Research Article | Open Access

# Hypervector Spaces Based on Intersectional Soft Sets

**Academic Editor:**Gerd Teschke

#### Abstract

The notion of int-soft subfields, int-soft algebras over int-soft subfields, and int-soft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, int-soft hypervector spaces are discussed.

#### 1. Introduction

The hyperstructure theory was introduced by Marty [1] at the 8th congress of Scandinavian Mathematicians in 1934. Since then many researchers have worked in these areas, for example, hyper-MV-algebras, hyper--algebras, hyper--algebras, hyperlattices, and so forth. As a generalization of fuzzy vector spaces, the fuzzy hypervector spaces are studied by Ameri and Dehghan (see [2, 3]).

Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [4]. In response to this situation Zadeh [5] 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 [6]. 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 consider 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 [7]. Maji et al. [8] and Molodtsov [7] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [7] 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. At present, works on the soft set theory are progressing rapidly. Maji et al. [8] described the application of soft set theory to a decision making problem. Maji et al. [9] also studied several operations on the theory of soft sets. Chen et al. [10] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. Çağman et al. [11] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision making method based on FP-soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [12] considered the application of soft rough approximations in multicriteria group decision making problems. Aktaş and Çağman [13] 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. After that, many algebraic properties of soft sets are studied (see [14–24]).

In this paper, we introduce the notion of int-soft subfields, int-soft algebras over int-soft subfields, and int-soft hypervector spaces. We study their properties and characterizations. In connection with linear transformations, we discuss int-soft hypervector spaces.

#### 2. Preliminaries

A soft set theory is introduced by Molodtsov [7], and Çağman and Enginoğlu [25] 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. Let denote the power set of and .

*Definition 1 (see [7, 25]). *A *soft set * over is defined to be the set of ordered pairs
where such that if .

For any sets and , let be a function and and be soft sets over .(1)The soft set
where , is called the *soft preimage* of under .(2)The soft set
where
is called the *soft image* of under .

A map is called a hyperoperation or join operation, where is the set of all nonempty subsets of . The join operation is extended to subsets of in natural way, so that is given by The notations and are used for and , respectively. Generally, the singleton is identified by its element .

*Definition 2 (see [26]). *Let be a field and be an abelian group. A *hypervector space* over is defined to be the quadruplet , where “” is a mapping
such that for all and the following conditions hold:(H1),
(H2),
(H3),
(H4),
(H5).

A hypervector space over a field is said to be *strongly left distributive* (see [2]) if it satisfies the following condition:

#### 3. Int-Soft Algebras over an Int-Soft Field

In what follows let be a field unless otherwise specified.

*Definition 3. *A soft set over is called an *int-soft subfield* of if the following conditions are satisfied:(1),
(2),
(3),
(4).

Proposition 4. *If is an int-soft subfield of , then*(1),
(2),
(3).

*Proof. *(1) For all , we have

(2) Let be such that . Then

(3) It is by (1).

It is easy to show that the following theorem holds.

Theorem 5. *A soft set over is an int-soft subfield of if and only if the nonempty -inclusive set
**
of is a subfield of for all .*

*Definition 6. *Let be an algebra over and let be an int-soft subfield of . A soft set is called an *int-soft algebra* over if it satisfies the following conditions:(1),
(2),
(3),
(4).

Proposition 7. *Let be an algebra over and let be an int-soft subfield of . If is an int-soft algebra over , then for all .*

*Proof. *For any , we have .

Theorem 8. *Let be an algebra over and let be an int-soft subfield of . Then a soft set is an int-soft algebra over if and only if the following conditions are valid:*(1)*,
*(2)*,
*(3)*. *

*Proof. *Assume that is an algebra over . Using (1) and (2) of Definition 6, we have
for all and .

Conditions (2) and (3) are by Definition 6 (3) and Definition 6 (4), respectively.

Conversely, suppose that the inclusions of Theorem 8 hold for all and . Then
The condition (3) and Proposition 4 (3) imply that for all . Thus
for all and . Therefore is an algebra over .

Theorem 9. *Let and be algebras over . For any algebraic homomorphism , we have the following.*(1)*If is an int-soft algebra over , then the soft preimage of under is also an int-soft algebra over .*(2)*If is an int-soft algebra over , then the soft image of under is also an int-soft algebra over .*

*Proof. *(1) For any and , we have
and . Therefore, by Theorem 8, is an algebra over .

(2) Let . If or , then
Assume that and . Then , and so
For any and , we have
For all , if at least one of and is empty, then the inclusion
is clear. Assume that and . Then
Since for all , it follows that
for all . Therefore is an algebra over .

#### 4. Int-Soft Hypervector Spaces

*Definition 10. *Let be a hypervector space over and an int-soft subfield of . A soft set over is called an *int-soft hypervector space* of related to if the following assertions are valid:(1),
(2),
(3),
(4) where is the zero of .

Proposition 11. *Let be a hypervector space over and an int-soft subfield of . If is an int-soft hypervector space of related to , then*(1),
(2),
(3).

*Proof. *It is an immediate consequence of Definition 10 and Proposition 4.

Proposition 12. *Let be a hypervector space over . If is an int-soft hypervector space of related to an int-soft subfield of , then
*

*Proof. *Let . Since by (H5), we have . Using Definition 10 (3)
Hence for all .

Theorem 13. *Assume that a hypervector space over is strongly left distributive. Let be an int-soft subfield of . Then a soft set over is an int-soft hypervector space of related to if and only if the following conditions are true:*(1)(2)*for all and all .*

*Proof. *Assume that is an hypervector space of related to . The second condition follows from Proposition 11 (2) and Definition 10 (4). Let and . Then
Conversely suppose the conditions (1) and (2) are true. For all , we have
Since is a subfield of , we have and . Note that for all . It follows that
for all . Let and . Then
Clearly, . Therefore is an int-soft hypervector space of related to .

Theorem 14. *Let be a hypervector space over and an int-soft subfield of . If a soft set over is an int-soft hypervector space of related to , then the nonempty -inclusive set
**
of is a subhypervector space of over the field for all .*

*Proof. *Let . Then and . It follows that
Hence . Note that is a subfield of (see Theorem 5). Let , and . Then
and so which shows that . Therefore is a hypervector space over the field for all .

Let and be hypervector spaces over . A mapping is called *linear transformation* (see [3]) if it satisfies the following:(i),
(ii).

Theorem 15. *Let and be hypervector spaces over and let be an int-soft subfield of . For any linear transformation , if is an int-soft hypervector space of related to , then is an int-soft hypervector space of related to .*

*Proof. *Let and . Since is a linear transformation, we have
Obviously, for all . It follows from Theorem 13 that is a hypervector space of related to .

Theorem 16. *Let and be hypervector spaces over and let be an int-soft subfield of . For any linear transformation , if is an int-soft hypervector space of related to , then is an int-soft hypervector space of related to .*

*Proof. *Let and . If at least one of and is empty, then the inclusion
is clear. Assume that and are nonempty. Then there exists such that and . Thus
since is linear. Hence . Then
Obviously, for all . Therefore is a hypervector space of related to by Theorem 13.

#### Conflict of Interests

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

#### Acknowledgment

This work (RPP-2012-021) was supported by the fund of Research Promotion Program, Gyeongsang National University, 2012. The third author would like to thank Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, for financial support.

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#### Copyright

Copyright © 2014 Young Bae Jun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.