Interval-Valued Fuzzy Congruences on a Semigroup

International Journal of Fuzzy Logic and Intelligent Systems.
2013.
Sep,
13(3):
231-244

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Received : January 02, 2013
- Accepted : September 10, 2013
- Published : September 30, 2013

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We introduce the concept of interval-valued fuzzy congruences on a semigroup
S
and we obtain some important results: First, for any interval-valued fuzzy congruence
R
on a group
G
, the interval-valued congruence class
R_{e}
is an interval-valued fuzzy normal subgroup of
G
. Second, for any interval-valued fuzzy congruence
R
on a groupoid
S
, we show that a binary operation * an
S/R
is well-defined and also we obtain some results related to additional conditions for
S
. Also we improve that for any two interval-valued fuzzy congruences
R
and
Q
on a semigroup
S
such that
R
⊂
Q
, there exists a unique semigroup homomorphism
g : S/R → S/G
.
S
and we obtain some important results:
(i) For any interval-valued fuzzy congruence
R
on a group
G
, the interval-valued congruence class
R_{e}
is an interval-valued fuzzy normal subgroup of
G
(Proposition 3.11).
(ii) For any interval-valued fuzzy congruence
R
on a groupoid
S
, we show that a binary operation * an
S/R
is well-defined (Proposition 3.20) and also we obtain some results related to additional conditions for
S
(Theorem 3.21, Corollaries 3.21-1, 3.21-2, and 3.21-3). Also we improve that for any two intervalvalued fuzzy congruences
R
and
Q
on a semigroup
S
such that
R
⊂
Q
, there exists a unique semigroup homomorphism
g : S/R → S/G
(Theorem 4.3).
D
(
I
) be the set of all closed subintervals of the unit interval [0, 1]. The elements of
D
(
I
) are generally denoted by capital letters
M
,
N
, …, and note that
M
= [
M^{L}, M^{U}
], where
M^{L}
and
M^{U}
are the lower and the upper end points respectively. Especially, we denoted ,
0
= [0, 0],
1
= [1, 1], and
a
= [
a, a
] for every
a
∈ (0, 1), We also note that
(i) (∀
M, N
∈
D
(
I
)) (
M
=
N
⇔
M^{L}
=
N^{L}
,
M^{U}
=
N^{U}
),
(ii) (∀
M, N
∈
D
(
I
)) (
M
=
N
≤
M^{L}
≤
N^{L}
,
M^{U}
≤
N^{U}
),
For every
M
∈
D
(
I
), the
complement
of
M
, denoted by
M^{C}
, is defined by
M^{C}
= 1 –
M
= [1 –
M^{U}
, 1 –
M^{L}
](
[7
,
14]
).
Definition 2.1
[4
,
10
,
14]
. A mapping
A
:
X
→
D
(
I
) is called an
interval-valued fuzzy set
(
IVFS
) in
X
, denoted by
A
= [
A^{L}, A^{U}
], if
A^{L}
,
A^{L}
∈
I^{X}
such that
A^{L}
≤
A^{U}
, i.e.,
A^{L}
(
x
) ≤
A^{U}
(
x
) for each
x
∈
X
, where
A^{L}
(
x
)[resp
A^{U}
(
x
)] is called the
lower
[resp
upper
]
end point of x to A.
For any [
a, b
] ∈
D
(
I
), the interval-valued fuzzy
A
in
X
defined by
A
(
x
) = [
A^{L}
(
x
),
A^{U}
(
x
)] = [
a, b
] for each
x
∈
X
is denoted by
and if
a
=
b
, then the IVFS
is denoted by simply
ã
. In particular,
denote the
interval-valued fuzzy empty set
and the
interval-valued fuzzy whole set
in
X
, respectively.
We will denote the set of all IVFSs in
X
as
D
(
I
)
^{X}
. It is clear that set
A
= [
A, A
] ∈
D
(
I
)
^{X}
for each
A
∈
I^{X}
.
Definition 2.2
[14]
. Let
A, B
∈
D
(
I
)
^{X}
and let ｛
A_{α}
｝
_{α∈Г}
⊂
D
(
I
)
^{X}
. Then
Result 2.A
[
14
, Theorem 1]. Let
A, B, C
∈
D
(
I
)
^{X}
and let {
A_{α}
}
_{α∈Г}
⊂
D
(
I
)
^{X}
. Then
Definition 2.3
[8]
. Let
X
be a set. Then a mapping
R
= [
R^{L}
,
R^{U}
] :
X
∏
X
→
D
(
I
) is called an
interval-valued fuzzy relation
(
IVFR
) on
X
.
Definition 2.4
[8]
. Let
R
∈ IVR(X). Then the
inverse
of
R
,
R
^{–1}
is defined by
R
^{–1}
(
x,y
) =
R
(
y,x
), for each
x, y
∈
X
.
Definition 2.5
[11]
. Let
X
be a set and let
R
,
Q
∈ IVR(X). Then the composition of
R
and
Q
,
Q
○
R
, is defined as follows : For any
x, y
∈
X
,
and
Result 2.B
[
11
, Proposition 3.4]. Let
X
be a set and let
R
,
R
_{1}
,
R
_{2}
,
R
_{3}
,
Q
_{1}
,
Q
_{2}
∈ IVR(
X
). Then
Definition 2.6
[11]
. An IVFR
R
on a set
X
is called an
interval-valued fuzzy equivalence relation
(
IV FER
) on
X
if it satisfies the following conditions :
(1) it is
interval-valued fuzzy reflexiv
, i.e.,
R
(
x, x
) = [1, 1], for each
x
∈
X
,
(2) it is
interval-valued fuzzy symmetric
, i.e.,
R
^{–1}
=
R
,
(3) it is
interval-valued fuzzy transitive
, i.e.,
R
○
R
⊂
R
.
We will denote the set of all IVFERS on
X
as IVE(
X
).
From Definition 2.6, we can easily see that the following hold.
Remark 2.7
(a) If
R
is an fuzzy equivalence relation on a set
X
, then [
R, R
] ∈ IVE(
X
).
(b) If
R
∈ IVE(
X
), then
R^{L}
and
R^{U}
are fuzzy equivalence relation on
X
.
(c) Let
R
be an ordinary relation on a set
X
. Then
R
is an equivalence relation on
X
if and only if [
Χ_{R}, Χ_{R}
] ∈ IVE(
X
).
Result 2.C
[
11
, Proposition 3.9]. Let
X
be a set and let
Q
,
R
∈ IVE(
X
). If
Q
○
R
=
R
○
Q
, then
R
○
Q
∈ IVE(
X
).
Let
R
be an IVFER on a set
X
and let
a
∈
X
. We define a mapping
Ra
:
X
→
D
(
I
) as follows : For each
a
∈
X
,
Then clearly
Ra
∈
D
(
I
)
^{X}
. In this case,
Ra
is called the
interval-valued fuzzy equivalence class
of
R containing a
∈
X
. The set {
Ra
:
a
∈
X
} is called the
interval-valued fuzzy quotient set of X by R
and denoted by
X/R
.
Result 2.D
[
11
, Proposition 3.10]. Let
R
be an IVFER on a set
X
. Then the following hold :
(
a
)
Ra
=
Rb
if and only if
R
(
a, b
) = [1, 1], for any
a, b
∈
X
.
(
b
)
R
(
a, b
) = [0, 0] if and only if
Ra
∩
Rb
=
for any
a, b
∈
X
.
(
d
) There exits the surjection
π
:
X
→
X/R
defined by
π
(
x
) =
Rx
for each
x
∈
X
.
Definition 2.8
[11]
. Let
X
be a set, let
R
∈ IVR(
X
) and let {
R_{α}
}
_{α∈Г}
be the family of all IVFERs on
X
containing
R
. Then ∩
_{α∈Г}
R_{α}
is called the IVFER
generated by R
and denoted by
R^{e}
.
It is easily seen that
R^{e}
is the smallest IVFER containing
R
.
Definition 2.9
[11]
. Let
X
be a set and let
R
∈ IVR(
X
). Then the
interval-valued fuzzy transitive closure
of
R
, denoted
R
^{∞}
, is defined as followings :
,where
R^{n}
=
R
○
R
○ … ○
R
(n factors).
Definition 2.10
[11]
. We define two mappings △, ▽ :
X
→
D
(
I
) as follows : For any
x, y
∈
X
,
and
It is clear that △, ▽ ∈ IVE(
X
) and
R
is an interval-valued fuzzy reflexive relation on
X
if and only if △ ⊂
R
.
Result 2.E
[
11
, Proposition 4.7]. If
R
is an IVFR on a set
X
, then
Definition 2.11
[17]
. Let (
X
, ·) be a groupoid and let
A, B
∈
D
(
I
)
^{X}
. Then the interval-valued fuzzy product of
A
and
B
,
A
○
B
is defined as follows : For each
a
∈
X
,
and
Definition 2.12
[17]
. Let (
X
, ·) be a groupoid and let
A
∈
D
(
I
)
^{X}
. Then
A
is called an i
interval-valued fuzzy subgroupoid
(
IVGP
) of
X
if for any
x, y
∈
X
,
and
We will denote the set of all IVGPs of
X
as IVGP(
X
). Then it is clear that
Definition 2.13
[17]
. Let
G
be a group and let
A
∈ IVGP(
G
). Then
A
is an i
interval-valued fuzzy subgroup
(
IVG
) of
G
if for each
x
∈
G
,
i.e.,
We will denote the set of all IVGs of
G
as IVG(
G
).
Definition 2.14
[17]
. Let
G
be a group and let
A
∈ IVG(
G
). Then
A
is said to be
normal
if
A
(
xy
) =
A
(
yx
), for any
x, y
∈
G
.
We will denote the set of all interval-valued fuzzy normal subgroups of
G
as IVNG(
G
). In particular, we will denote the set {
N
∈ IVNG(
G
) :
N
(
e
) = [1, 1]} as IVN(
G
).
Result 2.F
[
17
, Proposition 5.2]. Let
G
be a group and let
A
∈
D
(
I
)
^{G}
. If
B
∈ IVNG(
G
), then
A
○
B
=
B
○
A
.
Definition 2.15
[18]
. Let
G
be a group, let
A
∈ IVG(
G
) and let
x
∈
G
. We define two mappings
and
as follows, respectively : For each
g
∈
G
,
Then
Ax
[resp:
xA
] is called the
interval-valued fuzzy right
[resp.
left
]
coset
of
G
determined by
x
and
A
.
It is obvious that if
A
∈ IVNG(
G
), then the interval-valued fuzzy left coset coincides with the interval-valued fuzzy right coset of
A
on
G
. In this case, we will call
interval-valued fuzzy coset
instead of interval-valued fuzzy left coset or intervalvalued fuzzy right coset.
Definition 3.1
[19]
. A relation
R
on a groupoid
S
is said to be:
(1)
left compatible
if (
a, b
) ∈
R
implies (
xa, xb
) ∈
R
, for any
a, b
∈
S
,
(2)
right compatible
if (
a, b
) ∈
R
implies (
ax, bx
) ∈
R
, for any
a, b
∈
S
,
(3)
compatible
if (
a, b
) ∈
R
and (
s, d
) ∈
R
imply (
ab, cd
) ∈
R
, for any
a, b, c, d
∈
S
,
(4) a
left
[resp.
right
]
congruence
on
S
if it is a left[resp. right] compatible equivalence relation.
(5) a
congruence
on
S
if it is both a left and a right congruence on
S
.
It is well-known [
19
, Proposition I.5.1] that a relation
R
on a groupoid
S
is congruence if and only if it is both a left and a right congruence on
S
. We will denote the set of all ordinary congruences on
S
as
C
(
S
).
Now we will introduce the concept of interval-valued fuzzy compatible relation on a groupoid.
Definition 3.2
An IVFR
R
on a groupoid
S
is said to be :
(1)
interval-valued fuzzy left compatible
if for any
x, y, z
∈
G
,
(2)
interval-valued fuzzy right compatible
if for any
x, y, z
∈
G
,
(3)
interval-valued fuzzy compatible
if for any
x, y, z, t
∈
G
,
and
Example 3.3
Let
S
=
e, a, b
be the groupoid with multiplication table :
(
a
) Let
R
_{1}
:
S
×
S
→
D
(
I
) be the mapping defined as the matrix :
where [
λ
_{ij}
,
μ
_{ij}
] ∈
D
(
I
) such that [
λ
_{1i}
,
μ
_{1i}
](
i
= 1, 2, 3),
[
λ
_{21}
,
μ
_{21}
] and [
λ
_{31}
,
μ
_{31}
] are arbitrary, and
Then we can see that
R
_{1}
is an interval-valued fuzzy left compatible relation on
S
.
(
b
) Let
R
_{2}
:
S
×
S
→
D
(
I
) be the mapping defined as the matrix :
where [
λ
_{ij}
,
μ
_{ij}
] ∈
D
(
I
) such that [
λ
_{ij}
,
μ
_{ij}
](
i, j
= 1, 2, 3) is arbitrary and
Then we can see that
R
_{2}
is an interval-valued fuzzy right compatible relation on
S
.
(
c
) Let
R
_{3}
:
S
×
S
→
D
(
I
) be the mapping defined as the matrix :
where [
λ
_{ij}
,
μ
_{ij}
] ∈
D
(
I
) such that
Then we can see that
R
_{3}
is an interval-valued fuzzy compatible relation on
S
.
Lemma 3.4
Let
R
be a relation on a groupoid
S
. Then
R
is left compatible if and only if [
Χ_{R}, Χ_{R}
] is interval-valued fuzzy left compatible.
Proof
. (⇒) : Suppose
R
is left compatible. Let
a, b, x
∈
S
.
Case(1)
Suppose (
a, b
) ∈
R
. Then
Χ_{R}
(
a, b
) = 1. Since
R
is left compatible, (
xa, xb
) ∈
R
, for each
x
∈
S
. Thus
Χ_{R}
(
xa, xb
) = 1 =
Χ_{R}
(
a, b
).
Case(2)
Suppose ¬(
a, b
) ∈
R
. Then, for each
x
∈
S
, it holds that
Χ_{R}
(
a, b
) = 0 ≤
Χ_{R}
(
xa, xb
). Thus, in either cases, [
Χ_{R}, Χ_{R}
].
(⇐) : Suppose [
Χ_{R}, Χ_{R}
] is interval-valued fuzzy compatible. Let
a, b, x
∈
S
and (
a, b
) ∈
R
. Then, by hypothesis,
Χ_{R}
(
xa, xb
) ≥
Χ_{R}
(
a, b
) = 1. Thus
Χ_{R}
(
xa, xb
) = 1. So (
xa, xb
) ∈
R
. Hence
R
is left compatible.
Lemma 3.5
[The dual of Lemma 3.4]. Let
R
be a relation on a groupoid
S
. Then
R
is right compatible if and only if [
Χ_{R}, Χ_{R}
] is interval-valued fuzzy right compatible.
Definition 3.6
An IVFER
R
on a groupoid
S
is called an :
(1)
interval-valued fuzzy left congruence
(
IVLC
) if it is intervalvalued fuzzy left compatible,
(2)
interval-valued fuzzy right congruence
(
IVRC
) if it is interval-valued fuzzy right compatible,
(3)
interval-valued fuzzy congruence
(
IVC
) if it is intervalvalued fuzzy compatible.
We will denote the set of all IVCs[resp. IVLCs and IVRCs] on
S
as IVC(
S
) [resp: IVLC(
S
) and IVRC(
S
)].
Example 3.7
Let
S
=
e, a, b
be the groupoid defined in Example 3.3. Let
R
_{1}
:
S
×
S
→
D
(
I
) be the mapping defined as the matrix :
Then it can easily be checked that
R
∈ IVE(
S
). Moreover we can see that
R
∈ IVC(
S
).
Proposition 3.8
Let
S
be a groupoid and let
R
∈ IVE(
S
). Then
R
∈ IVC(
S
) if and only if it is both an IVLC and an IVRC.
Proof. (⇒) : Suppose
R
∈ IVC(
S
) and let
x, y, z
∈
S
. Then
and
Also,
and
Thus
R
is both an IVLC and an IVRC.
(⇐) : Suppose
R
is both an IVLC and an IVRC. and let
x, y, z, t
∈
S
. Then
By the similar arguments, we have that
So
R
is interval-valued fuzzy compatible. Hence
R
∈ IVC(
S
).
The following is the immediate result of Remark 2.7(c), Lemmas 3.4 and 3.5, and Proposition 3.5.
Theorem 3.9
Let
R
be a relation on a groupoid
S
. Then
R
∈ C(
S
) if and only if [
Χ_{R}, Χ_{R}
] ∈ IVC(
S
).
For any interval-valued fuzzy left[resp. right] compatible relation
R
, it is obvious that if
G
is a group, then
R
(
x, y
) =
R
(
tx, ty
)[resp:
R
(
x, y
) =
R
(
xt, yt
)], for any
x, y, t
∈
G
. Thus we have following result.
Lemma 3.10
Let
R
be an IVC on a group
G
. Then
for any
a, b, x, y
∈
G
.
Example 3.11
Let
V
be the Klein 4-group with multiplication table :
Let
R
:
V
×
V
→
D
(
I
) be the mapping defined as the matrix :
Then we can see that
R
∈ IVC(
V
). Furthermore, it is easily checked that Lemma 3.10 holds : For any
s, t, x, y
∈
V
,
R
(
xsy, xty
) =
R
(
xs, xt
) =
R
(
sy, ty
) =
R
(
s, t
)
The following is the immediate result of Proposition 3.8 and Lemma 3.10.
Theorem 3.12
Let
R
be an IVFR on a group
G
. Then
R
∈ IVC(
G
) if and only if it is interval-valued fuzzy left(right) compatible equivalence relation.
Lemma 3.13
Let
P
and
Q
be interval-valued fuzzy compatible relations on a groupoid
S
. Then
Q
○
P
is also an interval-valued fuzzy compatible relation on
S
.
Proof.
Let
a, b, x
∈
S
. Then
By the similar arguments, we have that
Thus
and
So
Q
○
P
is interval-valued fuzzy right compatible. Similarly, we can see that
Q
○
P
is interval-valued fuzzy left compatible. Hence
Q
○
P
is interval-valued fuzzy compatible.
Theorem 3.14
Let
P
and
Q
be IVC on a groupoid
S
. Then the following are equivalent :
Proof.
It is obvious that (
a
) ⇒ (
b
) ⇒ (
c
).
(
c
) ⇒ (
d
) : Suppose the condition (c) holds and let
a, b
∈
S
. Then
Similarly, we have that
Hence
Q
○
P
=
P
○
Q
(
d
) ⇒ (
a
) : Suppose the condition (d) holds. Then , by Result 2.C,
Q
○
P
∈ IVE(
S
). Since
P
and
Q
are interval-valued fuzzy compatible, by Lemma 3.13,
Q
○
P
is interval-valued fuzzy compatible. So
Q
○
P
∈ IVC(
S
). This completes the proof.
Proposition 3.15
Let
S
be a groupoid and let
Q, P
∈ IVC(
S
). If
Q
○
P
=
P
○
Q
, then
P
○
Q
∈ IVC(
S
).
Proof.
By Result 2.C, it is clear that
P
○
Q
∈ IVE(
S
). Let
x, y, t
∈
S
. Then, since
P
and
Q
are interval-valued fuzzy right compatible,
Similarly, we have that
By the similar arguments, we have that
and
So
P
○
Q
is interval-valued fuzzy left and right compatible.
Hence
P
○
Q
∈ IVC(
S
).
Let
R
be an IVC on a groupoid
S
and let
a
∈
S
. Then
Ra
∈
D
(
I
)
^{S}
is called an
interval-valued fuzzy congruence class of R containing a
∈
S
and we will denote the set of all interval-valued fuzzy congruence classes of
R
as
S/R
.
Proposition 3.16
If
R
is an IVC on a groupoid
S
, then
Ra
○
Rb
⊂
Rab
, for any
a, b
∈
S
.
Proof.
Let
x
∈
S
. If
x
is not expressible as
x
=
yz
, then clearly (
Ra
○
Rb
)(
x
) = [0, 0]. Thus
Ra
○
Rb
⊂
Rab
. Suppose
x
is expressible as
x
=
yz
. Then
Similarly, we have that
(
Ra
○
Rb
)
^{U}
(
x
) ≤ (
Rab
)
^{U}
(
x
).
Thus
Ra
○
Rb
⊂
Rab
. This completes the proof.
Proposition 3.17
Let
G
be a group with the identity
e
and let
R
∈ IVC(
G
). We define the mapping
A_{R}
:
G
→
D
(
I
) as follows : For each
a
∈
G
,
Then
A_{R}
=
R_{e}
∈ IVNG(
G
).
Proof.
From the definition of
A_{R}
, it is obvious that
A_{R}
∈
D
(
I
)
^{G}
. Let
a, b
∈
G
. Then
Similarly, we have that
On the other hand,
Moreover,
So
A_{R}
∈ IVG(
G
) such that
A_{R}
(
e
) = [1, 1].
Finally,
Hence
A_{R}
∈ IVNG(
G
). This completes the proof.
The following is the immediate result of Proposition 3.17 and Result 2.F.
Proposition 3.18
Let
G
be a group with the identity
e
. If
P,Q
∈ IVNG(
G
), then
Pe
○
Qe
=
Qe
○
Pe
.
Proposition 3.19
Let
G
be a group with the identity
e
. If
R
∈ IVC(
G
), then any interval-valued fuzzy congruence class
Rx
of
x
∈
G
by
R
is an interval-valued fuzzy coset of
Re
. Conversely, each interval-valued fuzzy coset of
Re
is an interval-valued fuzzy congruence class by
R
.
Proof.
Suppose
R
∈ IVC(
G
) and let
x.g
∈
G
. Then
Rx
(
g
) =
R
(
x, g
). Since
R
is interval-valued fuzzy left compatible, by Lemma 3.10,
R
(
x, g
) =
R
(
e, x
^{–1}
g
). Thus
So
Rx
=
xRe
. Hence
Rx
is an interval-valued fuzzy coset of
Re
.
Conversely, let
A
be any interval-valued fuzzy coset of
Re
. Then there exists an
x
∈
G
such that
A
=
xRe
. Let
g
∈
G
.
Then
Since
R
is interval-valued fuzzy left compatible,
So
A
=
Rx
. Hence
A
is an interval-valued fuzzy congruence class of
x
by
R
.
Proposition 3.20
Let
R
be an IVC on a groupoid
S
. We define the binary operation * on
S/R
as follows : For any
a, b
∈
S
,
Then * is well-defined.
Proof.
Suppose
Ra
=
Rx
and
Rb
=
Ry
, where
a, b, x, y
∈
S
. Then, by Result 2.D(
a
),
Thus
Similarly, we have that
Thus
R
(
ab, xy
) = [1, 1]. By Result 2.D(
a
),
Rab
=
Rxy
. So
Ra
*
Rb
=
Rx
*
Ry
. Hence * is well-defined.
From Proposition 3.20 and the definition of semigroup, we obtain the following result.
Theorem 3.21
Let
R
be an IVC on a semigroup
S
. Then (
S/R
, *) is a semigroup.
A semigroup
S
is called an
inverse semigroup
[7]
if each
a
∈
S
has a unique inverse, i.e., there exists a unique
a
^{–1}
∈
S
such that
aa
^{–1}
a
=
a
and
a
^{–1}
=
a
^{–1}
aa
^{–1}
.
Corollary 3.21-1
Let
R
be an IVC on an inverse semigroup
S
. Then (
S/R
, *) is an inverse semigroup.
Proof.
By Theorem 3.21, (
S/R
, *) is a semigroup. Let
a
∈
S
. Since
S
is an inverse semigroup, there exists a unique
a
^{–1}
∈
S
such that
aa
^{–1}
a
=
a
and
a
^{–1}
=
a
^{–1}
aa
^{–1}
. Moreover, it is clear that (
Ra
)
^{–1}
=
Ra
^{–1}
. Then (
Ra
)
^{–1}
*
Ra
* (
Ra
)
^{–1}
=
Ra
^{–1}
*
Ra
*
Ra
^{–1}
=
Ra
^{–1}
aa
^{–1}
=
Ra
^{–1}
and
Ra
* (
Ra
)
^{–1}
*
Ra
=
Ra
*
Ra
^{–1}
*
Ra
=
Raa
^{–1}
a
=
Ra
.
So
Ra
^{–1}
is an inverse of Ra for each
a
∈
S
.
An element
a
of a semigroup
S
is said to be
regular
if
a
∈
aSa
, i.e., there exists an
x
∈
S
such that
a
=
axa
. The semigroup
S
is said to be
regular
if for each
a
∈
S
,
a
is a regular element. Corresponding to a regular element
a
, there exists at least one
á
∈
S
such that
a
=
aáa and á
=
áaá
. Such an
á
is called an
inverse
of
a
.
Corollary 3.21-2
Let
R
be an IVC on a regular semigroup
S
.
Then (
S/R
, *) is a regular semigroup.
Proof.
By Theorem 3.21, (
S/R
, *) is a semigroup. Let
a
∈
S
. Since
S
is a regular semigroup, there exists an
x
∈
S
such that
a
=
axa
. It is obvious that
Rx
∈
S/R
. Moreover,
Ra
*
Rx
*
Ra
=
Raxa
=
Ra
. So
Ra
is an regular element of
S/R
. Hence
S/R
is a regular semigroup.
Corollary 3.21-3
Let
R
be an IVC on a group
G
. Then (
G/R
, *) is a group.
Proof.
By Theorem 3.21, (
G/R
, *) is a semigroup. Let
x
∈
G
. Then
Thus
Re
is the identity in
G/R
with respect to *. Moreover,
So
Rx
^{–1}
is the inverse of
Rx
with respect to *. Hence
G/R
is a group.
Proposition 3.22
Let
G
be a group and let
R
∈ IVC(
G
). We define the mapping
π
:
G/R
→
D
(
I
) as follows : For each
x
∈
G
,
Then
π
∈ IVG(
G/R
).
Proof.
From the definition of
π
, it is clear that
π
= [
π
^{L}
,
π
^{U}
] ∈
D
(
I
)
^{G/R}
. Let
x, y
∈
G
. Then
Similarly, we have that
By the process of the proof of Corollary 3.21-1, (
Rx
)
^{–1}
=
Rx
^{–1}
. Thus
So
π
((
Rx
)
^{–1}
) =
π
(
Rx
) for each
x
∈
G
. Hence
π
∈ IVG(
G/R
).
Proposition 3.23
If
R
is an IVC on an inverse semigroup
S
. Then
R
(
x
^{–1}
,
y
^{–1}
) =
R
(
x, y
) for any
x, y
∈
S
.
Proof.
By Corollary 3.21-1,
S/R
is an inverse semigroup with (
Rx
)
^{–1}
=
Rx
^{–1}
for each
x
∈
S
. Let
x, y
∈
S
. Then
Hence
R
(
x
^{–1}
,
y
^{–1}
) =
R
(
x, y
).
The following is the immediate result of Proposition 3.22
Corollary 3.23
Let
R
be an IVC on a group
G
. Then
for any
x, y
∈
G
.
Proposition 3.24
Let
R
be an IVC on a semigroup
S
. Then
is a congruence on
S
.
Proof.
It is clear that
R
^{–1}
([1, 1]) is reflexive and symmetric. Let (
a, b
), (
b, c
) ∈
R
^{–1}
([1, 1]). Then
R
(
a, b
) =
R
(
b, c
) = [1, 1]. Thus
Similarly, we have that
R^{U}
(
a, c
) ≥ 1. So
R
(
a, c
) = [1, 1], i.e., (
a, c
) ∈
R
^{–1}
([1, 1]). Hence
R
^{–1}
([1, 1]) is an equivalence relation on
S
.
Now let (
a, b
) ∈
R
^{–1}
([1, 1]) and let
x
∈
S
. Since
R
is an IVC on
S
,
R^{L}
(
ax, bx
) ≥
R^{L}
(
a, b
) = 1 and
R^{U}
(
ax, bx
) ≥
R^{U}
(
a, b
) = 1.
Then
R
(
ax, bx
) = [1, 1]. Thus (
ax, bx
) ∈
R
^{–1}
([1, 1]). Similarly, (
xa, xb
) ∈
R
^{–1}
([1, 1]). So
R
^{–1}
([1, 1]) is compatible. Hence
R
^{–1}
([1, 1]) is a congruence on
S
.
Let
S
be a semigroup. Then
S
^{1}
denotes the monoid defined as follows :
Definition 3.25
Let
S
be a semigroup and let
R
∈ IVR(
S
). Then we define a mapping
R
* :
S
×
S
→
D
(
I
) as follows : For any
c, d
∈
S
,
and
It is obvious that
R
* ∈ IVR(
S
).
Proposition 3.26
Let
S
be a semigroup and let
R, P,Q
∈ IVR(
S
). Then :
Proof.
From Definition 3.25, the proofs of (
a
), (
b
) and (
c
) are clear.
(
d
) By (
a
) and (
c
), it is clear that
R
* ⊂ (
R
*)*. Let
c, d
∈
S
. Then
By the similar arguments, we have that
Thus (
R
*)* ⊂
R
*. So (
R
*)* =
R
*.
(
e
) By (
c
),
R
* ⊂ (
P
∪
Q
)* and
Q
* ⊂ (
P
∪
Q
)*. Thus
P
* ∪
Q
* ⊂ (
P
∪
Q
)*. Let
c, d
∈
S
. Then
Similarly, we have that
Thus (
P
∪
Q
)* ⊂
P
* ∪
Q
*. So (
P
∪
Q
)* =
P
* ∪
Q
*.
(
f
) (⇒) : Suppose
R
=
R
* and let
c, d, e
∈
S
. Then
Similarly, we have that
By the similar arguments, we have that
(⇐) : Suppose
R
is interval-valued fuzzy left and right compatible. Let
c, d
∈
S
. Then
Similarly, we have that
Thus
R
* ⊂
R
. So
R
* =
R
. This completes the proof.
Proposition 3.27
If
R
is an IVFR on a semigroup
S
such that is interval-valued fuzzy left and right compatible, then so is
R
^{∞}
.
Proof.
Let
a, b, c
∈
S
and let
n
≥ 1. Then
Similarly, we have that
By the similar arguments, we have that
and
So
R^{n}
is interval-valued fuzzy left and right compatible for each
n
≥ 1. Hence
R
^{∞}
is interval-valued fuzzy left and right compatible.
Let
R
∈ IVR(
S
)and let {
Rα
}
_{α∈Г}
be the family of all IVCs on a semigroup
S
containing
R
. Then the IVFR
defined by
is clearly the least IVC on
S
. In this case,
is called the IVC on
S generated by R
.
Theorem 3.28
If
R
is an IVFR on a semigroup
S
, then
Proof.
By Definition 2.8, (
R
*)
^{e}
∈ IVE(
S
) such that
R
* ⊂ (
R
*)
^{e}
. Then, by Proposition 3.26(
a
),
R
⊂ (
R
*)
^{e}
. Also, by (
a
) and (
b
) of Proposition 3.26
R
* ∪ (
R
*)
^{–1}
∪ △ = (
R
∪
R
^{–1}
∪ △)*. Thus, by Proposition 3.26(
f
) and Result 2.E,
R
* ∪ (
R
*)
^{–1}
∪ △ is left and right compatible. So, by Proposition 3.27, (
R
*)
^{e}
= [
R
* ∪ (
R
*)
^{–1}
∪ △]
^{∞}
is left and right compatible. Hence, by Proposition 3.8, (
R
*)
^{e}
∈ IVC(
S
) . Now suppose
Q
∈ IVC(
S
) such that
R
⊂
Q
. Then, by (
c
) and (
d
) of Proposition 3.26,
R
* ⊂
Q
* =
Q
. Thus (
R
*)
^{e}
⊂
Q
. So
This completes the proof.
f
:
S
→
T
be a semigroup homomorphism. Then it is well-known that the relation
is a congruence on
S
.
The following is the immediate result of Theorem 3.9.
Proposition 4.1
Let
f
:
S
→
T
be a semigroup homomorphism. Then
R
= [
Χ
_{Ker(f)}
,
Χ
_{Ker(f)}
] ∈ IVC(
S
).
In this case,
R
is called the
interval-valued fuzzy kernel
of
f
and denoted by IVK(
f
). In fact, for any
a, b
∈
S
,
Theorem 4.2
(
a
) Let
R
be an interval-valued fuzzy congruence on a semigroup
S
. Then the mapping
π
:
S
→
S/R
defined same as in Result 2.D(
d
) is an epimorphism.
(
b
) If
f
:
S
→
T
is a semigroup homomorphism, then there is a monomorphism
g
:
S
/IVK(
f
) →
T
such that the diagram
commutes, where [IVK(
f
)]
^{#}
denotes the natural mapping.
Proof.
(
a
) Let
a, b
∈
S
. Then, by the definition of
R
^{#}
and Theorem 3.21,
So
π
is a homomorphism. By Result 2.D(
d
),
π
is surjective. Hence
π
is an epimorphism.
(
b
) We define
g
:
S
/IVK(
f
) →
T
by
g
([IFK(
f
)]
a
) =
f
(
a
) for each
a
∈
S
. Suppose [IVK(
f
)]
a
= IVK(
f
)]
b
for any
a, b
∈
S
. Since IVK(
f
)(
a, b
) = [1, 1], i.e.
Χ
_{IVK(f)}
(
a, b
) = 1. Thus (
a, b
) ∈ Ker(
f
). So (
a, b
) ∈
Ker
(
f
). So
g
([IVK(
f
)]
a
) =
f
(
a
) =
f
(
b
) =
g
([IVK(
f
)]
b
). Hence
g
is well-defined.
Suppose
f
(
a
) =
f
(
b
). Then IVK(
f
)(
a, b
) = [1, 1]. Thus, by Result 2.D(
a
), [IVK(
f
)]
a
= IVK(
f
)]
b
. So
g
is injective. Now let
a, b
∈
S
, Then
So
g
is a homomorphism. Let
a
∈
S
. Then
g
([IVK(
f
)]
^{#}
(
a
)) =
g
([IVK(
f
)]
a
) =
f
(
a
). So
g
○ [IVK(
f
)]
^{#}
=
f
. This completes the proof.
Theorem 4.3
Let
R
and
Q
be IVCs on a semigroup such that
R
⊂
Q
. Then there exists a unique semigroup
S
homomorphism
g
:
S/R
→
S/Q
such that the diagram
commutes and (
S/R
)/IVK(
g
) is isomorphic to
S/R
, where
R
^{#}
and
Q
^{#}
denote the natural mappings, respectively.
Proof.
Define
g
:
S/R
→
S/Q
by
g
(
Ra
) =
Qa
for each
a
∈
S
. Suppose
Ra
=
Rb
. Then, by Result 2.D(
a
),
R
(
a, b
) = [1, 1]. Since
R
⊂
Q
,
Then
Q
(
a, b
) = [1, 1]. Thus
Qa
=
Qb
, i.e.,
g
(
Ra
) =
g
(
Rb
). So
g
is well- defined.
Let
a, b
∈
S
. Then
So
g
is a semigroup homomorphism. The remainders of the proofs are easy. This completes the proof.
for the IVC
on
S
generated by
R
for each IVFR
R
on a semigroup
S
(See Theorem 3.28). Finally, for any IVCs
R
and
Q
on a semigroup
S
such that
R
⊂
Q
, there exists a unique semigroup homomorphism
g
:
S/K
→
S/Q
such that (
S
?
R
)/IVK(
g
) is isomorphic to
S/Q
(See Theorem 4.3).
In the future, we will investigate interval-valued fuzzy congruences on a semiring.

Interval-valued fuzzy set
;
Interval-valued fuzzy (normal) subgroup
;
Interval-valued fuzzy congruence

1. Introduction

As a generalization of fuzzy sets introduced by Zadeh
[1]
, Zadeh
[2]
also suggested the concept of interval-valued fuzzy sets. After that time, Biswas
[3]
applied it to group theory, and Gorzalczany
[4]
introduced a method of inference in approximate reasoning by using interval-valued fuzzy sets. Moreover, Mondal and Samanta
[5]
introduced the concept of interval-valued fuzzy topology and investigated some of it’s properties. In particular, Roy and Biswas
[6]
introduced the notion of interval-valued fuzzy relations and studied some of it’s properties. Recently, Jun et al.
[7]
investigated strong semi-openness and strong semicontinuity in interval-valued fuzzy topology. Moreover, Min
[8]
studied characterizations for interval-valued fuzzy m-semicontinuous mappings, Min and Kim
[9
,
10]
investigated intervalvalued fuzzy m*-continuity and m*-open mappings. Hur et al.
[11]
studied interval-valued fuzzy relations in the sense of a lattice theory. Also, Choi et al.
[12]
introduced the concept of interval-valued smooth topological spaces and investigated some of it’s properties.
On the other hand, Cheong and Hur
[13]
, and Lee et al.
[14]
studied interval-valued fuzzy ideals/(generalized)bi-ideals in a semigroup. In particular, Kim and Hur
[15]
investigated interval-valued fuzzy quasi-ideals in a semigroup. Kang
[16]
, Kang and Hur
[17]
applied the notion of interval-valued fuzzy sets to algebra. Jang et al.
[18]
investigated interval-valued fuzzy normal subgroups.
In this paper, we introduce the concept of interval-valued fuzzy congruences on a semigroup
2. Preliminaries

In this section, we list some concepts and well-known results which are needed in later sections.
Let
Lager Image

Lager Image

Lager Image

- (i)A⊂BiffAL≤BLandAU≤BU.
- (ii)A=BiffA⊂BandB⊂A.
- (iii)AC= [1 –AU, 1 –AL].
- (iv)A∪B= [AL˅BL,AU˅BU].

Lager Image

- (v)A∩B= [AL˄BL,AU˄BU].

Lager Image

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- (b)A∪B=B∪A,A∩B=B∩A.
- (c)A∪ (B∪C) = (A∪B) ∪C,
- A∩ (B∩C) = (A∩B) ∩C.
- (d)A,B⊂A∪B,A∩B⊂A,B.

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- (h) (Ac)c=A.

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- We will denote the set of all IVFRs onXas IVR(X).

Lager Image

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- (a) (R1○R2) ○R3=R1○ (R2○R3).
- (b) IfR1⊂R2andQ1⊂Q2, thenR1○Q1⊂R2○Q2.
- In particular, ifQ1⊂Q2, thenR1○Q1⊂R1○Q2.
- (c)R1(R2∪R3) = (R1○R2) ∪ (R1○R3),
- R1(R2∩R3) = (R1○R2) ∩ (R1○R3).

Lager Image

- Ra(x) =R(a, x).

Lager Image

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- ▽(x, y) = [1, 1].

- Re= [R∪R–1∪ △]∞.

Lager Image

Lager Image

- AL≥AL(x) ∧AL(y)

- AU≥AU(x) ∧AU(y).

Lager Image

- A(x–1) ≥A(x),

- AL(x–1) ≥AL(x) andAU(x–1) ≥AU(x).

- Ax:G→D(I)

- xA:G→D(I)

- Ax(g) =A(gx–1) andxA(g) =A(x–1g).

3. Interval-Valued Fuzzy Congruences

- RL(x, y) ≤RL(zx, zy) andRU(x, y) ≤RU(zx, zy),

- RL(x, y) ≤RL(xz, yz) andRU(x, y) ≤RU(xz, yz),

- RL(x, y) ∧RL(z, t) ≤RL(xz, yz)

- RU(x, y∧RU(z, t) ≤RU(xz, yz).

Lager Image

Lager Image

- [λ23,μ23] = [λ32,μ32], [λ22,μ22] = [λ33,μ33],
- [λ11,μ11] ≤ [λ22,μ22],
- [λ12,μ12] ≤ [λ23,μ23] ∧ [λ22,μ22],
- [λ13,μ13] ≤ [λ23,μ23] ∧ [λ22,μ22],
- [λ21,μ21] ≤ [λ23,μ23] ∧ [λ22,μ22],
- [λ31,μ31] ≤ [λ23,μ23] ∧ [λ22,μ22].

Lager Image

- [λ11,μ11] ≤ [λ21,μ21], [λ12,μ12] ≤ [λ31,μ31],
- [λ13,μ13] ≤ [λ31,μ31], [λ21,μ21] ≤ [λ31,μ31],
- [λ32,μ32] ≤ [λ22,μ22],
- [λ33,μ33] ≤ [λ23,μ23] = [λ22,μ22].

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- λ11∧λ12≤λ12,μ11∧μ12≤μ12,λ11∧λ13≤λ13,
- μ11∧μ13≤μ13,λ12∧λ13≤λ12,μ12∧μ13≤μ12,
- λ21∧λ22≤λ32,μ21∧μ22≤μ32,λ21∧λ23≤λ33,
- μ21∧μ23≤μ33,λ22∧λ23≤λ32,μ22∧μ23≤μ32,
- λ31∧λ32≤λ22,μ31∧μ32≤μ22,λ31∧λ33≤λ23,
- μ31∧μ33≤μ23,λ32∧λ33≤λ22,μ32∧μ33≤μ22,

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- RL(x, y) =RL(x, y) ∧RL(z, z) ≤RL(xz, yz)

- RU(x, y) =RU(x, y) ∧RU(z, z) ≤RU(xz, yz).

- RL(x, y) =RL(z, z) ∧RL(x, y) ≤RL(zx, zy)

- RU(x, y) =RU(z, z) ∧RU(x, y) ≤RU(zx, zy).

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- RU(x, y) ∧RU(z, t) ≤RU(xz, yt)

- R(xay, xby) =R(xa, xb) =R(ay, by) =R(a, b),

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- (Q○P)U(ax, bx) ≥PU(a, c) ∧QU(c, b) for eachc∈S:

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- (a)Q○P∈ IVC(S).
- (b)Q○P∈ IVE(S).
- (c)Q○Pis interval-valued fuzzy symmetric.
- (d)Q○P=P○Q

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- (Q○P)U(a, b) = (P○Q)U(a, b).

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- (P○Q)U(x, y) ≤ (P○Q)U(xt, yt).

- (P○Q)L(x, y) ≤ (P○Q)L(tx, ty)

- (P○Q)U(x, y) ≤ (P○Q)U(tx, ty).

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- AR(a) =R(a, e) =Re(a).

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- Rx(g) =R(e, x–1g) =Re(–1g) = (xRe)(g).

- A(g) = (xRe)(g) =Re(x–1g) =R(e, x–1g).

- R(e, x–1g) =R(x, g) =Rx(g).

- Ra*Rb=Rab.

- R(a, x) =R(b, y) = [1, 1].

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- RU(ab, xy) ≥ 1.

- Rx*Re=Rxe=Rx=Rex=Re*Rx.

- Rx*Rx–1=Rxx–1=Re=Rx–1x=Rx–1*Rx.

- π(Rx) = [(Rx)L(e),Rx)U(e)].

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- πU(Rx*Ry) ≥πU(Rx) ∧πU(Ry).

- π((Rx)–1) =π(Rx–1) =R(x–1,e) =R(e, x) =π(Rx).

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- R(x–1,y–1) =R(x, y)

- R–1([1, 1]) = {(a, b) ∈S×S:R(a, b) = [1; 1]}

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- (a)R⊂R*.
- (b) (R*)–1= (R–1)*.
- (c) IfP⊂Q, thenP* ⊂Q*.
- (d) (P*)* =P*.
- (e) (P∪Q)* =P* ∪Q*.
- (f)R=R* if and only ifRis left and right compatible.

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- ((R*)*)U(c, d) ≤ (R*)U(c, d).

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- ((P∪Q)*)U(c, d) ≤ (P*)U(a, b) ∧ (Q)*)U(c, d).

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- RU(ec, ed) ≥RU(c, d).

- RL(ce, de) ≥RL(c, d) andRU(ce, de) ≥RU(c, d).

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- (R*)U(c, d) ≤RU(c, d)

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- (Rn)U(a, b) ≤ (Rn)U(ac, bc).

- (Rn)L(a, b) ≤ (Rn)U(ca, cb)

- (Rn)U(a, b) ≤ (Rn)U(ca, cb).

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4. Homomorphisms

Let
- Ker(f) = {(a, b) ∈S×S:f(a) =f(b)}

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- π(ab) =Rab=Ra*Rb=π(a) *π(b).

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- 1 =RL(a, b) ≤QL(a, b) and 1 =RU(a, b) ≤QU(a, b).

- g(Ra*Rb) =g(Rab) =Qab=Qa*Qb=g(Ra) *g(Rb).

5. Conclusion

Hur et al.
[11]
studied interval-valued fuzzy relations in the sense of a lattice. Cheong and Hur
[13]
, Hur et al.
[14]
, and Kim et al.
[15]
investigated interval-valued fuzzy ideals/(generalized) bi-ideas and quasi-ideals in a semigroup, respectively.
In this paper, we mainly study interval-valued fuzzy congruences on a semigroup. In particular, we obtain the result that
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- Conflict of Interest

No potential conflict of interest relevant to this article was reported.
Acknowledgements

This work was supported by the research grant of theWonkwang University in 2013.

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Citing 'Interval-Valued Fuzzy Congruences on a Semigroup
'

@article{ E1FLA5_2013_v13n3_231}
,title={Interval-Valued Fuzzy Congruences on a Semigroup}
,volume={3}
, url={http://dx.doi.org/10.5391/IJFIS.2013.13.3.231}, DOI={10.5391/IJFIS.2013.13.3.231}
, number= {3}
, journal={International Journal of Fuzzy Logic and Intelligent Systems}
, publisher={Korean Institute of Intelligent Systems}
, author={Lee, Jeong Gon
and
Hur, Kul
and
Lim, Pyung Ki}
, year={2013}
, month={Sep}