The notion of soft
α
ideals and
α
idealistic soft BCIalgebras is introduced and their basic properties are discussed. Relations between soft ideals and soft
α
ideals of soft BCIalgebras are provided. Also idealistic soft BCIalgebras and
α
idealistic soft BCIalgebras are being related. The restricted intersection, union, restricted union, restricted difference and “AND” operation of soft
α
ideals and
α
idealistic soft BCIalgebras are established. The characterizations of (fuzzy)
α
ideals in BCIalgebras are given by using the concept of soft sets. Relations between fuzzy
α
ideals and
α
idealistic soft BCIalgebras are discussed.
AMS Mathematics Subject Classification : 06D35, 06D72.
1. Introduction
classical methods, out of which the inability of the parametrization tool for these methods is one of he main reasons, Molodtsov
[16]
gave the concept of soft set theory as a new tool for dealing with such type of problems. Now a days, soft set theory is considered to be the one of the most reliable method for dealing with uncertainties. Here, first of all we discuss some developments in different fields of life which have been done by using the soft set theory.
De Morgans laws have been verified in soft set theory. Sezgin and Atagun
[17]
proved that certain De Morgans law holds in soft set theory with respect to different operations on soft sets. Herawan et al.
[8]
defined an attribute reduction based on the notion of multisoft sets and “AND” operation. Gong
[7]
et al. introduced the concept of bijective soft set and also defined restricted “AND”, relaxed “AND” operations on bijective soft set. This work revealed an application of bijective soft set in decision making problems. Babitha and Sunil
[3]
introduced soft set relations as a subsoft set of cartesian product of the soft sets. They also discussed related concepts like equivalent soft set relation, partition, composition and function. Kharal and Ahmad
[12]
discussed mappings on soft classes and also images and inverse images of soft sets which are used for medical diagnosis in medical expert systems. Çağman et al.
[4]
defined soft matrices and their operations and constructed a maxmin decision making method. Chen et al.
[5]
proposed parametrization reduction of soft sets and compared it with the concept of attributes reduction in rough sets theory. Aktaş and Çağman
[2]
discussed the basics of soft set theory and its differences with fuzzy and rough set theories. They also derived the notion of soft groups. Feng et al.
[6]
established a connection between rough sets and soft sets. Yang et al.
[18]
defined the operation on fuzzy soft sets , which are based on three fuzzy logic operations: negation, triangular norm and triangular conorm.
Jun
[10]
was the first person who discussed the applications of soft sets in BCK/BCIalgebras. He introduced the notion of soft BCK/BCIalgebras and soft subalgebras. Ali et al.
[1]
disproved certain definitions and results discussed by Maji et al. in
[15]
and defined some new operations. In this paper, we introduce the notion of soft
α
ideals and
α
idealistic soft BCIalgebras and discuss various operations introduced in
[1]
on these concepts. Using soft sets, we give characterizations of (fuzzy)
α
ideals in BCIalgebras. We provide relations between fuzzy
α
ideals and
α
idealistic soft BCIalgebras.
2. BCIalgebras
Y. Imai and K. Iséki
[9]
, introduced the idea of BCKalgebras in 1966, after exploring the properties of set difference. Y. Imai generalized the concept of BCKalgebras and gave the idea of BCIalgebras.
We define BCIalgebra as an algebra (
X
, ∗, 0) of type (2, 0), in which the following axioms hold:
(I) (
x
∗
y
) ∗ (
x
∗
z
) ≤ (
z
∗
y
)
(II)
x
∗ (
x
∗
y
) ≤
y
(III)
x
≤
x
(IV)
x
≤
y
and
y
≤
x
imply
x
=
y
for all
x
,
y
,
z
∈
X
. Here a partial ordering ” ≤ ” is defined by putting,
x
≤
y
if and only if
x
∗
y
= 0.
If a BCIalgebra
X
satisfies the identity:
(V) 0 ≤
x
= 0,
for all
x
∈
X
, then
X
is called a BCKalgebra.
In any BCIalgebra the following hold:
(VI) (
x
∗
y
) ∗
z
= (
x
∗
z
) ∗
y
(VII)
x
∗ 0 =
x
(VIII)
x
≤
y
implies
x
∗
z
≤
y
∗
z
and
z
∗
y
≤
z
∗
x
(IX) 0 ∗ (
x
∗
y
) = (0 ∗
x
) ∗ (0 ∗
y
)
for all
x
,
y
,
z
∈
X
.
A nonempty subset
S
of a BCIalgebras
X
is called a subalgebra of
X
if
x
∗
y
∈
S
for all
x
,
y
∈
S
. A nonempty subset
I
of a BCIalgebra
X
is called an ideal of
X
if for any
u
∈
X
(I1) 0 ∈
I
(I2)
u
∗
v
∈
I
and
v
∈
I
implies
u
∈
I
A nonempty subset I of a BCIalgebra X is called an
α
ideal of
X
if it satisfies (I1) and

(I3) (u∗w) ∗ (0 ∗v) ∈Iand w ∈I⇒v∗u∈Ifor allu,v∈X.
It can be observed that every
α
ideal of a BCIalgebra
X
is an ideal of
X
.
3. Preliminaries
Molodtsov in
[16]
defined the soft sets as under: “Let
U
be an universal set and
P
be a set of parameters. Let
denotes the power set of
U
and
A
⊂
P
”.
Definition 3.1
(Molodtsov
[16]
). A pair (
F
,
A
) is called a soft set over
U
, where
F
is a mapping given by
In other words, a soft set over
U
is a family of parameters of subsets of universal set
U
.
Definition 3.2
(Maji et al.
[15]
). For two soft sets (
F
,
A
) and (
g
,
B
) over a common universe
U
, the union of (
F
,
A
) and (
g
,
B
), denoted by (
F
,
A
) Ũ (
g
,
B
), is defined to be the soft set (
H
,
C
) satisfying the following conditions:
(i)
C
=
A
∪
B
(ii) for all
c
∈
C
,
Definition 3.3
(Ali et al.
[1]
). For two soft sets (
F
,
A
) and (
g
,
B
) over a common universe
U
, the extended intersection of (
F
,
A
) and (
g
,
B
), denoted by (
F
,
A
)∩
_{ξ}
(
g
,
B
), is defined to be the soft set (
H
,
C
) satisfying the following conditions:
(i)
C
=
A
∪
B
(ii) for all
c
∈
C
,
Definition 3.4
(Ali et al.
[1]
). For two soft sets (
F
,
A
) and (
g
,
B
) over a common universe
U
such that
A
∩
B
≠ ∅, the restricted intersection of (
F
,
A
) and (
g
,
B
), denoted by (
F
,
A
) ∩
_{ℜ}
(
g
,
B
), is defined to be the soft set (
H
,
C
) satisfying the following conditions:
(i)
C
=
A
∩
B
(ii)
H
(
c
) =
F
(
c
) ∩
g
(
c
) for all
c
∈
C
.
Definition 3.5
(Ali et al.
[1]
). For two soft sets (
F
,
A
) and (
g
,
B
) over a common universe
U
such that
A
∩
B
≠ ∅, the restricted union of (
F
,
A
) and (
g
,
B
), denoted by (
F
,
A
) ∪
_{ℜ}
(
g
,
B
), is defined to be the soft set (
H
,
C
) satisfying the following conditions:
(i)
C
=
A
∩
B
(ii)
H
(
c
) =
F
(
c
) ∪
g
(
c
) for all
c
∈
C
.
Definition 3.6
(Ali et al.
[1]
). For two soft sets (
F
,
A
) and (
g
,
B
) over a common universe
U
such that
A
∩
B
≠ ∅, the restricted difference of (
F
,
A
) and (
g
,
B
), denoted by (
F
,
A
)
(
g
,
B
), is defined to be the soft set (
H
,
C
) satisfying the following conditions:
(i)
C
=
A
∩
B
(ii)
H
(
c
) =
F
(
c
) −
g
(
c
) for all
c
∈
C
(set difference of
F
(
c
) and
g
(
c
)).
Definition 3.7
(Maji et al.
[15]
). Let (
F
,
A
) and (
g
,
B
) be two soft sets over a common universe
U
. Then “(
F
,
A
)
AND
(
g
,
B
)” denoted by (
F
,
A
)
(
g
,
B
) is defined as (
F
,
A
)
(
g
,
B
) = (
H
,
A
×
B
), where
H
(
x
,
y
) =
F
(
x
) ∩
g
(
y
) for all (
x
,
y
) ∈
A
×
B
.
Definition 3.8
(Maji et al.
[15]
). Let (
F
,
A
) and (
g
,
B
) be two soft sets over a common universe
U
. Then “(
F
,
A
)
OR
(
g
,
B
)” denoted by (
F
,
A
)
(
g
,
B
) is defined as (
F
,
A
)
(
g
,
B
) = (
H
,
A
×
B
), where
H
(
x
,
y
) =
F
(
x
) ∪
g
(
y
) for all (
x
,
y
) ∈
A
×
B
.
4. Softαideals
Let
U
and
M
be a BCIalgebra and a nonempty set, respectively and
R
will refer to an arbitrary binary relation between an element of
M
and an element of
U
, that is,
R
is a subset of
M
×
U
without otherwise specified. “A set valued function
F
:
M
→
can be defined as
F
(
x
) = {
y
∈
U

xRy
} for all
x
∈
M
. The pair (
F
,
M
) is then a soft set over
U
”.
Definition 4.1
(Jun and Park
[11]
). Let
V
be a subalgebra of
U
. A subset
I
of
U
is called an ideal of
U
related to
V
(briefly,
V
ideal of
U
), denoted by
I
◁
V
, if it satisfies:
(i) 0 ∈
I
(ii)
x
∗
y
∈
I
and
y
∈
I
⇒
x
∈
I
for all
x
∈
V
.
Definition 4.2.
Let
V
be a subalgebra of
U
. A subset
I
of
U
is called an
α
ideal of
U
related to
V
(briefly,
V

α
ideal of
U
), denoted by
I
◁
_{α}
V
, if it satisfies:
(i) 0 ∈
I
(ii) (
x
∗
z
) ∗ (0 ∗
y
)) ∈
I
and
z
∈
I
⇒
y
∗
x
∈
I
for all
x
,
y
∈
V
.
Example 4.3.
Consider a BCIalgebra
X
= {0,
a
,
b
,
c
} defined by the following Cayley table:
Then
Q
= {0,
a
} is a subalgebra of
X
and
I
= {0,
a
,
b
} is an
Q

α
ideal of
X
.
Note that every
S

α
ideal of
X
is an
S
ideal of
X
.
Definition 4.4
(Jun
[10]
). Let (
F
,
V
) be a soft set over
U
. Then (
F
,
V
) is called a soft BCIalgebra over
U
if
F
(
x
) is a subalgebra of
U
for all
x
∈
V
.
Definition 4.5
(Jun and Park
[11]
). Let (
F
,
V
) be a soft BCIalgebra over
U
. A soft set (
g
,
I
) over
U
is called a soft ideal of (
F
,
V
), denoted (
g
,
I
)
(
F
,
V
), if it satisfies:
(i)
I
⊂
V
(ii)
g
(
x
) ◁
F
(
x
) for all
x
∈
I
.
Definition 4.6.
Let (
F
,
V
) be a soft BCIalgebra over
U
. A soft set (
g
,
I
) over
U
is called a soft
α
ideal of (
F
,
V
), denoted (
g
,
I
)
(
F
,
V
), if it satisfies:
(i)
I
⊂
V
(ii)
g
(
x
) ◁
_{α}
F
(
x
) for all
x
∈
I
.
The following example will be helpful to understand the above example.
Example 4.7.
Consider a BCIalgebra
X
= {0,
a
,
b
,
c
} which is given in Example 4.3. Let (
F
,
A
) be a soft set over
X
, where
A
=
X
and
F
:
A
→
is a setvalued function defined by:
for all
x
∈
A
. Then
F
(0) =
F
(
a
) =
X
,
F
(
b
) =
F
(
c
) = {0}, which are subalgebras of
X
. Hence (
F
,
A
) is a soft BCIalgebra over
X
. Let
I
= {0,
a
,
b
} ⊂
A
and
g
:
I
→
be a setvalued function defined by:
where
Z
({0,
a
}) = {
x
∈
X
 0 ∗ (0 ∗
x
) ∈ {0,
a
}}. Then
g
(0) = {0} ◁
_{α}
X
=
F
(0),
g
(
a
) = {0} ◁
_{α}
X
=
F
(
a
),
g
(
b
) = {0,
a
} ◁
_{α}
{0} =
F
(
b
). Hence (
g
,
I
) is a soft
α
ideal of (
F
,
A
).
Note that every soft
α
ideal is a soft ideal but the converse is not true as seen in the following example.
Example 4.8.
Consider a BCIalgebra
X
= {0,
a
,
b
,
c
,
d
} defined by the following Cayley table:
Let (
F
,
A
) be a soft set over
X
, where
A
=
X
and
F
:
A
→
is a setvalued function defined by:
for all
u
∈
A
. Then
F
(0) =
X
,
F
(
a
) =
F
(
b
) = {0,
b
,
c
,
d
},
F
(
c
) =
F
(
d
) = {0,
b
}, which are subalgebras of
X
. Hence (
F
,
A
) is a soft BCIalgebra over
X
.
Let (
g
,
I
) be a soft set over
X
, where
I
= {
b
,
c
,
d
} ⊂
A
and
g
:
I
→
be a setvalued function defined by:
for all
u
∈
I
. Then
g
(
b
) = {0,
a
,
b
} ◁ {0,
b
,
c
,
d
} =
F
(
b
),
g
(
c
) = {0,
a
,
c
} ◁ {0,
b
} =
F
(
c
),
g
(
d
) =
X
◁ {0,
b
} =
F
(
d
). Hence (
g
,
I
) is a soft ideal of (
F
,
A
) but it is not a soft
α
ideal of (
F
,
A
) because
g
(
b
) is not an
F
(
b
)
α
ideal of
X
since (
b
∗
b
) ∗ (0 ∗
d
) = 0 ∈
g
(
b
) and
b
∈
g
(
b
) but
d
∗
b
=
c
∉
g
(
b
).
Theorem 4.9.
Let
(
F
,
A
)
be a soft BCIalgebra over X and
(
g
,
I
)
and
(
H
,
J
)
are two soft sets over X such that I
∩
J
≠ ∅.
If
(
g
,
I
)
(
F
,
A
), (
H
,
J
)
(
F
,
A
)
, then
((
g
,
I
) ∩
_{ℜ}
(
H
,
J
))
(
F
,
A
).
Proof
. Using Definition 3.4, we can write
where
U
=
I
∩
J
and
R
(
e
) =
g
(
e
) ∩
H
(
e
) for all
e
∈
U
. Obviously,
U
⊂
A
and
R
:
U
→
is a mapping. Hence (
R
,
U
) is a soft set over
X
. Since (
g
,
I
)
(
F
,
A
) and (
H
,
J
)
(
F
,
A
), it follows that
g
(
e
) ◁
_{α}
F
(
e
) for all
e
∈
I
and
H
(
e
) ◁
_{α}
F
(
e
) for all
e
∈
J
. Therefore
R
(
e
) =
g
(
e
) ∩
H
(
e
) ◁
_{α}
F
(
e
) for all
e
∈
U
=
I
∩
J
. Hence
This completes the proof. □
Corollary 4.10.
Let
(
F
,
A
)
be a soft BCIalgebra over X and
(
g
,
I
)
and
(
H
,
I
)
are two soft sets over X. If
(
g
,
I
)
(
F
,
A
), (
H
,
I
)
(
F
,
A
)
, then
(
g
,
I
) ∩
_{ℜ}
(
H
,
I
)
(
F
,
A
).
Proof
. Straightforward.
Theorem 4.11.
Let
(
F
,
A
)
be a soft BCIalgebra over X and
(
g
,
I
)
and
(
H
,
J
)
are two soft sets over X. If
(
g
,
I
)
(
F
,
A
), (
H
,
J
)
(
F
,
A
)
, then
((
g
,
I
) ∩
_{ξ}
(
H
,
J
))
(
F
,
A
).
Proof
. By means of Definition 3.3, we can write (
g
,
I
) ∩
_{ξ}
(
H
,
J
) = (
R
,
U
), where
U
=
I
∪
J
⊂
A
and for every
e
∈
U
,
For every
e
∈
U
such that
e
∈
I
╲
J
,
R
(
e
) =
g
(
e
) ◁
_{α}
F
(
e
) since (
g
,
I
)
(
F
,
A
). Similarly for any
e
∈
U
such that
e
∈
J
╲
I
,
R
(
e
) =
H
(
e
) ◁
_{α}
F
(
e
) since (
H
,
J
)
(
F
,
A
). Moreover for some
e
∈
U
such that
e
∈
I
∩
J
,
R
(
e
) =
g
(
e
)∩
H
(
e
). Since
g
(
e
) is an
α
ideal of
X
related to
F
(
e
) for all
e
∈
I
and
H
(
e
) is an
α
ideal of
X
related to
F
(
e
) for all
e
∈
J
, it follows that
g
(
e
)∩
H
(
e
) =
R
(
e
) is an
α
ideal of
X
related to
F
(
e
) for all
e
∈
I
∩
J
=
U
. Thus
R
(
e
) ◁
_{α}
F
(
e
) for all
e
∈
U
. Hence (
g
,
I
) ∩
_{ξ}
(
H
,
J
) = (
R
,
U
)
(
F
,
A
). □
Theorem 4.12.
Let
(
F
,
A
)
be a soft BCIalgebra over X. For any soft sets
(
g
,
I
)
and
(
H
,
J
)
over X in which I and J are disjoint, we have
Proof
. Assume that (
g
,
I
)
(
F
,
A
) and (
H
,
J
)
(
F
,
A
). By means of Definition 3.2, we can write (
g
,
I
) Ũ (
H
,
J
) = (
R
,
U
), where
U
=
I
∪
J
and for every
e
∈
U
,
Since
I
∩
J
= ∅, either
e
∈
I
╲
J
or
e
∈
J
╲
I
for all
e
∈
U
. If
e
∈
I
╲
J
, then
R
(
e
) =
g
(
e
) ◁
_{α}
F
(
e
) since (
g
,
I
)
(
F
,
A
). If
e
∈
J
╲
I
, then
R
(
e
) =
H
(
e
) ◁
_{α}
F
(
e
) since (
H
,
J
)
(
F
,
A
). Thus
R
(
e
) ◁
_{α}
F
(
e
) for all
e
∈
U
and so
It
I
and
J
are not disjoint in Theorem 4.12, then Theorem 4.12 is not true in general as seen in the following example.
Example 4.13.
Consider a BCIalgebra
X
= {0, 1,
a
,
b
,
c
} defined by the following Cayley table:
Let (
F
,
A
) be a soft set over
X
, where
A
= {0, 1} and
F
:
A
→
is a setvalued function defined by:
for all
u
∈
A
. Then
F
(0) =
X
and
F
(1) = {0,
a
,
b
,
c
}, which are subalgebras of
X
. Hence (
F
,
A
) is a soft BCIalgebra over
X
.
If we take
I
=
A
and define a set valued function
g
:
I
→
by:
for all
u
∈
I
. Then
g
(0) = {0, 1,
b
} ◁
_{α}
X
=
F
(0) and
g
(1) = {0,
b
} ◁
_{α}
{0,
a
,
b
,
c
} =
F
(1). Hence (
g
,
I
) is a soft
α
ideal of (
F
,
A
).
Now consider
J
= {0} which is not disjoint with
I
and let
H
:
J
→
be a set valued function by:
for all
u
∈
J
. Then
H
(0) = {0, 1,
c
} ◁
_{α}
X
=
F
(0). Hence (
H
,
J
) is a soft
α
ideal of (
F
,
A
). But if (
R
,
U
) = (
g
,
I
) Ũ (
H
,
J
), then
R
(0) =
g
(0) ∪
H
(0) = {0, 1,
b
,
c
}, which is not an
α
ideal of
X
related to
F
(0), since (
b
∗
b
) ∗ (0 ∗
c
) =
c
∈
R
(0) and
b
∈
R
(0) but
c
∗
b
=
a
∉
R
(0). Hence (
R
,
U
) = (
g
,
I
) Ũ (
H
,
J
) is not a soft
α
ideal of (
F
,
A
).
Remark.
(
i
) It should be noted that the restricted difference of two soft
α
ideals is not a soft
α
ideal in general as is the case in the above example, i.e,
We define (
g
,
I
)
(
H
,
J
) = R,
U
), where
U
=
I
∩
J
and
R
(
u
) =
g
(
u
)−
H
(
u
) for all
u
∈
U
. Then
R
(0) =
g
(0) −
H
(0) = {
b
}, which is not an
α
ideal of
X
related to
F
(0). Hence (
g
,
I
)
(
H
,
J
) = (
R
,
U
) is not a soft
α
ideal of (
F
,
A
).
(
ii
) From the Example 4.13, it is also clear that the restricted union of any two soft
α
ideals is not a soft
α
ideal in general.
Since by Definition 3.5, (
g
,
I
) ∪
_{ℜ}
(
H
,
J
) =
R
,
U
), where
U
=
I
∩
J
and
R
(
u
) =
g
(
u
) ∪
H
(
u
) for all
u
∈
U
. Then
R
(0) =
g
(0) ∪
H
(0) = {0, 1,
b
,
c
}, which is not an
α
ideal of
X
related to
F
(0) since (
b
∗
b
) ∗ (0 ∗
c
) =
c
∈
R
(0) and
b
∈
R
(0) but
c
∗
b
=
a
∉
R
(0). So (
g
,
I
) ∪
_{ℜ}
(
H
,
J
) =
R
,
U
) is not a soft
α
ideal of (
F
,
A
).
5.αidealistic soft BCIalgebras
Definition 5.1
(Jun and Park
[11]
). Let (
F
,
V
) be soft set over
U
. Then (
F
,
V
) is called an idealistic soft BCIalgebra over
U
if
F
(
x
) is an ideal of
U
for all
x
∈
V
.
Definition 5.2.
A soft set (
F
,
V
) over
U
is called an
α
idealistic soft BCIalgebra over
U
if
F
(
v
) is an
α
ideal of
U
for all
v
∈
V
.
Example 5.3.
Let
X
= {0,
a
,
b
,
c
} be the BCIalgebra which is defined by the Cayley table given in Example 4.3. Let (
F
,
A
) be a soft set over
X
, where
A
=
X
and define a set valued function,
F
:
A
→
as:
where
Z
({0,
a
}) = {
u
∈
X
 0 ∗ (0 ∗
u
) ∈ {0,
a
}}. Then (
F
,
A
) will be an
α
idealistic soft BCIalgebra over
X
.
It is easy to see that
α
idealistic soft BCIalgebra over
X
is an idealistic soft BCIalgebra over
X
. But in general, the converse is not true and it can be observed by the following example.
Example 5.4.
Le
X
= {0,
a
,
b
,
c
,
d
} be the BCIalgebra defined by the Cayley table given in Example 4.8. Let (
F
,
B
) be a soft set over
X
, where
B
= {
b
,
c
,
d
} and the set valued functon
F
:
B
→
is defined as:
for all
u
∈
B
. Then
F
(
b
) = {0,
a
,
b
},
F
(
c
) = {0,
a
,
c
},
F
(
d
) =
X
, which are ideals of
X
. Hence (
F
,
B
) is an idealistic soft BCIalgebra over
X
but it is not an
α
idealistic soft BCIalgebra over
X
because
F
(
b
) is not an
α
ideal of
X
since, (
b
∗
b
) ∗ (0 ∗
d
) = 0 ∈
F
(
b
) and
b
∈
F
(
b
) but
d
∗
b
=
c
∉
F
(
b
).
Theorem 5.5.
For two αidealistic soft BCIalgebras
(
F
,
A
)
and
(
g
,
B
)
over X such that A
∩
B
≠ ∅
, the restricted intersection
(
F
,
A
) ∩
_{ℜ}
(
g
,
B
)
is also an αidealistic soft BCIalgebra over X
.
Proof
. Using Definition 3.4, we can write
where
C
=
A
∩
B
and
H
(
e
) =
F
(
e
)∩
g
(
e
) for all
e
∈
C
. Note that
H
:
C
→
is a mapping, therefore (
H
,
C
) is a soft set over
X
. Since (
F
,
A
) and (
g
,
B
) are
α
idealistic soft BCIalgebras over
X
, it follows that
F
(
e
) is an
α
ideal of
X
for all
e
∈
A
and
g
(
e
) is an
α
ideal of
X
for all
e
∈
b
. Then
F
(
e
) ∩
g
(
e
) =
H
(
e
) is also an
α
ideal of
X
for all
e
∈
A
∩
B
=
C
. Hence (
H
,
C
) = (
F
,
A
)∩
_{ℜ}
(
g
,
B
) is an
α
idealistic soft BCIalgebra over
X
. □
Corollary 5.6.
Let
(
F
,
A
)
and
(
g
,
A
)
be two αidealistic soft BCIalgebras over X. Then their restricted intersection
(
F
,
A
) ∩
_{ℜ}
(
g
,
A
)
is also an αidealistic soft BCIalgebra over X
.
Proof
. Straightforward. □
Theorem 5.7.
Let
(
F
,
A
)
and
(
g
,
B
)
be two αidealistic soft BCIalgebras over X. Then the extended intersection
(
F
,
A
)∩
_{ξ}
(
g
,
B
)
is also an αidealistic soft BCIalgebra over X
.
Proof
. By means of Definition 3.3, we can write (
F
,
A
) ∩
_{ξ}
(
g
,
B
) = (
H
,
C
), where
C
=
A
∪
B
and for every
e
∈
C
,
For every
e
∈
C
such that
e
∈
A
╲
B
, we get
H
(
e
) =
F
(
e
) which is an
α
ideal of
X
as (
F
,
A
) is an
α
idealistic soft BCIalgebra over
X
. Similarly for any
e
∈
C
such that
e
∈
B
╲
A
,
H
(
e
) =
g
(
e
), which is an
α
ideal of
X
since (
g
,
B
) is an
α
idealistic soft BCIalgebra over
X
. Moreover for some
e
∈
C
such that
e
∈
A
∩
B
,
H
(
e
) =
F
(
e
) ∩
g
(
e
). Since
F
(
e
) is an
α
ideal of
X
for all
e
∈
A
and
H
(
e
) is an
α
ideal of
X
for all
e
∈
B
, it follows that
F
(
e
) ∩
g
(
e
) =
H
(
e
) is an
α
ideal of
X
for all
e
∈
A
∩
B
. Thus
H
(
e
) is an
α
ideal of
X
for all
e
∈
C
. Hence (
F
,
A
) ∩
_{ξ}
(
g
,
B
) = (
H
,
C
) is an
α
idealistic soft BCIalgebra over
X
. □
Theorem 5.8.
Let
(
F
,
A
)
and
(
g
,
B
)
be two αidealistic soft BCIalgebras over X. If A and B are disjoint, then the union
(
F
,
A
) Ũ (
g
,
B
)
is also an αidealistic soft BCIalgebra over X
.
Proof
. By means of Definition 3.2, we can write (
F
,
A
) Ũ (
g
,
B
) = (
H
,
C
), where
C
=
A
∪
B
and for every
e
∈
C
,
Since
A
∩
B
= ∅, either
e
∈
A
╲
B
or
e
∈
B
╲
A
for all
e
∈
C
. If
e
∈
A
╲
B
, then
H
(
e
) =
F
(
e
) is an
α
ideal of
X
since (
F
,
A
) is an
α
idealistic soft BCIalgebra over
X
. If
e
∈
B
╲
A
, then
H
(
e
) =
g
(
e
) is an
α
ideal of
X
since (
g
,
B
) is an
α
idealistic soft BCIalgebra over
X
. Hence (
H
,
C
) = (
F
,
A
) Ũ (
g
,
B
) is an
α
idealistic soft BCIalgebra over
X
.
Example 5.9.
We consider the case when we have a nonempty intersection of the set of parameters
A
and
B
. Let
X
= {0,
a
,
b
,
c
} be the BCIalgebra defined by the Cayley table given in Example 4.3. Let (
F
,
A
) be a soft set over
X
, where
A
= {0,
a
,
b
} and
F
:
A
→
is a setvalued function defined by:
where
Z
({0,
a
}) = {
x
∈
X
 0 ∗ (0 ∗
x
) ∈ {0,
a
}}. Then
F
(0) = {0},
F
(
a
) = {0} and
F
(
b
) = {0,
a
}, which are
α
ideals of
X
. Thus (
F
,
A
) is an
α
idealistic soft BCIalgebra over
X
.
Now take (
g
,
B
) as another soft set over
X
, where
B
= {
b
} and
g
:
B
→
is a setvalued function defined by:
g
(
u
) = {0} ∪ {
y
∈
X

x
≤
y
} for all
u
∈
B
. Then
g
(
b
) = {0,
b
}, which is an
α
ideal of
X
. Hence (
g
,
B
) is also an
α
idealistic soft BCIalgebra over
X
.
By Definition 3.2, for the union, (
F
,
A
) Ũ (
g
,
B
) = (
H
,
C
), we get
H
(
b
) =
F
(
b
)∪
g
(
b
) = {0,
a
,
b
}, which is not an
α
ideal of
X
since (
a
∗
b
)∗(0∗
b
) =
a
∈
H
(
b
) and
b
∈
H
(
b
) but
b
∗
a
=
c
∉
H
(
b
).
Thus the union two
α
idealistic soft BCIalgebras will be an
α
idealistic soft BCIalgebra provided that the set of parameters of these soft sets are disjoint.
Remark.
(
i
) It should be noted from the above example that the restricted difference of two
α
idealistic soft BCIalgebras is not an
α
idealistic soft BCIalgebra in general, i.e,
(
F
,
A
)
(
g
,
B
) = (
H
,
C
), where
C
=
A
∩
B
≠ Φ and
H
(
c
) =
F
(
c
) −
g
(
c
) for all
c
∈
C
. Then
H
(
b
) =
F
(
b
) −
g
(
b
) = {
a
}, which is not an
α
ideal of
X
. Therefore (
F
,
A
)
(
g
,
B
) = (
H
,
C
) is not an
α
idealistic soft BCIalgebra over
X
.
(
ii
) It can also be observed that, in general, the restricted union of two
α
idealistic soft BCIalgebras is not an
α
idealistic soft BCIalgebra, i.e,
(
F, A
) ∪
_{ℜ}
(
g, B
) = (
H, C
), where
C
=
A
∩
B
≠ Φ and
H
(
c
) =
F
(
c
) ∪
g
(
c
) for all
c
∈
C
. Then
H
(
b
) =
F
(
b
) ∪
g
(
b
) = {0,
a
,
b
}, which is not an
α
ideal of
X
since (
a
∗
b
) ∗ (0 ∗
b
) =
a
∈
H
(
b
) and
b
∈
H
(
b
) but
b
∗
a
=
c
∉
H
(
b
). Hence (
F
,
A
) ∪
_{ℜ}
(
g
,
B
) = (
H
,
C
) is not an
α
idealistic soft BCIalgebra over
X
.
Theorem 5.10.
Let
(
F
,
A
)
and
(
g
,
B
)
be two αidealistic soft BCIalgebras over X, then
(
F
,
A
)
(
g
,
B
)
is an αidealistic soft BCIalgebra over X
.
Proof
. Since by Definition 3.4, we know that
where
H
(
x
,
y
) =
F
(
x
) ∩
g
(
y
) for all (
x
,
y
) ∈
A
×
B
. Since
F
(
x
) and
g
(
y
) are
α
ideals of
X
, the intersection
F
(
x
) ∩
g
(
y
) is also an
α
ideal of
X
. Hence
H
(
x
,
y
) is an
α
ideal of
X
for all (
x
,
y
) ∈
A
×
B
.
Hence (
F
,
A
)
(
g
,
B
) = (
H
,
A
×
B
) is an
α
idealistic soft BCIalgebra over
X
. □
Definition 5.11
(Liu and Zhang
[14]
). A fuzzy set
μ
in
X
is called a fuzzy
α
ideal of
X
, if for all
x
,
y
,
z
∈
X
,
(i)
μ
(0) ≥
μ
(
x
)
(ii)
μ
(
y
∗
x
) ≥
min
{
μ
((
x
∗
z
) ∗ (0 ∗
y
)),
μ
(
z
)}
The transfer principle for fuzzy sets described in
[13]
suggest’s the following theorem.
Lemma 5.12
(Liu and Zhang
[14]
).
A fuzzy set μ in X is a fuzzy αideal of X if and only if for any t
∈ [0,1]
, the level subset
U
(
μ
;
t
) := {
x
∈
X

μ
(
x
) ≥
t
}
is either empty or an αideal of X
.
Theorem 5.13.
For every fuzzy αideal μ of X, there exists an αidealistic soft BCIalgebra
(
F
,
A
)
over X
.
Proof
. Let
μ
be a fuzzy
α
ideal of
X
. Then
U
(
μ
;
t
) := {
x
∈
X

μ
(
x
) ≥
t
} is an
α
ideal of
X
for all
t
∈
Im
(
μ
). If we take
A
=
Im
(
μ
) and consider a set valued function
F
:
A
→
given by
F
(
t
) =
U
(
μ
;
t
) for all
t
∈
A
, then (
F
,
A
) is an
α
idealistic soft BCIalgebra over
X
.
Conversely, it can be easily observed that the following theorem holds.
Theorem 5.14.
For any fuzzy set μ in X, if an αidealistic soft BCIalgebra
(
F
,
A
)
over X is given by A
=
Im
(
μ
)
and F
(
t
) =
U
(
μ
;
t
)
for all t
∈
A, then μ is a fuzzy αideal of X
.
Let
μ
be a fuzzy set in
X
and let (
F
,
A
) be a soft set over
X
in which
A
=
Im
(
μ
) and
F
:
A
→
is a setvalued function defined by
for all
t
∈
A
. Then there exists
t
∈
A
such that
F
(
t
) is not an
α
ideal of
X
as seen in the following example.
Example 5.15.
For any BCIalgebra
X
, define a fuzzy set
μ
in
X
by
μ
(0) =
t
_{o}
< 0.5 and
μ
(
x
) = 1 −
t
_{o}
for all
x
≠ 0. Let
A
=
Im
(
μ
) and
F
:
A
→
be a setvalued function defined by (5.2). Then
F
(1 −
t
_{o}
) =
X
╲ {0}, which is not an
α
ideal of
X
.
Theorem 5.16.
Let μ be a fuzzy set in X and let
(
F
,
A
)
be a soft set over X in which A
= [0, 1]
and F
:
A
→
)
is given by (5.2). Then the following assertions are equivalent:
(1) μ is a fuzzy αideal of X
.
(2) for every t
∈
A with F
(
t
) ≠ ∅,
F
(
t
)
is an αideal of X
.
Proof
. Assume that
μ
is a fuzzy
α
ideal of
X
. Let
t
∈
A
be such that
F
(
t
) ≠ ∅. Then for any
x
∈
F
(
t
), we have
μ
(0) +
t
≥
μ
(
x
) +
t
> 1, that is, 0 ∈
F
(
t
). Let (
x
∗
z
) ∗ (0 ∗
y
) ∈
F
(
t
) and
z
∈
F
(
t
) for any
t
∈
A
and
x
,
y
,
z
∈
X
. Then
μ
((
x
∗
z
) ∗ (0 ∗
y
)) +
t
> 1 and
μ
(
z
) +
t
> 1. Since
μ
is a fuzzy
α
ideal of
X
, it follows that
so that
y
∗
x
∈
F
(
t
). Hence
F
(
t
) is an
α
ideal of
X
for all
t
∈
A
such that
F
(
t
) ≠ ∅.
Conversely, suppose that (2) is valid. If there exists
x
_{o}
∈
X
such that
μ
(0) <
μ
(
x
_{o}
), then there exists
t
_{o}
∈
A
such that
μ
(0) +
t
_{o}
≤ 1 <
μ
(
x
_{o}
) +
t
_{o}
.
It follows that
x
_{o}
∈
F
(
t
_{o}
) and 0 ∉
F
(
t
_{o}
), which is a contradiction. Hence
μ
(0) ≥
μ
(
x
) for all
x
∈
X
. Now assume that
for some
x
_{o}
,
y
_{o}
,
z
_{o}
∈
X
. Then there exists some
s
_{o}
∈
A
such that
which implies that (
x
_{o}
∗
z
_{o}
)∗(0∗
y
_{o}
) ∈
F
(
s
_{o}
) and
z
_{o}
∈
F
(
s
_{o}
) but
y
_{o}
∗
x
_{o}
∉
F
(
s
_{o}
). This is a contradiction. Therefore
for all
x
,
y
,
z
∈
X
and thus
μ
is fuzzy
α
ideal of
X
. □
Corollary 5.17.
Let μ be a fuzzy set in X such that μ
(
x
) > 0.5
for all x
∈
X and let
(
F
,
A
)
be a soft set over X in which
and F
:
A
→
is given by (5.2). If μ is a fuzzy αideal of X, then
(
F
,
A
)
is an αidealistic soft BCIalgebra over X
.
Proof
. Straightforward.
Theorem 5.18.
Let μ be a fuzzy set in X and let
(
F
,
A
)
be a soft set over X in which A
= (0.5, 1] and
F
:
A
→
is defined by
Then F
(
t
)
is an αideal of X for all t
∈
A with F
(
t
) ≠ ∅
if and only if the following assertions are valid:
(1) max
{
μ
(0), 0.5} ≥
μ
(
x
)
for all x
∈
X
.
(2) max
{
μ
(
y
∗
x
), 0.5} ≥
min
{
μ
((
x
∗
z
) ∗ (0 ∗
y
)),
μ
(
z
)}
for all x
,
y
,
z
∈
X
.
Proof
. Assume that
F
(
t
) is an
α
ideal of
X
for all
t
∈
A
with
F
(
t
) ≠ ∅. If there exists
x
_{o}
∈
X
such that
max
{
μ
(0), 0.5} <
μ
(
x
_{o}
), then there exists
t
_{o}
∈
A
such that
max
{
μ
(0), 0.5} <
t
_{o}
≤
μ
(
x
_{o}
). It follows that
μ
(0) <
t
_{o}
, so that
x
_{o}
∈
F
(
t
_{o}
) and 0 ∉
F
(
t
_{o}
). This is a contradiction. Therefore (1) is valid. Suppose that there exist
a, b, c
∈
X
such that
Then there exists
s
_{o}
∈
A
such that
which implies (
a
∗
c
) ∗ (0 ∗
b
) ∈
F
(
s
_{o}
) and
c
∈
F
(
s
_{o}
), but
b
∗
a
∉
F
(
s
_{o}
). This is a contradiction. Hence (2) is valid.
Conversely, suppose that (1) and (2) are valid. Let
t
∈
A
with
F
(
t
) ≠ ∅. Then for any
x
∈
F
(
t
), we have
which implies
μ
(0) ≥
t
and thus 0 ∈
F
(
t
). Let (
x
∗
z
) ∗ (0 ∗
y
)) ∈
F
(
t
) and
z
∈
F
(
t
), for any
x
,
y
,
z
∈
X
. Then
μ
((
x
∗
z
) ∗ (0 ∗
y
)) ≥
t
and
μ
(
z
) ≥
t
. It follows from the second condition that
So that
μ
(
y
∗
x
) ≥
t
, i.e.,
y
∗
x
∈
F
(
t
). Therefore
F
(
t
) is an
α
ideal of
X
for all
t
∈
A
with
F
(
t
) ≠ ∅. □
6. CONCLUSION

• The union of two softαideals (F,A) and (g,B) is a softαideal ifA∩B=ϕ. Similarly the union of twoαidealistic soft BCIalgebras (F,A) and (g,B) is anαidealistic soft BCIalgebra provided thatA∩B=ϕ

• The restricted intersection and restricted difference of any two softαideals are not softαideals in general. The same is the case for the restricted intersection and restricted difference of any twoαidealistic soft BCIalgebras.

• Fuzzyαideals can be characterized using the concept of soft sets.

• For every fuzzyαideal, there exists anαidealistic soft BCIalgebra.

• For a soft set (F,A) overX, a fuzzy setμinXis a fuzzyαideal ofXif and only if for everyt∈AwithF(t) = {x∈Xμ(x) +t> 1} ≠ ∅,F(t) is anαideal ofX.
BIO
M. Touqeer Ph.D. student at the Department of Mathematics, University of the Punjab, Lahore, Pakistan. His research interests include fuzzy set theory and soft set theory.
Department of Mathematics, University of the Punjab, QuaideAzam Campus, Lahore54590, Pakistan.
email: touqeerfareed@yahoo.com
M. Aslam Malik Ph.D. from University of the Punjab, Lahore, Pakistan. He is currently a professor at Department of Matheamtics, University of the Punjab, Lahore, Pakistan. His research interests orbits and stabilizer groups, fuzzy set theory and soft set theory.
Department of Mathematics, University of the Punjab, QuaideAzam Campus, Lahore 54590, Pakistan.
email: malikpu@yahoo.com
Ali M.I.
,
Feng F.
,
Liu X.
,
Min W.K.
,
Shabir M.
(2009)
On some new operations in soft set theory
Comp. Math. appl.
57
1547 
1553
DOI : 10.1016/j.camwa.2008.11.009
Çağman N.
,
Citakand F.
,
Enǵinoglu S.
(2010)
Fuzzy parameterized fuzzy soft set theory and its applications
Turkish Journal of Fuzzy Systems
1
21 
35
Chen D.
,
Tsang E.C.C.
,
Yeung D.S.
,
Wang X.
(2005)
The parameterization reduction of soft sets and its application
Comp. Math. appl.
45
757 
763
DOI : 10.1016/j.camwa.2004.10.036
Herawan T.
,
Ghazali R.
,
Deris M.M.
(2010)
Soft set theoretic approach for dimensionality reduction
International Journal of Database Theory and Applic ation
3
47 
60
Imai Y.
,
Iseki K.
On axioms of propositional calculi
XIV Proc., Jpn. Acad.
(1966)
19 
22
Jun Y.B.
,
Park C.H.
(2008)
Applications of soft sets in ideal theory of BCK/BCIalgerbas
Inform. Sci.
178
2466 
2475
Kharal A.
,
Ahmad B.
(2010)
Mappings on soft classes
Inf. Sci.
INSD081231 by ESS
1 
11
Kondo M.
,
Dudek W.A.
(2005)
On the transfer principle in fuzzy theory
Mathware and Soft Computing,
12
41 
55
Liu Y.L.
,
Zhang X.H.
(2002)
Fuzzy αideals of BCIalgebras
Advances in Mathematics
31
65 
73
Yang X.
,
Yu D.
,
Yang J.
,
Wu C.
(2007)
Generalization of soft set theory: from crisp to fuzzy case, In: BingYuan Cao ,eds., Fuzzy Information and Engineering: Proceedings of ICFIE 2007, Advances in Soft Computing 40
Springer
345 
355