# Closed Int SoftBCI-Ideals and Int Soft c-BCI-Ideals

Authors: Young Bae Jun; Kyoung Ja Lee; Eun Hwan Roh
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notions of closed intersectional softBCI-ideals and intersectional soft commutative BCI-ideals are introduced, and related properties are investigated. Conditions for an intersectional soft BCI-ideal to be closed are provided. Characterizations of an intersectional soft commutative BCI-ideal are established, and a new intersectional soft c-BCI-ideal from an old one is constructed.


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## Body

## 1. Introduction

The real world is inherently uncertain, imprecise, and vague. Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [1]. In response to this situation Zadeh [2] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [3]. To solve complicated problem in economics, engineering, and environment, we cannot 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 be considered as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties 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 [4]. Maji et al. [5] and Molodtsov [4] suggested that one reason for these difficulties may be the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [4] 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 out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [5] 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. Aktaş and Çağman [7] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. Jun and Park [8] studied applications of soft sets in ideal theory of BCK/BCI-algebras. In 2012, Jun et al. [9, 10] introduced the notion of intersectional soft sets, and considered its applications to BCK/BCI-algebras. Independent of Jun et al.'s introduction, Çağman and Çitak [11] also studied soft int-group and its applications to group theory. Also, Jun [12] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [13–26] for further information regarding algebraic structures/properties of soft set theory. Present authors [10] introduced the notion of  int soft BCK/BCI-ideals in BCK/BCI-algebras. As a continuation of the paper [10], we introduce the notion of closed  int soft BCI-ideals and  int soft c-BCI-ideals in BCI-algebras and investigate related properties. We discuss relations between a closed  int soft BCI-ideal and an  int soft BCI-ideal and provide conditions for an  int soft BCI-ideal to be closed. We establish characterizations of an  int soft c-BCI-ideal and construct a new intersectional soft c-BCI-ideal from an old one.


## 2. Preliminaries

ABCK/BCI-algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers.An algebra(X;*,0) of type (2,0) is called a BCI-algebra if it satisfies the following conditions:  (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 aBCI-algebra X satisfies the following identity:(V)
(∀x∈X)(0*x=0),thenX is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following axioms:(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),wherex≤y  if and only if x*y=0. In a BCI-algebra X, the following hold:(b1)
(∀x,y∈X)(x*(x*(x*y))=x*y);(b2)
(∀x,y∈X)(0*(x*y)=(0*x)*(0*y)).ABCI-algebra X is said to be commutative (see [27]) if
(2.1)(∀x,y∈X)(x≤y⇒x=y*(y*x)).Proposition 2.1.
ABCI-algebra X is commutative  if and only if it satisfies
(2.2)(∀x,y∈X)(x*(x*y)=y*(y*(x*(x*y)))).A nonempty subsetS of a BCK/BCI-algebra X is called a subalgebra of X if x*y∈S for all x,y∈S. A subset I of a BCI-algebra X is called a BCI-ideal of X if it satisfies
(2.3)0∈I,(2.4)(∀x∈X)(∀y∈I)(x*y∈I⇒x∈I).
A BCI-ideal I of a BCI-algebra X satisfies
(2.5)(∀x∈X)(∀y∈I)(x≤y⇒x∈I).
A BCI-ideal I of a BCI-algebra X is said to be closed if it satisfies
(2.6)(∀x∈X)(x∈I⇒0*x∈I).
A subset I of a BCI-algebra X is called a commutativeBCI-ideal (briefly, c-BCI-ideal) of X (see [28]) if it satisfies (2.3) and
(2.7)(x*y)*z∈I,z∈I⇒x*((y*(y*x))*(0*(0*(x*y))))∈I
for all x,y,z∈X.Proposition 2.2 (see [28]).
ABCI-ideal I of a BCI-algebra X is commutative if and only if x*y∈I implies x*((y*(y*x))*(0*(0*(x*y))))∈I.Proposition 2.3 (see [28]).
LetI be a closed BCI-ideal of a BCI-algebra X. Then I is commutative  if and only if it satisfies
(2.8)(∀x,y∈X)(x*y∈I⇒x*(y*(y*x))∈I).Observe that every c-BCI-ideal is a BCI-ideal, but the converse is not true (see [28]).We refer the reader to the books [29, 30] for further information regarding BCK/BCI-algebras.A soft set theory is introduced by Molodtsov [4], and Çağman and Enginoğlu [31] provided new definitions and various results on soft set theory.In what follows, letU be an initial universe set and E be a set of parameters. We say that the pair (U,E) is a soft universe. Let 𝒫(U) denote the power set of U and A,B,C,…⊆E.Definition 2.4 (see [4, 31]).
A soft setℱA over U is defined to be the set of ordered pairs
(2.9)ℱA:={(x,fA(x)):x∈E,fA(x)∈𝒫(U)},
where fA:E→𝒫(U) such that fA(x)=∅ if x∉A.
The functionfA is called the approximate function of the soft set ℱA. The subscript A in the notation fA indicates that fA is the approximate function of ℱA.
In what follows, denote byS(U) the set of all soft sets over U.
LetℱA∈S(U). For any subset γ of U, the γ-inclusive set of ℱA, denoted by ℱAγ, is defined to be the set
(2.10)ℱAγ:={x∈A∣γ⊆fA(x)}.


## 3. Closed  Int SoftBCI-Ideals and  Int Soft c-BCI-Ideals

Definition 3.1 (see [10]).
Assume thatE has a binary operation ↪. For any nonempty subset A of E, a soft set ℱA over U is said to be  intersectional over U if its approximate function fA satisfies
(3.1)(∀x,y∈A)(x↪y∈A⇒fA(x)∩fA(y)⊆fA(x↪y)).Definition 3.2 (see [12]).
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). Then ℱA is called an  intersectional softBCI-ideal (briefly,   int softBCI-ideal) over U if the approximate function fA of ℱA satisfies
(3.2)(∀x∈A)(fA(0)⊇fA(x)),(3.3)(∀x,y∈A)(fA(x)⊇fA(x*y)∩fA(y)).Definition 3.3.
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). Then ℱA is called an  intersectional soft commutativeBCI-ideal (briefly,   int softc-BCI-ideal) over U if the approximate function fA of ℱA satisfies (3.2) and
(3.4)fA((x*y)*z)∩fA(z)⊆fA(x*((y*(y*x))*(0*(0*(x*y)))))
for all x,y,z∈A.Example 3.4.
Let(U,E)=(U,X) where X={0,a,1,2,3} is a BCI-algebra with the following Cayley table:
(3.5)*0a123000321aa0321111032222103333210
For subsets γ1,γ2, and γ3 of U with γ1⊋γ2⊋γ3, let ℱE∈S(U) in which its approximation function fE is defined as follows:
(3.6)fE:E→𝒫(U),x↦{γ1,ifx=0,γ2,ifx=a,γ3,ifx∈{1,2,3}.
Then ℱE is an  int soft c-BCI-ideal over U.Theorem 3.5.
Let(U,E)=(U,X) where X is a BCI-algebra. Then every  int soft c-BCI-ideal is an  int soft BCI-ideal.Proof.
LetℱA be an  int soft c-BCI-ideal over U where A is a subalgebra of E. Taking y=0 in (3.4) and using (a1) and (III) imply that
(3.7)fA(x)=fA(x*0)=fA(x*((0*(0*x))*(0*(0*(x*0)))))⊇fA((x*0)*z)∩fA(z)=fA(x*z)∩fA(z)
for all x,z∈A. Therefore ℱA is an  int soft BCI-ideal over U.The following example shows that the converse of Theorem3.5 is not true.Example 3.6.
Let(U,E)=(U,X) where X={0,1,2,3,4} is a BCI-algebra with the following Cayley table:
(3.8)*01234000000110100222000333300444430
Let γ1,γ2, and γ3 be subsets of U such that γ1⊋γ2⊋γ3. Let ℱE∈S(U) in which its approximation function fE is defined as follows:
(3.9)fE:E→𝒫(U),x↦{γ1,ifx=0,γ2,ifx=1,γ3,ifx∈{2,3,4}.
Routine calculations show that ℱE is an  int soft BCI-ideal over U. But it is not an  int soft c-BCI-ideal over U since
(3.10)fE(2*((3*(3*2))*(0*(0*(2*3)))))=γ3⊉γ1=fE((2*3)*0)∩fE(0).
We provide conditions for an  int softBCI-ideal to be an  int soft c-BCI-ideal.Theorem 3.7.
Let(U,E)=(U,X) where X is a BCI-algebra. For a subalgebra A of E, let ℱA∈S(U). Then the following are equivalent:  (1)
ℱA is an  int soft c-BCI-ideal over U;(2)
ℱA is an  int soft BCI-ideal over U and its approximate function fA satisfies:
(3.11)(∀x,y∈A)(fA(x*((y*(y*x))*(0*(0*(x*y)))))⊇fA(x*y)).Proof.
Assume thatℱA is an  int soft c-BCI-ideal over U. Then ℱA is an  int soft BCI-ideal over U (see Theorem 3.5). If we take z=0 in (3.4) and use (a1) and (3.2), then we have (3.11).
Conversely, letℱA be an  int soft BCI-ideal over U such that its approximate function fA satisfies (3.11). Then fA(x*y)⊇fA((x*y)*z)∩fA(z) for all x,y,z∈A by (3.3), which implies from (3.11) that
(3.12)fA(x*((y*(y*x))*(0*(0*(x*y)))))⊇fA((x*y)*z)∩fA(z)
for all x,y,z∈A. Therefore ℱA is an  int soft c-BCI-ideal over U.Definition 3.8.
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). An  int soft BCI-ideal ℱA over U is said to be  closed if the approximate function fA of ℱA satisfies
(3.13)(∀x∈A)(fA(0*x)⊇fA(x)).Example 3.9.
Let(U,E)=(U,X) where X={0,1,2,a,b} is a BCI-algebra with the following Cayley table:
(3.14)*012ab0000aa1101ba2220aaaaaa00bbab10
Let {γ1,γ2,γ3,γ4,γ5} be a class of subsets of U which is a poset with the following Hasse diagram:
(3.15)
Let ℱE∈S(U) in which its approximation function fE is defined as follows:
(3.16)fE:E→𝒫(U),x↦{γ5,ifx=0,γ2,ifx=1,γ4,ifx=2,γ3,ifx=a,γ1,ifx=b.
Then ℱE is a closed  int soft BCI-ideal over U.Example 3.10.
Let(U,E)=(U,X) where X={2n∣n∈ℤ} is a BCI-algebra with a binary operation “÷” (usual division). Let ℱE∈S(U) in which its approximation function fE is defined as follows:
(3.17)fE:E→𝒫(U),x↦{γ1,ifn≥0,γ2,ifn<0,
where γ1 and γ2 are subsets of U with γ1⊋γ2. Then ℱE is an  int soft BCI-ideal over U which is not closed since
(3.18)fE(1÷23)=fE(2-3)=γ2⊉γ1=fE(23).Theorem 3.11.
Let(U,E)=(U,X) where X is a BCI-algebra. Then an  int soft BCI-ideal over U is closed  if and only if it is an  int soft algebra over U.Proof.
LetℱA be an  int soft BCI-ideal over U. If ℱA is closed, then fA(0*x)⊇fA(x) for all x∈A. It follows from (3.3) that
(3.19)fA(x*y)⊇fA((x*y)*x)∩fA(x)=fA(0*y)∩fA(x)⊇fA(x)∩fA(y)
for all x,y∈A. Hence ℱA is an  int soft algebra over U.
Conversely, letℱA be an  int soft BCI-ideal over U which is also an  int soft algebra over U. Then
(3.20)fA(0*x)⊇fA(0)∩fA(x)=fA(x)
for all x∈A. Therefore ℱA is closed.LetX be a BCI-algebra and B(X):={x∈X∣0≤x}. For any x∈X and n∈ℕ, we define xn by
(3.21)x1=x,xn+1=x*(0*xn).
If there is an n∈ℕ such that xn∈B(X), then we say that x is of finite periodic (see [32]), and we denote its period |x| by
(3.22)|x|=min{n∈ℕ∣xn∈B(X)}.
Otherwise, x is of infinite period and denoted by |x|=∞.Theorem 3.12.
Let(U,E)=(U,X) where X is a BCI-algebra in which every element is of finite period. Then every  int soft BCI-ideal over U is closed.Proof.
LetℱE be an  int soft BCI-ideal over U. For any x∈E, assume that |x|=n. Then xn∈B(X). Note that
(3.23)(0*xn-1)*x=(0*(0*(0*xn-1)))*x=(0*x)*(0*(0*xn-1))=0*(x*(0*xn-1))=0*xn=0,
and so fE((0*xn-1)*x)=fE(0)⊇fE(x) by (3.2). It follows from (3.3) that
(3.24)fE(0*xn-1)⊇fE((0*xn-1)*x)∩fE(x)⊇fE(x).
Also, note that
(3.25)(0*xn-2)*x=(0*(0*(0*xn-2)))*x=(0*x)*(0*(0*xn-2))=0*(x*(0*xn-2))=0*xn-1,
which implies from (3.24) that
(3.26)fE((0*xn-2)*x)=fE(0*xn-1)⊇fE(x).
Using (3.3), we have
(3.27)fE(0*xn-2)⊇fE((0*xn-2)*x)∩fE(x)⊇fE(x).
Continuing this process, we have fE(0*x)⊇fE(x) for all x∈E. Therefore ℱE is closed.Lemma 3.13 (see [10]).
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). If ℱA is an  int soft BCI-ideal over U, then the approximate function fA satisfies the following condition:
(3.28)(∀x,y,z∈A)(x*y≤z⇒fA(x)⊇fA(y)∩fA(z)).Proposition 3.14.
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). If the approximate function fA of ℱA satisfies (3.2) and (3.28), then ℱA is an  int soft BCI-ideal over U.Proof.
Note thatx*(x*y)≤y by (II), and thus fA(x)⊇fA(x*y)∩fA(y) by (3.28). Therefore ℱA is an  int soft BCI-ideal over U.Theorem 3.15.
Let(U,E)=(U,X) where X is a BCI-algebra. For a subalgebra A of E, let ℱA be a closed  int soft BCI-ideal over U. Then the following are equivalent:  (1)
ℱA is an  int soft c-BCI-ideal over U;(2)
the approximate functionfA of ℱA satisfies:
(3.29)(∀x,y∈A)(fA(x*(y*(y*x)))⊇fA(x*y)).Proof.
Assume thatℱA is an  int soft c-BCI-ideal over U. Note that
(3.30)(x*(y*(y*x)))*(x*((y*(y*x))*(0*(0*(x*y)))))≤((y*(y*x))*(0*(0*(x*y))))*(y*(y*x))=((y*(y*x))*(y*(y*x)))*(0*(0*(x*y)))=0*(0*(0*(x*y)))=0*(x*y)
for all x,y∈A. Using Lemma 3.13, (3.11), and (3.13), we have
(3.31)fA(x*(y*(y*x)))⊇fA(x*((y*(y*x))*(0*(0*(x*y)))))∩fA(0*(x*y))⊇fA(x*y)∩fA(0*(x*y))=fA(x*y)
for all x,y∈A. Now suppose that the approximate function fA of ℱA satisfies (3.29). Since
(3.32)(x*((y*(y*x))*(0*(0*(x*y)))))*(x*(y*(y*x)))≤(y*(y*x))*((y*(y*x))*(0*(0*(x*y))))≤0*(0*(x*y)),
it follows from Lemma 3.13, (3.13), and (3.29) that
(3.33)fA(x*((y*(y*x))*(0*(0*(x*y)))))⊇fA(x*(y*(y*x)))∩fA(0*(0*(x*y)))⊇fA(x*y)∩fA(0*(0*(x*y)))=fA(x*y)
for all x,y∈A. By Theorem 3.7, ℱA is an  int soft c-BCI-ideal over U.Theorem 3.16.
Let(U,E)=(U,X) where X is a commutative BCI-algebra. Then every closed  int soft BCI-ideal is an  int soft c-BCI-ideal.Proof.
LetℱA be a closed  int soft BCI-ideal over U where A is a subalgebra of E. Using (a3), (b1), (I), (III), and Proposition 2.1, we have
(3.34)(x*(y*(y*x)))*(x*y)=(x*(x*y))*(y*(y*x))=(y*(y*(x*(x*y))))*(y*(y*x))=(y*(y*(y*x)))*(y*(x*(x*y)))=(y*x)*(y*(x*(x*y)))≤(x*(x*y))*x=0*(x*y).
It follows from Lemma 3.13 and (3.13) that
(3.35)fA(x*(y*(y*x)))⊇fA(x*y)∩fA(0*(x*y))=fA(x*y)
for all x,y∈A. Therefore, by Theorem 3.15, ℱA is an  int soft c-BCI-ideal over U.Using the notion ofγ-inclusive sets, we consider a characterization of an  int soft c-BCI-ideal.Lemma 3.17 (see [25]).
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). Then the following are equivalent:  (1)
ℱA is an  int soft BCI-ideal over U;(2)
the nonemptyγ-inclusive set of ℱA is a BCI-ideal of A for any γ⊆U.Theorem 3.18.
Let(U,E)=(U,X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱA∈S(U). Then the following are equivalent: (1)
ℱA is an  int soft c-BCI-ideal over U;(2)
the nonemptyγ-inclusive set of ℱA is a c-BCI-ideal of A for any γ⊆U.Proof.
Assume thatℱA is an  int soft c-BCI-ideal over U. Then ℱA is an  int soft BCI-ideal over U by Theorem 3.5. Hence ℱAγ is a BCI-ideal of A for all γ⊆U by Lemma 3.17. Let γ⊆U and x,y∈A be such that x*y∈ℱAγ. Then fA(x*y)⊇γ, and so
(3.36)fA(x*((y*(y*x))*(0*(0*(x*y)))))⊇fA(x*y)⊇γ
by Theorem 3.7. Thus
(3.37)x*((y*(y*x))*(0*(0*(x*y))))∈ℱAγ.
It follows from Proposition 2.2 that ℱAγ is a c-BCI-ideal of A.
Conversely, suppose that the nonemptyγ-inclusive set of ℱA is a c-BCI-ideal of A for any γ⊆U. Then ℱAγ is a BCI-ideal of A for all γ⊆U. Hence ℱA is an  int soft BCI-ideal over U by Lemma 3.17. Let x,y∈A be such that fA(x*y)=γ. Then x*y∈ℱAγ, and so
(3.38)x*((y*(y*x))*(0*(0*(x*y))))∈ℱAγ
by Proposition 2.2. Hence
(3.39)fA(x*((y*(y*x))*(0*(0*(x*y)))))⊇γ=fA(x*y).
It follows from Theorem 3.7 that ℱA is an  int soft c-BCI-ideal over U.The c-BCI-ideals ℱAγ in Theorem 3.18 are called the inclusivec-BCI-ideals of ℱA.Theorem 3.19.
Let(U,E)=(U,X) where X is a BCI-algebra. Let ℱE,𝒢E∈S(U) such that  (i)
(∀x∈E)(fE(x)⊇gE(x));(ii)
ℱE and 𝒢E are  int soft BCI-ideals over U. If ℱE is closed and 𝒢E is an  int soft c-BCI-ideal over U, then ℱE is also an  int soft c-BCI-ideal over U.Proof.
Assume thatℱE is closed and 𝒢E is an  int soft c-BCI-ideal over U. Let γ be a subset of U such that ℱEγ≠∅≠𝒢Eγ. Then ℱEγ and 𝒢Eγ are BCI-ideals of E and obviously ℱEγ⊇𝒢Eγ. Let x∈ℱEγ. Then fE(x)⊇γ, and so fE(0*x)⊇fE(x)⊇γ since ℱE is closed. Thus 0*x∈ℱEγ, and thus ℱEγ is a closed BCI-ideal of E. Since 𝒢E is an  int soft c-BCI-ideal over U, it follows from Theorem 3.18 that 𝒢Eγ is a c-BCI-ideal of E. Let x,y∈E be such that x*y∈ℱEγ. Then 0*(x*y)∈ℱEγ. Since (x*(x*y))*y=0∈𝒢Eγ, it follows from Proposition 2.2 that
(3.40)(x*(x*y))*(y*(y*(x*(x*y))))=(x*(x*y))*((y*(y*(x*(x*y))))*(0*(0*((x*(x*y))*y))))∈𝒢Eγ⊆ℱEγ,and so from (a3) that (3.41)(x*(y*(y*(x*(x*y)))))*(x*y)∈ℱEγ.
Hence x*(y*(y*(x*(x*y))))∈ℱEγ by (2.4). Note that
(3.42)(x*(y*(y*x)))*(x*(y*(y*(x*(x*y)))))≤(y*(y*(x*(x*y))))*(y*(y*x))≤(y*x)*(y*(x*(x*y)))≤(x*(x*y))*x=0*(x*y)∈ℱEγ.
Using (2.5) and (2.4), we have x*(y*(y*x))∈ℱEγ. Hence ℱEγ is a c-BCI-ideal of E. Therefore ℱE is an  int soft c-BCI-ideal over U by Theorem 3.18.Theorem 3.20.
Let(U,E)=(U,X) where X is a BCI-algebra. Let ℱE∈S(U) and define a soft set ℱE* over U by
(3.43)fE*:E→𝒫(U),x↦{fE(x),ifx∈ℱEγ,δ,otherwise,
where γ and δ are subset of U with δ⊊fE(x). If ℱE is an  int soft c-BCI-ideal over U, then so is ℱE*.Proof.
IfℱE is an  int soft c-BCI-ideal over U, then ℱEγ is a c-BCI-ideal of A for any γ⊆U. Hence 0∈ℱEγ, and so fE*(0)=fE(0)⊇fE(x)⊇fE*(x) for all x∈A. Let x,y,z∈A. If (x*y)*z∈ℱEγ and z∈ℱEγ, then x*((y*(y*x))*(0*(0*(x*y))))∈ℱEγ and so
(3.44)fE*(x*((y*(y*x))*(0*(0*(x*y)))))=fE(x*((y*(y*x))*(0*(0*(x*y)))))⊇fE((x*y)*z)∩fE(z)=fE*((x*y)*z)∩fE*(z).
If (x*y)*z∉ℱEγ or z∉ℱEγ, then fE*((x*y)*z)=δ or fE*(z)=δ. Hence
(3.45)fE*(x*((y*(y*x))*(0*(0*(x*y)))))⊇δ=fE*((x*y)*z)∩fE*(z).
This shows that ℱE* is an  int soft c-BCI-ideal over U.Theorem 3.21.
Let(U,E)=(U,X) where X is a BCI-algebra. Then any c-BCI-ideal of E can be realized as an inclusive c-BCI-ideal of some  int soft c-BCI-ideal over U.Proof.
LetA be a c-BCI-ideal of E. For any subset γ⊊U, let ℱA be a soft set over U defined by
(3.46)fA:E→𝒫(U),x↦{γ,ifx∈A,∅,ifx∉A.
Obviously, fA(0)⊇fA(x) for all x∈E. For any x,y,z∈E, if (x*y)*z∈A and z∈A then x*((y*(y*x))*(0*(0*(x*y))))∈A. Hence
(3.47)fA((x*y)*z)∩fA(z)=γ=fA(x*((y*(y*x))*(0*(0*(x*y))))).
If (x*y)*z∉A or z∉A then fA((x*y)*z)=∅ or fA(z)=∅. It follows that
(3.48)fA(x*((y*(y*x))*(0*(0*(x*y)))))⊇∅=fA((x*y)*z)∩fA(z).
Therefore ℱA is an  int soft c-BCI-ideal over U, and clearly ℱAγ=A. This completes the proof.


## 4. Conclusion

We have introduced the notions of closed  int softBCI-ideals and  int soft commutative BCI-ideals, and investigated related properties. We have provided conditions for an  int soft BCI-ideal to be closed, and established characterizations of an  int soft commutative BCI-ideal. We have constructed a new  int soft c-BCI-ideal from old one.On the basis of these results, we will apply the theory of  int soft sets to the another type of ideals, filters, and deductive systems inBCK/BCI-algebras, Hilbert algebras, MV-algebras, MTL-algebras, BL-algebras, and so forth, in future study.



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*Source: 125614-2012-12-29.xml*