#### Abstract

The main motivation of this paper is to introduce the notion of cubic linear space. This inspiration is received from the structure of cubic sets. The notions of R-intersection, R-union, P-intersection, and P-union of cubic linear spaces are defined and we provide some results on these. We further introduce the notion of internal cubic linear space and external cubic linear space and establish some results on them.

#### 1. Introduction

The notion of fuzzy sets introduced by Zadeh [1] in 1965 laid the foundation for the development of fuzzy Mathematics. This theory has a wide range of application in several branches of Mathematics such as logic, set theory, group theory, semigroup theory, real analysis, measure theory, and topology. After a decade, the notion of interval-valued fuzzy sets was introduced by Zadeh [2] in 1975, as an extension of fuzzy sets, that is, fuzzy sets with interval-valued membership functions. Lubczonok and Murali [3] introduced an interesting theory of flags and fuzzy subspaces of vector spaces. Katsaras and Liu [4] introduced the concepts of fuzzy vector and fuzzy topological vector spaces. Fuzzy bases of vector spaces and fuzzy vector spaces have been studied in [5, 6]. Nanda [7] introduced the notion of fuzzy field and fuzzy linear space over a fuzzy field. Wenxiang and Lu [8] redefined the concepts of fuzzy field and fuzzy linear space. Vijayabalaji et al. [9] introduced the notion of interval-valued fuzzy linear subspace and interval-valued fuzzy -normed linear space. They have also proved that the intersection of two interval-valued fuzzy linear spaces is again an interval-valued fuzzy linear space. Atanassov [10] introduced the notion of intuitionistic fuzzy sets as a generalization of fuzzy sets.

Jun et al. [11] have introduced a remarkable theory, namely, the theory of cubic sets. This structure is comprised of an interval-valued fuzzy set and a fuzzy set. In the same paper they introduced the notion of cubic subalgebras/ideals in BCK/BCI algebras and investigated some of their properties. Moreover, Jun et al. [12] introduced the notion of cubic subgroups. They also studied images or inverse images of cubic subgroups. Furthermore, Jun et al. [13] introduced the concept of an internal cubit set and an external cubic set. Recently, Yaqoob et al. [14] introduced the notion of cubic ideals of -algebras.

Attracted by the theory of cubic sets we introduce the notion of cubic linear space. The concept of -intersection, -union, -intersection, and -union of cubic linear space are introduced and some properties are studied. We prove that the -intersection of two cubic linear spaces is again a cubic linear space. It is shown by means of counter examples that the -union, -intersection, and -union of two cubic linear spaces need not be a cubic linear space. We also introduce the notions of internal cubic linear space and external cubic linear space. It is established that the -intersection of two internal (resp., external) cubic linear spaces is again an internal (resp., external) cubic linear space. We conclude the paper by providing examples to show that the -intersection, -union, and the -union of two internal (resp., external) cubic linear spaces are not internal (resp., external) cubic linear spaces.

#### 2. Preliminaries

In the following we provide the essential definitions and results necessary for the development of our theory.

Definition 1 (see [2]). An interval number on , say , is a closed subinterval of ; that is, , where . Let denote the set of all closed subintervals of ; that is,

Definition 2 (see [2]). Let for all , , the index set. Define(a);(b).In particular, whenever , in , one defines(i) if and only if and ;(ii) if and only if and ;(iii) if and only if and ;(iv);(v).

Definition 3 (see [2]). Let be a set. A mapping is called an interval-valued fuzzy set (briefly, an i-v fuzzy set) of , where , for all , and and are fuzzy sets in such that for all .

Definition 4 (see [4]). A fuzzy linear space is a pair , where is a vector space over a field , and is a mapping satisfying for any , . Here stands for intersection.

Definition 5 (see [9]). Let denote a vector space over a field . Let be an interval-valued fuzzy subset of . Then is said to be an interval-valued fuzzy linear space if ; and .

Theorem 6 (see [9]). The intersection of two interval-valued fuzzy linear spaces is again an interval-valued fuzzy linear space.

Definition 7 (see [11]). Let be a nonempty set. A cubic set is a structure of the form and denoted by . is an interval-valued fuzzy set (briefly, IVF) in and is a fuzzy set in .

Definition 8 (see [13]). Let be a nonempty set. A cubic set is said to be an internal cubic set (briefly, ICS) if for all .

Definition 9 (see [13]). Let be a nonempty set. A cubic set is said to be an external cubic set (briefly, ECS) if for all .

Definition 10 (see [13]). For any where (index set), one defines(i) (-union);(ii) (-intersection);(iii) (-union);(iv) (-intersection).

Definition 11 (see [13]). The complement of is defined to be the cubic set

Theorem 12 (see [14]). Let = be a cubic subset in , then = in a cubic -ideal of if and only if for all and , the set is either empty or a -ideal of .

#### 3. A Cubic Set Theoretical Approach to Linear Space

In this section, we introduce the notion of cubic linear space as follows.

Definition 13. Let be a linear space over a field , an interval-valued fuzzy linear space, and a fuzzy linear space. A cubic set in is called a cubic linear space of if for all and ,(i),(ii).

Example 14. Let be the Klein 4-group defined by the binary operation as follows:Let be the field GF(2). Let , for all . Then is a linear space over .
Define an interval-valued fuzzy set in byThen is an interval-valued fuzzy linear space.
Define a fuzzy set in byNote that is a fuzzy linear space of .
Hence is a cubic linear space of .

Theorem 15. Let and be two cubic linear spaces. Then their -intersection is a cubic linear space.

Proof. Define as followsNow,Also define bySo,Thus is a cubic linear space.

Remark 16. (i) Let and be two cubic linear spaces. Then their -union need not be a cubic linear space.
(ii) Let and be two cubic linear spaces. Then their -intersection need not be a cubic linear space.
(iii) Let and be two cubic linear spaces. Then their -union need not be a cubic linear space.

Proof. We will prove the above three statements by means of an example.
(i) Let be the Klein -group as in Example 14.
Let be the field GF(2). Let , for all . Then is a linear space over .
Define two interval-valued fuzzy sets and as follows:Observe that and are interval-valued fuzzy linear spaces of .
Define by for all .
So, , , , .
Thus is an interval-valued fuzzy subset of .
When we haveBut , which is absurd.
This shows that the union of two interval-valued fuzzy linear spaces need not be an interval-valued fuzzy linear space.
Now define two fuzzy sets and in byWe observe that and are fuzzy linear spaces over .
Define by .
Then , , , and .
So is fuzzy subset of .
When we haveBut , which is absurd.
So, the intersection of two fuzzy linear spaces need not be a fuzzy linear space.
Hence the -union need not be a cubic linear space.
(ii) Let , , and be as in (i).
Define by for all .
So, , , and .
One can verify that is an interval-valued fuzzy linear space.
Also by (i), is not a fuzzy linear space.
Hence the -intersection is not a cubic linear space.
(iii) Let , , , and be as in (i).
Define by .
Then , , and .
By verification it can be seen that is a fuzzy linear space.
By (i), is not an interval-valued fuzzy linear space.
Thus the -union is not a cubic linear space.

Definition 17. Let be a cubic linear space of . Define , where , , called the cubic level set of .

Theorem 18. Let be a linear space over a field . A cubic set is a cubic linear space of if and only if for all and , the set is either empty or a linear space of over a field .

Proof. Assume that is a cubic linear space of over a field , let and be such that , and let be such that ; then , and , .
Therefore,Moreover,Therefore, is a linear space over a field .
Conversely, suppose that is a linear space over a field and let and be such that , .
Taking   and , we have , .
Also , .
It follows that and .
This is a contradiction and hence is a cubic linear space of over a field .

Definition 19. Let and be two cubic linear spaces of over the field . The Cartesian product of cubic linear spaces and is denoted by defined as(i),(ii), for all and .

Theorem 20. Let and be cubic linear spaces of and over the field . Then is a cubic linear space of over .

Proof. Let , , , and . ConsiderLet . ThenTherefore is a cubic linear space of over .

#### 4. Internal and External Cubic Linear Spaces

In this section, we introduce the notion of internal and external cubic linear spaces and establish some of their properties.

Definition 21. Let be a linear space over a field . A cubic set in is called an internal cubic linear space (briefly, ICLS) if for all and . It is denoted by

Example 22. Let be the Klein 4-group as in Example 14.
Define an interval-valued fuzzy set in by Then is an interval-valued fuzzy linear space.
Define a fuzzy set in byIt is easy to verify that is a fuzzy linear space of .
When , we have for all . So, is an ICLS.

Definition 23. Let be a linear space over a field . A cubic set in is called an external cubic linear space (briefly, ECLS) if for all and . It is denoted by .

Example 24. Let be the Klein 4-group as in Example 14.
Define an interval-valued fuzzy set in byThen is an interval-valued fuzzy linear space.
Define a fuzzy set in byObserve that is a fuzzy linear space of .
So, is a cubic linear space of .
For all and , we have .
Thus, is an ECLS.

Theorem 25. Let be a cubic linear space of which is not an ECLS. Then there exist such that .

Proof. The proof is straightforward.

Theorem 26. Let be a cubic linear space of . If is both an ICLS and an ECLS, then , where and .

Proof. Assume that is both an ICLS and an ECLS. Using Definitions 21 and 23, we have and for all . Thus or , and so .

Theorem 27. Let and be two ICLSs. Then their -intersection is again an ICLS.

Proof. Since is an ICLS in , we have for all and .
Since is an ICLS in , we have for all and .
This implies thatHence is an ICLS.

Theorem 28. Let and be two ECLSs. Then their -intersection is again an ECLS.

Proof. Since is an ECLS in , we have for all and .
Since is an ECLS in , we have for all and .
This implies thatHence is an ECLS.

Remark 29. (i) Let and be two ICLSs. Then their -intersection = need not be an ICLS.
(ii) Let and be two ECLSs. Then their -intersection need not be an ECLS.
(iii) Let and be two ICLSs. Then their -union need not be an ICLS.
(iv) Let and be two ECLSs. Then their -union need not be an ECLS.
(v) Let and be two ICLSs. Then their -union need not be an ICLS.
(vi) Let and be two ECLSs. Then their -union need not be an ECLS.

Proof. We will prove the above six statements by means of an example.
(i) Let be the Klein -group as in Example 14.
Define two interval-valued fuzzy sets and as follows:Observe that and are interval-valued fuzzy linear space of .
Define for all .
Therefore, , , and .
By routine calculations it can be seen that is an interval-valued fuzzy linear space of .
Now define two fuzzy sets and in byWe observe that and are fuzzy linear spaces over .
Define .
Then , , , and .
So is fuzzy set of .
Here we note that for all andBut , which is absurd.
So, the intersection of two ICLSs need not be an ICLS.
That is, the -intersection of two ICLSs need not be an ICLS.
(ii) Let be the Klein 4-group as in Example 14.
Define two interval-valued fuzzy sets and as follows:Observe that and are interval-valued fuzzy linear space of .
Define for all .
Therefore, , , and .
By routine calculations it can be seen that the intersection of two interval-valued fuzzy linear spaces is again an interval-valued fuzzy linear space.
Now define two fuzzy sets and in byWe observe that and are fuzzy linear spaces over .
Here we note that for all and Now define .
Then, , , .
So is fuzzy subset of . Consider But , which is absurd.
So, the intersection of two ECLSs need not be an ECLS.
That is, -intersection of two ECLSs need not be an ECLS.
(iii) Let be the Klein 4-group as in Example 14.
Define two interval-valued fuzzy sets and as follows:Observe that and are interval-valued fuzzy linear space of .
Define for all .
Therefore, , , , and .
Thus is an i-v fuzzy subset of . ConsiderBut , which is absurd.
This shows that the union of two interval-valued fuzzy linear spaces need not be an interval-valued fuzzy linear space.
Now define two fuzzy sets and in byWe observe that and are fuzzy linear spaces over .
Define .
Then , , , and .
So is fuzzy subset of .
Here we note that for all andBut , which is absurd.
So, the intersection of two ICLSs need not be an ICLS.
That is, the union of two ICLSs, need not be an ICLS.
(iv) Let be the Klein 4-group as in Example 14.
Define two interval-valued fuzzy sets and as follows:Observe that and are interval-valued fuzzy linear space of .
Define for all .
Therefore, , , , .
Thus is an i-v fuzzy subset of But , which is absurd.
Now define two fuzzy sets and in byWe observe that and are fuzzy linear spaces over .
Here we note that for all and Now define
Then, , , .
So is fuzzy subset of By routine calculations it can be seen that the union of two fuzzy linear spaces is also a fuzzy linear space.
That is, -union of two ECLSs need not be an ECLS.
(v) From (i) and (ii), -union of two ICLSs need not be an ICLS.
(vi) From (ii) and (iv), -union of two ECLSs need not be an ECLS.

Theorem 30. Let be a cubic linear space of . If is an ICLS (resp., ECLS), then is an ICLS (resp., ECLS).

Proof. Since is an ICLS (resp., ECLS) in , we have () for all . This implies that Hence is an ICLS (resp., ECLS) in .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors wish to express their sincere thanks to the referee for giving insightful comments for improving the quality of the paper.