Intuitionistic Fuzzy Hv-subgroups

International Journal of Advanced Engineering Research and Science (IJAERS) [Vol-1, Issue-3, Aug- 2014]ISSN: 2349-6495

Intuitionistic Fuzzy Hv-subgroupsArvind Kumar Sinha1, Manoj Kumar Dewangan2

1Department of Mathematics, NIT, Raipur, Chhattisgarh, INDIA2Department of Mathematics, NIT, Raipur, Chhattisgarh, INDIA

Abstract— In this paper we introduce the concept of intuitionistic fuzzy Hv-subgroup and prove some related

results. We also consider the fundamental relation defined on Hv-group H and for an intuitonistic fuzzy subset

of H, we define an intuitonistic fuzzy subset

of and we prove a theorem concerning the

fundamental relation . Mathematics Subject Classification: 20N20Keywords— Hv-group, Hv-subgroup, fuzzy Hv-group, fuzzy Hv-subgroup, intuitionistic fuzzy Hv-subgroup.

I. INTRODUCTIONThe concept of hyperstructure was introduced in 1934 by Marty [10]. Hyperstructures have many applications to several branches of pure and applied sciences. Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Vougiouklis [26] introduced a new class of hyperstructures, the so-called Hv-structures, in which equality is replaced by non-empty intersection. After the introduction of fuzzy sets by Zadeh [18], there have been a number of generalizations came in existence of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov [17] is one of them.At the beginning according to Atanassov the term intuitionistic means that the sum of the degree of membership and the degree of non-membership is less than 1 but after some time this condition is extended and supposed to satisfy the constraint that the sum of the degree of membership and the degree of non-membership is less than or equal to 1.Basically the algebraic structure of intuitionistic fuzzy set is one of the interval-valued set not the intuitionistic logic [31]. In the same time the idea came in existence that the interval-valued sets are mathematically redundant up-to which level so by discussion [32] it is mathematically redundant up-to every level like power set level, fiber level,

and categorical level. This naturally leads to interval-valued sets in a first step of departure away from standard fuzzy set. Indeed this is used in a long tradition in the field of Economics, Engineering and Science, etc. where the intervals were used to represent values of quantities in case of uncertainty. Currently it is studied in various domains of Information Technology, including preference modeling, learning and reasoning [33-34].In 1971 the concept of fuzzy subgroup was introduced by Rosenfeld [1]. After that B. Davvaz [4] has given fuzzy Hv-submodules and also he [6] has given redefined fuzzy Hv-submodules and many valued implications. Zhan [12-15] et al. has given some results on fuzzy hypermodules. After that Violeta Leoreanu fotea [28] introduced fuzzy hypermodules. Zhan et al. [29-30] has given some results on L-fuzzy hypermodules. After some year M. Asghari Larimi [20-21] has given homomorphism of intuitionistic (α, β)-fuzzy Hv-submodules of Hv-modules. In 2012 Zhan et al. [16] has given intuitionistic (S, T) fuzzy Hv-submodules of Hv-modules. Recently in 2013 M. Aliakbarnia et al. [19] has given fuzzy isomorphism theorem of hyper-near modules. This paper continues this line of research on intuitionistic fuzzy Hv-subgroups. Here it is very important to note that Intuitionistic fuzzy set is to be combined with the study of hyperstructures for more generalization of the generalized concept. In this paper, we generalize the concept of fuzzy Hv-groups [3] by using the notion of intuitionistic fuzzy set and prove some results in this respect. We also consider the fundamental

relation defined on Hv-group H and for an intuitonistic

fuzzy subset of H, we define an intuitonistic

fuzzy subset of and prove a fundamental

theorem concerning the group . Throughout this paper left reproduction axiom for the hypergroups is verified.

II. BASIC DEFINITIONS

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In this section first we give some basic definitions for proving the further results.

Definition 2.1[7] Let be a non-empty set. A mapping

is called a fuzzy set in .Definition 2.2[7] An intuitionistic fuzzy set A in a non-

empty set is an object having the form

, where the functions

and denote the degree of membership and degree of non membership of each

element to the set A respectively and

for all . We shall use the

symbol for the intuitionistic fuzzy set

.

Definition 2.3[27] Let be a non–empty set and

be a hyperoperation, where

is the set of all the non-empty subsets of .

The is called weak associative if

.

Where

The is called weak commutative if

.

is called an Hv–group if

(i) is weak associative.

(ii) (Reproduction axiom).

Definition 2.4[3] Let be a hypergroup (or Hv-group)

and let be a fuzzy subset of . Then is said to be a

fuzzy subhypergroup (or fuzzy Hv-subgroup) of if the following axioms hold:

(ii) For all there exists such that

and III. INTUITIONISTIC FUZZY HV-SUBGROUP

In this section we define intuitionistic fuzzy Hv-subgroup of a hypergroup and then we obtain the relation between a intuitionistic fuzzy subhypergroup and level subhypergroup. This relation is expressed in terms of a necessary and sufficient condition.

Definition 3.1 Let be a hypergroup (or Hv-group). An

intuitionistic fuzzy set of is called intuitionistic fuzzy subhypergroup (or intuitionistic fuzzy

Hv-subgroup) of if the following axioms hold:

(ii) For all there exists such that

and

(iv) For all there exists such that

and

Proposition 3.2 Let be a group and be an intuitionistic fuzzy subgroup of H. If we define the

following hyperoperation on ; ,

then is an Hv-

group and is an intuitionistic fuzzy Hv-subgroup of H.

Proof. For all in we have

and .

Since is associative, therefore is weak

associative, because

and the left reproduction axiom is satisfied,

i.e. for all ,

.

Now we prove that is an intuitionistic fuzzy Hv-subgroup.Here the conditions (i) and (ii) of definition 3.1 can be

easily proved by taking in [3].

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(iii)

.(iv) , hence

.Therefore it is enough to put

and in this case

.

Now suppose that is a set and is an

intuitionistic fuzzy subset of . We define the

hyperoperation as follows:

Let if , then

. We will prove the following proposition.

Preposition 3.3 Let be a group and

be an intuitionistic fuzzy subgroup of , then

is an intuitionistic fuzzy Hv- subgroup of .Proof. In order to prove the proposition, it is sufficient to prove the conditions (iii) and (iv) of definition 3.1.

(iii) Since we have

.(iv) Now suppose , if , then

which implies and

if , then

implying , therefore if we put ,

then in any case .

Now let be a set and be an intuitionistic

fuzzy subset of . Then the set

is called the level set of . We will prove the following results for Hv-subgroups because they are more general than hypergroups.

Theorem 3.4 Let be an Hv-group and

be an intuitionistic fuzzy subset of . Then

is an intuitionistic fuzzy Hv-subgroup of

if and only if for every t, is an Hv-

subgroup of .

Proof. Let be an intuitionistic fuzzy Hv-

subgroup of . we have

and so .

Therefore for every we have so

. Hence for every we have .

Now let , then there exists such that

and . From

and we get and

so , and this proves .

Conversely, assume that is an

Hv-subgroup of .

(i) , we can write

and

and if we put

, then and ,

so . Therefore for every we have

implying

.

(ii) If we put then

and , so there exists , such that

. On the other hand since , then

and hence .

(iii) , we can write

and

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and if we put

, then and , so

. Therefore for every we have

implying

.

(iv) If we put then

and , so there exists , such that

. On the other hand since , then

and hence .We can obtain the following two corollaries from Theorem 3.4.

Corollary 3.5 Let be an Hv-group and

be an intuitionistic fuzzy Hv-subgroup of

. If , then if and only if there is

no in such that .

Corollary 3.6 Let be an Hv-group and

be an intuitionistic fuzzy Hv-subgroup of

. If the range of is the finite set

, then the set contains all

the level Hv-subgroups of . Moreover if

, then all the level Hv-subgroups

form the following chain .

Theorem 3.7 Let be an Hv-group. Then every Hv-

subgroup of is a level Hv-subgroup of an intuitionistic

fuzzy Hv-subgroup of .

Proof. Let be an Hv-subgroup of . For a fixed real

number , , the intuitionistic fuzzy subset is

defined as follows:

We have and by theorem 3.4, it is adequate to

prove that is an intuitionistic fuzzy Hv-subgroup. This is simple and we leave out for readers.

Corollary 3.8 Let be an Hv-group and be a non-

empty subset of . Then a necessary and sufficient

condition for to be an Hv-subgroup is that ,

where is an intuitionistic fuzzy Hv-subgroup and

.Proof. This is obvious from Theorems 3.4 and 3.7.

Definition 3.9 Let be an Hv-group and be an

intuitionistic fuzzy Hv-subgroup of . is called right

fuzzy closed with respect to if all the in

satisfy . We call

left fuzzy closed with respect to if all

the y in satisfy .

If is left and right fuzzy closed, then is called fuzzy closed.Theorem 3.10 If the intuitionistic fuzzy Hv-subgroup

is right fuzzy closed, then

.

Proof. If , then there exists and

such that . Therefore

and since is right fuzzy closed we

get . Hence

which implies . So we have

proved .

On the other hand if , then for every

by the reproduction axiom there exists such that

and so it is enough to prove . Since

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is an intuitionistic fuzzy Hv-subgroup of H, by definition

we have which implies

. Since is right fuzzy

closed so . Now from

we get and so .

Using (ii) we obtain . Therefore

and by (i) the relation

is obtained. But

and hence .

So and since we get

and the theorem is proved.IV. THE FUNDAMENTAL RELATION

In this section we will prove a theorem concerning the

fundamental relation . Let be an Hv-group. The

relation is the smallest equivalence relation on such

that the quotient is a group. is called the

fundamental equivalence relation on . This relation is studied by Corsini [22] concerning hypergroups, see also [9,

24, 27]. According to [27] if denotes the set of all the

finite products of elements of , then a relation can be

defined on whose transitive closure is the fundamental

relation . The relation is as follows: for x and y in

we write if and only if , for some

.

Suppose (a) is the equivalence class containing .

Then the product on , the set of all the equivalence classes, is defined as follows:

. It is proved in [27] that is the singleton

{ (c)} for all . In this way becomes a hypergroup. If we put

, then becomes a group.

Definition 4.1 Let be an Hv-group and

be an intuitionistic fuzzy subset of . The

intuitionistic fuzzy subset on is defined as follows:

The concept of T-norm has been studied in [8], and definition of T-fuzzy subgroups of a group G has been introduced in [11]. Now we define T- intuitionistic fuzzy Hv-subgroup as follows:

Definition 4.2 Let be an Hv-group and let

be an intuitonistic fuzzy subset of . Then

is said to be a T-intuitionistic fuzzy Hv-subgroup of with respect to T-norm T if the following axioms hold:

there exists such that

and .

there exists such that

and .

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Now we give a more general proof of the following theorem by using the concept of T-norm.

Theorem 4.3 Let T be a continuous T-norm and

be a T-intuitionistic fuzzy Hv-subgroup of

. Considering as a hypergroup, then is a T-

intuitionistic fuzzy Hv-subgroup of .

Proof. If then the conditions (i) and (ii) of definition 4.2 can be easily proved by [3].

(iii) Let

and be two elements of .

We can write:

(iv) Now suppose and are two arbitrary

elements of . Since is a T-

intuitionistic fuzzy Hv-subgroup of , it follows that for

all there exists such that

and . From

it follows that

which implies . Now if

and , then there exists

such that and since

we get

and therefore

. So all the satisfying

belong to the same equivalence class. Now we have:

Corollary 4.4 Let be an intuitionistic fuzzy

Hv-subgroup of . Considering as a hypergroup,

then is an intuitionistic fuzzy Hv-subgroup of .

Proof. This is obvious from Theorem 4.3, because minimum function is a continuous T-norm.

Theorem 4.5 Let be an Hv-group and be

an intuitionistic fuzzy Hv-subgroup of . Then is an

inuitionistic fuzzy subgroup of .

Proof. Since is an intuitionistic fuzzy Hv-subgroup, by Corollary 4.4, the first and second conditions of Definition 3.1 are satisfied, therefore

there exists

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such that and

.

there exists

such that and

.

Now for all in we prove that

. Since , by

considering which is obtained from the

second condition there exists in such that

and

. From

we obtain ,

where denotes the unit of the group .

Therefore, we get (I) .

Now considering , in , by

condition (ii) above there exists in such

that and

. From

we obtain , so we get

(II)

By (I) and (II) the inequality

is obtained.

Now for all in we prove that

. Since by

considering which is obtained from the

second condition there exists in such that

and

. From we obtain

, where denotes the unit of the group

.Therefore, we get (III) .

Now considering , in , by condition (iv)

above there exists in such that

and

. From

we obtain , so we get

(IV)

By (III) and (IV) the inequality

is obtained.REFERENCES

[1] Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517.

[2] Davvaz, A brief survey of the theory of Hv-structures, in: Proceedings of the 8th International Congress on AHA, Greece 2002, Spanids Press, 2003, pp. 39-70.

Page | 36


[3] Davvaz, Fuzzy Hv-groups, Fuzzy Sets and Systems 101 (1999) 191-195.

[4] B. Davvaz, Fuzzy Hv-submodules, Fuzzy Sets and Systems 117 (2001) 477-484.

[5] B. Davvaz, J. M. Zhan, K. H. Kim, Fuzzy Γ-hypernear-rings, Computers and Mathematics with Applications 59 (2010) 2846-2853.

[6] B. Davvaz, P. Corsini, Redefined fuzzy Hv-submodules and many valued implications, Inform. Sci. 177 (2007) 865-875.

[7] B. Davvaz, W. A. Dudek, Y. B. Jun, Intuitionistic fuzzy Hv–submodules, Inform. Sci. 176 (2006) 285-300)

[8] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960) 313 -334.

[9] Freni, Una nota sul coure di un ipergruppo e sulla

chiu transitiva di , Rivista Math. Pura Appl. 8 (1991)153-156.

[10] Marty, Sur une generalization de la notion de group, in: 8th congress Math.Skandenaves, Stockholm, 1934, pp. 45-49.

[11] J.M. Anthony, H. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl. 69 (1979) 124 –130.

[12] J. M. Zhan, B. Davvaz, K. P. Shum, A New View on Fuzzy Hypermodules, Acta Mathematica Sinica, English series, Aug., 2007, Vol. 23, No 8, pp.1345-1356.

[13] J. M. Zhan, B. Davvaz, K. P. Shum, Generalized fuzzy Hv-submodules endowed with interval valued membership functions, Inform. Sci. 178 (2008) 3147-3159.

[14] J. M. Zhan, B. Davvaz, K. P. Shum, Generalized fuzzy hyperideals of hyperrings, Computers and Mathematics with Applications 56 (2008) 1732-1740.

[15] J. M. Zhan, B. Davvaz, K. P. Shum, On fuzzy isomorphism theorems of Hypermodules, Soft Comput (2007) 11: 1053-1057.

[16] J. M. Zhan, B. Davvaz, P. Corsini, On Intuitionistic (S, T)-Fuzzy Hv-Submodules of Hv-Modules, Southeast Asian Bulletin of Mathematics (2012) 36:589-601.

[17] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96.

[18] L. A. Zadeh, Fuzzy sets, Inform. And Control 8 (1965) 338-353.

[19] M. Aliakbarnia. Omran, Y. Nasabi, E. Hendukolaie, On Fuzzy Isomorphism Theorem Of Hypernear-modules, Journal of mathematics and computer science, 7 (2013), 112-120.

[20] M. Asghari-Larimi, Homomorphism of intuitionistic (α, β)-fuzzy Hv-submodule, The journal of Mathematics and Computer science Vol.3No.3 (2011) 287-300.

[21] M. Asghari-Larimi, Intuitionistic (α, β)-Fuzzy Hv-Submodules, J. Math. Comput. Sci. 2 (2012), No. 1, 1-14.P. Corsini, in: Aviani (Ed.), Prolegomena of Hypergroup, 2nd ed., 1993.

[22] P. Corsini, Joint spaces, power sets, fuzzy sets, in: M. Stefanescu (Ed.), Proc. 5th Internat. Congress, Algebraic Hyperstructures & Appl. Jasi, Rumani, 1993, Hadronic Press, Florida, 1994.

[23] [24] P. Corsini, T. Vougiouklis, From groupoids to groups through hypergroups, Rendiconti Math. S. VII (9) (1989) 173-181.

[24] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269.

[25] T. Vougiouklis, A new class of hyperstructures, J. Combin. Inf. System Sci., to appear.

[26] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Florida, 1994.

[27] L. Fotea, Fuzzy hypermodules, Computers and Mathematics with Applications 57 (2009) 466-475.

[28] X. Ma, J. M. Zhan, V. L. Fotea, On (fuzzy) isomorphism theorems of Γ-hyperrings, Computers and Mathematics with Applications 60 (2010) 2594-2600.

[29] Y. Yin, J. M. Zhan, D. Xu, J. Wang, The L-fuzzy hypermodules, Computers and Mathematics with Applications 59 (2010) 953-963.

[30] Didier Dubois, Siegfried Gottwald, Peter Hajak, Janusz Kacprzyk and Henri Prade: Terminological difficulties in fuzzy set theory – The case of “Intuitionistic Fuzzy Sets”,

[31] Fuzzy Sets and System, Volume 156, Issue 3, 2005, pages 485-491.

[32] J.Gutierrez Garcia and S.E.Rodabaugh: Order-theoretic, topological,categorical redundancies of interval-valued sets,grey sets,vague sets,interval-valued “intuitionistic” sets,

[33] “intuitionistic” fuzzy sets and topologies, Fuzzy Sets and System, Volume 156, Issue 3, 2005, pages 445-484.

Page | 37


[34] M. Grabisch,C. Labreuche, Bi-capacities (Part 1 and 2) Fuzzy Sets and System 151 (2005) 211-259.

[35] D. Dubois, P.Hajek, H.Prade, Knowledge-driven versus data-driven logics, J. Logic Language Inform. 9 (2000) 65-89.

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Intuitionistic Fuzzy Hv-subgroups