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 soft BCI-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.

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

A BCK/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:

(∀x,y,z∈X)(((x*y)*(x*z))*(z*y)=0);

(∀x,y∈X)((x*(x*y))*y=0);

(∀x∈X)(x*x=0);

(∀x,y∈X)(x*y=0,y*x=0⇒x=y).

If a BCI-algebra X satisfies the following identity:

(∀x∈X)(0*x=0),

then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following axioms:

(∀x∈X)(x*0=x);

(∀x,y,z∈X)(x≤y⇒x*z≤y*z,z*y≤z*x);

(∀x,y,z∈X)((x*y)*z=(x*z)*y);

(∀x,y,z∈X)((x*z)*(y*z)≤x*y),

where x≤y if and only if x*y=0. In a BCI-algebra X, the following hold:

(∀x,y∈X)(x*(x*(x*y))=x*y);

(∀x,y∈X)(0*(x*y)=(0*x)*(0*y)).

A BCI-algebra X is said to be commutative (see [27]) if
(2.1)(∀x,y∈X)(x≤y⇒x=y*(y*x)).

Proposition 2.1.

A BCI-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 subset S 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 commutative BCI-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 [<xref ref-type="bibr" rid="B22">28</xref>]).

A BCI-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 [<xref ref-type="bibr" rid="B22">28</xref>]).

Let I 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, let U 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 [<xref ref-type="bibr" rid="B26">4</xref>, <xref ref-type="bibr" rid="B4">31</xref>]).

A soft set ℱA 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 function fA 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 by S(U) the set of all soft sets over U.

Let ℱA∈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 Soft <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M124"><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>I</mml:mi></mml:math></inline-formula>-Ideals and Int Soft c-<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M125"><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>I</mml:mi></mml:math></inline-formula>-IdealsDefinition 3.1 (see [<xref ref-type="bibr" rid="B16">10</xref>]).

Assume that E 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 [<xref ref-type="bibr" rid="B11">12</xref>]).

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 BCI-ideal (briefly, int soft BCI-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 commutative BCI-ideal (briefly, int soft c-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.

Let ℱA 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 Theorem 3.5 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 soft BCI-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:

ℱA is an int soft c-BCI-ideal over U;

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 that ℱA 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, let ℱA 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.

Let ℱA 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, let ℱA 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.

Let X 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.

Let ℱE 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 [<xref ref-type="bibr" rid="B16">10</xref>]).

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 that x*(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:

ℱA is an int soft c-BCI-ideal over U;

the approximate function fA of ℱA satisfies:
(3.29)(∀x,y∈A)(fA(x*(y*(y*x)))⊇fA(x*y)).

Proof.

Assume that ℱA 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.

Let ℱA 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 of γ-inclusive sets, we consider a characterization of an int soft c-BCI-ideal.

Lemma 3.17 (see [<xref ref-type="bibr" rid="B28">25</xref>]).

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:

ℱA is an int soft BCI-ideal over U;

the nonempty γ-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:

ℱA is an int soft c-BCI-ideal over U;

the nonempty γ-inclusive set of ℱA is a c-BCI-ideal of A for any γ⊆U.

Proof.

Assume that ℱA 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 nonempty γ-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 inclusive c-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

(∀x∈E)(fE(x)⊇gE(x));

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 that ℱE 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.

If ℱE 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.

Let A 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 soft BCI-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 in BCK/BCI-algebras, Hilbert algebras, MV-algebras, MTL-algebras, BL-algebras, and so forth, in future study.

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions.

ZadehL. A.From circuit theory to system theoryZadehL. A.Fuzzy setsZadehL. A.Toward a generalized theory of uncertainty (GTU)—an outlineMolodtsovD.Soft set theory—first resultsMajiP. K.RoyA. R.BiswasR.An application of soft sets in a decision making problemMajiP. K.BiswasR.RoyA. R.Soft set theoryAktaşH.ÇağmanN.Soft sets and soft groupsJunY. B.ParkC. H.Applications of soft sets in ideal theory of BCK/BCI-algebrasJunY. B.LeeK. J.KangM. S.Intersectional soft sets and applications to BCK/BCI-algebrasCommunications of the Korean Mathematical Society. In pressJunY. B.LeeK. J.RohE. H.Intersectional soft BCK/BCI-idealsÇağmanN.ÇitakF.Soft int-group and its applications to group theoryJunY. B.Union soft sets with applications in BCK/BCI-algebrasSubmitted to Bulletin of the Korean Mathematical SocietyAcarU.KoyuncuF.TanayB.Soft sets and soft ringsAtagünA. O.SezginA.Soft substructures of rings, fields and modulesÇağmanN.ÇitakF.EnginogluS.FP-soft set theory and its applicationsFengF.Soft rough sets applied to multicriteria group decision makingFengF.JunY. B.ZhaoX.Soft semiringsJunY. B.Soft BCK/BCI-algebrasJunY. B.KimH. S.NeggersJ.Pseudo d-algebrasJunY. B.LeeK. J.KhanA.Soft ordered semigroupsJunY. B.LeeK. J.ParkC. H.Soft set theory applied to ideals in d-algebrasJunY. B.LeeK. J.ZhanJ.Soft p-ideals of soft BCI-algebrasJunY. B.ZhanJ.Soft ideals of BCK/BCI-algebras based on fuzzy set theoryParkC. H.JunY. B.ÖztürkM. A.Soft WS-algebrasSongS. Z.LeeK. J.JunY. B.Intersectional soft sets applied to subalgebras/ideals in BCK/BCI-algebrasSubmitted to Knowledge-Based SystemsZhanJ.JunY. B.Soft BL-algebras based on fuzzy setsMengJ.XinX. L.Commutative BCI-algebrasMengJ.An ideal characterization of commutative BCI-algebrasHuangY.MengJ.JunY. B.ÇağmanN.EnginoğluS.Soft set theory and uni-int decision makingMengJ.WeiS. M.Periods of elements in BCI-algebras