#### 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 soft -ideals and intersectional soft commutative -ideals are introduced, and related properties are investigated. Conditions for an intersectional soft -ideal to be closed are provided. Characterizations of an intersectional soft commutative -ideal are established, and a new intersectional soft c--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 -algebras. In 2012, Jun et al. [9, 10] introduced the notion of intersectional soft sets, and considered its applications to -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 -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 -ideals in -algebras. As a continuation of the paper [10], we introduce the notion of closed int soft -ideals and int soft c--ideals in -algebras and investigate related properties. We discuss relations between a closed int soft -ideal and an int soft -ideal and provide conditions for an int soft -ideal to be closed. We establish characterizations of an int soft c--ideal and construct a new intersectional soft c--ideal from an old one.

#### 2. Preliminaries

A -algebra is an important class of logical algebras introduced by Iséki and was extensively investigated by several researchers.

An algebra of type is called a *-algebra* if it satisfies the following conditions: (I);
(II);
(III);
(IV). If a -algebra satisfies the following identity: (V),
then is called a *-algebra*. Any -algebra satisfies the following axioms: (a1);
(a2);
(a3);
(a4),
where if and only if . In a -algebra , the following hold: (b1);
(b2) .

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

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

A nonempty subset of a -algebra is called a *subalgebra* of if for all . A subset of a -algebra is called a *-ideal* of if it satisfies
A -ideal of a -algebra satisfies
A -ideal of a -algebra is said to be *closed* if it satisfies
A subset of a -algebra is called a *commutative **-ideal* (briefly, *-ideal*) of (see [28]) if it satisfies (2.3) and
for all .

Proposition 2.2 (see [28]). * A -ideal of a -algebra is commutative if and only if implies .*

Proposition 2.3 (see [28]). * Let be a closed -ideal of a -algebra . Then is commutative if and only if it satisfies
*

Observe that every c--ideal is a -ideal, but the converse is not true (see [28]).

We refer the reader to the books [29, 30] for further information regarding -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 be an initial universe set and be a set of parameters. We say that the pair is a *soft universe.* Let denote the power set of and .

*Definition 2.4 (see [4, 31]). *A soft set over is defined to be the set of ordered pairs
where such that if .

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

In what follows, denote by the set of all soft sets over .

Let . For any subset of , the *-inclusive set* of , denoted by , is defined to be the set

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

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

*Definition 3.2 (see [12]). *Let where is a -algebra. Given a subalgebra of , let . Then is called an *intersectional soft *-*ideal* (briefly, *int soft *-*ideal*) over if the approximate function of satisfies

*Definition 3.3. *Let where is a -algebra. Given a subalgebra of , let . Then is called an *intersectional soft commutative *-*ideal* (briefly, *int soft *c--*ideal*) over if the approximate function of satisfies (3.2) and
for all .

*Example 3.4. *Let where is a -algebra with the following Cayley table:
For subsets , and of with , let in which its approximation function is defined as follows:
Then is an int soft c--ideal over .

Theorem 3.5. * Let where is a -algebra. Then every int soft c--ideal is an int soft -ideal. *

*Proof. *Let be an int soft c--ideal over where is a subalgebra of . Taking in (3.4) and using (a1) and (III) imply that
for all . Therefore is an int soft -ideal over .

The following example shows that the converse of Theorem 3.5 is not true.

*Example 3.6. *Let where is a -algebra with the following Cayley table:
Let , and be subsets of such that . Let in which its approximation function is defined as follows:
Routine calculations show that is an int soft -ideal over . But it is not an int soft c--ideal over since

We provide conditions for an int soft -ideal to be an int soft c--ideal.

Theorem 3.7. * Let where is a -algebra. For a subalgebra of , let . Then the following are equivalent: *(1)* is an int soft c--ideal over ;*(2)* is an int soft -ideal over and its approximate function satisfies:
*

*Proof. *Assume that is an int soft c--ideal over . Then is an int soft -ideal over (see Theorem 3.5). If we take in (3.4) and use (a1) and (3.2), then we have (3.11).

Conversely, let be an int soft -ideal over such that its approximate function satisfies (3.11). Then for all by (3.3), which implies from (3.11) that
for all . Therefore is an int soft c--ideal over .

*Definition 3.8. *Let where is a -algebra. Given a subalgebra of , let . An int soft -ideal over is said to be *closed* if the approximate function of satisfies

*Example 3.9. * Let where is a -algebra with the following Cayley table:
Let be a class of subsets of which is a poset with the following Hasse diagram:
(3.15)
Let in which its approximation function is defined as follows:
Then is a closed int soft -ideal over .

*Example 3.10. *Let where is a -algebra with a binary operation “” (usual division). Let in which its approximation function is defined as follows:
where and are subsets of with . Then is an int soft -ideal over which is not closed since

Theorem 3.11. *Let where is a -algebra. Then an int soft -ideal over is closed if and only if it is an int soft algebra over .*

*Proof. *Let be an int soft -ideal over . If is closed, then for all . It follows from (3.3) that
for all . Hence is an int soft algebra over .

Conversely, let be an int soft -ideal over which is also an int soft algebra over . Then
for all . Therefore is closed.

Let be a -algebra and . For any and , we define by
If there is an such that , then we say that is of * finite periodic* (see [32]), and we denote its period by
Otherwise, is of infinite period and denoted by .

Theorem 3.12. *Let where is a -algebra in which every element is of finite period. Then every int soft -ideal over is closed. *

*Proof. *Let be an int soft -ideal over . For any , assume that . Then . Note that
and so 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 for all . Therefore is closed.

Lemma 3.13 (see [10]). *Let where is a -algebra. Given a subalgebra of , let . If is an int soft -ideal over , then the approximate function satisfies the following condition:
*

Proposition 3.14. *Let where is a -algebra. Given a subalgebra of , let . If the approximate function of satisfies (3.2) and (3.28), then is an int soft -ideal over .*

*Proof. *Note that by (II), and thus by (3.28). Therefore is an int soft -ideal over .

Theorem 3.15. * Let where is a -algebra. For a subalgebra of , let be a closed int soft -ideal over . Then the following are equivalent: *(1)* is an int soft c--ideal over ;*(2)*the approximate function of satisfies:
*

*Proof. *Assume that is an int soft c--ideal over . Note that
for all . Using Lemma 3.13, (3.11), and (3.13), we have
for all . Now suppose that the approximate function of satisfies (3.29). Since
it follows from Lemma 3.13, (3.13), and (3.29) that
for all . By Theorem 3.7, is an int soft c--ideal over .

Theorem 3.16. *Let where is a commutative -algebra. Then every closed int soft -ideal is an int soft c--ideal. *

*Proof. * Let be a closed int soft -ideal over where is a subalgebra of . Using (a3), (b1), (I), (III), and Proposition 2.1, we have
It follows from Lemma 3.13 and (3.13) that
for all . Therefore, by Theorem 3.15, is an int soft c--ideal over .

Using the notion of -inclusive sets, we consider a characterization of an int soft c--ideal.

Lemma 3.17 (see [25]). *Let where is a -algebra. Given a subalgebra of , let . Then the following are equivalent: *(1)* is an int soft -ideal over ;*(2)*the nonempty -inclusive set of is a -ideal of for any .*

Theorem 3.18. * Let where is a -algebra. Given a subalgebra of , let . Then the following are equivalent: *(1)* is an int soft c--ideal over ;*(2)*the nonempty -inclusive set of is a c--ideal of for any .*

* Proof. *Assume that is an int soft c--ideal over . Then is an int soft -ideal over by Theorem 3.5. Hence is a -ideal of for all by Lemma 3.17. Let and be such that . Then , and so
by Theorem 3.7. Thus
It follows from Proposition 2.2 that is a c--ideal of .

Conversely, suppose that the nonempty -inclusive set of is a c--ideal of for any . Then is a -ideal of for all . Hence is an int soft -ideal over by Lemma 3.17. Let be such that . Then , and so
by Proposition 2.2. Hence
It follows from Theorem 3.7 that is an int soft c--ideal over .

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

Theorem 3.19. *Let where is a -algebra. Let such that *(i)*;
*(ii)* and are int soft -ideals over .** If is closed and is an int soft c--ideal over , then is also an int soft c--ideal over .*

*Proof. * Assume that is closed and is an int soft c--ideal over . Let be a subset of such that . Then and are -ideals of and obviously . Let . Then , and so since is closed. Thus , and thus is a closed -ideal of . Since is an int soft c--ideal over , it follows from Theorem 3.18 that is a c--ideal of . Let 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--ideal of . Therefore is an int soft c--ideal over by Theorem 3.18.

Theorem 3.20. *Let where is a -algebra. Let and define a soft set over by
**
where and are subset of with . If is an int soft c--ideal over , then so is .*

*Proof. *If is an int soft c--ideal over , then is a c--ideal of for any . Hence , and so for all . Let . If and , then and so
If or , then or . Hence
This shows that is an int soft c--ideal over .

Theorem 3.21. *Let where is a -algebra. Then any c--ideal of can be realized as an inclusive c--ideal of some int soft c--ideal over .*

*Proof. *Let be a c--ideal of . For any subset , let be a soft set over defined by
Obviously, for all . For any , if and then . Hence
If or then or . It follows that
Therefore is an int soft c--ideal over , and clearly . This completes the proof.

#### 4. Conclusion

We have introduced the notions of closed int soft -ideals and int soft commutative -ideals, and investigated related properties. We have provided conditions for an int soft -ideal to be closed, and established characterizations of an int soft commutative -ideal. We have constructed a new int soft c--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 -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.