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:

(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-algebra X satisfies the following identity:
(V)
(∀x ∈ X) (0*x = 0),
then X 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),
where x ≤ 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)).

A BCI-algebra X is said to be commutative (see [27]) if

()

Proposition 2.1. A BCI-algebra X is commutative if and only if it satisfies

()

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

()

A BCI-ideal I of a BCI-algebra X satisfies

()

A BCI-ideal I of a BCI-algebra X is said to be closed if it satisfies

()

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

()

for all x, y, z ∈ X.

Proposition 2.2 (see [28].)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 [28].)Let I be a closed BCI-ideal of a BCI-algebra X. Then I is commutative if and only if it satisfies

()

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 [4], [31].)A soft set ℱ_A over U is defined to be the set of ordered pairs

()

where f_A : E → 𝒫(U) such that f_A(x) = ∅ if x ∉ A.

The function f_A is called the approximate function of the soft set ℱ_A. The subscript A in the notation f_A indicates that f_A is the approximate function of ℱ_A.

In what follows, denote by S(U) the set of all soft sets over U.

Let ℱ_A ∈ S(U). For any subset γ of U, the γ-inclusive set of ℱ_A, denoted by , is defined to be the set

()

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

Definition 3.1 (see [10].)Assume that E has a binary operation ↪. For any nonempty subset A of E, a soft set ℱ_A over U is said to be intersectional over U if its approximate function f_A satisfies

()

Definition 3.2 (see [12].)Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱ_A ∈ S(U). Then ℱ_A is called an intersectional soft BCI-ideal (briefly, int soft BCI-ideal) over U if the approximate function f_A of ℱ_A satisfies

()

Definition 3.3. Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱ_A ∈ S(U). Then ℱ_A is called an intersectional soft commutative BCI-ideal (briefly, int soft c-BCI-ideal) over U if the approximate function f_A of ℱ_A satisfies (3.2) and

()

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:

()

For subsets γ₁, γ₂, and γ₃ of U with γ₁⊋γ₂⊋γ₃, let ℱ_E ∈ S(U) in which its approximation function f_E is defined as follows:

()

Then ℱ_E 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 ℱ_A 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

()

for all x, z ∈ A. Therefore ℱ_A 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:

()

Let γ₁, γ₂, and γ₃ be subsets of U such that γ₁⊋γ₂⊋γ₃. Let ℱ_E ∈ S(U) in which its approximation function f_E is defined as follows:

()

Routine calculations show that ℱ_E is an int soft BCI-ideal over U. But it is not an int soft c-BCI-ideal over U since

()

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 ℱ_A ∈ S(U). Then the following are equivalent:

(1)
ℱ_A is an int soft c-BCI-ideal over U;
(2)
ℱ_A is an int soft BCI-ideal over U and its approximate function f_A satisfies:
()

Proof. Assume that ℱ_A is an int soft c-BCI-ideal over U. Then ℱ_A 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 ℱ_A be an int soft BCI-ideal over U such that its approximate function f_A satisfies (3.11). Then f_A(x*y)⊇f_A((x*y)*z)∩f_A(z) for all x, y, z ∈ A by (3.3), which implies from (3.11) that

()

for all x, y, z ∈ A. Therefore ℱ_A 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 ℱ_A ∈ S(U). An int soft BCI-ideal ℱ_A over U is said to be closed if the approximate function f_A of ℱ_A satisfies

()

Example 3.9. Let (U, E) = (U, X) where X = {0,1, 2, a, b} is a BCI-algebra with the following Cayley table:

()

Let {γ₁, γ₂, γ₃, γ₄, γ₅} be a class of subsets of U which is a poset with the following Hasse diagram:

()

Let ℱ_E ∈ S(U) in which its approximation function f_E is defined as follows:

()

Then ℱ_E 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 ℱ_E ∈ S(U) in which its approximation function f_E is defined as follows:

()

where γ₁ and γ₂ are subsets of U with γ₁⊋γ₂. Then ℱ_E is an int soft BCI-ideal over U which is not closed since

()

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 ℱ_A be an int soft BCI-ideal over U. If ℱ_A is closed, then f_A(0*x)⊇f_A(x) for all x ∈ A. It follows from (3.3) that

()

for all x, y ∈ A. Hence ℱ_A is an int soft algebra over U.

Conversely, let ℱ_A be an int soft BCI-ideal over U which is also an int soft algebra over U. Then

()

for all x ∈ A. Therefore ℱ_A 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

()

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

()

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 ℱ_E be an int soft BCI-ideal over U. For any x ∈ E, assume that |x | = n. Then xⁿ ∈ B(X). Note that

()

and so f_E((0*xⁿ⁻¹)*x) = f_E(0)⊇f_E(x) by (3.2). It follows from (3.3) that

()

Also, note that

()

which implies from (3.24) that

()

Using (3.3), we have

()

Continuing this process, we have f_E(0*x)⊇f_E(x) for all x ∈ E. Therefore ℱ_E 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 ℱ_A ∈ S(U). If ℱ_A is an int soft BCI-ideal over U, then the approximate function f_A satisfies the following condition:

()

Proposition 3.14. Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱ_A ∈ S(U). If the approximate function f_A of ℱ_A satisfies (3.2) and (3.28), then ℱ_A is an int soft BCI-ideal over U.

Proof. Note that x*(x*y) ≤ y by (II), and thus f_A(x)⊇f_A(x*y)∩f_A(y) by (3.28). Therefore ℱ_A 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 ℱ_A be a closed int soft BCI-ideal over U. Then the following are equivalent:

(1)
ℱ_A is an int soft c-BCI-ideal over U;
(2)
the approximate function f_A of ℱ_A satisfies:
()

Proof. Assume that ℱ_A is an int soft c-BCI-ideal over U. Note that

()

for all x, y ∈ A. Using Lemma 3.13, (3.11), and (3.13), we have

()

for all x, y ∈ A. Now suppose that the approximate function f_A of ℱ_A satisfies (3.29). Since

()

it follows from Lemma 3.13, (3.13), and (3.29) that

()

for all x, y ∈ A. By Theorem 3.7, ℱ_A 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 ℱ_A 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

()

It follows from Lemma 3.13 and (3.13) that

()

for all x, y ∈ A. Therefore, by Theorem 3.15, ℱ_A 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 [25].)Let (U, E) = (U, X) where X is a BCI-algebra. Given a subalgebra A of E, let ℱ_A ∈ S(U). Then the following are equivalent:

(1)
ℱ_A is an int soft BCI-ideal over U;
(2)
the nonempty γ-inclusive set of ℱ_A 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 ℱ_A ∈ S(U). Then the following are equivalent:

(1)
ℱ_A is an int soft c-BCI-ideal over U;
(2)
the nonempty γ-inclusive set of ℱ_A is a c-BCI-ideal of A for any γ⊆U.

Proof. Assume that ℱ_A is an int soft c-BCI-ideal over U. Then ℱ_A is an int soft BCI-ideal over U by Theorem 3.5. Hence is a BCI-ideal of A for all γ⊆U by Lemma 3.17. Let γ⊆U and x, y ∈ A be such that . Then f_A(x*y)⊇γ, and so

()

by Theorem 3.7. Thus

()

It follows from Proposition 2.2 that

is a c-BCI-ideal of A.

Conversely, suppose that the nonempty γ-inclusive set of ℱ_A is a c-BCI-ideal of A for any γ⊆U. Then is a BCI-ideal of A for all γ⊆U. Hence ℱ_A is an int soft BCI-ideal over U by Lemma 3.17. Let x, y ∈ A be such that f_A(x*y) = γ. Then , and so

()

by Proposition 2.2. Hence

()

It follows from Theorem 3.7 that ℱ_A is an int soft c-BCI-ideal over U.

The c-BCI-ideals in Theorem 3.18 are called the inclusive c-BCI-ideals of ℱ_A.

Theorem 3.19. Let (U, E) = (U, X) where X is a BCI-algebra. Let ℱ_E, 𝒢_E ∈ S(U) such that

(i)
(∀x ∈ E) (f_E(x)⊇g_E(x));
(ii)
ℱ_E and 𝒢_E are int soft BCI-ideals over U.

If ℱ_E is closed and 𝒢_E is an int soft c-BCI-ideal over U, then ℱ_E is also an int soft c-BCI-ideal over U.

Proof. Assume that ℱ_E is closed and 𝒢_E is an int soft c-BCI-ideal over U. Let γ be a subset of U such that . Then and are BCI-ideals of E and obviously . Let . Then f_E(x)⊇γ, and so f_E(0*x)⊇f_E(x)⊇γ since ℱ_E is closed. Thus , and thus is a closed BCI-ideal of E. Since 𝒢_E is an int soft c-BCI-ideal over U, it follows from Theorem 3.18 that is a c-BCI-ideal of E. Let x, y ∈ E be such that . Then . Since , it follows from Proposition 2.2 that

()

and so from (a3) that

()

Hence

by (2.4). Note that

()

Using (2.5) and (2.4), we have

. Hence

is a c-BCI-ideal of E. Therefore ℱ_E 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 ℱ_E ∈ S(U) and define a soft set over U by

()

where γ and δ are subset of U with δ⊊f_E(x). If ℱ_E is an int soft c-BCI-ideal over U, then so is

Proof. If ℱ_E is an int soft c-BCI-ideal over U, then is a c-BCI-ideal of A for any γ⊆U. Hence , and so for all x ∈ A. Let x, y, z ∈ A. If and , then and so

()

, then

. Hence

()

This shows that

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 ℱ_A be a soft set over U defined by

()

Obviously, f_A(0)⊇f_A(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

()

If (x*y)*z ∉ A or z ∉ A then f_A((x*y)*z) = ∅ or f_A(z) = ∅. It follows that

()

Therefore ℱ_A is an int soft c-BCI-ideal over U, and clearly

. 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.

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

Abstract

1. Introduction

2. Preliminaries

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

4. Conclusion

Acknowledgments

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

Abstract

1. Introduction

2. Preliminaries

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

4. Conclusion

Acknowledgments

References

Citing Literature

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