Research Article | Open Access

# Unisoft Hypervector Spaces

**Academic Editor:**M. Mursaleen

#### Abstract

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

#### 1. Introduction

The hyperstructure theory was introduced by Marty [1] at the 8th Congress of Scandinavian Mathematicians in 1934. As a generalization of fuzzy vector spaces, the fuzzy hypervector spaces are studied by Ameri and Dehghan (see [2, 3]). 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 to several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. 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. Chen et al. [7] 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. [8] 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 [9] considered the application of soft rough approximations in multicriteria group decision-making problems. Aktaş and Çağman [10] 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 [11–21]).

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

#### 2. Preliminaries

A soft set theory introduced by Molodtsov [4] and Çağman and Enginoğlu [22] provided new definitions and various results on soft set theory.

In what follows, let be an initial universe set and let be a set of parameters. Let denote the power set of and .

*Definition 1 (see [4, 22]) . *A soft set over is defined to be the set of ordered pairs
where such that if .

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 [23]) . *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. Unisoft Algebras over a Unisoft Field

In what follows let be a field unless otherwise specified.

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

Proposition 4. *If is a unisoft subfield of , then *(1),
(2),
(3).

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

(2) Let be such that . Then

(3) It follows from (1).

It is easy to show that the following theorem holds.

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

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

Proposition 7. *Let be an algebra over and let be a unisoft subfield of . If is a unisoft algebra over , then for all .*

*Proof. *For any , we have .

We provide a characterization of a unisoft algebra over .

Theorem 8. *For any algebra over , let be a unisoft subfield of . Then a soft set is a unisoft algebra over if and only if it satisfies (3) and (4) of Definition 6 and
*

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

Conversely, suppose that satisfies (3) and (4) of Definition 6 and (8). Then

By using Definition 6(3) and Proposition 4(3), we obtain for all . Thus
for all and . Therefore is a unisoft algebra over .

For any sets and , let be a function and and be soft sets over .

(1) The soft set
where , is called the *unisoft preimage* of under .

(2) The soft set
where
is called the *unisoft image* of under .

Theorem 9. *Let and be algebras over . For any algebraic homomorphism ,*(1)*if is a unisoft algebra over , then the unisoft preimage of under is also a unisoft algebra over .*(2)*If is a unisoft algebra over , then the unisoft image of under is also a unisoft algebra over .*

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

#### 4. Unisoft Hypervector Spaces

*Definition 10. *Let be a hypervector space over and a unisoft subfield of . A soft set over is called a *unisoft 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 a unisoft subfield of . If is a unisoft 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 a unisoft hypervector space of related to a unisoft subfield of , then
*

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

Hence for all .

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

*Proof. *Assume that is a unisoft 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 unisoft subfield of , we have and . Note that for all . It follows that
for all . Let and . Then

Clearly, . Therefore is a unisoft hypervector space of related to .

Theorem 14. *Let be a hypervector space over and a unisoft subfield of . If a soft set over is a unisoft hypervector space of related to , then the nonempty -exclusive 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(i),
(ii).

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

Theorem 16. *Let and be hypervector spaces over and let be a unisoft subfield of . For any linear transformation , if is a unisoft hypervector space of related to , then is a unisoft 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 exist such that and . Thus,
since is linear. Hence, . Then

Obviously, for all . Therefore is a unisoft hypervector space of related to by Theorem 13.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author would like to express his sincere thanks to the anonymous referees for their valuable suggestions and comments that improved this paper.

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#### Copyright

Copyright © 2014 Abdullah M. Al-Roqi. 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.