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Positive Implicative Ideals of BCK-Algebras Based on a Soft Set Theory
1Young Bae Jun, 2Hee Sik Kim and3Chul Hwan Park
1Department of Mathematics Education (and RINS) Gyeongsang National University, Chinju 660–701, Korea
2Department of Mathematics Hanyang University, Seoul 133–791, Korea
3School of Digital, Mechanics, Ulsan College, Namgu, Ulsan, 680–749, Korea
1[email protected],2[email protected],3[email protected]
Abstract. Molodtsov [8] introduced the concept of soft set as a new mathemat- ical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to the theory of BCK-algebras. The notions of positive implicative soft ideals and positive implicative idealistic soft BCK-algebras are introduced, and their basic properties are derived.
2010 Mathematics Subject Classification: 06D72, 06F35, 03G25
Keywords and phrases: (Positive implicative) ideal, soft BCK-algebra, (posi- tive implicative) idealistic soft BCK-algebra, positive implicative soft ideal.
1. Introduction
To solve complicated problem in economics, engineering, and environment, we can- not successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncer- tainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [8]. Majiet al. [7] and Molodtsov [8] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed
Communicated byLee See Keong.
Received:July 11, 2008;Revised: November 12, 2009.
out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Majiet al. [7] described the application of soft set theory to a decision making problem. Maji et al. [6] also studied several operations on the theory of soft sets. Chen et al. [1] presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. By using the notions of fuzzy sets, Jun et al. [3] studied the fuzzy ideals in BE-algebras. Lee [5] discussed bipolar fuzzy subalgebras/ideals of BCK/BCI-algebras by using the notion of bipolar fuzzy sets. Zhan and Jun [10] applied the notion of fuzzy points to ideal theory of BCI-algebras, and generalized well-known fuzzy algebraic structures in BCI-algebras.
In this paper, we deal with the algebraic structure of BCK-algebras by applying soft set theory. We discussed the algebraic properties of soft sets in BCK-algebras.
We introduced the notion of positive implicative soft ideals and positive implica- tive idealistic soft BCK-algebras, and gave several examples. We investigated re- lations between idealistic soft BCK-algebras and positive implicative idealistic soft BCK/BCI-algebras. We established the intersection, union, “AND” operation, and
“OR” operation of positive implicative soft ideals and positive implicative idealistic soft BCK/BCI-algebras.
2. Basic Results on BCK-algebras
A BCK-algebra is an important class of logical algebras introduced by K. Is´eki and was extensively investigated by several researchers.
An algebra (X;∗,0) of type (2,0) is called aBCI-algebraif it satisfies the following axioms:
(I) (∀x, y, z∈X) (((x∗y)∗(x∗z))∗(z∗y) = 0), (II) (∀x, y∈X) ((x∗(x∗y))∗y= 0),
(III) (∀x∈X) (x∗x= 0),
(IV) (∀x, y∈X) (x∗y= 0, y∗x= 0 ⇒ x=y).
If a BCI-algebraX satisfies the following identity:
(V) (∀x∈X) (0∗x= 0), thenX is called aBCK-algebra.
Any BCK-algebraX satisfies the following conditions:
(a1) (∀x∈X) (x∗0 =x),
(a2) (∀x, y, z∈X) (x≤y ⇒ x∗z≤y∗z, z∗y≤z∗x), (a3) (∀x, y, z∈X) ((x∗y)∗z= (x∗z)∗y),
(a4) (∀x, y, z∈X) ((x∗z)∗(y∗z)≤x∗y)
where x ≤ y if and only if x∗y = 0. A BCK-algebra X is said to be positive implicative if it satisfies the following identity:
(2.1) (∀x, y, z∈X) ((x∗y)∗z= (x∗z)∗(y∗z)).
A nonempty subsetS of a BCK-algebraX is called a subalgebraof X ifx∗y ∈S for allx, y∈S.A subsetH of a BCK-algebraX is called anidealofX,denoted by HCX,if it satisfies the following axioms:
(I1) 0∈H,
(I2) (∀x∈X) (∀y∈H) (x∗y∈H ⇒ x∈H).
Any idealH of a BCK-algebraX satisfies the following implication:
(2.2) (∀x∈X) (∀y∈H) (x≤y ⇒ x∈H).
A subsetH of a BCK-algebraX is called apositive implicative idealofX,denoted byHCpiX,if it satisfies the following axioms:
(I1) 0∈H,
(I3) (∀x, y, z∈X) ((x∗y)∗z∈H, y∗z∈H ⇒ x∗z∈H).
We refer the reader to the book [9] for further information regarding BCK-algebras.
3. Basic results on soft sets
Molodtsov [8] defined the soft set in the following way: LetU be an initial universe set andEbe a set of parameters. LetP(U) denotes the power set ofU andA⊂E.
Definition 3.1. [8]A pair(A, A)is called a soft set overU,whereA is a mapping given by
A :A→P(U).
In other words, a soft set over U is a parameterized family of subsets of the universeU.Forε∈A,A(ε) may be considered as the set ofε-approximate elements of the soft set (A, A). Clearly, a soft set is not a classical set. For illustration, Molodtsov considered several examples in [8].
Definition 3.2. [6]Let(A, A)and(B, B)be two soft sets over a common universe U.The intersection of(A, A)and(B, B)is defined to be the soft set(G, C)satisfying the following conditions:
(i) C=A∩B,
(ii) (∀e∈C) (G(e) =A(e) or B(e),(as both are same set)).
In this case, we write(A, A)∩(e B, B) = (G, C).
Definition 3.3. [6]Let(A, A)and(B, B)be two soft sets over a common universe U.The union of(A, A)and(B, B)is defined to be the soft set(G, C)satisfying the following conditions:
(i) C=A∪B, (ii) for alle∈C,
G(e) =
A(e) ife∈A\B, B(e) ife∈B\A, A(e)∪B(e) ife∈A∩B.
In this case, we write(A, A)∪(e B, B) = (G, C).
Definition 3.4. [6]If(A, A)and(B, B)are two soft sets over a common universe U,then “(A, A)AND(B, B)” denoted by(A, A)e∧(B, B)is defined by(A, A)∧(e B, B)
= (G, A×B), whereG(x, y) =A(x)∩B(y)for all (x, y)∈A×B.
Definition 3.5. [6]If(A, A)and(B, B)are two soft sets over a common universe U,then “(A, A)OR(B, B)” denoted by(A, A)∨(e B, B)is defined by(A, A)∨(e B, B)
= (G, A×B), whereG(x, y) =A(x)∪B(y)for all (x, y)∈A×B.
Definition 3.6. [6] For two soft sets (A, A)and (B, B)over a common universe U, we say that (A, A)is a soft subset of (B, B), denoted by (A, A)⊂e (B, B), if it satisfies:
(i) A⊂B,
(ii) For everyε∈A,A(ε)andB(ε)are identical approximations.
4. Positive implicative idealistic soft BCK-algebras
In what follows let X and A be a BCK-algebra and a nonempty set, respectively, and R will refer to an arbitrary binary relation between an element of A and an element of X, that is, R is a subset of A×X without otherwise specified. A set- valued functionA :A→P(X) can be defined asA(x) ={y ∈X |(x, y)∈R}for allx∈A.The pair (A, A) is then a soft set over X.
Definition 4.1. [2] Let (A, A) be a soft set over X. Then (A, A) is called a soft BCK-algebra over X ifA(x)is a subalgebra ofX for all x∈A.
Example 4.1. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 a a a
b b b 0 b b
c c c c 0 c
d d d d d 0
Let (A, A) be a soft set overX,where A=X and A :A→P(X) is a set-valued function defined by
A(x) ={y∈X |xRy ⇔ y∈x−1Φ}
for allx∈Awhere Φ ={0, a}and x−1Φ ={y ∈X |x∧y∈Φ}.Then (A, A) is a soft BCK-algebra overX (see [2]).
For anya∈X and a subsetD ofX,let
a
D :={x∈X |x∗a∈D}, aD2 :={x∈X|x∗(x∗a)∈D}.
Lemma 4.1. For any a∈X andD⊂X, we have
(4.1) DCpiX ⇒ Da CX.
Proof. Assume thatDCpiX.Obviously 0∈ Da.Letx, y∈X be such thatx∗y∈ Da andy∈ Da.Then (x∗y)∗a∈D andy∗a∈D.SinceDCpiX,it follows from (I3) thatx∗a∈D,that is,x∈ Da so that Da CX.
Definition 4.2. [4] Let (A, A) be a soft set over X. Then (A, A) is called an idealistic soft BCK-algebra over X if it satisfies:
(4.2) (∀x∈A) (α(x)CX).
Definition 4.3. Let (A, A)be a soft set over X. Then(A, A) is called a positive implicative idealistic soft BCK-algebra over X if it satisfies:
(4.3) (∀x∈A) (α(x)CpiX).
Let us illustrate this definition using the following examples.
Example 4.2. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 a 0 0
b b b 0 b 0
c c c c 0 0
d d d c b 0
Let (A, A) be a soft set over X, where A = {0, b, c, d} and A : A → P(X) is a set-valued function defined byA(x) ={0,b}x2 for allx∈A.It is routine to check that (A, A) is a positive implicative idealistic soft BCK-algebra overX.Now, let (A, B) be a soft set over X where B = X and A : B → P(X) is a set-valued function defined by A(x) = {0,b}x2 for all x ∈ B. Then A(a) = {0, b, c, d} is not a positive implicative ideal of X since (a∗c)∗b = 0∗b = 0∈ A(a) andc∗b =c ∈A(a), but a∗b = a /∈ A(a). Hence (A, B) is not a positive implicative idealistic soft BCK-algebra overX.
Note that every positive implicative idealistic soft BCK-algebra over X is an idealistic soft BCK-algebra over X,but the converse is not true in general as seen in the following example.
Example 4.3. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 0 a 0
b b a 0 b 0
c c c c 0 0
d d d d d 0
Let (A, A) be a soft set over X, where A = X and A : A → P(X) is a set- valued function defined by A(x) = {0,a,c}x2 for all x∈ A. It is easy to verify that (A, A) is a soft BCK-algebra over X.Now let (B, I) be a soft set over X, where I={0, b, c, d} ⊂A andB:I →P(X) is a set-valued function defined byB(x) =
x
{0,c} for all x∈ I. Then B(0) = B(c) = {0, c}CX, B(b) = {0, a, b, c}CX and B(d) =XCX.Hence (B, I) is an idealistic soft BCK-algebra overX.Now we have (b∗a)∗a=a∗a= 0 ∈ {0, c} and b∗a=a /∈ {0, c}. ThusB(0) =B(c) ={0, c}
is not a positive implicative ideal ofX, and so (B, I) is not a positive implicative idealistic soft BCK-algebra overX.
Proposition 4.1. Let (A, A)and (A, B) be soft sets overX whereB ⊆A ⊆X.
If (A, A) is a positive implicative idealistic soft BCK-algebra over X, then so is (A, B).
Proof. Straightforward.
By means of Example 4.2, we know that the converse of Proposition 4.1 is not true in general.
Theorem 4.1. Let (A, A) and (B, B) be two positive implicative idealistic soft BCK-algebras over X. If A∩B 6= ∅, then the intersection (A, A)∩(e B, B) is a positive implicative idealistic soft BCK-algebra over X.
Proof. Using Definition 3.2, we can write (A, A)∩(e B, B) = (D, C),whereC=A∩B andD(x) =A(x) orB(x) for all x∈C.Note thatD :C →P(X) is a mapping, and therefore (D, C) is a soft set over X. Since (A, A) and (B, B) are positive implicative idealistic soft BCK-algebras over X,it follows that D(x) = A(x) is a positive implicative ideal ofX,orD(x) =B(x) is a positive implicative ideal of X for allx∈C.Hence (D, C) = (A, A)∩(e B, B) is a positive implicative idealistic soft BCK-algebra overX.
Corollary 4.1. Let (A, A) and (B, A) be two positive implicative idealistic soft BCK-algebras over X.Then their intersection (A, A)e∩(B, A) is a positive implica- tive idealistic soft BCK-algebra overX.
Proof. Straightforward.
Theorem 4.2. Let (A, A) and (B, B) be two positive implicative idealistic soft BCK-algebras overX. If AandB are disjoint, then the union(A, A)∪(e B, B)is a positive implicative idealistic soft BCK-algebra over X.
Proof. Using Definition 3.3, we can write (A, A)∪(e B, B) = (D, C),whereC=A∪B and for everye∈C,
D(e) =
A(e) ife∈A\B, B(e) ife∈B\A, A(e)∪B(e) ife∈A∩B.
Since A∩B = ∅, either x ∈ A\B or x ∈ B \A for all x ∈ C. If x ∈ A\B, then D(x) = A(x) is a positive implicative ideal of X since (A, A) is a positive implicative idealistic soft BCK-algebra over X. If x ∈ B \A, then D(x) = B(x) is a positive implicative ideal of X since (B, B) is a positive implicative idealistic soft BCK-algebra overX. Hence (D, C) = (A, A)∪(e B, B) is a positive implicative idealistic soft BCK-algebra overX.
Theorem 4.3. If (A, A) and (B, B) are positive implicative idealistic soft BCK- algebras over X, then (A, A)∧(e B, B) is a positive implicative idealistic soft BCK- algebra over X.
Proof. By means of Definition 3.4, we know that
(A, A)∧(e B, B) = (D, A×B),
whereD(x, y) =A(x)∩B(y) for all (x, y)∈A×B.SinceA(x) andB(y) are positive implicative ideals of X,the intersectionA(x)∩B(y) is also a positive implicative ideal ofX.HenceD(x, y) is a positive implicative ideal ofX for all (x, y)∈A×B, and therefore (A, A)∧(e B, B) = (D, A×B) is a positive implicative idealistic soft BCK-algebra overX.
Definition 4.4. A positive implicative idealistic soft BCK-algebra (A, A) over X is said to be trivial (resp., whole)if A(x) ={0} (resp., A(x) =X)for allx∈A.
Example 4.4. Let X = {0, a, b, c, d} be the BCK-algebra which is described in Example 4.2. Consider A={c, d} ⊂X and a set-valued functionA :A→P(X) defined byA(x) ={0,b}x for allx∈A.ThenA(c) ={0,b}c =X andA(d) = {0,b}d = X.Hence (A, A) is a whole positive implicative idealistic soft BCK-algebra overX.
Example 4.5. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 0 a 0
b b b 0 b 0
c c c c 0 c
d d d d d 0
LetB:{0} →P(X) be a set-valued function given byB(0) ={0}0 .Then (B,{0}) is a zero positive implicative idealistic soft BCK-algebra overX.
The following example shows that there exists a BCK-algebraX such that a soft set (A,{0}) may not be a zero positive implicative idealistic soft BCK-algebra over X,whereA :{0} →P(X) is given byA(0) = {0}0 .
Example 4.6. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 a 0 a
b b b 0 b 0
c c a c 0 c
d d d d d 0
LetA :{0} →P(X) be a set-valued function given byA(0) = {0}0 .ThenA(0) = {0}is not a positive implicative ideal of X since (c∗a)∗a=a∗a= 0∈A(0) and c∗a= a /∈ A(0). Hence (A,{0}) is not a zero positive implicative idealistic soft BCK-algebra overX.
Since the zero ideal{0} in a positive implicative BCK-algebra is a positive im- plicative ideal, we have the following proposition.
Proposition 4.2. For any positive implicative BCK-algebra X,a soft set(B,{0}) over X, where B : {0} → P(X) is given by B(0) = {0}0 , is the zero positive implicative idealistic soft BCK-algebra over X.
Let f : X →Y be a mapping of BCK-algebras. For a soft set (A, A) overX, (f(A), A) is a soft set overY where f(A) : A →P(Y) is defined by f(A)(x) = f(A(x)) for allx∈A.
Lemma 4.2. Let f : X → Y be an onto homomorphism of BCK-algebras. If (A, A)is a positive implicative idealistic soft BCK-algebra over X,then (f(A), A) is a positive implicative idealistic soft BCK-algebra over Y.
Proof. For everyx∈A,we havef(A)(x) =f(A(x)) is a positive implicative ideal ofY sinceA(x) is a positive implicative ideal ofX and its onto homomorphic image is also a positive implicative ideal of Y. Hence (f(A), A) is a positive implicative idealistic soft BCK-algebra overY.
Theorem 4.4. Let f :X →Y be an onto homomorphism of BCK-algebras and let (A, A)be a positive implicative idealistic soft BCK-algebra overX.
(i) IfY is positive implicative andA(x) = ker(f)for allx∈A,then(f(A), A) is the trivial positive implicative idealistic soft BCK-algebra over Y.
(ii) If(A, A)is whole, then (f(A), A)is the whole positive implicative idealistic soft BCK-algebra overY.
Proof. (i) Assume thatA(x) = ker(f) for all x∈A. Then f(A)(x) =f(A(x)) = {0} for all x∈A. Hence (f(A), A) is the trivial positive implicative idealistic soft BCK-algebra overY.
(ii) Suppose that (A, A) is whole. Then A(x) = X for all x ∈ A, and so f(A)(x) = f(A(x)) = f(X) = Y for all x ∈ A. It follows from Lemma 4.2 and Definition 4.4 that (f(A), A) is the whole positive implicative idealistic soft BCK- algebra overY.
Definition 4.5. Let S be a subalgebra of X. A subset I of X is called a positive implicative ideal ofX related toS(briefly, positive implicativeS-ideal ofX), denoted byICpiS,if it satisfies:
(i) 0∈I,
(ii) (∀x, y, z∈S) ((x∗y)∗z∈I, y∗z∈I ⇒ x∗z∈I).
Note that a positive implicativeX-ideal means a positive implicative ideal.
Definition 4.6. Let (A, A) be a soft BCK-algebra overX. A soft set (B, I) over X is called a positive implicative soft ideal of(A, A),denoted by(B, I)Cepi (A, A), if it satisfies:
(i) I⊂A,
(ii) (∀x∈I) (B(x)CpiA(x)).
Let us illustrate this definition using the following examples.
Example 4.7. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 a 0 0
b b b 0 b 0
c c a c 0 a
d d d d d 0
Let (A, A) be a soft set over X, where A = X and A : A → P(X) is a set- valued function defined byA(x) ={0,a}x for allx∈A.Then (A, A) is a soft BCK- algebra over X. Now take I ={a, c} ⊂A and let B: I →P(X) be a set-valued function defined byB(x) = {0,a}x for allx∈I.ThenB(a) ={0, a, c}CpiA(a) and B(c) ={0, a, c}CpiA(c).Hence (B, I)Cepi (A, A).If we takeJ ={a, b, c} ⊂Aand
define a set-valued function B : J → P(X) by B(x) = {0,a}x for all x∈ J, then B(b) ={0, a, b}is not a positive implicative ideal ofA(b) since (c∗a)∗b=a∗b= a∈A(b) anda∗b=a∈A(b),butc∗b=c /∈A(b).Hence (B, J) is not a positive implicative soft ideal of (A, A).
Theorem 4.5. Let (B, I)and(B, J)be soft sets overX such thatI⊂J.If(B, J) is a positive implicative soft ideal of a soft BCK-algebra (A, A)over X,then so is (B, I).
Proof. Straightforward.
Example 4.7 shows that the converse of Theorem 4.5 is not valid in general.
Theorem 4.6. Let(A, A)be a soft BCK-algebra overX.For any soft sets(B1, I1) and(B2, I2)overX whereI1∩I26=∅,we have
(B1, I1)Cepi(A, A),(B2, I2)Cepi(A, A) ⇒ (B1, I1)∩(e B2, I2)Cepi(A, A).
Proof. Using Definition 3.2, we can write
(B1, I1)∩(e B2, I2) = (B, I),
where I = I1∩I2 and B(x) = B1(x) or B2(x) for all x ∈ I. Obviously, I ⊂ A and B : I → P(X) is a mapping. Hence (B, I) is a soft set over X. Since (B1, I1)Cepi(A, A) and (B2, I2)Cepi(A, A),we know thatB(x) =B1(x)CpiA(x) or B(x) =B2(x)CpiA(x) for all x∈I.Hence
(B1, I1)∩(e B2, I2) = (B, I)Cepi(A, A).
This completes the proof.
Corollary 4.2. Let(A, A) be a soft BCK-algebra overX.For any soft sets(B, I) and(D, I)overX, we have
(B, I)Cepi(A, A),(D, I)Cepi(A, A) ⇒ (B, I)∩(e D, I)Cepi(A, A).
Proof. Straightforward.
Theorem 4.7. Let(A, A)be a soft BCK-algebra overX. For any soft sets(B, I) and(D, J)overX in which I andJ are disjoint, we have
(B, I)Cepi(A, A),(D, J)Cepi(A, A) ⇒ (B, I)e∪(D, J)Cepi(A, A).
Proof. Assume that (B, I)Cepi(A, A) and (D, J)Cepi(A, A).By means of Definition 3.3, we can write (B, I)∪(e D, J) = (H, U) whereU =I∪J and for everyx∈U,
H(x) =
B(x) ifx∈I\J, D(x) ifx∈J\I, B(x)∪D(x) ifx∈I∩J.
SinceI∩J =∅,eitherx∈I\J orx∈J\Ifor allx∈U.Ifx∈I\J,thenH(x) = B(x)CpiA(x) since (B, I)Cepi(A, A). If x ∈ J \I, then H(x) = D(x)CpiA(x) since (D, J)Cepi(A, A).ThusH(x)CpiA(x) for all x∈U, and so (B, I)∪(e D, J) = (H, U)Cepi(A, A).
IfIandJare not disjoint in Theorem 4.7, then Theorem 4.7 is not true in general as seen in the following example.
Example 4.8. LetX ={0, a, b, c, d} be a BCK-algebra with the following Cayley table:
∗ 0 a b c d
0 0 0 0 0 0
a a 0 0 0 0
b b b 0 b 0
c c c c 0 0
d d d c b 0
Let (A, A) be a soft set overX,where A=X and A :A→P(X) is a set-valued function defined byA(x) = {0,b}x2 for allx∈A. ThenA(0) =X, A(a) =A(b) = {0, b, c, d}, and A(c) = A(d) = {0, b} which are subalgebras of X. Hence (A, A) is a soft BCK-algebra over X. Let (B, I) be a soft set over X,where I ={b, c, d}
and B: I → P(X) is a set-valued function defined by B(x) = {0}x for all x∈I.
ThenB(b) ={0, a, b}Cpi{0, b, c, d} =A(b), B(c) ={0, a, c}Cpi{0, b} =A(c),and B(d) =XCpiA(d),and so (B, I) is a positive implicative soft ideal of (A, A).Let (G, J) be a soft set overX,whereJ ={b}andG :J →P(X) is a set-valued function defined byG(x) = {0}x2 for allx∈J.ThenG(b) ={0, c}Cpi{0, b, c, d}=A(b),and so (G, J) is a positive implicative soft ideal of (A, A).Then (S, U) = (B, I)∪(e G, J) is not a positive implicative soft ideal of (A, A) sinceS(b) =B(b)∪G(b) ={0, a, b, c}
is not anA(b)-ideal because d∗c=b∈ {0, a, b, c}andd /∈ {0, a, b, c},and hence it is not a positive implicativeA(b)-ideal ofX.
Acknowledgement. The authors are highly grateful to the referees for their valu- able comments and helpful suggestions in improving this paper.
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