Generalized -fuzzy rough rings via -fuzzy relational morphisms
© Ekiz et al.; licensee Springer 2013
Received: 14 December 2012
Accepted: 20 May 2013
Published: 3 June 2013
In this paper, we introduce the notion of -fuzzy (full) relational morphisms of rings and investigate some properties of ℐ-lower and -upper fuzzy rough approximations on rings with respect to them.
MSC:20M99, 03E72, 16W20.
The theory of rough sets proposed by Pawlak  is a new mathematical approach for the study of incomplete or imprecise information. The usefulness of rough sets has been demonstrated by some successful applications in many areas such as knowledge discovery, machine learning, data analysis, approximate classification, conflict analysis, and so on. Equivalence relation is a key notion in Pawlak’s rough set model. The equivalence classes are employed to construct lower and upper approximations. Some constructive and algebraic properties of rough sets are investigated by using equivalence relations. However, the requirement of an equivalence relation in Pawlak’s rough set models seems to be a very restrictive condition that may limit the applications. Thus one of the aims of many researchers working on rough set theory has been to generalize this theory. Pawlak’s model has been generalized by use of arbitrary relations in place of equivalence relations, or by use of covering or neighborhood systems or set-valued mappings in place of equivalence classes (see [2–8]).
Fuzzy set theory which was introduced by Zadeh  in 1965 and was generalized by Goguen  in 1967 is another mathematical tool to cope with the vague concepts via grading the turbidity. Dubois and Prade  introduced the problem communicating with the fuzzy sets and the rough sets. Many researchers have dealt with the integration of rough sets and fuzzy sets which are two distinct and complementary theories. For a deep study of fuzzy rough sets, the reader is referred to [11–16].
Biswas and Nanda  applied the notion of rough sets to algebra and introduced the notion of rough subgroups. Kuroki  introduced the notion of a rough ideal in a semigroup. In [19, 20], Davvaz analyzed a relationship between rough sets and ring theory considering a ring as a universal set and introduced the notion of rough ideals and rough subrings with respect to an ideal of a ring. By considering a ring as a universal set, Li et al.  studied -fuzzy rough approximation operators with respect to a -fuzzy ideal of the ring. Li and Yin  generalized lower and upper approximations to ν-lower and -upper fuzzy rough approximations with respect to -congruence L-fuzzy relation on a semigroup. In order to have a more flexible tool for analysis of an information system, recently, Davvaz has studied the concept of generalized rough sets called by him -rough sets . This is another generalization of rough sets. In this type of generalized rough sets, instead of equivalence relations, we require set-valued maps. This technique is useful, where it is difficult to find an equivalence relation among the elements of the universe set. In this generalized rough sets, a set-valued map gives rise to lower and upper generalized approximation operators. Ali et al.  studied some topological properties of the sets which are fixed by these operators. They also studied the degree of accuracy (DAG) for generalized rough sets and some properties of fuzzy sets which are induced by DAG. Davvaz, in , also introduced the concept of set-valued homomorphisms for groups, which is a generalization of an ordinary homomorphism. Yamak et al. [25, 26] investigated some properties of rough approximations with respect to set-valued homomorphisms of rings and modules in the perspective of set-valued homomorphisms. In , Ali et al. initiated the study of roughness in hemirings with respect to the Pawlak approximation space and also with respect to the generalized approximation space.
Recently, Ignjatović et al.  have introduced the notion of a (relational morphism) fuzzy relational morphism which is more general than a (congruence relation) fuzzy congruence relation. Set-valued homomorphisms and relational morphisms are related closely because a set-valued homomorphism defines a relational morphism and vice versa. This paper, in one respect, is an attendance in the sense of fuzzy generalization of the ideas presented in references . This study offers a wider perspective than  in terms of using -fuzzy relational morphisms which are L-fuzzy relations from any ring to any other ring instead of -congruence L-fuzzy relation on a semigroup. In this paper, we investigate some properties of the -fuzzy relational morphisms looking from the side of homomorphisms of rings and designate them to construct an L-fuzzy approximation space and investigate some properties of this L-fuzzy approximation space with respect to rings.
The following definitions and preliminaries are required in the sequel of our work and hence they are presented in brief.
Throughout this paper, unless otherwise indicated, L is referred to as any lattice, and and ℐ are referred to as any t-norm and any implicator on L, respectively.
2.1 L-fuzzy subsets
2.2 L-fuzzy relations
Let X, Y and Z be non-empty sets.
Definition 2.1 An L-fuzzy subset is referred to as an L-fuzzy relation from X to Y, is the degree of relation between x and y, where . If for each , there exists such that , then Θ is referred to as a serial L-fuzzy relation from X to Y. If , then Θ is referred to as an L-fuzzy relation on X; Θ is referred to as a reflexive L-fuzzy relation if for all ; Θ is referred to as a symmetric L-fuzzy relation if for all ; Θ is referred to as a -transitive L-fuzzy relation if for all . Let and be L-fuzzy relations which satisfy the condition for all and . Then is called the inverse L-fuzzy relation of Θ.
The -compositions of the L-fuzzy relations and is an L-fuzzy relation defined by for all (see ).
2.3 Generalized L-fuzzy rough approximation operators
The operators and from to are referred to as -upper and ℐ-lower L-fuzzy rough approximation operators of respectively, and the pair is called the -L-fuzzy rough set of μ with respect to .
In , Wu et al. studied some properties of -upper and ℐ-lower fuzzy rough approximation operators of the generalized approximation space , where is a continuous t-norm and ℐ is a hybrid monotonic implicator on . The same properties for the -upper and the ℐ-lower L-fuzzy rough approximation operators can be obtained.
2.4 -fuzzy subrings
Let be a ring and I be a non-empty subset of R. Then we say that I is a subring whenever then . A subring I of R which satisfies the condition and for all , is said to be an ideal of R. Let R and S be rings. A function is a homomorphism provided and for all .
then μ is called a -fuzzy subring of R. Moreover, a -fuzzy subring μ is called a -fuzzy ideal of R if .
The L-fuzzy sets , and are called the -sum, -product and -product, respectively.
3 -fuzzy relational morphism of rings
Throughout this paper, unless otherwise stated, R, S, and K will be referred to as rings. Recall that a fuzzy relation Θ on a semigroup A is called fuzzy compatible iff for all (see [30, 34]). We will give a more general definition for rings in the light of the reference .
then Θ is called a -fuzzy relational morphism from R to S. Moreover, a -fuzzy relational morphism Θ is said to be full if . The set of all the -fuzzy (full) relational morphisms from R to S is denoted by (respectively, ). It is obvious that and if , then .
Let μ and ν be -fuzzy subrings of R and S, respectively. Let a -fuzzy relation Θ be defined by for all . Then Θ is a -fuzzy relational morphism from R to S. Moreover, if is a ∨-distributive t-norm and μ and ν are -fuzzy ideals of R and S, respectively, then Θ is full.
Let and and be homomorphisms of rings. Then defined by for all is a -fuzzy relational morphism from to . In particular, if Φ is full, then Θ is full.
Let be a homomorphism of rings and μ be a -fuzzy subring of . Then defined by for all is a -fuzzy relational morphism from to .
- (4)Let be a homomorphism of rings and such that . Then defined by
- (5)ℤ is a ring under the usual operations of addition and multiplication. Let L be any lattice with the least element 0 and the greatest element 1. defined by
Let be the lattice, see Figure 2.
is a -fuzzy full relational morphism.
If , then for all and , .
Let and . Then .
Let . Then .
Proof It is straightforward. □
By Proposition 3.3(iii), it is clear that is a monoid.
Theorem 3.4 Let be an infinitely ∨-distributive t-norm. If and , then .
So, is a -fuzzy relational morphism of rings. It is clear to see that if and , then . □
Definition 3.5 Let . Then the kernel and image of Θ (denoted by KerΘ and ImΘ, respectively) are defined to be the L-fuzzy subsets of R and S, respectively that satisfy and for all , .
Then and and .
KerΘ is a -fuzzy subring of R.
If , then KerΘ is a -fuzzy ideal of R.
ImΘ is a -fuzzy subring of S if is an infinitely ∨-distributive t-norm on L.
- (i)Let . Then
If , then it follows immediately from (i).
- (iii)Let . Then
Thus ImΘ is a -fuzzy subring of S.
If , then is a subring of S.
If , then is a subring of R.
If , then is a subring of .
If , then is a subring of .
Proof Let . Then there exist such that and . So, we have . Since , then by Proposition 3.3(i) and . Therefore . Hence is a subring of S. For (ii), the proof is similar to (i). For (iii), it is similar to (i) that is a subring of S. Since , then is a subring of . For (iv), the proof is similar to (iii). □
Definition 3.8 Let μ be a -fuzzy ideal of R. The L-fuzzy subset of R defined by is called a coset of the -fuzzy ideal μ (see ).
Proof It is similar to the proof of Proposition 3.4 in . □
It is straightforward to see that is a ring under these binary operations with additive identity μ, multiplicative identity and .
Theorem 3.10 Let μ be a -fuzzy ideal of R which satisfies and be defined by for all . Then .
So, Θ is a -fuzzy relational morphism of rings. □
and are called the - and -restriction of Θ, respectively (see ).
Theorem 3.11 Let be an infinitely ∨-distributive t-norm. If Θ is a -fuzzy (full) relational morphism from R to S, then and are -fuzzy subrings (ideals) of S and R, respectively.
Proof Since , then it is obvious that is a -fuzzy subgroup of R by Proposition 3.6. Similarly, is -fuzzy subgroup of S since and . Thus if , then it is easy to see and are -fuzzy ideals of S and R, respectively. □
4 ℐ-lower and -upper fuzzy rough approximations on a ring
In this section, we study the properties of ℐ-lower and -upper fuzzy rough approximation operators with respect to a -fuzzy (full) relational morphism of rings.
- (i)Let . Then
It is similar to the proof of (i).
- (iii)Let . Then
The following example shows that replacing ‘=’ by ‘≤’ is not true in general in Theorem 4.1.
Then and . Thus, .
Definition 4.3 Let be a -fuzzy approximation space and μ be an L-fuzzy subset of S. μ is called a ℐ-lower -fuzzy rough ring of S if is a -fuzzy subring of R and μ is called a -upper -fuzzy rough ring of S if is a -fuzzy subring of R. The -L-fuzzy rough set is called a -fuzzy rough ring of μ if both and are -fuzzy subrings of R.
Theorem 4.4 Let be an infinitely ∨-distributive t-norm, and μ be a -fuzzy subring of S. Then μ is a -upper -fuzzy rough ring of S.
Hence we obtain is a -fuzzy subring of R. Thus μ is a -upper -fuzzy rough ring of S. □
Theorem 4.5 Let be an infinitely ∨-distributive t-norm, and μ be a -fuzzy ideal of S. Then is a -fuzzy ideal of R.
So, is a -fuzzy ideal of R in conjunction with Theorem 4.4. □
The following example shows that μ is not a ℐ-lower -fuzzy rough ring of S in general under the condition of Theorem 4.4, and Theorem 4.5 may not be true for the ℐ-lower L-fuzzy rough approximation of μ even if Θ is full.
Since , then is not a -fuzzy subring of ℤ.
The generalized rough sets on algebraic sets such as group, ring and module were mainly studied by a set-valued homomorphism [23, 25, 26] which is a generalization of a congruence relation. In this paper, a definition of -fuzzy (full) relational morphism is considered as a generalization of fuzzy congruence relations (or -congruence L-fuzzy relations) for rings. Then we obtained some new properties of a -fuzzy (full) relational morphism to provide opportunity of putting reasonable interpretations and explored the features of generalized ℐ-lower and -upper fuzzy rough approximations of an L-fuzzy subset on rings. From these points of view, taking a fresh look at the generalized -L-fuzzy rough sets on rings is an interesting research topic. Our further work on this topic will focus on the properties of generalized -L-fuzzy rough sets on modules with respect to the -fuzzy (full) relational morphism.
Dedicated to Professor Hari M Srivastava.
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