Abstract: In this study, the concept of TH-interval-valued fuzzy ring is first introduced based on the notion of fuzzy ring, then some meaningful properties of TH-interval-valued fuzzy ring are investigated. The results show that fuzzy ring and interval-valued fuzzy ring are special cases of TH-interval-valued fuzzy ring.
INTRODUCTION
The research on interval-valued fuzzy set has attracted many authors’ attention. For example, Biswas (1994) first introduced the concept of interval-valued fuzzy set to discuss fuzzy algebra and studied interval-valued fuzzy group. Sun and Gu (1998) investigated the properties of fuzzy algebra based on interval-valued fuzzy set. Wang and Zhou (1997) and Li and Wang (2000) proposed the concept of TH-interval-valued fuzzy group and SH-interval-valued fuzzy group and studied their properties.
In this study, we first introduce the concept of TH-interval-valued fuzzy ring. Then, the corresponding properties of TH-interval-valued fuzzy ring are studied. Through the remarks below, one would find that the results of this study are an interesting and meaningful extension of fuzzy ring and interval -valued fuzzy ring.
PRELIMINARIES
Here, some basic notions and notations of fuzzy set are reviewed, the detailed descriptions could be found by Zadeh (1965), Meng (1993) and Kumar et al. (1992).
Definition 1: Let X be a crisp set. The mapping A: X→[I] is called an interval-valued fuzzy set based on X. We denote by (X) the set of all the fuzzy set on X. For any A∈IF(X), if we define that A(X) = [A(x), A+(x)], A¯(x)≤A+(x), x∈X, then A¯: X→I and A+: X→I are called the lower fuzzy set and the upper fuzzy set of X, respectively.
For any x∈X, A,B∈IF(X), one can define that (A∪B)(x) = A(x)∨B(x), (A∩B)(x) = A(x)∧B(x).
For any A∈IF(X), [λ1, λ2], we have that A[λ1, λ2] = {x∈X|A¯(x)≥λ1, A¯(x)≥λ2}, A[λ1, λ2] is called the [λ1, λ2]- cut sets of A. Obviously, we have that A[λ1, λ2] = A¯λ1∩A+λ2, A[0, 0] = X.
For any A, B∈IF(X) and [λ1, λ2]∈[I], the following two equations are satisfied:
Definition 2: A mapping T : IxI→I is said to be an idempotent norm, if the following conditions satisfy, where a, b, c, d∈I :
(1) |
If a≤c,b≤d then T(a,b)≤T(c,d) |
| (2) | T(a,b) = T(b,a) |
| (3) | T(T(a,b),c) = T(a,T(b,c)) |
| (4) | T(a,0) = 0, T(a,1) = a |
| (5) | T(a,a) = a |
Proposition 1: If T is an idempotent norm, then for any a, b, c, d∈I we have:
T(a,b)∧T(c,d)≥T(a∧c,b∧d) |
Proof: Noting that a≥a∧c, b≥b∧d according to (1) in Definition 2, T(a,b)≥T(a∧c,b∧d) is satisfied; Analogically, T(c,d)≥T(a∧c,b∧d); Thus, T(a,b)≥T(a∧c,b∧d).
Definition 3: A mapping TH : [I]x[I]→[I] is said to
be an idempotent interval norm, if for any
Proposition 2: For any
Proof: If
[T(a¯,b¯), T(a+,b+)]≤[T(c¯,d¯),
T(c+,d+)] |
Thus, we have that TH(
Proposition 3: For any
Proof: Noting that
TH-INTERVAL-VALUED FUZZY RING
Here, we shall introduce the concept of TH-Interval-valued fuzzy ring based on the concept of fuzzy j ring.
Definition 4: Let R be a ring and A is a fuzzy set on R. A is said to be a fuzzy subring of R, if the following two conditions are satisfied.
|
(1) |
For any a, b∈R, A(a+b)≥A(a)∧A(b), A(ab)≥A(a)∧A(b) |
| (2) | For any a∈R, A(-a)≥A(a) |
Definition 5: Let R be a ring and TH an idempotent interval norm. For any A∈IF(X), if the following two conditions are satisfied, then A is said to be a TH-interval-valued fuzzy ring of R:
| (1) |
For any x,y∈R, A(x+y)≥TH(A(x),A(y)) A(xy)≥TH(A(x),A(y)) |
(2) |
For any x∈R, A(-x)≥A(x) |
Theorem 1: Let R be a ring and A an interval-valued fuzzy set of R. Then A¯and A+ are T-type fuzzy ring of R if and only if A is a TH-interval-valued fuzzy ring of R.
Proof: If A is a TH- interval-valued fuzzy ring of R, then, according to Definition 5, for any x,y∈R, we have A(xy)≥TH(A(x),A(y)) that is to say, [A¯(x+y),A+(x+y)]≥TH ([A¯(x),A+(x)], [A¯(y),A+(y)]) = [T(A¯(x),A¯(y)), T(A+(x), A+(y))].
Then, we have that A¯(x+y)≥T(A¯(x),A¯(y)), A+(x+y)≥T(A+(x),A+(y). Thus, A¯ and A+ are T-type fuzzy ring of R. On the contrary, if A(xy)≥TH(A(x), A(y)) then, we have A¯(xy)≥T(A¯(x),A¯(y))A+(xy)≥T(A+(x),A+(y)); if A(-x)≥A(x) then, we have that A¯(-x)≥A¯(-x) and A+(-x)≥A+(-x). Thus, A is a TH-interval-valued fuzzy ring of R.
Theorem 2: Let R be a ring. If A1 and A2 are two interval-valued fuzzy rings of R, then A1∩A2 is a TH interval-valued fuzzy ring of R.
Proof:
| • | For any a,b∈R,we have that: (A1∩A2)(a+b) = A1(a+b)∧A2(a+b): |
≥TH(A1(a),A1(b))∧TH(A2(a),
A2(b)) ≥TH(A1(a),A2(a), A1(b), A2(b)) = TH((A1∩,A2(a), (A1∩A2(b)) |
and (A1∩A2)(ab) = A1(ab)∧A2(ab):
≥TH(A1(a),A1(b))∧TH(A2(a),
A2(b)) ≥TH(A1(a),A2(a), A1(b)∧A2(b)) = TH((A1∩,A2(a), (A1∩A2(b)) |
| • | For any a∈R,we have that: |
| (A1∩A2)(-a) = A1(-a)∧A2(-a)≥A1(a)∧A2(a)
= (A1∩A2)(-a) |
Thus, according to Definition 5, A1∩A2 is a TH interval-valued fuzzy ring of R.
HOMOMORPHISM PROPERTIES
Here, we will discuss the homomorphism properties about the TH interval-valued fuzzy ring.
Definition 6: Let R1 and R2 be two rings and a
mapping φ between R1 and R2 is given as
where, A∈IF(R1), B∈IF(R2),
|
|
Theorem 3: Let R1 and R2 be two rings, φ:
R1→R2 a homomorphism mapping from R1to
R2 and
Proof:
| • | Firstly, if for any yεR1, there is φ-1 (y) = φ, then we can conclude that φ-1 (-y) = φ. In fact, if there exists x0εφ-1 (-y), then there must have φ(x0) = -y. Since n is a homomorphism mapping, we have that φ(-x0) = -φ(x0) = y. Consequently, -x0εφ-1 (-y). This is contradicted with hypotheses. By Definition 4.1, it follows that: |
Secondly, if φ-1 (-y)≠φ, we have that:
| • | Suppose φ: R1→R2 be an epimorphism mapping. If there exist y1, y2∈R2 such that: |
Thus, we can conclude that there exist x1, x2∈R1 satisfying φ(x1) = y1, φ(x2) - y2 and A2 (y1+y2):
Noting that φ is an epimorphism mapping, then we have that φ(x1+x2) = φ(x1)+φ(x2) = y1+y2 such that:
Hence, A1(x2+x1)<TH(A1(x1), A1(x2)).
This is contradicted with the fact that A1 is a TH-interval-valued fuzzy ring of R1. Therefore, for all y1, y2εR2 we have:
| • | If there exist y1, y2εR2 such that: |
then, we can obtain that there exist x1, x2εR1 satisfying φ(x1) = y1, φ(x2) = y2 and:
Noting that φ is an epimorphism mapping, then we have φ(x1 x2) = φ(x1) φ(x2) = y1 y2 such that:
Consequently, A1(x2+x1)<TH(A1(x1), A1(x2)).
This is contradicted with the fact that A1 is a TH- interval-valued fuzzy ring of R1.
Hence, for all y1, y2εR2 we have:
Therefore, according to Definition 3.2, we have that A2 is a TH-interval-valued fuzzy ring of R2.
Theorem 4: Let R1 and R2 be two rings, φ:
R1→R2 a homomorphismz mapping and
Proof: For any x, yεR, we have:
Therefore, according to Definition 5, we conclude that A1 is a TH-interval-valued fuzzy ring of R1.
CONCLUSION
In this paper we have studied the problem of fuzzy ring. We have proposed the notion of TH-interval-valued fuzzy ring. And based on the notion, the corresponding homomorphism properties have been researched and some interesting results have been got. In the future we may involve in the investigation of the isomorphic properties of TH-interval-valued fuzzy ring.
ACKNOWLEDGMENT
The research is supported by Natural Science Foundation of Gannan Normal University under grant No: 10 kyz03 and Natural Science Foundation of Jiangxi Province, under grant No: 20114BAB211021.