The notion of intersectionsoft ideal of
CI
algebras is introduced, and related properties are investigated. A characterization of an intersectionsoft ideal is provided, and a new intersectionsoft ideal from the old one is established.
1. INTRODUCTION
Mathematics requires that all mathematical notions (including set) must be exact, otherwise precise reasoning would be impossible. However, philosophers and recently computer scientists as well as other researcher have become interested in vague concepts
[1

4
,
10

12]
. One of them, Hájek
[3]
introduced a BLalgebra which is an algebraic structure for many valued logic. Many researchers investigated the various algebraic structures as MValgebras, BCKalgebras, BEalgebras and CIalgebras
[1

6
,
12]
. As a generalization of a BCKalgebra, Kim and Kim
[6]
introduced the notion of a
BE
algebra, and investigated several properties. The notion of
CI
algebras is introduced by Meng
[8]
as a generalization of BEalgebras. Ideal theory and properties in
CI
algebras are studied by Kim
[5]
.
On the hand, rough set theory was introduced by Pawlak
[11
,
12]
to generalize the classical set theory. Rough approximations are defined by the equivalence relation. There has been a rapid growth in interest in rough set theory in recent years. Its applications are decision system modeling and analysis of complex systems, neural networks, evolutionary computing, data mining and knowledge discovery, pattern recognition, machine learning, business and finance, chemistry, computer engineering, environment, medicine, etc. As a generalization of a rough set, Molodtsov
[9]
introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the di±culties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. In
[7]
, Lee applied soft set theory to
CI
algebras.
In this paper, we introduce the notion of intsoft ideal in
CI
algebras, and investigate related properties. We provide a characterization of an intsoft ideal. We make a new intsoft ideal from the old one.
2. PRELIMINARIES
An algebra (
X
; *, 1) of type (2, 0) is called a
CI

algebra
if it satisfies the following properties:

(CI1)x*x= 1,

(CI2) 1 *x=x,

(CI3)x* (y*z) =y* (x*z),
for all
x
,
y
,
z
∈
X
. A
CI
algebra (
X
; *, 1) is said to be
transitive
if it satisfies:
A
CI
algebra (
X
; *, 1) is said to be
selfdistributive
if it satisfies:
Note that every selfdistributive
CI
algebra is a transitive
CI
algebra (see
[5]
).
A nonempty subset
I
of a
CI
algebra (
X
; *, 1) is called an
ideal
of
X
(see
[5]
) if it satisfies:

(I1) (∀x,y∈X) (y∈I⇒x*y∈I),

(I2) (∀x,a,b∈X) (a,b∈I⇒ (a* (b*x)) *x∈I) .
Molodtsov
[9]
defined the soft set in the following way: Let
U
be an initial universe set and
E
be a set of parameters. Let
P
(
U
) denotes the power set of
U
and
A
⊂
E
:
A pair(
,
A
) is called a
soft set
(see
[9]
) over
U
, where
is a mapping given by
In other words, a soft set over
U
is a parameterized family of subsets of the universe
U
. For ε∈
A
,
(ε) may be considered as the set of εapproximate elements of the soft set(
, A). Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in
[9]
.
For a soft set (
,
X
) over
U
and a subset 𝛾 of
U
, the 𝛾
inclusive
set of (
,
X
), denoted by (
; 𝛾)
^{⊇}
, is defined to be the set
3. INTERSECTIONSOFT IDEALS
In what follows, denote by S(
U
,
X
) the set of all soft sets of
X
over
U
where
X
is a
CI
algebra unless otherwise specified.
Definition 3.1.
A soft set (
,
X
)∈
S
(
U
,
X
) is called an
intersectionsoft ideal
(briefiy,
intsoft ideal
) (of
X
) over
U
if it satisfies the following conditions:
Example 3.2.
Let
X
={1,
a
,
b
,
c
,
d
, 0} be a
CI
algebra with the following Cayley table:
(1) Let (
,
X
)∈
S
(
U
,
X
) be given as follows:
where 𝛾
_{1}
and 𝛾
_{2}
are subsets of
U
with 𝛾
_{2}
⊈ 𝛾
_{1}
. Then(
,
X
) is an intsoft ideal over
U
.
(2) For
U
=
(the set of integers), let (
,
X
) ∈
S
(
U
,
X
) be given as follows:
Then (
,
X
) is not an intsoft ideal over
U
since
Proposition 3.3.
Every intsoft ideal
(
,
X
)
over U satisfies the following assertion
:
Proof
. Using (CI2), (CI1) and (3.2), we have
for all
x
,
y
∈
X
. □
Lemma 3.4.
Every intsoft ideal
(
,
X
)
over U satisfies the following assertion
:
Proof
. Using (CI1) and (3.1), we have
(1) =
(
x
*
x
)⊇
(
x
) for all
x
∈
X
. □
Proposition 3.5.
Every intsoft ideal
(
,
X
)
over U satisfies the following assertion
:
Proof
. Taking
y
= 1 and
z
=
y
in (3.2) and using (CI2) and Lemma 3.4, we get
for all
x
,
y
∈
X
. □
Corollary 3.6.
Every intsoft ideal(
,
X
)over
U
satisfies the following assertion:
Proof
. Let
x
,
y
∈
X
be such that
x
*
y
= 1. Then
by (CI2) and (3.5). □
If a soft set (
,
X
)∈
S
(
U
,
X
) satisfies the condition (3.6), we say that(
,
X
) is
order preserving
. Hence every intsoft ideal is order preserving.
Proposition 3.7.
If X is a transitive CIalgebra, then every intsoft ideal
(
,
X
)
over U satisfies the condition
Proof
. Let (
,
X
) be an intsoft ideal over
U
. Since
X
is transitive, we have
for all
x
,
y
,
z
∈
X
. It follows from (CI2), (3.2) and (3.5) that

(x*z)

=(1*(x*z))

=(((y*z)*z)*((x*(y*z))*(x*z))*(x*z))

⊇((y*z)*z)∩(x*(y*z))

⊇(x*(y*z))∩(y)
Therefore (3.7) is valid. □
Corollary 3.8.
If X is a selfdistributive CIalgebra
,
then every intsoft ideal
(
,
X
)
over U satisfies the condition
(3.7).
Proposition 3.9.
If a soft set
(
,
X
)∈
S
(
U
,
X
)
satisfies two conditions
(3.4)
and
(3.7),
then
(
,
X
)
is order preserving
.
Proof
. Let
x
,
y
∈
X
be such that
x
*
y
= 1. Then
by (CI1), (CI2), (3.7) and (3.4). Therefore (
,
X
) is order preserving. □
Theorem 3.10.
If
(
,
X
)
is an intsoft ideal over U, then the set
is an ideal of X
.
Proof
. Let
x
∈
X
and
a
∈
I
. Then
(
a
) =
(1), and so
by (3.1). Combining this and (3.4), we have
(
x
*
a
) =
(1), that is,
x
*
a
∈
I
. For any
x
,
a
,
b
∈
X
, if
a
,
b
∈
I
, then
(
a
) =
(1) =
(
b
). It follows from (3.2) that
and so that
((
a
* (
b
*
x
)) *
x
) =
(1): Thus (
a
* (
b
*
x
)) *
x
∈
I
. Therefore
I
is an ideal of
X
. □
We provide characterizations of an intsoft ideal.
Theorem 3.11.
A soft set
(
,
X
)∈
S
(
U
,
X
)
is an intsoft ideal over U if and only if the
𝛾
inclusive set
(
; 𝛾)
^{⊇}
is an ideal of X for all
𝛾 ∈
P
(
U
)
with
(
; 𝛾)
^{⊇}
≠ Ø.
The ideal (
; 𝛾)
^{⊇ }
in Theorem 3.11 is called the
inclusive
ideal of
X
.
Proof
. Assume that(
,
X
) is an intsoft ideal over
U
. Let 𝛾 ∈
P
(
U
) be such that (
; 𝛾)
^{⊇}
≠ Ø. Let
x
∈
X
and
a
∈ (
; 𝛾)
^{⊇}
. Then
(
a
) ⊇ 𝛾. It follows from (3.1) that
(
x
*
a
) ⊇
(
a
) ⊇ 𝛾. Hence
x
*
a
∈ (
; 𝛾)
^{⊇}
. Let
x
∈
X
and
a
,
b
∈(
; 𝛾)
^{⊇}
. Then
(
a
) ⊇ 𝛾 and
(
b
) ⊇ 𝛾. Using (3.2), we have
((
a
* (
b
*
x
)) *
x
) ⊇
(
a
) ⊇
(
b
) ⊇ 𝛾, and thus (
a
* (
b
*
x
)) *
x
2 (
; 𝛾)
^{⊇}
. Therefore (
; 𝛾)
^{⊇}
is an ideal of X for all 𝛾 ∈
P
(
U
) with (
; 𝛾)
^{⊇}
≠ Ø.
Conversely, suppose that (
; 𝛾)
^{⊇}
is an ideal of
X
for all 𝛾 ∈
P
(
U
) with (
; 𝛾)
^{⊇}
≠ Ø. For any
a
∈
X
, let
(
a
) = 𝛾. Then
a
∈ (
; 𝛾)
^{⊇}
. Since (
; 𝛾)
^{⊇}
is an ideal of
X
, we have
x
*
a
∈ (
; 𝛾)
^{⊇}
for all
x
∈
X
. Thus
(
x
*
a
) ⊇ 𝛾 =
(
a
) for all
x
,
a
∈
X
. For any
x
,
y
∈
X
, let
(
x
) = 𝛾
x
and
(
y
) = 𝛾y. Take 𝛾 = 𝛾
_{x}
∩ 𝛾
_{y}
. Then
x
,
y
∈ (
; 𝛾)
^{⊇}
which implies that (
x
* (
y
*
z
)) *
z
∈ (
; 𝛾)
^{⊇}
for all
z
∈
X
. Hence
for all
x
,
y
,
z
∈
X
. Thus(
,
X
) is an intsoft ideal over
U
. □
Theorem 3.12.
For any soft set
(
,
X
) ∈
S
(
U
,
X
),
let
(
,
X
) ∈ S(
U
,
X
)
be defined by
where
𝛾
and
𝛿
are subsets of U with
𝛿 ⊈
(
x
).
If
(
,
X
)
is an intsoft ideal over U
,
then so is
(
,
X
).
Proof.
It is straightforward by Theorem 3.11. □
Theorem 3.13.
Every ideal of X can be realized as an inclusive ideal of some intsoft ideal over X.
Proof
. Let
I
be an ideal of
X
. Define a soft set(
,
X
)∈
S
(
U
,
X
) as follows:
where 𝛾 is a nonempty subset of
U
. Let
x
,
y
∈
X
. If
y
∈
I
, then
x
*
y
∈
I
and so
(
y
) = 𝛾 =
(
x
*
y
). If
y
=∉
I
, then
(
y
) Ø ⊆
(
x
*
y
). For any
x
,
a
,
b
∈
X
, let
a
,
b
∈
I
. Then (
a
* (
b
*
x
)) *
x
∈
I
and thus
(
a
)∩
(
b
) = 𝛾 =
((
a
* (
b
*
x
)) *
x
). If
a
=∉
I
or
b
=∉
I
, then
(
a
) Ø or
(
b
) Ø. Hence
(
a
)∩
(
b
) Ø ⊆
((
a
* (
b
*
x
)) *
x
). Obviously, (
; 𝛾)
^{⊇}
=
I
. This completes the proof. □
For any
a
,
b
∈
X
, consider the following set:
Theorem 3.14.
Every intsoft ideal
(
,
X
)
over U satisfies the following assertion
:
Proof.
Assume that (
,
X
) is an intsoft ideal over
U
.Let
a
,
b
∈
X
be such that
a
,
b
∈ (
; 𝛾)
^{⊇}
. Then
(
a
) ⊇ 𝛾 and
(
b
) ⊇ 𝛾. If
y
∈
C
(
a
,
b
), then
a
* (
b
*
y
) = 1 and so
by (CI2) and (3.2). Hence
y
∈ (
;𝛾)
^{⊇}
.Therefore
C
(
a
,
b
) ⊆(
; 𝛾)
^{⊇}
. □
Corollary 3.15.
For every intsoft ideal
(
,
X
)over
U
,
we have
Acknowledgements
This work was supported by the Research Institute of Natural Science of GangneungWonju National University.
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