ternary semihypergroups in terms of bipolar-valued … · hyperstructure theory was introduced in...
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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 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
IJRRAS 16 (1) ● July 2013 Masmali ● Bipolar-Valued Fuzzy Sets
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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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