J. Appl. Environ. Biol. Sci., 7(10)134-142, 2017 | ISSN: 2090-4274 |
© 2017, TextRoad Publication | Journal of Applied Environmental and Biological Sciences |
www.textroad.com |
Tariq Mahmood1, M. Ibrar2, Asghar Khan2,*, Hidayat Ullah Khan3 and Fatima Abbas4
1Department of Electronic Engineering, University of Engineering and Technology, Taxila, Sub Campus Chakwal, Pakistan 2Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan 3Department of Mathematics, University of Malakand, at Chakdara, District Dir (L), Khyber Pakhtunkhwa Pakistan 4Department of Mathematics, Gomal University, D. I. Khan, Khyber Pakhtunkhwa, Pakistan
Received: May 11, 2017 Accepted: July 30, 2017
ABSTRACT
Bipolar fuzzy set is an extension of fuzzy set. In bipolar fuzzy set the range of the membership function is
[−1,1] , whereas, in fuzzy set it is [0,1] . We employed the concept of bipolar fuzzy set theory in the structure of ordered semigroup and introduced a generalization of bipolar fuzzy bi-ideals. This generalized form of bipolar fuzzy bi-ideals is called (∈,∈∨q) -bipolar fuzzy bi-ideals of ordered semigroups. We
obtained some interesting characterization results of ordered semigroup in terms of this new concept. KEYWORDS: Bipolar fuzzy bi-ideals, bipolar fuzzy point, (α, β ) -bipolar fuzzy bi-ideals, positive t -cut,
negative s -cut, (s,t) -cut.
The list of abbreviations that are used frequently in this paper are given in see Table 1.
Table 1: NAMES AND ABBREVIATIONS
Name | Abbreviation |
Bipolar fuzzy bi-ideals | BFBI |
Bipolar fuzzy set | BFS |
Fuzzy Bi-ideals | FBI |
Generalised Fuzzy Bi-ideals | GFBI |
Ordered Semigroups | OS |
If and only if | iff |
The set of all bipolar fuzzy subsets of S. | BF(S) |
Ordered Bipolar Fuzzy Point | OBFP |
Bipolar fuzzy bi-ideals | ),(α β -BFBI |
),(∈ ∈ ∨q -bipolar fuzzy generalized bi-ideal | ),(∈ ∈ ∨q -BFGBI |
),(∈ ∈ -bipolar fuzzy bi-ideal | ),(∈ ∈ -BFBI |
),(∈ ∈ ∨q -bipolar fuzzy bi-ideal | ),(∈ ∈ ∨q -BFBI |
Bipolar fuzzy left (right) ideal | BFL (R) I |
Bipolar fuzzy ideal | BFI |
Zhang (Zhang, 1994, 1998) further generalized the theory of fuzzy set (Zadeh, 1965) and defined BFS. In BFS the range of the membership function is [−1,1] rather than [0,1] (as in case of fuzzy set). In BFS, we split the range [−1,1] into (0,1] , 0 and [−1,0) . If the membership degree of an element x (say) belongs to the interval (0,1] , then it indicates that x to some extent satisfies the property, whereas, if the membership
Corresponding Author: Asghar Khan, Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan. Email: azhar4set@yahoo.com
degree x is equal to 0 , then we say that x is irrelevant to the corresponding property. On the other hand and the membership degree of x belongs to [−1,0) , then x satisfies the implicit counter property to a certain degree. Although BFS look like similar to intuitionistic fuzzy sets but they are totally different from each other as mentioned in (Lee, 2004).
Kuroki (Kuroki, 1979, 1981,1991) was the first to employed the theory of fuzzy set in the structure of semigroups. In (Bhakat and Das, 1992) a generalized form of fuzzy subgroup (Rosenfeld,
1971) is presented. In addition, the generalization of FBI in semigroups is given in (Kazanci and Yamak, 2008). Further, Jun et al (Jun et al., 2005) generalized fuzzy sub-algebra in BCK/BCI algebra and presented the idea of (α , β ) -fuzzy sub-algebra, where α and β are in {∈, q,∈∨q,∈∧q} and α ≠∈∧q . In this
regard for further study in different branches of algebra see (Criste, 2010, Davvaz, 2010, Kazanci, 2008, Shabir, 2010, Ma, 2008). A BFS in a set S is denoted by:
f = {(x,( f (x): S → [−1,0], f (x): S → [0,1])) | x ∈ S}, where f and f are called the negative
np np
membership and positive membership mappings respectively. If for an element x in S we have f (x) ≠ 0, f (x) = 0 , then x has a positive satisfaction of a BFS f in S . On other hand if f (x) ≠ 0,
pn n
fp (x) = 0 , then x satisfies the counter property of a BFS f in S up to some extent. There is a possibility
for an element (s) x such that f (x) ≠ 0 ≠ f (x) . For short write f = (S; f , f ) to represent a BFS.
pn np
This study aims to introduce a more generalized form of BFBI of OS and investigate several characterization of OS in terms of this new idea. We further discuss relations between (∈,∈) -BFBI,
(∈,∈∨q) -BFBI and a (q,∈∨q) -BFBI.
Let S be a non-empty set, “ ⋅ ” denotes operation of multiplication and “ ≤ ” denotes partial order relation. If S satisfy the following conditions:
(3) (∀a,b, x ∈ S)(a ≤ b ⇒ ax ≤ bx and xa ≤ xb).
Then (S,⋅, ≤) is called an OS and is denoted by S . If φ≠ A, B ⊆ S and x, y, z ∈ S , then we denote and define the following:
(A]:= {t ∈ S : t ≤ h ∈ A}, AB = {ab : a ∈ A and b ∈ B},
Ax = {( y, z) ∈ S × S | x ≤ yz},
A ⊆ B ⇒ A ⊆ (A],
(A] ⊆ (B],
(A](B] ⊆ (AB],
((A]] = (A]. The characteristic function of A is denoted by χ= (S, χ , χ ) where χ and χ are defined as:
A nApA nA pA
⎧−1, if x ∈ A, ⎧1, if x ∈ A,
χ (x) = , χ (x) =
nA pA
0, if x ∉ A. 0, if x ∉ A.
⎩ If A2 ⊆ A , then A is called a subsemigroup. A subsemigroup A of S is a bi-ideal if:
represented by P( f ;t):= {x ∈ S | f (x) ≥ t} for a BFS f = (S; f , f ) . We represent and define the (s,t)
p np
cut of f as C( f ;(s,t)) = N ( f ; s) ∩ P( f ;t).
J. Appl. Environ. Biol. Sci., 7(10)134-142, 2017
For two BFSs f = (S; f , f ) and g = (S; g , g ) and for all x ∈ S we say that:
np np
i. f pg iff f (x) ≥ g (x) and f (x) ≤ g (x),
nn pp
ii. f = g iff f pg and gp f ,
iii. f ∩ g = (S; f ∨ g , f ∧ g ),
nnp p
iv. f ∪ g = (S; f ∧ g , f ∨ g ),
nnp p
For all x ∈ S , the BFS 0 = (S;0,0 )and 1 = (S;1 ,1 ) are defined as follows:
np np
0(x) = 0 = 0(x), 1 (x) =−1and 1 (x) = 1.
npn p
The product of f = (S; f , f ) and g = (S; g , g ) is defined and denoted as:
np np
f o g = (S; f o g , f o g ), where
nnp p
⎧∧{ f ( y) ∨ g (z)} if A ≠φ,
nn x
( fn o gn )(x) = 0 if A =φ.
x
and ⎧∨{ f (y) ∧ g (z)} if A ≠φ,
pp x
( fp o gp )(x) = 0 if A =φ.
x
The multiplication “ o ” is well defined and associative and clearly (BF(S),o,p) is an ordered semigroup. Definition 2.1 (Shabir, 2013): A BFS f = (S; fn , fp ) in S is called a BFL (R) I of S if: 1) x ≤ y ⇒ f (x) ≤ f ( y) and f (x) ≥ f ( y))
nn pp
2) f (xy) ≤ f ( y) and f (xy) ≥ f (y)
nn pp
( f (xy) ≤ f (x) and f (xy) ≤ f (x))
nn pp
for all x, y ∈ S . A BFS is called BFI of S if it is both BFLI and BFRI. Definition 2.2 (Shabir, 2013): A BFS f is called BFBI of S, if:
(1) x ≤ y ⇒ f (x) ≤ f ( y) and f (x) ≥ f ( y)
nn pp
(2) f (xy) ≤∨{ f (x), f ( y)}, and f (xy) ≥∧{ f (x), f ( y)}
nnn ppp
(3) f (xyz) ≤∨{ f (x), f (z)} and f (xyz) ≥∧{ f (x), f (z)}
nnn ppp
for all x, y, z ∈ S . Theorem 2.3 Let f be a BFS in S . Then f is a BFBI iff C( f ;(s,t))≠φ is a bi-ideal of S for all (s,t)∈[−1,0) × (0,1]. Proposition 2.4 The following hold for A, B ⊆ S .
ABA∩B
(iii) χ o χ =χ .
AB ( AB]
Proof. (i) It is straightforward to prove and thus omitted.
(ii) First we consider A ∩ B =φ . Then clearly χ ∩χ =χ . On the other hand if A ∩ B ≠φ , then we
ABA∩B
let x ∈ A ∩ B and therefore χ (x) =−1 = (χ ∧χ )(x),
A∩B AB
n nn
and χ (x) = 1 = (χ ∧χ )(x).
A∩B AB
p pp
Hence χ ∩χ =χ .
ABA∩B
(iii) Now we prove χ o χ =χ . For this it is enough to show that χ o χ =χ and
AB ( AB] AB ( AB]
nn n
χ o χ =χ . Let x ∈ S : If x ∈ (AB] , then χ (x) =−1, χ (x) = 1and x ≤ ab for some a ∈ A
pApB p( AB]( AB] p ( AB ]
n
and b ∈ B . Hence (a,b)∈ Ax and therefore Ax ≠φ and we have
(χ o χ )(x) = {χ ( y) ∨χ (z)}
AB AB
∧
nn nn
( y , z )∈Ax
≤ {χ A (a) ∨χ B (b)} =−1,
nn
and (χ o χ )(x) = {χ ( y) ∧χ (z)}
AB AB
∨
pp pp
( y, z )∈Ax
≥ {χ A (a) ∧χ B (b)} = 1.
pp
It follows that χ (x) =−1 = (χ o χ )(x) and χ (x) = 1 = (χ o χ )(x).
n ( AB] AB p ( AB] A pB
nn p
3. Generalized BFBI of OS
The idea of BFBI is generalized to introduce the notion of (α, β ) -BFBI in OS in this section. If (s,t)∈[−1,0) × (0,1] , then a BFS f = (S; fn , fp ) of the form:
⎧s, if y ∈ (x], ⎧t, if y ∈ (x],
fn ( y):= , fp ( y):=
0, if y ∉ (x], 0,if y ∉ (x].
⎩
is called an OBFP with x and (s,t) are the value and support of the OBFP and is denoted by x . An
( s, t )
xx ( s, t )(s , t ) n pn
OBFP is said to belongs to f (denoted as ∈ f ) if f (x) ≤ s and f (x) ≥ t . If f (x) + s <−1 and
x xxx
f (x) + t > 1, then is said to be quasi-coincidence with f (denoted as qf ). If ∈ f or qf ,
p ( s, t )( s,t )(s , t )( s,t )
xx xxx
then we write ∈∨qf . By ∈∧qf we mean ∈ f and qf . We write α f if the relation
( s,t )(s,t )(s , t )( s,t )( s,t )
x
αf does not hold.
( s,t )
x
Let f (x) ≥−0.5 and f (x) ≤ 0.5 for all x ∈ S (s,t)∈[−1,0) × (0,1] such that ∈∧qf . Then f (x) ≤ s ,
np (s,t ) n
f (x) ≥ t , f (x) + s <−1 and f (x) + t > 1. It follows that −1 > f (x) + s ≥ f (x) + f (x) and
pnp nnn
xx
1 < f (x) + t ≤ f (x) + f (x) that is f (x) <−0.5 and f (x) > 0.5 . It follows that { | ∈∧q} =φ .
pppn p ( s,t )( s,t )
From the above discussion we conclude that the case α =∈∧q will be omitted in this study. Definition 3.1 A BFS f in S is called (α, β ) -bipolar fuzzy left (right) ideal of S where α ≠∈∧q if the following conditions hold for all (s,t)∈[−1,0) × (0,1] and x, y ∈ S :
y x
(1) If x ≤ y , then αf implies βf ,
( s, t )( s,t )
y xy xy
x (s,t )( s,t )(s ,t )(s ,t )
(2) αf ⇒βf ( αf ⇒βf ). Definition 3.2 A BFS f in S is called (α, β ) -BFBI the following conditions hold for all (s,t)∈[−1,0) × (0,1] and x, y, z ∈ S :
y x
(1) If x ≤ y , then αf ⇒βf ,
( s,t )( s,t )
y xyx
(2) αf and αf ⇒βf ,
( s ,t )(s , t )(∨{s , s }, ∧{t ,t })
11 22 1212
xyz ( s ,t )( s ,t )(∨{s , s },∧{t , t })
xz
(3) αf and αf ⇒βf .
11 22 1212
Theorem 3.3 Let f be a BFS in S . Then f is a BFBI if and only if for all (s,t) ∈[−1,0)× (0,1] and x, y, z ∈ S the following conditions hold simultaneously:
y x
(1) If x ≤ y , then ∈ f ⇒∈ f ),
( s,t )( s,t )
y xyx
(2) ∈ f and ∈ f ⇒∈ f ,
(s ,t )(s , t )(∨{s , s }, ∧{t ,t })
11 22 1212
xyzxz
(3) ∈ f and ∈ f ⇒∈ f .
(s ,t )( s,t )(s,t )
y
Proof. Suppose that f is a BFBI. If x, y ∈ S such that x ≤ y and ∈ f , then f ( y) ≤ s and f ( y) ≥ t .
(s ,t ) np
By Definition 2.2 we have f (x) ≤ f ( y) and f (x) ≥ f ( y) . This implies that f (x) ≤ s and f (x) ≥ t
nnpp np
x
and so ∈ f .
(s, t )
yx
If ∈ f and ∈ f , then f (x) ≤ s , f (x) ≥ t and f ( y) ≤ s , f (y) ≥ t . By Definition 2.2 we
(s1, t1) (s2, t2) n 1 p 1 n 2 p 2
J. Appl. Environ. Biol. Sci., 7(10)134-142, 2017
have
f (xy){ f (x), f ( y){s , s }
n ≤∨ nn ≤∨ 12
and f (xy) ≥∧{ f (x), f ( y)} ≥∧{t ,t },
p pp 12
xy
it follows that {s 2 }, ∧ t , 2}) ∈ f .
(∨ 1, s {1 t
xz
If x, y, z ∈ S such that (s , t ∈ f and ( s ,) ∈ f , then fn (x) ≤ s , fp (x) ≥ t1 and f (z) ≤ s , fp (z) ≥ t2. By
) t 1 n 2
11 22
Definition 2.2 (3) we have fn (xyz) ≤∨{ fn (x), f (z){s1, s }
n ≤∨ 2
and fp (xyz) ≥∧{ fp (x), fp (z)} ≥∧{t1,t2},
xyz
this implies (∨{s , s }, ∧{t ,t }) ∈ f .
12 12
y np (s , t )
Conversely let x, y ∈ S such that x ≤ y . If f ( y) = s and f ( y) = t , then ∈ f and hence by (1) we have x ∈ f . This implies f (x) ≤ s = f (y) and f (x) ≥ t = f ( y) . Hence we have f (x) ≤ f ( y)
(s , t ) nnpp nn
fp (x) ≥ fp ( y).
x
If ( fn ( x ), fp ( x )) ∈ f and ( fn ( y ), yf p ( y )) ∈ f , then by (2) we have (∨{ f ( x), fn ( y )}, xy ∧{ fp ( x ), fp ( y )}) ∈ f . This implies that
n
f (xy) ≤∨{ f (x), f ( y) and f (xy) ≥∧{ f (x), f (y)} .
n nnp pp xyzxz
Let x, y, z ∈ S such that ( f ( x ), f ( x )) ∈ f and ( f ( z ), f ( z )) ∈ f , then by (3) we have (∨{ f ( x ), f ( z )},∧{ f ( x ), f ( z )}) ∈ f . It
np np nnpp
follows that f (xyz) ≤∨{ f (x), f (z), f (xyz) ≥∧{ f (x), f (z)}.
n nnp pp
Proposition 3.4 (Ibrar et al., 2016) A BFS f in S isan (∈,∈∨q) -BFGBI iff the following conditions hold simultaneously for all x, y, z ∈ S :
nn pp
nnn ppp
Theorem 3.5 A BFS f in S isan (∈,∈∨q) -BFBI iff the following conditions hold simultaneously for all x, y, z ∈ S :
nn pp
nnn ppp
(3) fn (xyz) ≤∨{ fn (x), fn (z),−0.5} and fp (xyz) ≥∧{ fp (x), fp (z),0.5}. Proof. The proof follows from Proposition 3.4. Theorem 3.6 Every (∈,∈) -BFBI is (∈,∈∨q) -BFBI.
Proof. It can be proved easily. The below given example shows that every (∈,∈∨q) -BFBI is not (∈,∈) -BFBI. Example 3.7 Let S = {a,b,c, d,e} be a set with the following multiplication table and order relation “ ≤ ”
Table1
. | a | b | c | d | e |
a | a | d | a | d | d |
b | a | b | a | d | d |
c | a | d | c | d | e |
d | a | d | a | d | d |
e | a | d | c | d | e |
≤:= {(a, a),(a,c),(a, d ),(a,e),(b,b),(c,c),(c,e),(d, d ),(e,e)}. Then S is an OS. Let f be a BFSin S defined by Table 2 below:
138
Table 2
S | a | b | c | d | e |
---|---|---|---|---|---|
n f | 0.75− | 0.35− | 0.72− | 0.58− | 0.65− |
p f | 0.8 | 0.3 | 0.7 | 0.5 | 0.6 |
ae
Then f is (∈,∈∨q) -BFBI but not an (∈,∈) -BFBI because (−0.7, 0.7) ∈ f and (−0.6, 0.55) ∈ f but
ae d
=∉ f .
(∨{−0.7, −0.6}, ∧{0.7,0.55}) (−0.6,0.55)
Theorem 3.8 Every (∈∨q,∈∨q) -BFBI is an (∈,∈∨q) -BFBI. Proof. Let f bean (∈∨q,∈∨q) -BFBI of S. Let x, y ∈ S such that x ≤ y and (s,t)∈[−1,0) × (0,1]. If
yy x
∈ f then ∈∨qf and hence by hypothesis we have ∈∨qf . Next we consider
(s , t )( s,t )( s,t )
yy
xx
(s ,t ),(s ,t )∈[−1,0) × (0,1] and ∈ f , ∈ f . Then ∈∨qf , ∈∨qf and hence by
1122 (s1, t1) ( s2, t2) ( s1,t1) ( s,t )
xy xz
hypothesis this implies (∨{s1, s2 },∧{t1,t2 }) ∈∨qf . Again we let x, y, z ∈ S such that (s1, t1) ∈ f and ( s2, t2) ∈ f
xyzxz
implies ( s ,t ) ∈∨qf and ( s ,t ) ∈∨qf . By hypothesis (∨{s , s },∧{t ,t }) ∈∨qf . Hence f = (S; fn , fp ) is an
11 22 1212
(∈,∈∨q) -BFBI. Theorem 3.9 If f is a nonzero (∈,∈∨q) -BFBI of S, then the set S = {x ∈ S | f (x) ≠ 0}∩{x ∈ S | f (x) ≠ 0} is a bi-ideal of S.
o np
Proof. Let x, y ∈ S such that x ≤ y and y ∈ S . Then f ( y) ≠ 0 and f ( y) ≠ 0 . Hence f ( y) < 0 and
o np n
f ( y) > 0. Suppose that f (x) = 0 or f (x) = 0. Since y ∈ f and f (x) > f ( y) or f (x) < f ( y).
p np ( fn ( y ), fp ( y )) nn pp
x ( fn ( y ), fp ( y )) nn n
This implies that ∈f , which is a contradiction. Also f (x) + f ( y) = f ( y) ≥−1 or
x pp p ( fn ( y), fp ( y)) n
f (x) + f ( y) = f ( y) ≤ 1. This implies that qf , which is a contradiction. Therefore, f (x) ≠ 0 and fp (x) ≠ 0. Hence, x ∈ S o . If x, y ∈ S , then f (x) ≠ 0, f (y) ≠ 0, f (x) ≠ 0 and f (y) ≠ 0. So f (x) < 0, f (y) < 0, f (x) > 0 and
o nnp p nnp
yx
f (y) > 0. Suppose that xy ∉ S , then f (xy) = 0 or f (xy) = 0. Clearly ∈ f and ∈ f .
p o p ( f ( x), f ( x)) ( f ( y), f ( y
n np np ))
xy
Since fn (xy) = 0 > fn (x)∨ fn ( y) or fp (xy) = 0 < fp (x)∧ fp (y) . This implies ( fn ( x)∨ fn ( y), fp ( x)∧ fp ( y)) ∈f ,a
contradiction. Also f (xy) + f (x)∨ f (y) = f (x)∨ f (y) ≥−1 or f (xy) + f (x)∧ f ( y) = f (x)∧ f (y) ≤ 1.
n nnnn p pp pp
This implies that xy qf , again a contradiction. Therefore, fn (xy) ≠ 0 and
( f (x) ∨ f (y), f (x) ∧ f ( y))
nnp p
fp (xy) ≠ 0 . Hence, xy ∈ S o . Let x, y, z ∈ S such that x, z ∈ S . Then f (x) ≠ 0, f (z) ≠ 0, f (x) ≠ 0 and f (z) ≠ 0. So f (x) < 0,
o nnp pn
x
f (z) < 0, f (x) > 0 and f (z) > 0. If xyz ∉ S , then f (xyz) = 0 or f (xyz) = 0 . Clearly ∈ f
np p o np ( fn ( x ), fp ( x ))
z ( fn ( z ), fp ( z )) n nnp pp
and ∈ f . Since f (xyz) = 0 > f (x)∨ f (z) or f (xyz) = 0 < f (x)∧ f (z) . This implies
xyz
( fn ( x )∨ fn ( z ), fp ( x )∧ fp ( z )) ∈f , a contradiction. Also fn (xyz) + fn (x)∨ fn (z) = fn (x)∨ fn (z) ≥−1 or
f (xyz) + f (x)∧ f (z) = f (x)∧ f (z) ≤ 1. This implies that xyz qf ,a
p pp pp
( f (x) ∨ f (z), f (x) ∧ f (z))
nnp p
contradiction. Therefore xyz ∈ S o . Theorem 3.10 Let f bean (∈,∈∨q) -BFBI of S and ( fn (x), fp (x))∈ (−0.5,0) × (0,0.5) for all x ∈ S . Then f isan (∈,∈) -BFBI. Proof. Suppose f bean (∈,∈∨q) -BFBI of S .
y
Let x, y ∈ S be such that x ≤ y and ∈ f for (s,t) ∈[−1,0) × (0,1] , then f (y) ≤ s and f ( y) ≥ t . By
(s , t ) np
J. Appl. Environ. Biol. Sci., 7(10)134-142, 2017
x
Theorem 3.5 (1) we have f (x) ≤ f ( y)∨− 0.5 ≤ s and f (x) ≥ f ( y)∧0.5 ≥ t andso (s ,t ) ∈ f .
nn pp yx
Let x, y ∈ S and (s ,t ),(s ,t ) ∈[−1,0)× (0,1] be such that ∈ f and ∈ f then f (x) ≤ s ,
1122 (s ,t )( s ,t ) n 1
11 22
f ( y) ≤ sf (x) ≥ t , and f (y) ≥ t . By Theorem 3.5 (2) we have
n 2 p 1 p 2
f (xy) ≤∨{ f (x), f ( y),−0.5} ≤∨{s ,s } and f (xy) ≥∧{ f (x), f ( y),0.5} ≥∧{t ,t } , which implies
n nn 12 p pp 12
xy
∈ f .
(∨{s , s },∧{t ,t })
12 12
xz
Let x, y, z ∈ S and (s ,t ),(s ,t ) ∈[−1,0)× (0,1] such that ∈ f and ∈ f then f (x) ≤ s ,
1122 (s ,t )( s ,t ) n 1
11 22
f (z) ≤ s , f (x) ≥ t and f (z) ≥ t . By Theorem 3.5 (3) we have
n 2 p 1 p 2
f (xyz) ≤∨{ f (x), f (z),−0.5} ≤∨{s , s } and f (xyz) ≥∧{ f (x), f (z),0.5} ≥∧{t ,t } . In which it follows
n nn 12 p pp 12
xyz
that (∨{s1, s2 },∧{t1,t2 }) ∈ f . Hence f isan (∈,∈) -BFBI.
Theorem 3.11 Let I be a bi-ideals of S and f a BFS in S such that
y
Proof. Let x, y ∈ S such that x ≤ y and (s,t)∈[−1,0) × (0,1] . Let qf then f (y) + s <−1 and
(s,t ) n
x
f ( y) + t > 1 . This implies that y ∈ I . Since I is bi-ideals of S , so x ∈ I . In order to check ∈∨qf ,
p ( s,t )
we consider the following four cases:
x
The first case induces f (x) ≤−0.5 ≤ s and f (x) ≥ 0.5 ≥ t and hence ∈ f . The second case implies
np (s ,t )
y np (s,t )(s,t )
x
that f (x) + s <−1 and f (x) + t > 1, it follows that qf . Since qf so case (3) and (4) do not occur.
x
Consequently, ∈∨qf .
( s,t )
yx
Let x, y ∈ S such that qf and qf for (s ,t ),(s ,t )∈[−1,0)× (0,1] , then f (x) + s <−1,
(s ,t )(s ,t ) 11 22 n 1
11 22
f ( y) + s <−1, f (x) + t > 1 and f (y) + t > 1 and so x, y ∈ I . Since I is a bi-ideals of S , therefore
n 2 p 1 p 2
xy
xy ∈ I . In order to check (∨{s1, s2},∧{t1,t }) ∈∨qf we consider the following four cases:
2
12 12
12 12
12 12
12 12
xy
The first case induces fn (xy) ≤−0.5 ≤ s1 ∨ s2 and fp (xy) ≥ 0.5 ≥ t1 ∧t2 and so ( s ∨ s ,t ∧t ) ∈ f . The second case
1 212
xy y
x
implies that fn (xy) + s1 ∨ s2 <−1 and fp (xy) + t1 ∧t2 > 1 that is ( s1 ∨ s2,t1 ∧t2) qf . Since (s1,t1) qf and (s2,t2) qf so
xy
cases (3) and (4) do not occur. Consequently, ( s1 ∨ s2,t1 ∧t2) ∈∨qf .
xz
Let x, y, z ∈ S such that qf and qf for (s ,t ),(s ,t ) ∈[−1,0) × (0,1] , then f (x) + s <−1,
(s ,t )(s ,t ) 11 22 n 1
11 22
f (z) + s <−1, f (x) + t > 1 and f (z) + t > 1, in which it follows that x, z ∈ I . Since I is bi-ideals of
n 2 p 1 p 2
xyz
S , therefore we have xyz ∈ I . Now to show that ( s1 ∨ s2,t1 ∧t2) ∈∨qf we consider the following four cases:
12 12
12 12
12 12
12 12
xyz
The first case induces fn (xyz) ≤−0.5 ≤ s1 ∨ s2 and fp (xyz) ≥ 0.5 ≥ t1 ∧t2 and so ( s ∨ s ,t ∧t ) ∈ f . The second
1 212
xy x
case implies that f (xyz) + s ∨ s <−1 and f (xyz) + t ∧t > 1 and therefore qf . Since qf and
( s ∨ s ,t ∧t )(s ,t )
n 12 p 1 2 1212 11
xyzz
(s ,t ) qf and therefore both Case (3) and Case (4) do not occur. Hence (∨{s , s ,∧{t ,t }) ∈∨qf .
22 12} 12
In general every (∈,∈∨q) -BFBI may not be (q,∈∨q) -BFBI as shown in the following example. Example 3.12 Consider the OS of Example 3.7 and define BFS f in S as in Table 3.
Table 3
S | a | b | c | d | e |
---|---|---|---|---|---|
n f | 0.9− | 0.35− | 0.72− | 0.58− | 0.65− |
p f | 0.8 | 0.3 | 0.7 | 0.5 | 0.6 |
Then f isan (∈,∈∨q) -BFBI of S but nota (q,∈∨q) -BFBI because a qf and e qf but
(−0.15,0.7) (−0.45,0.52)
ae d
. In the following theorem we provide a condition for an (∈,∈∨q) -BFBI to be (q,∈∨q) -BFBI. Theorem 3.13 Every (∈,∈∨q) -BFBI is an (q,∈∨q) -BFBI of S for condition that (s,t) ∈[−0.5,0) × (0,0.5] .
(−0.15∨−0.45,0.7∧0.52) = (−0.15,0.52) ∈∨qf
y
Proof. Let f be an (∈,∈∨q) -BFBI of S and x, y ∈ S such that x ≤ y and (s,t ) qf for (s,t) ∈[−0.5,0)× (0,0.5] . Then f (y) + s <−1 and f ( y) + t > 1 that is f ( y) <−1− s ≤ s and
np n
y x
f ( y) > 1− t ≥ t and so ∈ f . Since f isan (∈,∈∨q) -BFBI of S , Therefore ∈∨qf .
p (s ,t ) ( s,t )
yx
Let x, y ∈ S such that (s ,t1) qf and (s2,t2) qf for some (s1,t ),(s2,t2) ∈[−0.5,0) × (0,0.5], then fn (x) + s <−1,
11 1
f ( y) + s2 <−1, f (x) + t > 1 and fp (y) + t2 > 1 . This implies that fn (x) <−1− s1 ≤ s1 fn ( y) <−1− s2 ≤ s ,
np 1 2
yx
f (x) > 1− t ≥ t and f ( y) > 1− t ≥ t . Hence ∈ f and ∈ f and since f isan (∈,∈∨q) -BFBI
p 11 p 22 (s1,t1) (s2,t2)
xy
of S , therefore ( s1 ∨ s2,t1 ∧t2) ∈∨qf .
Let x, y, z ∈ S such that x qf and z qf for (s ,t ),(s ,t ) ∈[−0.5,0) × (0,0.5], then f (x) + s <−1, f (z) + s2 <−1, fp (x) + t > 1 and f (z) + t2 > 1. This implies that fn (x) <−1− s1 ≤ s , fn (z) <−1− s ≤ s2,
(s1,t1) (s2,t2) 1122 n 1
n 1 p 12
xz
fp (x) > 1− t1 ≥ t1 and fp (z) > 1− t2 ≥ t2 that is (s ,t ) ∈ f and (s ,t ) ∈ f . Since f isan (∈,∈∨q) -BFBI of S ,
11 22
xyz
therefore ( s1 ∨ s2,t1 ∧t2) ∈∨qf and hence f is (q,∈∨q) -BFBI of S .
REFERENCES
[1] Bhakat. S. K., and P. Das, (1992). On the definition of a fuzzy subgroup. Fuzzy Sets and Systems. Vol. 51: 235-241.
[2] Criste I., and B. Davvaz, (2010). Atanassov's intuitionistic fuzzy grade of hypergroups. Inform. Sci. Vol. 180: 1506-1517.
[3] Davvaz. B., M. Fathi., and A. R. Salleh, (2010). Fuzzy hyperrings (Hv-rings) based on fuzzy universal sets. Inform. Sci. Vol. 180: 3021-3032.
[4] Ibrar. M., A. Khan., and B. Davvaz, (2016). Characterizations of regular ordered semigroup in terms of (α , β ) -bipolar fuzzy generalized bi-ideals. Intelligent and fuzzy systems, (submitted 2016).
[5] Jun. Y. B., (2005). On (α , β ) -fuzzy subalgebras of BCK / BCI-algebras. Bull. Korean Math. Soc. Vol. 42 (4): 703-711.
[6] Kazanci. O., and B. Davvaz, (2008). On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings. Inform. Sci. Vol. 178: 1343-1354.
[7] Kazanci. O., and S. Yamak, (2008). Generalized fuzzy bi-ideals of semigroups. Soft Comput. Vol.
12: 1119-1124.
J. Appl. Environ. Biol. Sci., 7(10)134-142, 2017
[8] | Kuroki. N., (1979). Fuzzy bi-ideals in semigroups. Comment. Math. Univ. St. Pauli. Vol. 28 (1): 17 | |
---|---|---|
21. | ||
[9] | Kuroki. N., (1981). On fuzzy ideals and fuzzy bi-ideals in semigroups. Fuzzy Sets System. Vol. 5: | |
203-215. | ||
[10] | Kuroki. N., (1991). On fuzzy semigroups. Inform. Sci. Vol. 53: 203-236. | |
[11] | Lee. K. M., (2004). Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets and bipolar- | |
valued fuzzy sets. J. Fuzzy Logic Intelligent Systems. Vol. 14 (2): 125-129. | ||
[12] | Ma. X., J. Zhan., and B. Davvaz., (2008). Some kinds of | ),(∈ ∈ ∨q -interval-valued fuzzy ideals of |
BCI-algebras. Inform. Sci. Vol. 178: 3738-3754. | ||
[13] | Rosenfeld. A., (1971). Fuzzy groups. J. Math. Anal. Appl. Vol. 35: 512-517. | |
[14] | Shabir. M., Y. B., Jun., and Y. Nawaz., (2010). Characterizations of regular semigroups by | ),(α β - |
fuzzy ideals, Comput. Math. Appl. Vol. 59: 161-175. | ||
[15] | Zadeh. L. A., (1965). Fuzzy sets. Information and Control. Vol. 8: 338-353. | |
[16] | Zhang. W. R., (1994). Bipolar fuzzy sets and relations: a computational framework for cognitive | |
modeling and multiagent decision analysis. Proc. of IEEE Conf. 305-309. | ||
[17] | Zhang. W. R., (1998). Bipolar fuzzy sets. Proc. of IEEE. 835-840. |