecutive join spaces are isomorphic. The length of the sequence is calledAtanassov’s intuitionistic fuzzy grade of H.

This construction is important for at least two reasons: first, it gives the possibility to study Atanassov’s intuitionistic fuz-zy sets from an algebraic point of view. Secondly, we may start the construction of the mentioned sequence from a hyperg-roupoid and also from a nonempty set H endowed with an Atanassov’s intuitionistic fuzzy set. The difference between thesetwo cases consists in the number of the sequences of join spaces associated with H. More exactly, when we start with ahypergroupoid we obtain only one sequence of join spaces (Example 4.5); if we start with a set endowed with an Atanassov’sintuitionistic fuzzy set, then we may obtain two distinct sequence of join spaces (see Example 4.3).

In a future work we will thoroughly consider the second case; it is interesting to find necessary and sufficient conditionsin order that the two sequences coincide.

Another objective of our research is to find possible connections between algebraic hypergroups and automata. By anautomata we mean a triple ðS;A; dÞ, where S;A are arbitrary sets, A – ;, called the set of states and the set of input symbols,and d : S A� ! S is the transition function which satisfies dðs; eÞ ¼ s and dðs; abÞ ¼ dðdðs; aÞ; bÞ, for any state s 2 S and any pairof words a; b 2 A�, where e stands for the empty word, and A� is the free monoid of words over A. Accordingly with [3], withany automaton ðS;A; dÞ we may associates a hyperoperation on S by s � t ¼ dðs;A�Þ [ dðt;A�Þ, for any pair of states s; t 2 S. It isproved that hS; �i is a commutative hypergroup, called the state hypergroup of the automaton ðS;A; dÞ. It is natural to ask whathappens if we associate with a state hypergroup S of an automaton the sequence of join spaces defined in this paper. We willcalculate Atanassov’s intuitionistic fuzzy grade of S and we will study the existence of an automaton with Atanassov’s intui-tionistic fuzzy grade of its state hypergroup equal to a given natural number n.

Acknowledgements

The authors are highly grateful to referees and to Professor Witold Pedrycz, Editor-in-Chief of the journal, for their valu-able comments and suggestions.

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Atanassov’s intuitionistic fuzzy grade of hypergroups