Double-Framed Soft Hypervector Spaces

Abstract

The notions of double-framed soft subfields, double-framed soft algebras over double-framed soft subfields, and double-framed soft hypervector spaces are introduced, and their properties and characterizations are considered.

1. Introduction

The concept of soft set has been introduced by Molodtsov in 1999 [1] as a new mathematical tool for dealing with uncertainties. Later on, some research papers have appeared on the algebraic structures of soft set theory dealing with uncertainties. In 2007, Aktaş and Çaman [2] 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. Çaman and Enginolu [3] introduced fuzzy parameterized (FP) soft sets and investigated 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 [4] considered the application of soft rough approximations in multicriteria group decision making problems. Recently, many algebraic properties of soft sets are studied (see [5–10]).

The hyperstructure theory was introduced by Marty [11] 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 [12, 13]).

In this paper, using the notion of DFS sets which is introduced in [14], we introduce the notions of double-framed soft subfields, double-framed soft algebras over double-framed soft subfields, and double-framed soft hypervector spaces, and then we investigate their properties.

2. Preliminaries

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 1 (see [15]). Let be a field and 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 [12]) if it satisfies the following condition:

Molodtsov [1] defined the soft set in the following way. Let be an initial universe set and let be a set of parameters. We say that the pair is a soft universe. Let denote the power set of and .

Definition 2 (see [1]). A pair is called a soft set over , where is a mapping given by

In other words, a soft set over is a parameterized family of subsets of the universe . For , may be considered as the set of -approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [1].

Let be a set of parameters, and are subsets of .

Definition 3 (see [14]). A double-framed pair is called a double-framed soft set (briefly, DFS-set) over where and are mappings from to .

For a DFS-set over and two subsets and of , the -inclusive set and the -exclusive set of , denoted by and , respectively, are defined as follows: respectively. The set is called a double-framed including set of . It is clear that

3. Double-Framed Soft Algebras over Double-Framed Soft Subfields

In what follows, let be a field unless otherwise specified.

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

Proposition 5. If is a double-framed soft subfield of , then (1), (2), (3) and .

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 6. A soft set over is a double-framed soft subfield of if and only if the nonempty -inclusive set and the nonempty -exclusive set of , are subfields of for all .

Definition 7. Let be algebra over and let be a double-framed soft subfield over . A double-framed soft set is called a double-framed soft algebra over if it satisfies the following conditions:(1), (2), (3), (4).

Proposition 8. Let be an algebra over and let be a double-framed soft subfield over . If is a double-framed soft algebra over , then and for all .

Proof. For any , we have and .

Theorem 9. Let be an algebra over and let be a double-framed soft subfield over . Then, a double-framed soft set is a double-framed soft algebra over   if and only if the following conditions are valid: (1), (2), (3).

Proof. Assume that is a double-framed soft algebra over . By using (1) and (2) of Definition 7, we have for all and . Also conditions (2) and (3) are hold by Definition 7(3) and Definition 7(4), respectively.
Conversely, suppose that satisfies three conditions (1), (2), and (3). Then, The condition (3) and Proposition 5(3) imply that and for all . Thus, for all and . Therefore, is a double-framed soft algebra over .

4. Double-Framed Soft Hypervector Spaces

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

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

Proof. It is an immediate consequence of Definition 10 and Proposition 5.

Proposition 12. Let be a hypervector space over . If is a double-framed soft hypervector space of related to a double-framed soft subfield of , then

Proof. Let . Since by (H5), we have and .
By using Definition 10(3) and we get and for all .

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

Proof. Assume that is a double-framed soft 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 double-framed soft subfield of , we have , , , and . Note that for all . It follows that for all . Let and . Then, Clearly, and . Therefore, is a double-framed soft hypervector space of related to .

Theorem 14. Let be a hypervector space over and a double-framed soft subfield of . If a DFS-set over is a double-framed soft hypervector space of related to , then the nonempty -inclusive set, and the nonempty -exclusive set, of are subhypervector spaces of over the fields and , respectively, for all .

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

Corollary 15. Let be a hypervector space over and a double-framed soft subfield of . If a DFS-set over is a double-framed soft hypervector space of related to , then the nonempty double-framed soft including set of , is a subhypervector space of over the field for all .

Conflict of Interests

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

Acknowledgment

The authors would like to express their sincere thanks to the anonymous referees.