New types of bipolar fuzzy sets in -semihypergroups Naveed Yaqoob1; , Muhammad Aslam2 ; Inayatur Rehman3 and M. M. Khalaf 4 1;4
Department of Mathematics, College of Science in Al-Zul…, Majmaah University, Al-Zul…, Saudi Arabia 2
Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia 3
Department of Mathematics and Science,
College of Arts and Applied Sciences, Dhofar University, Oman
Abstract: The notion of bipolar fuzzy set was initiated by Lee (2000) as a generalization of the notion fuzzy sets and intuitionistic fuzzy sets, which have drawn attention of many mathematicians and computer scientists. In this paper, we initiate a study on bipolar ( ; )-fuzzy sets in -semihypergroups. By using the concept of bipolar ( ; )-fuzzy sets (Yaqoob and Ansari, 2013), we introduce the notion of bipolar ( ; )fuzzy sub -semihypergroups ( -hyperideals and bi- -hyperideals) and discuss some basic results on bipolar D E + ( ; )-fuzzy sets in -semihypergroups. Furthermore, we de…ne the bipolar fuzzy subset B = ; B B and prove that if B =
+ B;
is a bipolar ( ; )-fuzzy sub -semihypergroup (resp., -hyperideal and D E + bi- -hyperideal) of H; then B = ; is also a bipolar ( ; )-fuzzy sub -semihypergroup (resp., B B B
-hyperideal and bi- -hyperideal) of H.
2010 AMS Classi…cation: 20N20, 03E72. Keywords:
-semihypergroups, bipolar fuzzy sets, bipolar ( ; )-fuzzy -hyperideals.
Corresponding author. E-mail address:
[email protected]
1
1
Introduction
Uncertainty is an attribute of information and uncertain data are presented in various domains. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh (1965), where he introduced the notion of a fuzzy subset of a non-empty set X, as a function from X to [0; 1]. Several researchers conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer science, logics and many branches of pure and applied mathematics. Rosenfeld (1971) de…ned the concept of fuzzy group. Since then many papers have been published in the …eld of fuzzy algebra, for instance, Kuroki (1979, 1981) applied fuzzy set theory to the ideal theory of semigroups. Shabir (2005) studied fully fuzzy prime semigroups. Kehayopulu and Tsingelis (2007) studied fuzzy ideals in ordered semigroups. Akram and Dar (2005) studied fuzzy d-algebras. 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 di¤erent from the above extensions. Lee (2000) 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 (Lee, 2004). Jun and park (2009) 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 (2009) 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 were introduced by Saeid (2009). Akram et al. (2011, 2012a, 2012b) applied bipolar fuzzy sets to Lie algebras. The theory of hyperstructure was born in 1934 when Marty (1934) de…ned hypergroups, began to analysis their properties and applied them to groups, rational algebraic functions. In the following decades and 2
nowadays, a number of di¤erent 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. 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 recent book on hyperstructures (Corsini and Leoreanu, 2003) points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Recently, Anvariyeh et al. (2010), Heidri et al. (2010), Mirvakili et al. (2013), Hila et al. (2012) and Abdullah et al. (2011, 2012) introduced the notion of -semihypergroups as a generalization of semigroups, semihypergroups and of
-semigroups. They proved some results in this respect and presented many interesting
examples of -semihypergroups. Fuzzy set theory has been well developed in the context of hyperalgebraic structure theory (Davvaz, 2010). Several authors investigated the di¤erent aspects of fuzzy semihypergroups and fuzzy -semihypergroups, for instance, Aslam et al. (2012), Corsini et al. (2011), Davvaz (2000, 2006, 2007), Ersoy et al. (2012, 2013a, 2013b), Shabir and Mahmood (2013, 2015) and Yaqoob et al. (2014). Yao introduced a new type of fuzzy sets called ( ; )-fuzzy sets, and studied ( ; )-fuzzy normal sub…elds (2005). Several authors extended Yao’s idea and continued their researches in applying ( ; )-fuzzy sets on di¤erent algebraic structures. Coumaressane (2010) characterized near-rings by their ( ; )-fuzzy quasiideals. Shabir et al. (2011) characterized semigroups by the properties of their fuzzy ideals with thresholds and Khan et al. (2012) characterized ordered semigroups by their ( ; )-fuzzy bi-ideals. Li and Feng (2013) extended the idea of ( ; )-fuzzy sets in intuitionistic fuzzy sets and studied intuitionistic fuzzy ( ; )-fuzzy sets in -semigroups. Yaqoob and Ansari (2013) also extended the idea of ( ; )-fuzzy sets in bipolar fuzzy sets and studied bipolar ( ; )-fuzzy ideals in ternary semigroups. In this paper, we study bipolar ( ; )-fuzzy sets in -semihypergroups and introduce the notion of bipolar ( ; )-fuzzy sub -semihypergroups ( -hyperideals and bi- -hyperideals).
3
2
Preliminaries and basic de…nitions
In this section, we will recall the basic terms and de…nitions from the hyperstructure theory and bipolar fuzzy sets.
De…nition 2.1 A map
:H
H ! P (H) is called hyperoperation or join operation on the set H, where
H is a non-empty set and P (H) = P(H)nf;g denotes the set of all non-empty subsets of H. A hypergroupoid is a set H with together a (binary) hyperoperation.
De…nition 2.2 A hypergroupoid (H; ), which is associative, that is x (y z) = (x y) z, 8x; y; z 2 H, is called a semihypergroup.
De…nition 2.3 (Anvariyeh et al., 2010) Let H and semihypergroup if every ;
2
2
be two non-empty sets. Then H is called a
is a hyperoperation on H, i:e:, x y
-
H for every x; y 2 H, and for every
and x; y; z 2 H we have x (y z) = (x y) z.
If every
2
is an operation, then H is a -semigroup. If (H; ) is a hypergroup for every
2 , then H
is called a -hypergroup. Let A and B be two non-empty subsets of H. Then we de…ne A B=
[
A B=
2
Let H be a -semihypergroup and H if x y
[
fa b j a 2 A; b 2 B and
2 g.
2 . A non-empty subset A of H is called a sub -semihypergroup of
A for every x; y 2 A. A -semihypergroup H is called commutative if for all x; y 2 H and
2 ,
we have x y = y x. A non-empty subset A of a -semihypergroup H is a right (left) -hyperideal of H if A H
A (H A
A), and is a -hyperideal of H if it is both a right and a left -hyperideal. A non-empty
subset B of a -semihypergroup H is called bi- -hyperideal of H if (1) B B
B; (2) B H B
B:
Lee (2000) introduced the concept of bipolar fuzzy set de…ned on a non-empty set X as objects having the form: B=
x;
+ B (x);
B (x)
4
:x2X :
Where
+ B
: X ! [0; 1] and
B
: X ! [ 1; 0]. The positive membership degree
+ B (x)
denotes the satisfaction
degree of an element x to the property corresponding to a bipolar fuzzy set B, and the negative membership degree
B (x)
denotes the satisfaction degree of x to some implicit counter property of B. For the sake of + B;
simplicity, we shall use the symbol B = De…nition 2.4 A =
+ A;
+ B;
and B =
A
for the bipolar fuzzy set B =
B
x;
+ B (x);
be two bipolar fuzzy subsets of a
B
B (x)
:x2X :
-semihypergroup H:
Then for all x 2 H; their intersection A \ B is de…ned by A\B =
x;
+ A\B (x);
A[B (x)
:x2H ;
where + A\B (x)
=
A[B (x)
=
+ A
^
+ B
A
_
B
(x) =
+ A (x)
^
+ B (x);
(x) =
A (x)
_
B (x).
Their union A [ B is de…ned by A[B =
x;
+ A[B (x);
A\B (x)
:x2H ;
where
+ A;
De…nition 2.5 Let A = Then their product A
+ A[B (x)
=
A\B (x)
=
A
and B =
+ A
_
+ B
A
^
B
+ B;
(x) =
+ A (x)
_
+ B (x);
(x) =
A (x)
^
B (x).
be two bipolar fuzzy subsets of a -semihypergroup H:
B
B is de…ned by A
where + A
B (x) =
and A
B (x) =
B=
x;
+ A
B (x);
8 > > < sup f
+ A (y)
8 > >
> : 0
inf f
x2y z
> > : 0
:x2H ;
if x 2 y z; 8 2 otherwise,
_
B (z)g
if x 2 y z; 8 2 otherwise,
5
for x; y; z 2 H:
+ B;
De…nition 2.6 Let B = 1) the sets Bt+ = fx 2 X j B=
+ B;
>
+ B; (t;s)
3) the set XB
3
H
+ B (x)
Bt+ = fx 2 X j
t-cut of B =
4) the set
be a bipolar fuzzy set and (s; t) 2 [ 1; 0] tg and Bs = fx 2 X j + B;
and the negative s-cut of B =
B
2) the sets
B
B
> tg and
t;
B (x)
sg; which are called positive t-cut of
, respectively, B (x)
+ B;
and the strong negative s-cut of B =
= fx 2 X j
(t;s)
XB
+ B (x)
B
B (x)
[0; 1]. De…ne:
B
< sg; which are called strong positive
, respectively,
sg is called an (s; t)-level subset of B, < sg is called a strong (s; t)-level subset of B.
Bipolar ( ; )-fuzzy -hyperideals in -semihypergroups
Yaqoob and Ansari (2013) introduced the notion bipolar ( ; )-fuzzy ideals in ternary semigroups. In this section, we introduce and study the notion bipolar ( ; )-fuzzy
-hyperideals (resp., interior
-hyperideals
and bi- -hyperideals) in -semihypergroups and discuss some related properties. In what follows, let 1
< 0:7 > > : 0:6
8 > > < > > :
+ B;
0:9 0:5 B
De…nition 3.3 A bipolar fuzzy subset B =
if 0 < x < if
1 2k
x > 0:5 if t 2 fx; yg > > > < + 0:6 if t = z B (t) = > > > > > > : 0:8 if t = w
Then by routine calculation, B =
+ B;
B
De…nition 3.6 A bipolar ( ; )-fuzzy sub
and
B
(t) =
8 > > > > > > < > > > > > > :
B
on M as:
0:6 if t 2 fx; yg 0:8 if t = z 0:9 if t = w
is a bipolar (0:3; 0:4)-fuzzy bi- -hyperideal of M:
-semihypergroup B = 8
+ B;
B
of H is called a bipolar (1; 2)-
fuzzy bi- -hyperideal of H if max and min
inf
(
a2x w (y z)
sup
a2x w (y z)
for all x; y; z; w 2 H and
; ;
)
B (a);
minf
+ B (x);
+ B (y);
maxf
B (x);
B (y);
+ + g B (z); B (z);
g;
2 . + B;
Theorem 3.7 A bipolar fuzzy subset B = -semihypergroup (resp., left
+
+ B (a);
-hyperideal, right (t;s)
-hyperideal) of H if and only if ; = 6 HB
of a
B
-semihypergroup H is a bipolar ( ; )-fuzzy sub
-hyperideal, interior
is a sub
-hyperideal, bi- -hyperideal, (1; 2)-
-semihypergroup (resp., left
-hyperideal, right
hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H for all (s; t) 2 [ + B;
Proof. Let B = [
;
B=
)
(
+ B;
B
+
;
+
be a bipolar ( ; )-fuzzy sub
B
(t;s)
] and x; y 2 HB
+ B (x)
: Then
;
) (
+
;
+
]:
-semihypergroup of H. Let x; y 2 H; (s; t) 2
t and
+ B (y)
t, also
B (x)
s and
B (y)
s: As
is a bipolar ( ; )-fuzzy sub -semihypergroup of H: Therefore, max
+ B (z);
inf
z2x y
+
minf
+ B (x);
+ + g B (y);
+
min t; t;
= t;
and min
sup z2x y
This implies that
inf
z2x y
B (z);
+ B (z)
maxf
t and sup
B (x);
B (z)
z2x y
B (y);
g
max s; s; (t;s)
s: Thus x y
HB
= s:
semihypergroup of H. (t;s)
Conversely, suppose that HB
is a sub -semihypergroup of H: Let x; y 2 H such that
max
inf
z2x y
+ B (z);
+
< minf
+ B (x);
+ + g; B (y);
> maxf
B (x);
B (y);
and min
sup
B (z);
z2x y
Then there exist (s; t) 2 [
; max
)
( inf
z2x y
+
;
+
g:
] such that
+ B (z);
+
s
maxf
t and inf
+ B (z)
< t, also
z2x y + B (x)
This shows that (t;s)
Thus x; y 2 HB
+ B (y)
t;
(t;s)
; since HB
z2x y
B (x);
B (y);
B (x)
g:
s;
B (y)
+ B (z)
z2x y
max
inf
z2x y
< t and sup z2x y +
+ B (z);
B (z)
z2x y (t;s)
is a sub -semihypergroup of H: Therefore x z
but this is a contradiction to inf
s and sup HB
B (z)
> s:
for some z 2 x y;
> s: Thus,
minf
+ B (x);
+ + g; B (y);
maxf
B (x);
B (y);
and min
sup z2x y
+
Hence B = (
;
B)
B (z);
g:
is a bipolar ( ; )-fuzzy sub -semihypergroup of H. The other cases can be seen in a
similar way.
Corollary 3.8 Every bipolar fuzzy sub
-semihypergroup (resp., left
rior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) B = (resp., left with
+
-hyperideal, right
= 0;
+
= 1 and
-hyperideal, interior
= 0;
=
+ B;
-hyperideal, right
-hyperideal, inte-
is a bipolar ( ; )-fuzzy sub -semihypergroup
B
-hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H
1:
Theorem 3.9 If a bipolar fuzzy subset B =
+ B;
left
-hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H: Then
-hyperideal, right
the set B =
>
-hyperideal, interior
B ++ ;< B
is a sub
is a bipolar ( ; )-fuzzy sub -semihypergroup (resp.,
B
-semihypergroup (resp., left
-hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H; where fx 2 H j
B (x)
-hyperideal, right
B ++ = fx 2 H j
-hyperideal, interior
+ B (x)
+
>
is a bipolar ( ; )-fuzzy sub -semihypergroup of H: Let x; y 2 H such
B +
+ B (y)
;
>
+
and
B (x)
B (z);
maxf
and sup
B (z)
z2x y
> + < if x 2 A + (x) = B > > : + if x 2 = A;
and
B (x)
=
8 > >
> :
if x 2 = A;
is a bipolar ( ; )-fuzzy sub -semihypergroup (resp., left -hyperideal, right -hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H:
Proof. Suppose that A is a sub -semihypergroup of H. Let x; y 2 H be such that x; y 2 A; then x y for
+
+ B (z)
2 : Hence inf
z2x y
and sup
B (z)
z2x y
max
+ B (z);
inf
z2x y
+
A
: Therefore +
= min
+ B (x);
+ + B (y);
;
= max
B (x);
B (y);
:
and min If x 2 = A or y 2 = A; then min
z2x y
sup
B (z);
+ B (x);
+ + B (y);
max
inf
z2x y
+ B (z);
=
+
+
and max
+
B (x);
B (y);
=
= min
+ B (x);
+ + B (y);
;
= max
B (x);
B (y);
:
: Thus
and min
sup z2x y
Consequently B =
+ B;
B
B (z);
is a bipolar ( ; )-fuzzy sub -semihypergroup of H:
Conversely: Let x; y 2 A: Then
+ B (x)
+
;
+
+ B (y)
and
B (x)
;
B (y)
: As B =
a bipolar ( ; )-fuzzy sub -semihypergroup of H; therefore max
inf
z2x y
+ B (z);
+
min
+ B (x);
11
+ + B (y);
min
+
;
+
;
+
=
+
;
+ B;
B
is
and min
sup z2x y
This implies that x y
B (z);
max
B (x);
A: Hence A is a sub
B (y);
max
;
;
=
:
-semihypergroup of H. The other cases can be seen in a
similar way.
Theorem 3.11 A non-empty subset A of H is a sub
-semihypergroup (resp., left
-hyperideal, right
hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H if and only if BA =
+ BA ;
BA
is
a bipolar ( ; )-fuzzy sub -semihypergroup (resp., left -hyperideal, right -hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H:
Proof.
Let A be a sub
+ BA ;
-semihypergroup of H: Then BA =
BA
is a bipolar fuzzy sub
-
semihypergroup of H and by Corollary 3.8, BA is a bipolar ( ; )-fuzzy sub -semihypergroup of H: + BA (x)
Conversely, let x; y 2 H be such that x; y 2 A: Then
=
+ BA (y)
= 1 and
BA (x)
=
BA (y)
=
1:
Since BA is a bipolar ( ; )-fuzzy sub -semihypergroup of H: Therefore max
inf
z2x y
+ B (z);
+
min =
+ + BA (y);
+ BA (x);
min 1; 1; 1;
+
=
+
;
and min
sup
B (z);
z2x y
= It implies that inf
z2x y
BA =
+ BA ;
BA
and sup z2x y
B (z)
BA (x);
BA (y);
max
1; 1; 1;
: Thus z 2 x y
=
:
A for z 2 x y and
2 : Therefore
is a sub -semihypergroup of H: The other cases can be seen in a similar way.
Theorem 3.12 Let A = ( ; )-fuzzy right
+
+ B (z)
max
+ A;
A
-hyperideal of a
be a bipolar ( ; )-fuzzy left -hyperideal and B = -semihypergroup H. Then A
of H.
Proof. The proof is straightforward. 12
+ B;
B
B is a bipolar ( ; )-fuzzy
be a bipolar -hyperideal
Lemma 3.13 Intersection of any family of bipolar ( ; )-fuzzy sub -semihypergroups (resp., left -hyperideals, right -hyperideals, interior -hyperideals, bi- -hyperideals, (1; 2)- -hyperideals) of a -semihypergroup H is a bipolar ( ; )-fuzzy sub -semihypergroup (resp., left -hyperideal, right -hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H:
Proof. The proof is straightforward. Now we prove that if B =
+ B;
is a bipolar ( ; )-fuzzy -hyperideal of H then B =
B
a bipolar ( ; )-fuzzy -hyperideal of H.
De…nition 3.14 Let B = that
+
+ B;
+
be a bipolar fuzzy subset of a -semihypergroup H, D E < + . We de…ne the bipolar fuzzy subset B = + ; of H as follows, B B + B
B
+ B
(x) =
+
(x) ^
_
+
and
B
(x) =
B
(x) _
^
D
+ B
;
;
+
B
E
is also
2 (0; 1] such
;
for all x 2 H. + A;
De…nition 3.15 Let A =
A
+ B;
and B =
be bipolar fuzzy subsets of a
B
-semihypergroup H.
Then we de…ne, (1) the bipolar fuzzy subset A ^ B = + (x) A^ B
=(
+ A^B (x)
(2) the bipolar fuzzy subset A _ B = + (x) A_ B
=(
B
(x) = (
D
+ A_B (x)
(3) the bipolar fuzzy subset A + A
D
B= + A
D
B (x)
+ ; A^ B
^
+
)_
+ ; A_ B
^ + A
^
+
)_
; B +
)_
A_ B +
and
A^ B +
A +
E
E
and
B
E
13
A_ B
(x) = (
A_B (x)
_
)^
A^B (x)
_
)^
B (x)
_
)^
as follows:
A^ B
(x) = (
as follows:
and
for all x 2 H.
as follows:
A
B
(x) = (
A
+ A;
Lemma 3.16 Let A =
A
and B =
+ B;
be bipolar fuzzy subsets of a -semihypergroup H. Then
B
the following holds: (1)
A^ B = A ^B
;
(2)
A_ B = A _B
;
(3)
A
:
B
A
B
Proof. The proof is straightforward.
+ B;
Theorem 3.17 A bipolar fuzzy subset B =
of a -semihypergroup H is a bipolar ( ; )-fuzzy,
B
(1) sub -semihypergroup of H if and only if B
B
(2) left -hyperideal of H if and only if H
B ;
B
(3) right -hyperideal of H if and only if B (4) bi- -hyperideal of H if and only if B
H B
Proof. (1) Let B =
+ B;
B
B ;
B and B
(5) interior -hyperideal of H if and only if H (6) (1; 2)- -hyperideal of H if and only if B
B ;
B B
H
H
B ;
B ;
B and B
be a bipolar ( ; )-fuzzy sub 14
B
H
B
B
B .
-semihypergroup of H and z 2 H. Let us
suppose that z 2 x y for x; y 2 H and + B
B
(z)
=
(
2 : Then + B
(
=
B (z)
_
+ B
z2x y
(
=
_
+ B
z2x y
(
_
=
_
+ B
(
z2x y
=
(
=
(
+ B
+ B
(x) ^
+ B
(x y) ^ +
(z) ^
+
(x) ^
inf + z2x y B
z2x y
_
)_
inf + z2x y B
z2x y
(
+
^
+
)_
)
(y)
(z) _ (z) ^
+
+
!
)
+ B
=
^
)
) +
_
!
)
+
(y) ^ +
+
^
^
+
_
+
+
_ +
! !
!
_
_
+
_
+
+
_
(z);
and
B
B
(z)
=
B (z)
B
(
_
^
B
z2x y
(
=
^
B
z2x y
(
^
=
sup
(
B
z2x y
If there do not exist any x; y 2 H and
(
B
(z) _
2
B
B
B
(y)
(z) ^
(x y) _
!
) =
B
) ^
B
B
B
(x) =
+
(z):
+ B
(x);
and
Hence B
B
(x) =
B
B : 15
)
(x):
_ ^
^
such that z 2 x y; then + B
!
_
)
(z) _
)^
)
(y) _
B
z2x y
z2x y
=
(x) _
sup
^
^
(x) _
z2x y
z2x y
(
)^
^
_ ! !
! ^ ^
^
+
Conversely, assume that B
+ B
max
B
B . If there exist m; n 2 H and
+
(x y) ;
+
( + B (x y) ^ 08 < _ = @ :
x y m n
+ B
(
+ B
max
inf + z2x y B
+
(z) ;
+ B
(x) ^
+ B
(x) ^ + B
min
such that x y
+ B
(x y) 9 = + + ^ B (m) ^ B (n) ;
)_
(y)) ^
(y)) ^
(x) ;
+
2
+ B
=
+
_
+
+ B
m n; then
B
1
+A
_
B
B
(x y) +
+
(y));
+
;
and
min
B
(x y) ;
( B (x y) _ 08 < ^ = @ :
)^
x y m n
(
B
B
min
sup z2x y
Hence B =
+ B;
B
B
(z) ;
(x) _
(x) _
max
is a bipolar ( ; )-fuzzy sub
B
B
B
=
(m) _
B
(y)) _
B
B
^
(x y) 9 = (n) _ ;
1
(x y)
A^
(y) _
(x) ;
B
(y) ;
:
-semihypergroup of H. The proofs of (2), (3), (4), (5)
and (6) are similar to the proof of (1).
+ B;
Proposition 3.18 If B =
B
is a bipolar ( ; )-fuzzy sub -semihypergroup (resp., left -hyperideal,
right -hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of a -semihypergroup H; then D E + B = ; is also a bipolar ( ; )-fuzzy sub -semihypergroup (resp., left -hyperideal, right B B hyperideal, interior -hyperideal, bi- -hyperideal, (1; 2)- -hyperideal) of H:
Proof. Suppose B =
+ B;
B
is a bipolar ( ; )-fuzzy sub -semihypergroup of H. Let x; y 2 H and 16
2 .
Then
max
inf + z2x y B
(z);
+
=
+ B (z)
inf
z2x y
+
^
=
inf + (z) z2x y B
^
+
_
=
inf + (z) z2x y B
_
+
^
=
inf + (z) z2x y B
_
+
^
inf + (z) z2x y B
=
=
+
_
^
+ B (y)
^
+
+ B (x)
^
+ B (y)
^
+
(
+ B (x)
^
+
=
((
+ B (x)
^
+
=
+ B
=
min
(x) ^ n
+ B
+ B
+ B
+
+
^
^
+
_
^
+
o
;
+
+
+
^
) _
+
+ B (y)
+
+
^
+
_
) ^ ((
(y);
+
_
+
+ B (y) +
)_
(y) ^
(x); 17
)^(
+
_
+
_
+ B (x)
=
+
_
^
+
+
^
+
)_
+
) ^
+
and
min
sup z2x y
B
(z);
=
sup
B (z)
z2x y
=
sup z2x y
=
sup z2x y
=
sup z2x y
=
_
^
B (z)
^
_
B (z)
^
_
sup
B (z)
+ B
in a similar way.
;
B
E
_
B (x)
_
B (y)
_
_ B (y)
)_(
=
(
B (x)
_
^
+ A;
A
(x) _
max
n
B
B
^
B
^ _
(y);
B (y)
o
_
)) ^
_
(y) _
(x);
_
_
)_ ^
)_
:
be a bipolar ( ; )-fuzzy right -hyperideal and B =
( ; )-fuzzy left -hyperideal of H. Then A
+ A;
B (y)
_
is a bipolar ( ; )-fuzzy sub -semihypergroup of H: The other cases can be seen
Theorem 3.19 Let A =
Proof. Let A =
_
_
+
^
B (x)
B (x)
B
^
^
(((
=
D
^
=
=
Hence B =
^
B (z)
z2x y
=
_
A
B
+ B;
B
be a bipolar
A ^ B:
be a bipolar ( ; )-fuzzy right 18
-hyperideal and B =
+ B;
B
be a bipolar
( ; )-fuzzy left -hyperideal of H. Then for all z 2 H, we have + A
B
(z)
+ A
= =
B (z)
(
_
=
_
=
_ _
(
=
(
=
((
=
(
+ A
+
_
+ A (z)
^
+
^
+ B (z)
+ B (z))
_
+
^
)
)
+
^
+
_
+
_
+
+
!
+
_ +
^
+
^
!
)
)
+
_ ^
+
inf + (z) z2x y B
^ +
)^
+
)^
_
^
+
+ B (y)
)^(
)^(
+ B )(z)
^
^
+ B (y)
inf + (z) z2x y A
z2x y
+ A (z)
^
+ B (y)
+ A (x)
z2x y
(
+
+ A (x)
z2x y
(
_
+ A (x)
z2x y
(
+
^
!
_
+
)
+
_
^
+
!
_
!
^
+
_ +
_
+ (z), A^ B
=
and
A
B
(z)
= =
B (z)
A
(
_
^
A (x)
z2x y
=
(
^
A (x)
z2x y
=
(
^
(
z2x y
(
^
(
=
((
=
(
A (z)
^
A (z) A
_
_
B
)
B (y)
_
)_(
z2x y
A (z)
^
)_(
B (z)
^
B (z))
^
)_
B )(z)
(x)
)
B (y)
_
A (x)
If there do not exist any x; y 2 H and + A
_
sup
z2x y
=
^
_
2 =
!
_ _
^ !
_
B (y)
_
_
)
sup z2x y
)_
^
^
)
!
_
B (z)
^ )
^
^
^ =
A^ B
(z).
such that z 2 x y; then + A
B (x)
+ A^B (x)
^
^ 19
+
+
_ _
+
+
=0_
=
+
=
+ (x); A^ B
+
_
+
and
A
B
(x)
=
A
B (x)
A_B (x)
Hence we get A
B
_
_
^ ^
=0^ =
=
A_ B
(x):
A ^ B:
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