# Unisoft Hypervector Spaces

Authors: Abdullah M. Al-Roqi
Journal: Journal of Applied Mathematics
 (2014)

---

## 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.


---

## Body

## 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, letU be an initial universe set and let E be a set of parameters. Let P
(
U
) denote the power set of U and A
,
B
,
C
,
…
⊆
E.Definition 1(see [4, 22]).
A soft set(
α
,
A
) over U is defined to be the set of ordered pairs
(1)
(
α
,
A
)
:
=
{
(
x
,
α
(
x
)
)
:
x
∈
E
,
α
(
x
)
∈
P
(
U
)
}
,
where α
:
E
→
P
(
U
) such that α
(
x
)
=
∅ if x
∉
A.
A map○
:
H
×
H
→
P
*
(
H
) is called a hyperoperation or join operation, where P
*
(
H
) is the set of all nonempty subsets of H. The join operation is extended to subsets of H in natural way, so that A
○
B is given by
(2)
A
○
B
=
⋃
‍
{
a
○
b
∣
a
∈
A
,
b
∈
B
}
.
The notationsa
○
A and A
○
a are used for {
a
}
○
A and A
○
{
a
}, respectively. Generally, the singleton {
a
} is identified by its element a.Definition 2(see [23]).
LetF be a field and (
V
,
+
) be an abelian group. A hypervector space over F is defined to be the quadruplet (
V
,
+
,
○
,
F
), where “○” is a mapping
(3)
○
:
F
×
V
⟶
P
*
(
V
)
such that for all a
,
b
∈
F and x
,
y
∈
V the following conditions hold: (H1)
a
○
(
x
+
y
)
⊆
a
○
x
+
a
○
y,
(H2)
(
a
+
b
)
○
x
⊆
a
○
x
+
b
○
x,
(H3)
a
○
(
b
○
x
)
=
(
a
b
)
○
x,
(H4)
a
○
(
-
x
)
=
(
-
a
)
○
x
=
-
(
a
○
x
),
(H5)
x
∈
1
○
x.
A hypervector space(
V
,
+
,
○
,
F
) over a field F is said to be strongly left distributive (see [2]) if it satisfies the following condition:
(4)
(
∀
a
∈
F
)
(
∀
x
,
y
∈
V
)
(
a
○
(
x
+
y
)
=
a
○
x
+
a
○
y
)
.


## 3.  Unisoft Algebras over a Unisoft Field

In what follows letF be a field unless otherwise specified.Definition 3.
A soft set(
α
,
F
) over F is called a unisoft subfield of F if the following conditions are satisfied: (1)
(
∀
a
,
b
∈
F
)
(
α
(
a
+
b
)
⊆
α
(
a
)
∪
α
(
b
)
),
(2)
(
∀
a
∈
F
)
(
α
(
-
a
)
⊆
α
(
a
)
),
(3)
(
∀
a
,
b
∈
F
)
(
α
(
a
)
∪
α
(
b
)
⊇
α
(
a
b
)
),
(4)
(
∀
a
∈
F
)
(
a
≠
0
⇒
α
(
a
)
⊇
α
(
a
-
1
)
).Proposition 4.
If(
α
,
F
) is a unisoft subfield of F, then (1)
(
∀
a
∈
F
)
(
α
(
a
)
⊇
α
(
0
)
),
(2)
(
∀
a
∈
F
)
(
a
≠
0
⇒
α
(
a
)
⊇
α
(
1
)
),
(3)
α
(
1
)
⊇
α
(
0
).Proof.
(1) For alla
∈
F, we have
(5)
α
(
0
)
=
α
(
a
+
(
-
a
)
)
⊆
α
(
a
)
∪
α
(
-
a
)
=
α
(
a
)
.
(2) Leta
∈
F be such that a
≠
0. Then
(6)
α
(
1
)
=
α
(
a
a
-
1
)
⊆
α
(
a
)
∪
α
(
a
-
1
)
=
α
(
a
)
.
(3) It follows from (1).It is easy to show that the following theorem holds.Theorem 5.
A soft set(
α
,
F
) over F is a unisoft subfield of F if and only if the nonempty γ-exclusive set
(7)
e
F
(
α
;
γ
)
:
=
{
a
∈
F
∣
γ
⊇
α
(
a
)
}
of (
α
,
F
) is a subfield of F for all γ
∈
P
(
U
).Definition 6.
LetV be an algebra over F and let (
α
,
F
) be a unisoft subfield of F. A soft set (
β
,
V
) is called an unisoft algebra over (
α
,
F
) if it satisfies the following conditions: (1)
(
∀
x
,
y
∈
V
)
(
β
(
x
+
y
)
⊆
β
(
x
)
∪
β
(
y
)
),
(2)
(
∀
a
∈
F
)
(
∀
x
∈
V
)
(
β
(
a
x
)
⊆
α
(
a
)
∪
β
(
x
)
),
(3)
(
∀
x
,
y
∈
V
)
(
β
(
x
y
)
⊆
β
(
x
)
∪
β
(
y
)
),
(4)
(
∀
x
∈
V
)
(
α
(
1
)
⊆
β
(
x
)
).Proposition 7.
LetV be an algebra over F and let (
α
,
F
) be a unisoft subfield of F. If (
β
,
V
) is a unisoft algebra over (
α
,
F
), then α
(
0
)
⊆
β
(
x
) for all x
∈
V.Proof.
For anyx
∈
V, we have α
(
0
)
⊆
α
(
1
)
⊆
β
(
x
).We provide a characterization of a unisoft algebra over(
α
,
F
).Theorem 8.
For any algebraV over F, let (
α
,
F
) be a unisoft subfield of F. Then a soft set (
β
,
V
) is a unisoft algebra over (
α
,
F
) if and only if it satisfies (3) and (4) of Definition 6 and
(8)
(
∀
a
,
b
∈
F
)
(
∀
x
,
y
∈
V
)
×
(
β
(
a
x
+
b
y
)
⊆
(
α
(
a
)
∪
β
(
x
)
)
∪
(
α
(
b
)
∪
β
(
y
)
)
)
.Proof.
Assume that(
β
,
V
) is a unisoft algebra over (
α
,
F
). Using (1) and (2) of Definition 6, we have
(9)
β
(
a
x
+
b
y
)
⊆
β
(
a
x
)
∪
β
(
b
y
)
⊆
(
α
(
a
)
∪
β
(
x
)
)
∪
(
α
(
b
)
∪
β
(
y
)
)
for all a
,
b
∈
F and x
,
y
∈
V.
Conversely, suppose that(
β
,
V
) satisfies (3) and (4) of Definition 6 and (8). Then
(10)
β
(
x
+
y
)
=
β
(
1
x
+
1
y
)
⊆
(
α
(
1
)
∪
β
(
x
)
)
∪
(
α
(
1
)
∪
β
(
y
)
)
=
β
(
x
)
∪
β
(
y
)
.
By using Definition6(3) and Proposition 4(3), we obtain α
(
0
)
⊆
α
(
1
)
⊆
β
(
x
) for all x
∈
V. Thus
(11)
β
(
a
x
)
=
β
(
a
x
+
0
x
)
⊆
(
α
(
a
)
∪
β
(
x
)
)
∪
(
α
(
0
)
∪
β
(
x
)
)
=
(
α
(
a
)
∪
β
(
x
)
)
∪
β
(
x
)
=
α
(
a
)
∪
β
(
x
)
for all a
∈
F and x
∈
V. Therefore (
β
,
V
) is a unisoft algebra over (
α
,
F
).For any setsX and Y, let μ
:
X
→
Y be a function and (
α
,
X
) and (
β
,
Y
) be soft sets over U.(1) The soft set(12)
μ
-
1
(
β
,
Y
)
=
{
(
x
,
μ
-
1
(
β
)
(
x
)
)
:
x
∈
X
,
μ
-
1
(
β
)
(
x
)
∈
P
(
U
)
}
,
where μ
-
1
(
β
)
(
x
)
=
β
(
μ
(
x
)
), is called the unisoft preimage of (
β
,
Y
) under μ.(2) The soft set(13)
μ
(
α
,
X
)
=
{
(
y
,
μ
(
α
)
(
y
)
)
:
y
∈
Y
,
μ
(
α
)
(
y
)
∈
P
(
U
)
}
,
where
(14)
μ
(
α
)
(
y
)
=
{
⋂
x
∈
μ
-
1
(
y
)
‍
α
(
x
)
if
μ
-
1
(
y
)
≠
∅
,
U
otherwise
,
is called the unisoft image of (
α
,
X
) under μ.Theorem 9.
LetV and W be algebras over F. For any algebraic homomorphism μ
:
V
→
W,(1)
if(
β
,
W
) is a unisoft algebra over (
α
,
F
), then the unisoft preimage μ
-
1
(
β
,
W
) of (
β
,
W
) under μ is also a unisoft algebra over (
α
,
F
).
(2)
If(
β
,
V
) is a unisoft algebra over (
α
,
F
), then the unisoft image μ
(
β
,
V
) of (
β
,
V
) under μ is also a unisoft algebra over (
α
,
F
).Proof.
(1) For anya
,
b
∈
F and x
,
y
∈
V, we have
(15)
μ
-
1
(
β
)
(
a
x
+
b
y
)
=
β
(
μ
(
a
x
+
b
y
)
)
=
β
(
a
μ
(
x
)
+
b
μ
(
y
)
)
⊆
(
α
(
a
)
∪
β
(
μ
(
x
)
)
)
∪
(
α
(
b
)
∪
β
(
μ
(
y
)
)
)
=
(
α
(
a
)
∪
μ
-
1
(
β
)
(
x
)
)
∪
(
α
(
b
)
∪
μ
-
1
(
β
)
(
y
)
)
,
μ
-
1
(
β
)
(
x
y
)
=
β
(
μ
(
x
y
)
)
=
β
(
μ
(
x
)
μ
(
y
)
)
⊆
β
(
μ
(
x
)
)
∪
β
(
μ
(
y
)
)
=
μ
-
1
(
β
)
(
x
)
∪
μ
-
1
(
β
)
(
y
)
,
and α
(
1
)
⊆
β
(
μ
(
x
)
)
=
μ
-
1
(
β
)
(
x
). Therefore, by Theorem 8, μ
-
1
(
β
,
W
) is a unisoft algebra over (
α
,
F
).
(2) Lety
1
,
y
2
∈
W. If μ
-
1
(
y
1
)
=
∅ or μ
-
1
(
y
2
)
=
∅, then
(16)
μ
(
β
)
(
y
1
)
∪
μ
(
β
)
(
y
2
)
=
U
⊇
μ
(
β
)
(
y
1
+
y
2
)
.
Assume thatμ
-
1
(
y
1
)
≠
∅ and μ
-
1
(
y
2
)
≠
∅. Then μ
-
1
(
y
1
+
y
2
)
≠
∅, and so
(17)
μ
(
β
)
(
y
1
+
y
2
)
=
⋂
‍
x
∈
μ
-
1
(
y
1
+
y
2
)
β
(
x
)
⊆
⋂
‍
x
1
∈
μ
-
1
(
y
1
)
x
2
∈
μ
-
1
(
y
2
)
β
(
x
1
+
x
2
)
⊆
⋂
‍
x
1
∈
μ
-
1
(
y
1
)
x
2
∈
μ
-
1
(
y
2
)
(
β
(
x
1
)
∪
β
(
x
2
)
)
=
(
⋂
‍
x
1
∈
μ
-
1
(
y
1
)
β
(
x
1
)
)
∪
(
⋂
‍
x
2
∈
μ
-
1
(
y
2
)
β
(
x
2
)
)
=
μ
(
β
)
(
y
1
)
∪
μ
(
β
)
(
y
2
)
.
For anyy
∈
W and a
∈
F, we have
(18)
μ
(
β
)
(
a
y
)
=
⋂
‍
μ
(
x
)
=
a
y
β
(
x
)
=
⋂
‍
μ
(
x
)
=
y
β
(
a
y
)
⊆
⋂
‍
μ
(
x
)
=
y
(
α
(
a
)
∪
β
(
y
)
)
=
α
(
a
)
∪
⋂
‍
μ
(
x
)
=
y
β
(
y
)
=
α
(
a
)
∪
μ
(
β
)
(
y
)
.
For ally
1
,
y
2
∈
W, if at least one of μ
-
1
(
y
1
) and μ
-
1
(
y
2
) is empty, then the inclusion
(19)
μ
(
β
)
(
y
1
)
∪
μ
(
β
)
(
y
2
)
⊇
μ
(
β
)
(
y
1
y
2
)
is clear. Assume that μ
-
1
(
y
1
)
≠
∅ and μ
-
1
(
y
2
)
≠
∅. Then
(20)
μ
(
β
)
(
y
1
)
∪
μ
(
β
)
(
y
2
)
=
(
⋂
x
1
∈
μ
-
1
(
y
1
)
‍
β
(
x
1
)
)
∪
(
⋂
x
2
∈
μ
-
1
(
y
2
)
‍
β
(
x
2
)
)
=
⋂
x
1
∈
μ
-
1
(
y
1
)
x
2
∈
μ
-
1
(
y
2
)
‍
(
β
(
x
1
)
∪
β
(
x
2
)
)
⊇
⋂
x
1
∈
μ
-
1
(
y
1
)
x
2
∈
μ
-
1
(
y
2
)
‍
β
(
x
1
x
2
)
=
⋂
x
∈
μ
-
1
(
y
1
y
2
)
‍
β
(
x
)
=
μ
(
β
)
(
y
1
y
2
)
.
Sinceα
(
1
)
⊆
β
(
x
) for all x
∈
V, it follows that
(21)
α
(
1
)
⊆
⋂
‍
x
∈
μ
-
1
(
y
)
β
(
x
)
=
μ
(
β
)
(
y
)
for all y
∈
W. Therefore μ
(
β
,
V
) is a unisoft algebra over (
α
,
F
).


## 4. Unisoft Hypervector Spaces

Definition 10.
LetV be a hypervector space over F and (
α
,
F
) a unisoft subfield of F. A soft set (
β
,
V
) over V is called a unisoft hypervector space of V related to (
α
,
F
) if the following assertions are valid: (1)
(
∀
x
,
y
∈
V
)
(
β
(
x
+
y
)
⊆
β
(
x
)
∪
β
(
y
)
),
(2)
(
∀
x
∈
V
)
(
β
(
-
x
)
⊆
β
(
x
)
),
(3)
(
∀
a
∈
F
)
(
∀
x
∈
V
)
(
⋃
y
∈
a
○
x
β
(
y
)
⊆
α
(
a
)
∪
β
(
x
)
),
(4)
α
(
1
)
⊆
β
(
θ
) where θ is the zero of (
V
,
+
).Proposition 11.
LetV be a hypervector space over F and (
α
,
F
) a unisoft subfield of F. If (
β
,
V
) is a unisoft hypervector space of V related to (
α
,
F
), then (1)
β
(
θ
)
⊇
α
(
0
),
(2)
(
∀
x
∈
V
)
(
β
(
x
)
⊇
β
(
θ
)
),
(3)
(
∀
x
∈
V
)
(
β
(
x
)
⊇
α
(
0
)
).Proof.
It is an immediate consequence of Definition10 and Proposition 4.Proposition 12.
LetV be a hypervector space over F. If (
β
,
V
) is a unisoft hypervector space of V related to a unisoft subfield (
α
,
F
) of F, then
(22)
(
∀
x
∈
V
)
(
β
(
x
)
=
⋃
‍
y
∈
1
○
x
β
(
y
)
)
.Proof.
Letx
∈
V. Since x
∈
1
○
x by (H5), we have β
(
x
)
⊆
⋃
y
∈
1
○
x
β
(
y
). Using Definition 10(3) we have
(23)
⋃
‍
y
∈
1
○
x
β
(
y
)
⊆
α
(
1
)
∪
β
(
x
)
⊆
β
(
x
)
.
Henceβ
(
x
)
=
⋃
y
∈
1
○
x
β
(
y
) for all x
∈
V.Theorem 13.
Assume that a hypervector spaceV over F is strongly left distributive. Let (
α
,
F
) be a unisoft subfield of F. Then a soft set (
β
,
V
) over V is a unisoft hypervector space of V related to (
α
,
F
) if and only if the following conditions are true: (1)
⋃
z
∈
a
○
x
+
b
○
y
β
(
z
)
⊆
(
α
(
a
)
∪
β
(
x
)
)
∪
(
α
(
b
)
∪
β
(
y
)
),
(2)
β
(
x
)
⊇
α
(
1
)for all a
,
b
∈
F and all x
,
y
∈
V.Proof.
Assume that(
β
,
V
) is a unisoft hypervector space of V related to (
α
,
F
). The second condition follows from Proposition 11(2) and Definition 10(4). Let a
,
b
∈
F and x
,
y
∈
V. Then
(24)
⋃
‍
z
∈
a
○
x
+
b
○
y
β
(
z
)
=
⋃
‍
z
∈
u
+
v
u
∈
a
○
x
,
v
∈
b
○
y
β
(
z
)
⊆
(
α
(
a
)
∪
β
(
x
)
)
∪
(
α
(
b
)
∪
β
(
y
)
)
.
Conversely suppose the conditions (1) and (2) are true. For allx
,
y
∈
V, we have
(25)
β
(
x
+
y
)
⊆
⋃
‍
z
∈
1
○
x
+
1
○
y
β
(
z
)
⊆
(
α
(
1
)
∪
β
(
x
)
)
∪
(
α
(
1
)
∪
β
(
y
)
)
⊆
β
(
x
)
∪
β
(
y
)
.
Since(
α
,
F
) is a unisoft subfield of F, we have β
(
a
)
⊇
α
(
1
)
⊇
α
(
0
) and β
(
a
)
⊇
α
(
1
)
⊇
α
(
-
1
). Note that 0
∈
0
○
x for all x
∈
V. It follows that
(26)
β
(
-
x
)
⊆
⋃
‍
y
∈
0
○
x
+
(
-
1
)
○
x
β
(
y
)
⊆
(
α
(
0
)
∪
β
(
x
)
)
∪
(
α
(
-
1
)
∪
β
(
x
)
)
=
β
(
x
)
∪
β
(
x
)
=
β
(
x
)
for all x
∈
V. Let a
∈
F and x
∈
V. Then
(27)
⋃
‍
y
∈
a
○
x
β
(
y
)
⊆
⋃
‍
y
∈
u
+
v
u
∈
0
○
x
,
v
∈
a
○
x
β
(
y
)
⊆
(
α
(
0
)
∪
β
(
x
)
)
∪
(
α
(
a
)
∪
β
(
x
)
)
=
β
(
x
)
∪
(
α
(
a
)
∪
β
(
x
)
)
=
α
(
a
)
∪
β
(
x
)
.
Clearly,α
(
1
)
⊆
β
(
θ
). Therefore (
β
,
V
) is a unisoft hypervector space of V related to (
α
,
F
).Theorem 14.
LetV be a hypervector space over F and (
α
,
F
) a unisoft subfield of F. If a soft set (
β
,
V
) over V is a unisoft hypervector space of V related to (
α
,
F
), then the nonempty γ-exclusive set
(28)
e
V
(
β
;
γ
)
:
=
{
x
∈
V
∣
γ
⊇
β
(
x
)
}
of (
β
,
V
) is a subhypervector space of V over the field e
F
(
α
;
γ
) for all γ
∈
P
(
U
).Proof.
Letx
,
y
∈
e
V
(
β
;
γ
). Then β
(
x
)
⊆
γ and β
(
y
)
⊆
γ. It follows that
(29)
β
(
x
-
y
)
=
β
(
x
+
(
-
y
)
)
⊆
β
(
x
)
∪
β
(
-
y
)
⊆
β
(
x
)
∪
β
(
y
)
⊆
γ
.
Hencex
-
y
∈
e
V
(
β
;
γ
). Note that e
F
(
α
;
γ
) is a subfield of F (see Theorem 5). Let a
∈
e
F
(
α
;
γ
)
,
x
∈
e
V
(
β
;
γ
) and y
∈
a
○
x. Then
(30)
β
(
y
)
⊆
⋃
‍
z
∈
a
○
x
β
(
z
)
⊆
α
(
a
)
∪
β
(
x
)
⊆
γ
,
and so y
∈
e
V
(
β
;
γ
) which shows that a
○
x
⊆
e
V
(
β
;
γ
). Therefore e
V
(
β
;
γ
) is a hypervector space over the field e
F
(
α
;
γ
) for all γ
∈
P
(
U
).LetV and W be hypervector spaces over F. A mapping T
:
V
→
W is called linear transformation (see [3]) if it satisfies(i)
(
∀
x
,
y
∈
V
)
(
T
(
x
+
y
)
=
T
(
x
)
+
T
(
y
)
),
(ii)
(
∀
a
∈
F
)
(
∀
x
∈
V
)
(
T
(
a
○
x
)
⊆
a
○
T
(
x
)
).Theorem 15.
LetV and W be hypervector spaces over F and let (
α
,
F
) be a unisoft subfield of F. For any linear transformation T
:
V
→
W, if (
β
,
W
) is a unisoft hypervector space of W related to (
α
,
F
), then T
-
1
(
β
,
W
) is a unisoft hypervector space of  V related to (
α
,
F
).Proof.
Leta
,
b
∈
F and x
,
y
∈
V. Since T is a linear transformation, we have
(31)
⋃
‍
z
∈
a
○
x
+
b
○
y
T
-
1
(
β
)
(
z
)
=
⋃
‍
z
∈
a
○
x
+
b
○
y
β
(
T
(
z
)
)
=
⋃
‍
u
∈
a
○
x
,
v
∈
b
○
y
β
(
T
(
u
+
v
)
)
⊆
⋃
‍
T
(
u
)
∈
a
○
T
(
x
)
T
(
v
)
∈
b
○
T
(
y
)
β
(
T
(
u
+
v
)
)
⊆
(
β
(
T
(
x
)
)
∪
α
(
a
)
)
∪
(
β
(
T
(
y
)
)
∪
α
(
b
)
)
=
(
T
-
1
(
β
)
(
x
)
∪
α
(
a
)
)
∪
(
T
-
1
(
β
)
(
y
)
∪
α
(
b
)
)
.
Obviously,α
(
1
)
⊆
T
-
1
(
β
)
(
x
) for all x
∈
V. It follows from Theorem 13 that T
-
1
(
β
,
W
) is a unisoft hypervector space of V related to (
α
,
F
).Theorem 16.
LetV and W be hypervector spaces over F and let (
α
,
F
) be a unisoft subfield of F. For any linear transformation T
:
V
→
W, if (
β
,
V
) is a unisoft hypervector space of V related to (
α
,
F
), then T
(
β
,
V
) is a unisoft hypervector space of W related to (
α
,
F
).Proof.
Leta
,
b
∈
F and x
,
y
∈
W. If at least one of T
-
1
(
x
) and T
-
1
(
y
) is empty, then the inclusion
(32)
⋃
‍
z
∈
a
○
x
+
b
○
y
T
(
β
)
(
z
)
⊆
(
α
(
a
)
∪
T
(
β
)
(
x
)
)
∪
(
α
(
b
)
∪
T
(
β
)
(
y
)
)
is clear. Assume that T
-
1
(
x
) and T
-
1
(
y
) are nonempty. Then there exist u
,
v
∈
V such that T
(
u
)
=
x and T
(
v
)
=
y. Thus,
(33)
a
○
x
+
b
○
y
=
a
○
T
(
u
)
+
b
○
T
(
v
)
⊇
T
(
a
○
u
)
+
T
(
b
○
v
)
=
T
(
a
○
u
+
b
○
v
)
since T is linear. Hence, a
○
u
+
b
○
v
⊆
T
-
1
(
a
○
x
+
b
○
y
). Then
(34)
⋃
‍
w
∈
a
○
x
+
b
○
y
T
(
β
)
(
w
)
=
⋃
‍
w
∈
a
○
x
+
b
○
y
T
(
z
)
=
w
β
(
z
)
⊆
⋃
‍
u
∈
T
-
1
(
x
)
,
v
∈
T
-
1
(
y
)
z
=
z
1
+
z
2
,
z
1
∈
a
○
u
,
z
2
∈
b
○
v
(
β
(
z
1
)
∪
β
(
z
2
)
)
⊆
(
α
(
a
)
∪
β
(
u
)
)
∪
(
α
(
b
)
∪
β
(
v
)
)
=
(
α
(
a
)
∪
T
(
β
)
(
x
)
)
∪
(
α
(
b
)
∪
T
(
β
)
(
y
)
)
.
Obviously,α
(
1
)
⊆
T
(
β
)
(
x
) for all x
∈
W. Therefore T
(
β
,
V
) is a unisoft hypervector space of W related to (
α
,
F
) by Theorem 13.



---

*Source: 418702-2014-02-23.xml*