14
IJRRAS 16 (1) July 2013 www.arpapress.com/Volumes/Vol16Issue1/IJRRAS_16_1_06.pdf 50 TERNARY SEMIHYPERGROUPS IN TERMS OF BIPOLAR-VALUED FUZZY SETS Ibtisam Masmali Department of Mathematics, College of Science, Jazan University, Jazan, Kingdom of Saudia Arabia E-mail: [email protected] ABSTRACT This paper represents the concept of bipolar-valued fuzzy ternary subsemihypergroups (left hyperideals, right hyperideals, lateral hyperideals, hyperideals) of ternary semihypergroups. Keywords: Ternary semihypergroups; Hyperideals; Bipolar-valued fuzzy sets; Bipolar-valued fuzzy left (right, lateral) hyperideals. 2010 AMS Classification: 20N20, 20N15, 20M17. 1. INTRODUCTION Hyperstructure theory was introduced in 1934, when F. Marty [18] defined hypergroups, began to analyze their properties and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science and they are studied in many countries of the world. The interesting fact in the hyper structure of any algebraic structure is that 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. A number of algebraist have contributed a lot of papers in this direction and several books have been written on hyperstructure theory, see [1], [2], [3], [23]. A recent book on hyperstructures [1] points out on their applications in rough set theory, cryptography, coding theory, automata, probability, algebraic geometry, lattices, binary relations, graphs and hypergraphs. Another book [3] is devoted especially to the study of hyperring theory and in the similar way many more in the study of hypernearring. We hope to study the concepts in case of hypersemirings. Several kinds of hyperrings are introduced and analyzed till now. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e -hyperstructures and transposition hypergroups. In 1932, Lehmer introduced the concept of a ternary semigroups [16]. A non-empty set X is called a ternary semigroup if there exists a ternary operation X X X X ; written as ) , , ( 3 2 1 x x x 3 2 1 x x x satisfying the following identity for any X x x x x x 5 4 3 2 1 , , , , , ]]. [ [ = ] ] [ [ = ] ] 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 x x x x x x x x x x x x x x x Any semigroup can be reduced to a ternary semigroup. However, Banach showed that a ternary semigroup does not necessarily reduce to a semigroup. For this we can consider some general examples as below } ,0, { = 1 i i T , } , { = 2 i i T are ternary semigroups with zero and without zero element while 1 T and 2 T are not a semigroup with zero or without zero element under complex multiplication. Another general example is the set of negative integers Z or set of negative integers with zero element 0 Z . Los showed that every ternary semigroup can be embedded into a semigroup [17] whereas Borowiec et al. [6] have given a connection between ternary semigroups and some ordinary semigroups. Further Dudek and Mukhin [7] have shown the criterion that an ordinary semigroup induces a ternary semigroup. Hila and Naka [19, 20] worked out on ternary semihypergroups and introduced some properties of hyperideals in ternary semihypergroups, also see [10]. The concept of a fuzzy set, introduced by Zadeh in his classic paper [25], provides a natural framework for generalizing some of the notions of classical algebraic structures, also see [26]. Fuzzy semigroups have been first considered by Kuroki [12]. After the introduction of the concept of fuzzy sets by Zadeh, several researches conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer, logics and many branches of pure and applied mathematics. Fuzzy set theory has been shown to be an useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. In 1971, Rosenfeld [21] defined the concept of fuzzy group. Since

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50

TERNARY SEMIHYPERGROUPS IN TERMS OF BIPOLAR-VALUED

FUZZY SETS

Ibtisam Masmali

Department of Mathematics, College of Science, Jazan University, Jazan,

Kingdom of Saudia Arabia

E-mail: [email protected]

ABSTRACT

This paper represents the concept of bipolar-valued fuzzy ternary subsemihypergroups (left hyperideals, right

hyperideals, lateral hyperideals, hyperideals) of ternary semihypergroups.

Keywords: Ternary semihypergroups; Hyperideals; Bipolar-valued fuzzy sets; Bipolar-valued fuzzy left (right,

lateral) hyperideals.

2010 AMS Classification: 20N20, 20N15, 20M17. 1. INTRODUCTION

Hyperstructure theory was introduced in 1934, when F. Marty [18] defined hypergroups, began to analyze their

properties and applied them to groups. In the following decades and nowadays, a number of different

hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of

pure and applied mathematics by many mathematicians. Nowadays, hyperstructures have a lot of applications to

several domains of mathematics and computer science and they are studied in many countries of the world. The

interesting fact in the hyper structure of any algebraic structure is that 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. A number of algebraist have contributed a lot of papers in this direction and several books have been written

on hyperstructure theory, see [1], [2], [3], [23]. A recent book on hyperstructures [1] points out on their applications

in rough set theory, cryptography, coding theory, automata, probability, algebraic geometry, lattices, binary

relations, graphs and hypergraphs. Another book [3] is devoted especially to the study of hyperring theory and in the

similar way many more in the study of hypernearring. We hope to study the concepts in case of hypersemirings.

Several kinds of hyperrings are introduced and analyzed till now. The volume ends with an outline of applications in

chemistry and physics, analyzing several special kinds of hyperstructures: e -hyperstructures and transposition

hypergroups.

In 1932, Lehmer introduced the concept of a ternary semigroups [16]. A non-empty set X is called a ternary

semigroup if there exists a ternary operation XXXX ; written as ),,( 321 xxx 321 xxx satisfying

the following identity for any Xxxxxx 54321 ,,,, ,

]].[[=]][[=]] 543215432154321 xxxxxxxxxxxxxxx

Any semigroup can be reduced to a ternary semigroup. However, Banach showed that a ternary semigroup does not

necessarily reduce to a semigroup. For this we can consider some general examples as below

},0,{=1 iiT , },{=2 iiT are ternary semigroups with zero and without zero element while 1T and 2T

are not a semigroup with zero or without zero element under complex multiplication. Another general example is the

set of negative integers Z or set of negative integers with zero element

0Z . Los showed that every ternary

semigroup can be embedded into a semigroup [17] whereas Borowiec et al. [6] have given a connection between

ternary semigroups and some ordinary semigroups. Further Dudek and Mukhin [7] have shown the criterion that an

ordinary semigroup induces a ternary semigroup. Hila and Naka [19, 20] worked out on ternary semihypergroups

and introduced some properties of hyperideals in ternary semihypergroups, also see [10].

The concept of a fuzzy set, introduced by Zadeh in his classic paper [25], provides a natural framework for

generalizing some of the notions of classical algebraic structures, also see [26]. Fuzzy semigroups have been first

considered by Kuroki [12]. After the introduction of the concept of fuzzy sets by Zadeh, several researches

conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer,

logics and many branches of pure and applied mathematics. Fuzzy set theory has been shown to be an useful tool to

describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a

degree to which a certain object belongs to a set. In 1971, Rosenfeld [21] defined the concept of fuzzy group. Since

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51

then many papers have been published in the field of fuzzy algebra. Recently fuzzy set theory has been well

developed in the context of hyperalgebraic structure theory. A recent book [1] contains an wealth of applications. In

[5], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup, also see [4, 8, 9]. Yaqoob and others

[24] introduced the concept of rough fuzzy hyperideals in ternary semihypergroups. A several papers are written on

fuzzy sets in several algebraic hyperstructures. The relationships between the fuzzy sets and algebraic

hyperstructures have been considered by Corsini, Davvaz, Leoreanu, Zhan, Zahedi, Ameri, Cristea and many other

researchers.

There are several kinds of fuzzy set extensions in the fuzzy set theory, for example, intuitionistic fuzzy sets, interval-

valued fuzzy sets, vague sets, etc. Bipolar-valued fuzzy set is another extension of fuzzy set whose membership

degree range is different from the above extensions. Lee [13] introduced the notion of bipolar-valued fuzzy sets.

Bipolar-valued fuzzy sets are an extension of fuzzy sets whose membership degree range is enlarged from the

interval [0,1] to 1,1][ . In a bipolar-valued fuzzy set, the membership degree 0 indicate that elements are

irrelevant to the corresponding property, the membership degrees on (0,1] assign that elements somewhat satisfy

the property, and the membership degrees on 1,0)[ assign that elements somewhat satisfy the implicit counter-

property [13, 14].

In [11], Jun and park applied the notion of bipolar-valued fuzzy sets to BCH-algebras. They introduced the concept

of bipolar fuzzy subalgebras and bipolar fuzzy ideals of a BCH-algebra. Lee [15] applied the notion of bipolar fuzzy

subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras. Also some results on bipolar-valued fuzzy BCK/BCI-

algebras are introduced by Saeid in [22].

In this paper, we study the concept of bipolar-valued fuzzy ternary subsemihypergroups (left hyperideals, right

hyperideals, lateral hyperideals, hyperideals) of ternary semihypergroups.

2. TERNARY SEMIHYPERGROUPS In this section we will present some basic definitions of ternary semihypergroups.

A map )(: HHH is called hyperoperation or join operation on the set H , where H is a non-empty

set and }{\)(=)( HH denotes the set of all non-empty subsets of H .

A hypergroupoid is a set H with together a (binary) hyperoperation.

Definition 2.1 A hypergroupoid ),( H , which is associative, that is zyxzyx )(=)( , Szyx ,, , is

called a semihypergroup.

Let A and B be two non-empty subsets of H . Then, we define

.=and=,=,

BaBaAaAabaBABbAa

Definition 2.2 A map )(: HHHHf is called ternary hyperoperation on the set H , where H is a

non-empty set and }{\)(=)( HH denotes the set of all non-empty subsets of H .

Definition 2.3 A ternary hypergroupoid is called the pair ),( fH where f is a ternary hyperoperation on the set

H .

Definition 2.4 A ternary hypergroupoid ),( fS is called a ternary semihypergroup if for all Saaa 521 ,...,, , we

have

)).,,(,,(=)),,,(,(=),),,,(( 543215432154321 aaafaafaaaafafaaaaaff

Definition 2.5 Let ),( fS be a ternary semihypergroup. Then S is called a ternary hypergroup if for all

Scba ,, , there exist Szyx ,, such that:

).,,(),,(),,( zbafbyafbaxfc

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Definition 2.6 Let ),( fS be a ternary semihypergroup and T a non-empty subset of S . Then T is called a

subsemihypergroup of S if and only if .),,( TTTTf

Definition 2.7 A non-empty subset I of a ternary semihypergroup S is called a left (right, lateral ) hyperideal of

S if

).),,(,),,((),,( ISISfISSIfIISSf

Example 2.2 [24] Let },,,,,{0,= gedcbaS and zyxzyxf )(=),,( for all Szyx ,, , where is

defined by the table:

ggddggg

geedccgeee

ddddddd

dccdccdccc

ggddbbb

geedccbaaa

gedcba

0

},{},{},{0

0

},{},{},{0

0

},{},{},{0

00000000

0

Then ),( fS is a ternary semihypergroup. Clearly, },,{= dcA },,,{= gedcB and S are lateral hyperideals

of S .

In what follows, let S denote a ternary semihypergroup unless otherwise specified. For simplicity we write

),,( cbaf as .abc

Definition 2.8 Let S be a ternary semihypergroup. A non-empty subset T of S is called prime subset of S if for

all ,,, Szyx Txyz implies Tx or Ty or Tz . A ternary subsemihypergroup T of S is called

prime ternary subsemihypergroup of S if T is a prime subset of S . Prime left hyperideals, prime right

hyperideals, prime lateral hyperideals and prime hyperideals of S are defined analogously.

3. BIPOLAR-VALUED FUZZY HYPERIDEALS OF TERNARY SEMIHYPERGROUPS

First we will recall the concept of bipolar-valued fuzzy sets.

Definition 3.1 [14] Let X be a nonempty set. A bipolar-valued fuzzy subset (BVF-subset, in short) of X is an

object having the form

.:)(),(,= Xxxxx

Where 0,1]: XB and 1,0]: X .

The positive membership degree )(x

B denotes the satisfaction degree of an element x to the property

corresponding to a bipolar-valued fuzzy set Xxxxx :)(),(,= , and the negative membership

degree )(x

denotes the satisfaction degree of x to some implicit counter property of

Xxxxx :)(),(,= . For the sake of simplicity, we shall use the symbol

,= for the

bipolar-valued fuzzy set .:)(),(,= Xxxxx

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Definition 3.2

111 ,= and

222 ,= be two BVF-subsets of a ternary semihypergroup .S

The symbol 21 will means the following

.forall)()(=)(2121

Sxxxx

.forall)()(=)(2121

Sxxxx

The symbol 21 will means the following

.forall)()(=)(2121

Sxxxx

.forall)()(=)(2121

Sxxxx

Definition 3.3 Let ,,=11

1

222 ,= and

333 ,= be three BVF-subsets of a

ternary semihypergroup .S Then their product 321 is defined by

,:)(),(,=321321

321 Stttt

where

otherwise0

if)}}(),(),({min{sup=)( 321

321

xyztzyxt xyzt

and

otherwise0

if)}}(),(),({max{inf=)( 321

321

xyztzyxt xyzt

for some Szyx ,, and for all .St

Definition 3.4 Let S be a ternary semihypergroup. A BVF-subset

,= of S is called

(1) a BVF-ternary subsemihypergroup of S if

)}(),(),({max)(supand)}(),(),({min)(inf zyxtzyxtxyztxyzt

for all Szyx ,, .

(2) a BVF-left hyperideal of S if

)()(supand)()(inf ztztxyztxyzt

for all Szyx ,, .

(3) a BVF-right hyperideal of S if

)()(supand)()(inf xtxtxyztxyzt

for all Szyx ,, .

(4) a BVF-lateral hyperideal of S if

)()(supand)()(inf ytytxyztxyzt

for all Szyx ,, .

(5) a BVF-hyperideal of S if

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)}(),(),({min)(supand)}(),(),({max)(inf zyxtzyxtxyztxyzt

for all Szyx ,, .

Example 3.1 Let },,,{0,= dcbaS and zyxzyxf )(=),,( for all Szyx ,, , where is defined by the

table:

bbddddccbaabbbbcbbaabaaaabcdcccccc 0},{},{00},{},{00000000|

Then ),( fS is a ternary semihypergroup. Define a bipolar-valued fuzzy subset

,= in S as follows:

},{0.2},{0.70=0.9=)(and},{}0.3,{00.5=0.8=)( dcifxbaifxifxllxdcifxbaifxifxllx

BB

By routine calculations it can be seen that the BVF-subset

,= is a BVF-hyperideal of .S

Theorem 3.5 If ii}{B is a family of BVF-ternary subsemihypergroups (BVF-left hyperideals, BVF-right

hyperideals, BVF-lateral hyperideals, BVF-hyperideals) of .S Then i

i

is a BVF-ternary subsemihypergroup

(BVF-left hyperideal, BVF-right hyperideal, BVF-lateral hyperideal, BVF-hyperideal) of ,S where

),(=

ii

ii

i

i

and

Sxixxii

i

,:)(inf=)(

.,:)(sup=)( Sxixxii

i

Proof. Consider ii}{ is a family of BVF-ternary subsemihypergroups of S . Let Hzyx ,, . Then for every

,xyzt we have

)(inf=)(inf tt

ixyztii

ixyzt

)(),(),(min zyxiii

i

)(),(),(min= zyx

ii

ii

ii

and

)(sup=)(sup zz

ixyzti

iixyzt

)(),(),(max zyxiii

i

.)(),(),(max=

zyx

ii

ii

ii

Hence this shows that i

i

is a BVF-ternary subsemihypergroups of .S The other cases can be seen in a similar

way.

Theorem 3.6 If ii}{ is a family of BVF-ternary subsemihypergroups (BVF-left hyperideals, BVF-right

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hyperideals, BVF-lateral hyperideals, BVF-hyperideals) of .S Then i

i

is a BVF-ternary subsemihypergroup

(BVF-left hyperideal, BVF-right hyperideal, BVF-lateral hyperideal, BVF-hyperideal) of ,S where

),(=

ii

ii

i

i

and

Sxixxii

i

,:)(sup=)(

.,:)(inf=)( Sxixxii

i

Proof. The proof is similar to the proof of Theorem 3.5.

Theorem 3.7 Let

111 ,= be a BVF-right hyperideal,

222 ,= a BVF-lateral hyperideal

and

333 ,= a BVF-left hyperideal of a ternary semihypergroup S . Then

.321321

Proof. Let

111 ,= be a BVF-right hyperideal,

222 ,= a BVF-lateral hyperideal and

333 ,= a BVF-left hyperideal of a ternary semihypergroup S . If there do not exist Szyx ,, such

that ,xyzt then

.0=321321

tt

and

.0=321321

tt

If there exist Szyx ,, such that ,xyzt then

)}}(),(),({min{sup=321321

zyxtxyzt

)}}(inf),(inf),(inf{min{sup321

tttxyztxyztxyztxyzt

)}(),(),({min=321

ttt

)()()(=321

ttt

,=321

t

and

)}}(),(),({max{inf=321321

zyxtxyzt

)}}(sup),(sup),(sup{max{inf321

tttxyztxyztxyztxyzt

)}(),(),({max=321

ttt

)()()(=321

ttt

.=321

t

Thus .321321

Let us consider

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SSS ,=

Sxxxxxx inforall1=)(and1=)(:)(),(,=

be a BVF-subset of a ternary semihypergroup S , and

,= will be carried out in operations with a

BVF-subset

,= such that

S and

S will be used in collaboration with

B and

B respectively.

Theorem 3.8 Let

111 ,= be a BVF-right hyperideal and

222 ,= a BVF-left hyperideal of

a ternary semihypergroup S . Then 2121 .

Proof. Let

111 ,= be a BVF-right hyperideal and

222 ,= a BVF-left hyperideal of a

ternary semihypergroup S . Let Sa . If there do not exist Szyx ,, such that ,xyzt then

,1=2121

tt

and

.1=2121

tt

If there exist Szyx ,, such that ,xyzt then

)}}(),(),({min{sup=3121

zyxtxyzt

)}}(),1,({min{sup=31

zxxyzt

)}}(),({min{sup=31

zxxyzt

)(inf),(infminsup31

ttxyztxyztxyzt

)}(),({min=31

tt

)()(=31

tt

),)((=31

t

and

)}}(),(),({max{inf=3121

zyxtxyzt

)}}(1,),({max{inf=31

zxxyzt

)}}(),({max{inf=31

zxxyzt

)(sup),(supmaxinf

31tt

xyztxyztxyzt

)}(),({max=31

tt

)()(=31

tt

),)((=31

t

Thus .2121

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Theorem 3.9 A BVF-subset

,= of a ternary semihypergroup S is a BVF-ternary subsemihypergroup

of S if and only if

.

Proof. Suppose

,= of a ternary subsemihypergroup S . If there do not exist Szyx ,, such that

,xyzt then

,0= tt

and

,0= tt

If there exist Szyx ,, such that ,xyzt then

)}}(),(),({min{sup= zyxtxyzt

),(=)(infsup ttxyztxyzt

and

)}}(),(),({max{inf= zyxtxyzt

).(=)(supinf ttxyztxyzt

Hence .

Conversely, assume that . Then for all Szyx ,, , we have xyzt such that

ttxyztxyzt

inf)(inf

},,{mininf= cbaabcxyz

},,,{min cba

and

ttxyztxyzt

sup)(sup

},,{maxsup= cbaabcxyz

}.,,{max cba

Hence

,= is a BVF-ternary subsemihypergroup .S

Theorem 3.10 A BVF-subset

,= of a ternary semihypergroup S is a BVF-left hyperideal (BVF-right

hyperideal, BVF-lateral hyperideal) of S if and only if ( , ).

Proof. Let

,= be a BVF-left hyperideal of S and St . Let us suppose that there exist Szyx ,,

such that xyzt . Then, since

,= is a BVF-left hyperideal of S , we have

)}](),(),({min[sup=))(( zyxtxyzt

),(sup=)}]({1,1,min[sup= zzxyztxyzt

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and

)}](),(),({max[inf=))(( zyxtxyzt

)(inf=)}](1,1,{max[inf= zzxyztxyzt

In case of

,= is a BVF-left hyperideal of S ,

).()(supand)()(inf zrzrxyzrxyzr

So, in particular, )()( az and )()( az for all xyza . Hence )()(sup azxyza

and

)()(inf azxyza

. Thus, ))(()( aa and ))(()( aa . If there do not

exist Szyx ,, such that yxa , then )(0=))(( aa and

)(0=))(( aa . Hence we get .

Conversely, let Szyx ,, and xyza . Then, ))(()(inf aaxyza

and

))(()(sup aaxyza

. We have,

)}](),(),({min[sup=))(( zyxaxyza

)}(),(),({min zyx

),(=)}({1,1,min= zz

and

)}](),(),({max[inf=))(( zyxaxyza

)}(),(),({max zyx

),(=)}(1,1,{max= zz

Consequently,

).()(supand)()(inf zazaxyzaxyza

Hence,

,= is a BVF-left hyperideal of S . The other case can be seen in a similar way.

Proposition 3.11 The product of three BVF-left hyperideals (BVF-right hyperideals) of a ternary semihypegroup

,S is again a BVF-left hyperideal (BVF-right hyperideal) of .S

Proof. Let ,,=11

1

222 ,= and

333 ,= be three BVF-left hyperideals of a

ternary semihypergroup ,S then by Theorem 3.10,

321321 )(=)( SSSS

321

This completes the proof. The other case can be seen in a similar way.

Theorem 3.12 Let S be a ternary semihypergroup and A a non-empty subset of S . The following statements hold

true:

(1) A is a ternary subsemihypergroup of S if and only if

AAA ,= is a BVF-ternary

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subsemihypergroup of S .

(2) A is a left hyperideal (right hyperideal, lateral hyperideal, hyperideal) of S if and only if

AAA ,=

is a BVF-left hyperideal (BVF-right hyperideal, BVF-lateral hyperideal, BVF-hyperideal) of S .

Proof. (1). Let us assume that A is a ternary subsemihypergroup of S . Let Szyx ,, .

Case 1. Azyx ,, . Since A is a ternary subsemihypergroup of S , we have Axyz . Then,

)},(),(),({min1=)(inf zyxtAxyzt

and

)}.(),(),({max1=)(sup zyxtA

xyzt

Case 2. Ax or Ay or Az . Thus 0=)(xA

or 0=)(yA

or 0=)(zA

B . Therefore,

),(inf0=)}(),(),({min tzyxAxyzt

Also 0=)(xA

B or 0=)(yA

B or 0=)(zA

B . Therefore,

).(sup0=)}(),(),({max tzyxA

xyzt

BBBB

Conversely, let Azyx ,, . We have 1=)(=)(=)( zyxAAA

and

1=)(=)(=)( zyxAAA

BBB . Since

AAA BBB ,= is a BVF-ternary subsemihypergroup of S , so

1,=)}(),(),({min)(inf zyxtAxyzt

and

1.=)}(),(),({max)(sup

zyxtA

xyzt

Hence Axyz .

(2). Let us assume that A is a left hyperideal of S . Let Szyx ,, .

Case 1. Az . Since A is a left hyperideal of S , then Axyz . Then

1=)(supand1=)(inf

ttA

xyztAxyzt

Therefore,

)()(supand)()(inf ztztAA

xyztAAxyzt

Case 2. Az . We have 0=)(zA

B and 0=)(zA

B . Hence,

)()(supand)()(inf ztztAA

xyztAAxyzt

BBBB

Conversely, let Syx , and Az . Since

AAA ,= is a fuzzy left hyperideal of S and ,Az

1.=)()(supand1=)()(inf

ztztAA

xyztAAxyzt

Thus Axyz . The remaining parts can be seen in similarly way.

For any 0,1]t and 1,0].s Let

,= be a BVF-set in S , the set

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})(,)(:{=),;( sxtxSxstU

is called the BVF-level set of .,=

Theorem 3.13 Let

,= be a BVF-subset of a ternary semihypergroup S . The following statements

hold true:

(1)

,= is a BVF-ternary subsemihypergroup of S if and only if for all 0,1]t and 1,0],s the

set ),;( stU is either empty or a ternary subsemihypergroup of S .

(2)

,= is a BVF-left hyperideal (BVF-right hyperideal, BVF-lateral hyperideal, BVF-hyperideal) of

S if and only if for all 0,1]t and 1,0],s the set ),;( stU is either empty or a left hyperideal (right

hyperideal, lateral hyperideal, hyperideal) of S .

Proof. (1). Let us assume that

,= is a BVF-ternary subsemihypergroup of S . Let 0,1]t and

1,0]s such that ),;( stU . Let ).,;(,, stUzyx So tzyx )(),(),( and

szyx )(),(),( . Thus,

.)(),(),(maxand)}(),(),({min szyxtzyx

Since

,= is a BVF-ternary subsemihypergroup of S ,

.)(supand)(inf shthxyzhxyzh

BB

Hence ),;( stUxyz B .

Conversely, let Szyx ,, . Let we take )}(),(),({min= zyxt

BBB and

)}(),(),({max= zyxs

BBB . Then tzyx )(),(),( BBB and szyx )(),(),( BBB . Thus

),;(,, stUzyx B . Since ),;( stU B is a ternary subsemihypergroup of S , ),;( stUxyz . Thus,

)},(),(),({min=)(inf zyxthxyzh

and

)}.(),(),({max=)(sup zyxshxyzh

(2). Let us assume that

,= is a BVF-left hyperideal of S . Let 0,1]t and 1,0]s such that

),;( stU . Let Syx , and ),;( stUz . Thus,

.)()(supand)()(inf szhtzhxyzhxyzh

Therefore ),;( stUxyz .

Conversely, let Szyx ,, . Let we take )(= zt

and )(= zs

. Thus ),;( stUz , this implies

),;( stU . By assumption, we have ),;( stU is a left hyperideal of S . So ),;( stUxyz . Therefore

thxyzh

)(inf and shxyzh

)(sup . Thus,

).()(supand)()(inf zhzhxyzhxyzh

The remain parts can be proved in a similar way.

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Definition 3.14 A BVF-subset

,= of S is called a prime BVF-subset of S if

)}(),(),({max)(inf zyxtxyzt

and

)}(),(),({min)(sup zyxtxyzt

for all Szyx ,, . A BVF-ternary subsemihypergroup

,= of S is called a prime BVF-ternary

subsemihypergroup of S if

,= is a prime BVF-subset of S . Prime BVF-left hyperideals, prime

BVF-right hyperideals, prime BVF-lateral hyperideals and prime BVF-hyperideals of S are defined analogously.

Theorem 3.15 Let S be a ternary semihypergroup and A a non-empty subset of S . The following statements hold

true:

(1) A is a prime subset of S if and only if

AAA ,= is a prime BVF-subset of S .

(2) A is a prime ternary subsemihypergroup (prime left hyperideal, prime right hyperideal, prime lateral hyperideal,

prime hyperideal) of S if and only if

AAA ,= is a prime BVF-ternary subsemihypergroup (prime

BVF-left hyperideal, prime BVF-right hyperideal, prime BVF-lateral hyperideal, prime BVF-hyperideal) of S .

Proof. (1). Let us assume that A is a prime subset of S . Let Szyx ,, .

Case 1. Axyz . Since A is prime, Ax or Ay or Az . Thus,

),(inf1=)}(),(),({max tzyxAxyztAAA

and

).(sup1=)}(),(),({min tzyxA

xyztAAA

Case 2. Axyz . Thus,

)},(),(),({max0=)(inf zyxtAAAAxyzt

and

)}.(),(),({min0=)(sup zyxtAAAA

xyzt

Conversely, let Szyx ,, such that Axyz . Thus 1=)(tA

and 1=)( tA

for all xyzt . Since

AAA ,= is prime, 1=)}(),(),({max zyx

AAA

, this implies 1=)(xA

or

1=)(yA

or 1=)(zA

. Also, 1,=)}(),(),({min zyxAAA

this implies 1=)( xA

or

1=)( yA

or 1=)( zA

. Hence Ax or Ay or Az .

(2) It follows from (1) and Theorem 3.12.

Theorem 3.16 Let S be a ternary semihypergroup and

,= be a BVF-subset of S . The following

statements hold true:

(1)

,= is prime BVF-subset of S if and only if for all 0,1]t and 1,0],s the set ),;( stU

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is either empty or a prime subset of S .

(2)

,= is a prime BVF-ternary subsemihypergroup (prime BVF-left hyperideal, prime BVF-right

hyperideal, prime BVF-lateral hyperideal, prime BVF-hyperideal) of S if and only if for all 0,1]t and

1,0],s the set ),;( stU is either empty or a prime ternary subsemihypergroup (prime left hyperideal, prime

right hyperideal, prime lateral hyperideal, prime hyperideal) of S .

Proof. (1) Let us assume that

,= is a prime BVF-subset of S . Let 0,1]t and 1,0]s . Let us

suppose that ),;( stU . Let Szyx ,, such that ),;( stUxyz . Thus,

.)(supand)(inf shthxyzhxyzh

Since

BBB ,= is prime, tx )( or ty )( or tz )( also sx )( or sy )( or

sz )( . This implies ),;( stUx or ),;( stUy or ),;( stUz .

Conversely, let Szyx ,, . Let we take )(inf= htxyzh

and )(sup= hs

xyzh

. Then ),;( stUxyz . Since

),;( stU is prime, ),;( stUx or ),;( stUy or ),;( stUz . Then tx )( or ty )( or

,)( tz

also sx )( or sy )( or sz )( . Hence,

),(inf=)}(),(),({max htzyxxyzh

and

).(sup=)}(),(),({min hszyxxyzh

(2). It follows from (1) and Theorem 3.15.

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