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Roughness in rings
Information Sciences 164 (2004) 147–163 www.elsevier.com/locate/ins
Roughness in rings B. Davvaz Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran Received 26 June 2003; received in revised form 29 September 2003; accepted 8 October 2003
Abstract In this paper basic notions of the rough set theory will be given. In fact, the paper concerns a relationship between rough sets and ring theory. We shall introduce the notion of rough subring (resp. ideal) with respect to an ideal of a ring which is an extended notion of a subring (resp. ideal) in a ring, and we shall give some properties of the lower and the upper approximations in a ring. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Rough set; Lower approximation; Upper approximation; Rough subring; Rough ideal
1. Introduction The concept of rough set was originally proposed by Pawlak [18,19] as a formal tool for modelling and processing incomplete information in information systems. Since then the subject has been investigated in many papers (see [11,20,21]). The theory of rough set is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. A key notion in Pawlak rough set model is an equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes which have a non-empty intersection with the set. Some authors, for example,
E-mail address: [email protected] (B. Davvaz). 0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.10.001
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Bonikowaski [2], Iwinski [9], and Pomykala and Pomykala [22] studied algebraic properties of rough sets. The lattice theoretical approach has been suggested by Iwinski [9]. Pomykala and Pomykala [22] showed that the set of rough sets forms a Stone algebra. Comer [3] presented an interesting discussion of rough sets and various algebras related to the study of algebraic logic, such as Stone algebras and relation algebras. It is a natural question to ask what does happen if we substitute an algebraic structure instead of the universe set. Biswas and Nanda [1] introduced the notion of rough subgroups. Kuroki in [12], introduced the notion of a rough ideal in a semigroup. Kuroki and Wang [14] gave some properties of the lower and upper approximations with respect to the normal subgroups. Also, Kuroki and Mordeson in [13] studied the structure of rough sets and rough groups. Jun applied the rough set theory to BCK-algebras [10]. In [4–6] the present author applied the concept of approximation spaces in the theory of algebraic hyperstructures, and in [7] investigated the similarity between rough membership functions and conditional probability. Dubois and Prade [8] began to investigate the problem of fuzzification of a rough set. Fuzzy rough sets are a notion introduced as a further extension of the idea of rough sets. Now, we propose a more general approach to this issue. In this paper basic notions of the rough set theory will be given. In fact, the paper concerns a relationship between rough sets and ring theory. We shall introduce the notion of rough subring (resp. ideal) with respect to an ideal of a ring which is an extended notion of a subring (resp. ideal) in a ring, and we shall give some properties of the lower and the upper approximations in a ring.
2. Rough sets Suppose that U is a non-empty set. A partition or classification of U is a family P of non-empty subsets of U such that each element of U is contained in exactly one element of P. Recall that an equivalence relation h on a set U is a reflexive, symmetric, and transitive binary relation on U . Each partition P induces an equivalence relation h on U by setting xhy () x and
y are in the same class of P:
Conversely, each equivalence relation h on U induces a partition P of U whose classes have the form ½xh ¼ fy 2 U jxhyg. The following notation will be used. Given a non-empty universe U , by PðU Þ we will denote the power-set on U . If h is an eqivalence relation on U then for every x 2 U , ½xh denotes the equivalence class of h determined by x. For any X U , we write X c to denote the complementation of X in U , that is the set U n X .
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Definition 2.1. A pair ðU ; hÞ where U 6¼ ; and h is an equivalence relation on U , is called an approximation space. Definition 2.2. For an approximation space ðU ; hÞ, by a rough approximation in ðU ; hÞ we mean a mapping Apr : PðU Þ ! PðU Þ PðU Þ defined by for every X 2 PðU Þ, AprðX Þ ¼ ðAprðX Þ; AprðX ÞÞ; where AprðX Þ ¼ fx 2 X j½xh X g;
AprðX Þ ¼ fx 2 X j½xh \ X 6¼ ;g:
AprðX Þ is called a lower rough approximation of X in ðU ; hÞ, where as AprðX Þ is called upper rough approximation of X in ðU ; hÞ. Definition 2.3. Given an approximation space ðU ; hÞ, a pair ðA; BÞ 2 PðU Þ PðU Þ is called a rough set in ðU ; hÞ iff ðA; BÞ ¼ AprðX Þ for some X 2 PðU Þ. For the sake of illustration, let ðU ; hÞ is an approximation space, where U ¼ fx1 ; x2 ; . . . ; x8 g and an equivalence relation h with the following equivalence classes: E1 ¼ fx1 ; x4 ; x5 g; E2 ¼ fx2 ; x5 ; x7 g; E3 ¼ fx3 g; E4 ¼ fx6 g: Let X ¼ fx3 ; x5 g, then AprðX Þ ¼ fx3 g and AprðX Þ ¼ fx2 ; x3 ; x5 ; x7 g and so ðfx3 g; fx2 ; x3 ; x5 ; x7 gÞ ¼ AprðX Þ is a rough set. The reader will find in [13,18–23] a deep study of rough set theory. Definition 2.4. A subset X of U is called definable if AprðX Þ ¼ AprðX Þ. If X U is given by a predicate P and x 2 U , then 1. x 2 AprðX Þ means that x certainly has property P , 2. x 2 AprðX Þ means that x possibly has property P , 3. x 2 U n AprðX Þ means that x definitely does not have property P .
Definition 2.5. Let AprðAÞ ¼ ðAprðAÞ, AprðAÞÞ and AprðBÞ ¼ ðAprðBÞ, AprðBÞÞ be any two rough sets in the approximation space ðU ; hÞ. Then
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ii(i) AprðAÞ t AprðBÞ ¼ ðAprðAÞ [ AprðBÞ; AprðAÞ [ AprðBÞÞ, i(ii) AprðAÞ u AprðBÞ ¼ ðAprðAÞ \ AprðBÞ; AprðAÞ \ AprðBÞÞ, (iii) AprðAÞ v AprðBÞ () AprðAÞ u AprðBÞ ¼ AprðAÞ. When AprðAÞ v AprðBÞ, we say that AprðAÞ is a rough subset of AprðBÞ. Thus in the case of rough sets AprðAÞ and AprðBÞ, AprðAÞ v AprðBÞ if and only if AprðAÞ AprðBÞ and AprðAÞ AprðBÞ: This property of rough inclusion has all the properties of set inclusion. The rough complement of AprðAÞ denoted by AprC ðAÞ is defined by AprC ðAÞ ¼ ðU n AprðAÞ; U n AprðAÞÞ: Also, we can define AprðAÞ n AprðBÞ as follows: AprðAÞ n AprðBÞ ¼ AprðAÞ u AprC ðBÞ ¼ ðAprðAÞ n AprðBÞ; AprðAÞ n AprðBÞÞ:
3. Rough ideals Throughout this paper R is a ring. Let I be an ideal of R and X be a nonempty subset of R. Then the sets AprI ðX Þ ¼ fx 2 Rjx þ I X g;
AprI ðX Þ ¼ fx 2 Rjðx þ IÞ \ X 6¼ ;g
are called, respectively, lower and upper approximations of the set X with respect to the ideal I. Let I be an ideal of R; for a, b 2 R we say a is congruent to b mod I, written as a bðmod IÞ if a b 2 I. It is easy to see that the relation a bðmod IÞ is an equivalence relation. Therefore, when U ¼ R and h is the above equivalence relation, then we use the pair ðR; IÞ instead of the approximation space ðU ; hÞ. Also, in this case we use the symbols AprI ðX Þ and AprI ðX Þ instead of AprðX Þ and AprðX Þ. Proposition 3.1. For every approximation space ðR; IÞ and every subsets A, B R, we have: (1) (2) (3) (4)
AprI ðAÞ A AprI ðAÞ; AprI ð;Þ ¼ ; ¼ AprI ð;Þ; AprI ðRÞ ¼ R ¼ AprI ðRÞ; If A B, then AprI ðAÞ AprI ðBÞ and AprI ðAÞ AprI ðBÞ;
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(5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
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AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAÞ ¼ ðAprI ðAC ÞÞC ; AprI ðAÞ ¼ ðAprI ðAC ÞÞC ; AprI ðA \ BÞ ¼ AprI ðAÞ \ AprI ðBÞ; AprI ðA \ BÞ AprI ðAÞ \ AprI ðBÞ; AprI ðA [ BÞ AprI ðAÞ [ AprI ðBÞ; AprI ðA [ BÞ ¼ AprI ðAÞ [ AprI ðBÞ; AprI ðx þ IÞ ¼ AprI ðx þ IÞ for all x 2 R.
Proof. The proof is similar to the [12, Theorem 2.1] and [4, Proposition 4.1]. h The following example shows that the converse of 12 and 13 in Proposition 3.1 is not true. Example 3.2. Let R ¼ Z15 , I ¼ f0; 5; 10g and A ¼ f1; 2; 3; 6; 7; 9; 11; 13g and B ¼ f1; 6; 7; 8; 10; 11g. Then AprI ðAÞ ¼ f1; 6; 11g, AprI ðBÞ ¼ f1; 6; 11g, AprI ðA [ BÞ ¼ f1; 3; 6; 8; 11; 13g, AprI ðAÞ ¼ f1; 2; 3; 4; 6; 7; 8; 9; 11; 12; 13; 14g, AprI ðBÞ ¼ f0; 1; 2; 3; 5; 6; 7; 8; 10; 11; 12; 13g, AprI ðA \ BÞ ¼ f1; 2; 6; 7; 11; 12g, and so AprI ðA [ BÞ 6 AprI ðAÞ [ AprI ðBÞ and AprI ðAÞ \ AprI ðBÞ 6 AprI ðA \ BÞ: By Proposition 3.1 (parts 5, 6, 7, 8 and 15) we immediately get: Corollary 3.3. For every approximation space ðR; IÞ, i(i) for every A R, AprI ðAÞ and AprI ðAÞ are definable sets, (ii) for every x 2 R, x þ I is definable set.
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If A and B are non-empty subsets of R, let AB denote the set of all finite sums fa1 b1 þ a2 b2 þ þ an bn jn 2 N; ai 2 A; bi 2 Bg. Proposition 3.4. Let I be an ideal of R, and A, B non-empty subsets of R, then AprI ðAÞ AprI ðBÞ ¼ AprI ðA BÞ: Pn Proof. Suppose x be any element of AprI ðAÞ AprI ðBÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðAÞ and bi 2 AprI ðBÞ. Hence ðai þ IÞ \ A 6¼ ; and ðbi þ IÞ \ B 6¼ ;, and so there exist xi 2 ðai þ IÞ \ A and yi 2 ðbi þ IÞ \ B for 1 6 i 6 n. Pn Pn Pn Therefore i¼1 xi yi 2 AB and i¼1 xi yi 2 i¼1 ai bi þ I. Thus ðx þ IÞ \ AB 6¼ ; which yields that x 2 AprI ðABÞ, and so AprI ðAÞ AprI ðBÞ AprI ðA BÞ. Conversely, let x 2 AprI ðABÞ,Pthen ðx þ IÞ \ AB 6¼ ;. Hence there exists n y 2 x þ I and y 2 AB, and so y ¼ i¼1 ai bi for some ai 2 A and bi 2 B. Now, we have x2yþI ¼
n X i¼1
ai bi þ I ¼
n X
ðai þ IÞðbi þ IÞ:
i¼1
P Then there exist xi 2 ai þ I and yi 2 bi þ I such that x ¼ ni¼1 xi yi . Since ai 2 ðxi þ IÞ \ A and bi 2 ðyi þ IÞ \ B, we get xi 2 AprI ðAÞ and yi 2 AprI ðBÞ, which yields that x 2 AprI ðAÞ AprI ðBÞ, and so AprI ðABÞ AprI ðAÞ AprI ðBÞ. h Proposition 3.5. Let I be an ideal of R, and A; B non-empty subsets of R, then AprI ðAÞ þ AprI ðBÞ ¼ AprI ðA þ BÞ: Proof. The proof is similar to the proof of Proposition 3.4, by considering the suitable modification by using the definition of A þ B. h Proposition 3.6. Let I be an ideal of R, and A; B non-empty subsets of R, then AprI ðAÞ AprI ðBÞ AprI ðA BÞ: Pn Proof. Suppose x be any element of AprI ðAÞ AprI ðBÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðAÞ and bi 2 AprI ðBÞ. Hence ai þ I A and bi þ I B for Pn Pn 1 6 i 6 n. Since i¼1 ðai þ IÞðbi þ IÞ AB, we get i¼1 ai bi þ I ¼ x þ I AB, and so x 2 AprI ðABÞ.
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The following example shows that the converse of Proposition 3.6 is not true. Example 3.7. Let R ¼ Z12 , I ¼ f0; 4; 8g, A ¼ f0; 2; 4; 6; 10g and B ¼ f0; 1; 4; 8g. Then AprI ðAÞ ¼ f2; 6; 10g, AprI ðBÞ ¼ f0; 4; 8g, AprI ðAÞ AprI ðBÞ ¼ f0; 4; 8g, A B ¼ f0; 2; 4; 6; 8; 10g, AprI ðA BÞ ¼ f0; 2; 4; 6; 8; 10g. Proposition 3.8. Let I be an ideal of R, and A, B non-empty subsets of R, then AprI ðAÞ þ AprI ðBÞ AprI ðA þ BÞ: Proof. The proof is similar to the proof of Proposition 3.6, by considering the suitable modification by using the definition of A þ B. h The following example shows that AprI ðA þ BÞ AprI ðAÞ þ AprI ðBÞ does not hold in general. Example 3.9. Let R ¼ Z12 , I ¼ f0; 6g, A ¼ f0; 1; 2; 5; 6; 8g and B ¼ f0; 3; 4; 6; 9g. Then AprI ðAÞ ¼ f0; 2; 6; 8g, AprI ðBÞ ¼ f0; 3; 6; 9g, AprI ðAÞ þ AprI ðBÞ ¼ f0; 2; 3; 5; 6; 8; 9; 11g, AprI ðA þ BÞ ¼ f0; 2; 3; 4; 5; 6; 8; 9; 10; 11g. Lemma 3.10. Let I, J be two ideals of R such that I J and let A be a non-empty subset of R, then i(i) AprJ ðAÞ AprI ðAÞ, (ii) AprI ðAÞ AprJ ðAÞ. Proof. It is straightforward.
h
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The following corollary follows from Lemma 3.10. Corollary 3.11. Let I, J be two ideals of R and A a non-empty subset of R, then i(i) AprI ðAÞ \ AprJ ðAÞ AprðI\J Þ ðAÞ, (ii) AprðI\J Þ ðAÞ AprI ðAÞ \ AprJ ðAÞ. Proposition 3.12. Let I, J be two ideals of R, then AprI ðJ Þ is an ideal of R. Proof. Suppose a; b 2 AprI ðJ Þ and r 2 R, then ða þ IÞ \ J 6¼ ; and ðb þ IÞ \ J 6¼ ;. So there exist x 2 ða þ IÞ \ J and y 2 ðb þ IÞ \ J . Since J is an ideal of R, we have x y 2 J and x y 2 ða þ IÞ ðb þ IÞ ¼ a b þ I. Hence ða b þ IÞ \ J 6¼ ;, which implies a b 2 AprI ðJ Þ. Also, we have rx 2 J and rx 2 rða þ IÞ ¼ ra þ I. So ðra þ IÞ \ J 6¼ ;, which implies ra 2 AprI ðJ Þ. Therefore AprI ðJ Þ is an ideal of R. h Similarly, if I is an ideal and J is a subring of R, then AprI ðJ Þ is a subring of R. Proposition 3.13. Let I, J be two ideals of R, then AprI ðJ Þ is an ideal of R. Proof. Suppose a; b 2 AprI ðJ Þ and r 2 R, then a þ I J and b þ I J . It is easy to see that ða b þ IÞ J and ðra þ IÞ J . Hence a b 2 AprI ðJ Þ and ra 2 AprI ðJ Þ. h Similarly, if I is an ideal and J is a subring of R, then AprI ðJ Þ is a subring of R. Definition 3.14. Let I be an ideal of R and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a rough set in the approximation space ðR; IÞ. If AprI ðAÞ and AprI ðAÞ are ideals (resp. subrings) of R, then we call AprI ðAÞ a rough ideal (resp. subring). Note that a rough subring also is called a rough ring. Corollary 3.15 i(i) Let I, J be two ideals of R, then AprI ðJ Þ and AprJ ðIÞ are rough ideals. (ii) Let I be an ideal and J is a subring of R, then AprI ðJ Þ is a rough ring. Proposition 3.16. Let I, J be two ideals of R and K a subring of R. Then i(i) AprI ðKÞ AprJ ðKÞ AprðIþJ Þ ðKÞ, (ii) AprI ðKÞ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ.
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Pn
Proof. (i) Suppose x be any element of AprI ðKÞ AprJ ðKÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðKÞ and bi 2 AprJ ðKÞ. Hence ðai þ IÞ \ K 6¼ ; and ðbi þ J Þ \ K 6¼ ;, and so there exist elements xi ; yi 2 R such that xi 2 ðai þ IÞ \ K and yi 2 ðbi þ J Þ \ K. Since K is a subring of R, we have P n i¼1 xi yi 2 K. On the other hand we have n n n X X X xi y i 2 ðai þ IÞðbi þ J Þ ¼ ai bi þ I þ J : i¼1
i¼1
Therefore n X
i¼1
! ai bi þ I þ J
\ K 6¼ ;;
i¼1
Pn which implies i¼1 ai bi 2 AprðIþJ Þ ðKÞ. Pn (ii) Let x be any element of AprI ðKÞ AprJ ðKÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðKÞ and bi 2 AprJ ðKÞ. Therefore ai þ I K and bi þ J K. Now, we have n n X X ðai þ IÞðbi þ J Þ K or ai bi þ I þ J K; i¼1
Pn
i¼1
which yields i¼1 ai bi 2 AprðIþJ Þ ðKÞ. Conversely, since I I þ J and J I þ J , then by Lemma 3.10, we have AprðIþJ Þ ðKÞ AprI ðKÞandAprðIþJ Þ ðKÞ AprJ ðKÞ: Thus AprðIþJ Þ ðKÞ AprðIþJ Þ ðKÞ AprI ðKÞ AprJ ðKÞ. Since AprðIþJ Þ ðKÞ is a subring of R, we obtain AprðIþJ Þ ðKÞ AprI ðKÞ AprJ ðKÞ:
The inclusion symbol in Proposition 3.16 (i) may not be replaced by an equals sign, as the next example shows. Example 3.17. Let R ¼ Z15 , I ¼ f0g, J ¼ f0; 5; 10g and K ¼ f0; 3; 6; 9; 12g. Then AprI ðKÞ ¼ f0; 3; 6; 9; 12g ¼ K
and
AprJ ðKÞ ¼ Z15 :
So AprI ðKÞ AprJ ðKÞ ¼ K Z15 ¼ K, whereas AprðIþJ Þ ðKÞ ¼ Z15 . This shows that AprI ðKÞ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ is not true in general. Proposition 3.18. Let I, J be two ideals of R and K a subring of R. Then AprI ðKÞ þ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ:
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Proof. Since I I þ J and J I þ J , by Lemma 3.10, we get AprðIþJ Þ ðKÞ AprI ðKÞ and AprðIþJ Þ ðKÞ AprJ ðKÞ and so AprðIþJ Þ ðKÞ AprI ðKÞþ AprJ ðKÞ. Now, let x 2 AprI ðKÞ þ AprJ ðKÞ, then x ¼ a þ b for some a 2 AprI ðKÞ and b 2 AprJ ðKÞ. Hence a þ I K and b þ J K. So x þ I þ J ¼ a þ b þ I þ J ¼ a þ I þ b þ J K þ K ¼ K; which yields x 2 AprðIþJ Þ ðKÞ.
h
Proposition 3.19. Let I, J be two ideals of R and K a subring of R. Then AprI ðKÞ þ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ: Proof. Since I I þ J and J I þ J , then by Lemma 3.10, we have AprI ðKÞ AprðIþJ Þ ðKÞ
and
AprJ ðKÞ AprðIþJ Þ ðKÞ:
Therefore AprI ðKÞ þ AprJ ðKÞ AprIþJ ðKÞ. Conversely, let x be an arbitrary element of AprIþJ ðKÞ, then ðx þ I þ J Þ \ K 6¼ ;. So there exists a 2 I such that ðx þ a þ J Þ \ K 6¼ ;, which yields x þ a 2 AprJ ðKÞ. On the other hand, since a 2 I, then ða þ IÞ \ K ¼ I \ K 6¼ ; which implies that a 2 AprI ðKÞ. Now, we have x ¼ a þ ðx þ aÞ 2 AprI ðKÞ þ AprJ ðKÞ:
Now, let R and R0 be two rings and f : R ! R0 a homomorphism from R to R . It is well known, ker f is an ideal of R. 0
Theorem 3.20. Let R and R0 be two rings and f a homomorphism from R to R0 . If A is a non-empty subset of R, then f ðAprker f ðAÞÞ ¼ f ðAÞ: Proof. Since A Aprker f ðAÞ, it follows that f ðAÞ f ðAprker f ðAÞÞ. Conversely, let y 2 f ðAprker f ðAÞÞ. Then there exists an element x 2 Aprker f ðAÞ such that f ðxÞ ¼ y, so we have ðx þ ker f Þ \ A 6¼ ;. Thus there exists an element a 2 ðx þ ker f Þ \ A. Then a ¼ x þ b for some b 2 ker f , that is, x ¼ a b. Then we have y ¼ f ðxÞ ¼ f ða bÞ ¼ f ðaÞ f ðbÞ ¼ f ðaÞ 2 f ðAÞ; and so f ðAprker f ðAÞÞ f ðAÞ. h
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Let f : R ! R0 be a homomorphism and A be a subset of R. Clearly we have f ðAprker f ðAÞÞ f ðAÞ. But, the following example shows that, in general, f ðAprker f ðAÞÞ 6¼ f ðAÞ. Example 3.21. The map f : Z12 ! Z4 given by f ð1Þ ¼ 3 defines a homomorphism. We obtain kerf ¼ f0; 4; 8g. Suppose A ¼ f1; 5; 6; 9g, then f ðAÞ ¼ f2; 3g, Aprker f ðAÞ ¼ f1; 5; 9g and f ðAprker f ðAÞÞ ¼ f3g. The lower and upper approximations can be presented in an equivalent form as follows: Let I be an ideal of R, and A a non-empty subset of R. Then Apr ðAÞ ¼ fa þ I 2 R=Ija þ I Ag; I
AprI ðAÞ ¼ fa þ I 2 R=Ijða þ IÞ \ A 6¼ ;g: Proposition 3.22. Let I, J be two ideals of R, then AprI ðJ Þ is an ideal of R=I. Proof. Assume that a þ I, b þ I 2 AprI ðJ Þ and r þ I 2 R=I. Then ða þ IÞ \ J 6¼ ; and ðb þ IÞ \ J 6¼ ;, so there exist x 2 ða þ IÞ \ J and y 2 ðb þ IÞ \ J . Since J is an ideal of R, we have x y 2 J and rx 2 J . Also, we have x y 2 ða þ IÞ ðb þ IÞ ¼ a b þ I;
rx 2 rða þ IÞ ¼ ra þ I:
Therefore ða b þ IÞ \ J 6¼ ; and ðra þ IÞ \ J 6¼ ;, which imply ða þ IÞ ðb þ IÞ 2 AprI ðJ Þ and ðr þ IÞða þ IÞ 2 AprI ðJ Þ. Therefore AprI ðJ Þ is an ideal of R=I. h Proposition 3.23. Let I, J be two ideals of R, then Apr ðJ Þ is an ideal of R=I. I
Proof. It is straightforward.
h
Similarly, if I is an ideal and J is a subring of R, then Apr ðJ Þ and AprI ðJ Þ I are subrings of R=I. 4. Fuzzy sets and fuzzy rough sets Zadeh in [24] introduced the notion of a fuzzy subset A of a non-empty set U as a membership function lA : U ! ½0; 1 which associates with each point x 2 U its ‘‘degree of membership’’ lA ðxÞ 2 ½0; 1. Let A and B be fuzzy subsets in U . Then (1) A ¼ B iff lA ðxÞ ¼ lB ðxÞ for all x 2 U ; (2) A B iff lA ðxÞ 6 lB ðxÞ for all x 2 U ; (3) C ¼ A [ B iff lC ðxÞ ¼ maxflA ðxÞ; lB ðxÞg for all x 2 U ;
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(4) D ¼ A \ B iff lD ðxÞ ¼ minflA ðxÞ; lB ðxÞg for all x 2 U ; (5) The complement of A denoted by AC , is defined by lAC ðxÞ ¼ 1 lA ðxÞ for all x 2 U . In 1971, Rosenfeld [23] introduced fuzzy sets in the realm of group theory and formulated the concept of a fuzzy subgroup of a group. Since then many researchers are engaged in extending the concepts of abstract algebra to the broader framework of the fuzzy setting. In 1982, Liu [15] defined and studied fuzzy subrings and fuzzy ideals of a ring. A fuzzy subset A of a ring R is called a fuzzy subring of R if, for all x; y 2 R i(i) lA ðx yÞ P minflA ðxÞ; lA ðyÞg, (ii) lA ðxyÞ P minflA ðxÞ; lA ðyÞg. If the condition (ii) is replaced by lA ðxyÞ P maxflA ðxÞ; lA ðyÞg; then A is called a fuzzy ideal of R. The reader will find in [11,16] some basic definitions and results about the fuzzy algebra. Let ðU ; hÞ is an approximation space and AprðX Þ a rough set in ðU ; hÞ. A fuzzy rough set AprðAÞ ¼ ðAprðAÞ; AprðAÞÞ in AprðX Þ is characterised by a pair of maps lAprðAÞ : AprðX Þ ! ½0; 1 and
lAprðAÞ : AprðX Þ ! ½0; 1
with the property that lAprðAÞ ðxÞ 6 lAprðAÞ ðxÞ
for all x 2 AprðX Þ:
Dubois and Prade [8] introduced the problem of fuzzification of a rough set. Also, Nanda and Majumdar in [17] investigated and discussed the concept of fuzzy rough sets. For two fuzzy rough sets AprðAÞ ¼ ðAprðAÞ; AprðAÞÞ and
AprðBÞ ¼ ðAprðBÞ; AprðBÞÞ
in AprðX Þ we define (1) AprðAÞ ¼ AprðBÞ iff lAprðAÞ ðxÞ ¼ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
lAprðAÞ ðxÞ ¼ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
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(2) AprðAÞ AprðBÞ iff lAprðAÞ ðxÞ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
lAprðAÞ ðxÞ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
(3) AprðCÞ ¼ AprðAÞ t AprðBÞ iff lAprðCÞ ðxÞ ¼ maxflAprðAÞ ðxÞ; lAprðBÞ ðxÞg
for all x 2 AprðX Þ;
lAprðCÞ ðxÞ ¼ maxflAprðAÞ ðxÞ; lAprðBÞ ðxÞg
for all x 2 AprðX Þ;
(4) AprðDÞ ¼ AprðAÞ u AprðBÞ iff lAprðDÞ ðxÞ ¼ minflAprðAÞ ðxÞ; lAprðBÞ ðxÞg for all x 2 AprðX Þ; lAprðDÞ ðxÞ ¼ minflAprðAÞ ðxÞ; lAprðBÞ ðxÞg
for all x 2 AprðX Þ;
(5) we define the complement AprC ðAÞ of AprðAÞ by the ordered pair C ðAprC ðAÞ; Apr ðAÞÞ of membership functions where lAprC ðAÞ ðxÞ ¼ 1 lAprðAÞ ðxÞ
for all x 2 AprðX Þ;
lAprC ðAÞ ðxÞ ¼ 1 lAprðAÞ ðxÞ
for all x 2 AprðX Þ:
5. Fuzzy rough ideals Let I be an ideal of R and AprI ðX Þ ¼ ðAprI ðX Þ; AprI ðX ÞÞ a rough ring. The d ðX Þ ¼ Apr ðX Þ n Apr ðX Þ is called the boundary region of X . difference Apr I I I Let AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ be a fuzzy rough set of AprI ðX Þ. We define lApr ðAÞ : AprI ðX Þ ! ½0; 1 as follows: I ( lApr ðAÞ ðxÞ if x 2 AprI ðX Þ; I lApr ðAÞ ðxÞ ¼ I d ðX Þ: 0 if Apr I
Definition 5.1. Let AprI ðX Þ is a rough ring. An interval-valued fuzzy subset A is given by A ¼ fðx; ½lApr ðAÞ ðxÞ; lAprI ðAÞ ðxÞ j x 2 AprI ðX Þg; I
where ðAprI ðAÞ; AprI ðAÞÞ is a fuzzy rough set of AprI ðX Þ. Denote lf A ðxÞ ¼ ½lApr ðAÞ ðxÞ; lAprI ðAÞ ðxÞ for all x 2 AprI ðX Þ, and Dð½0; 1Þ I denotes the family of all closed subintervals of ½0; 1. If lApr ðAÞ ðxÞ ¼ I lAprI ðAÞ ðxÞ ¼ c where 0 6 c 6 1, then we have lf A ðxÞ ¼ ½c; c which we also assume, for the sake of convenience, to belong to Dð½0; 1Þ. Thus lf A ðxÞ 2 Dð½0; 1Þ for all x 2 AprI ðX Þ.
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Definition 5.2. Let D1 ¼ ½a1 ; b1 , D2 ¼ ½a2 ; b2 be elements of Dð½0; 1Þ then we define rmaxðD1 ; D2 Þ ¼ ½a1 _ a2 ; b1 _ b2 ; rminðD1 ; D2 Þ ¼ ½a1 ^ a2 ; b1 ^ b2 : We call D2 6 D1 if and only if a2 6 a1 and b2 6 b1 . We write D1 P D2 if D2 6 D1 . Definition 5.3. Let AprI ðX Þ is a rough ring. A fuzzy rough set AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ in AprI ðX Þ is called a fuzzy rough subring if for all x; y 2 AprI ðX Þ, the following hold: f f i(i) lf A ðx þ yÞ P rminf l A ðxÞ; l A ðyÞg, f f (ii) lf A ðxyÞ P rminf l A ðxÞ; l A ðyÞg. If the condition (ii) is replaced by f f lf A ðxyÞ P rmaxf l A ðxÞ; l A ðyÞg; then A is called a fuzzy rough ideal. In Definition 5.3, if X be a definable set, i.e., AprI ðX Þ ¼ AprI ðX Þ ¼ X , then X is a ring. Suppose lf A ðxÞ ¼ ½lApr ðAÞ ðxÞ; lAprI ðAÞ ðxÞ. If lApr ðAÞ ðxÞ ¼ I I lAprI ðAÞ ðxÞ ¼ c where 0 6 c 6 1, then we have lf A ðxÞ ¼ ½c; c ¼ c. Therefore if we consider AprI ðAÞ ¼ AprI ðAÞ, then Definition 5.3 is the ordinary definition of fuzzy subring and fuzzy ideal, and so a non-empty subset A of X is a subring (ideal) of X if and only if the characteristic function of A is a fuzzy subring (ideal) of X . Lemma 5.4. Let AprI ðX Þ be a rough ring. If AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ and AprI ðBÞ ¼ ðAprI ðBÞ; AprI ðBÞÞ are two fuzzy rough subrings (resp. ideals) of AprI ðX Þ then A \ B is a fuzzy rough subring (resp. ideal) of AprI ðX Þ. Proof. The proof is straightforward.
h
Definition 5.5. Let AprðX Þ be a rough ring and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a fuzzy rough set of AprðX Þ. Then we define At ¼ fx 2 AprI ðX ÞjlApr ðAÞ ðxÞ P tg; I
At ¼ fx 2 AprI ðX ÞjlAprI ðAÞ ðxÞ P tg: ðAt ; At Þ is called a level rough set.
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Theorem 5.6. Let AprI ðX Þ be a rough ring and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a fuzzy rough set of AprI ðX Þ. Then AprI ðAÞ is a fuzzy rough subring of AprI ðX Þ if and only if for every t 2 Im lApr ðAÞ \ Im lAprI ðAÞ , ðAt ; At Þ is a rough subring of I AprI ðX Þ. Proof. Suppose for every 0 6 t 6 1, ðAt ; At Þ is a rough subring of AprI ðX Þ. For every x; y 2 AprI ðX Þ we must prove all of the conditions in Definition 5.3. We f put rminf lf A ðxÞ; l A ðyÞg ¼ ½t0 ; t1 , then minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg ¼ t0 ; I
minflAprI ðAÞ ðxÞ; lAprI ðAÞ ðyÞg ¼ t1 :
I
We have x; y 2 At1 then x y 2 At1 and xy 2 At1 . On the other hand if d ðX Þ or y 2 Apr d ðX Þ, then t0 ¼ 0 and so l x 2 Apr Apr ðAÞ ðx yÞ P 0 ¼ t0 and I I I d d l ðxyÞ P 0 ¼ t0 . If x 62 Apr ðX Þ or y 62 Apr ðX Þ then l ðxÞ ¼ Apr ðAÞ
I
I
Apr ðAÞ
I
I
lApr ðAÞ ðxÞ and lApr ðAÞ ðyÞ ¼ lApr ðAÞ ðyÞ. So t0 ¼ minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg. I
I
I
I
I
Therefore x 2 At0 , y 2 At0 , and so x y 2 At0 and xy 2 At0 which imply lApr ðAÞ ðx yÞ P t0 and lApr ðAÞ ðxyÞ P t0 . Therefore I
I
lf A ðx yÞ ¼ ½lApr ðAÞ ðx yÞ; lAprI ðAÞ ðx yÞ P ½t0 ; t1 ; I
lf A ðxyÞ ¼ ½lApr ðAÞ ðxyÞ; lAprI ðAÞ ðxyÞ P ½t0 ; t1 ; I
thus f f lf A ðx yÞ P rminf l A ðxÞ; l A ðyÞg; f f lf A ðxyÞ P rminf l A ðxÞ; l A ðyÞg: Therefore AprI ðAÞ is a fuzzy rough subring of AprI ðX Þ. Conversely, assume that AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ is a fuzzy rough subring of AprI ðX Þ. We show that for every 0 6 t 6 1, At and At are subrings. For every x; y 2 At , we have lApr ðAÞ ðxÞ P t and lApr ðAÞ ðyÞ P t, so I
I
f rminf lf A ðxÞ; l A ðyÞg P ½t; minflAprI ðAÞ ðxÞ; lAprI ðAÞ ðyÞg P ½t; minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg: I
I
Therefore lf A ðx yÞ P ½t; minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg; I
I
lf A ðxyÞ P ½t; minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg: I
I
Since x; y 2 AprI ðX Þ we have x y 2 AprI ðX Þ and xy 2 AprI ðX Þ and so lApr ðAÞ ðx yÞ P t I
and
lApr ðAÞ ðxyÞ P t: I
Therefore x y 2 At and xy 2 At . Now, let x; y 2 At , then we have
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lAprI ðAÞ ðxÞ P t
and
lAprI ðAÞ ðyÞ P t;
f then rminf lf A ðxÞ; l A ðyÞg P ½0; t, so and lf A ðx yÞg P ½0; t
lf A ðxyÞg P ½0; t:
Hence lAprI ðAÞ ðx yÞ P t
and
lAprI ðAÞ ðxyÞ P t:
Therefore x y 2 At and xy 2 At .
h
Theorem 5.7. Let AprI ðX Þ is a rough ring and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a fuzzy rough set of AprI ðX Þ. Then AprI ðAÞ is a fuzzy rough ideal of AprI ðX Þ if and only if for every 0 6 t 6 1, ðAt ; At Þ is a rough ideal of AprI ðX Þ. Proof. The proof is similar to the proof of Theorem 5.6.
h
Acknowledgements The author would like to thank the anonymous referees for their very constructive comments.
References [1] R. Biswas, S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42 (1994) 251–254. [2] Z. Bonikowaski, Algebraic structures of rough sets, in: W.P. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, Berlin, 1995, pp. 242–247. [3] S.D. Comer, On connections between information systems, rough sets and algebraic logic, in: C. Rauszer (Ed.), Algebraic Methods in Logic and Computer Science, Banach Center Publications 28, Warsaw, 1993, pp. 117–124. [4] B. Davvaz, Rough sets in a fundamental ring, Bull. Iranian Math. Soc. 24 (2) (1998) 49–61. [5] B. Davvaz, Lower and upper approximations in Hv -groups, Ratio Math. 13 (1999) 71–86. [6] B. Davvaz, Approximations in Hv -modules, Taiwanese J. Math. 6 (4) (2002) 499–505. [7] B. Davvaz, Fuzzy sets and probabilistic rough sets, Int. J. Sci. Technol. Univ. Kashan 1 (1) (2000) 23–29. [8] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. General Syst. 17 (2–3) (1990) 191–209. [9] T. Iwinski, Algebraic approach to rough sets, Bull. Polish Acad. Sci. Math. 35 (1987) 673–683. [10] Y.B. Jun, Roughness of ideals in BCK-algebras, Sci. Math. Jpn. 57 (1) (2003) 165–169. [11] R. Kumar, Fuzzy Algebra I, University of Delhi, Publ. Division, 1993. [12] N. Kuroki, Rough ideals in semigroups, Inform. Sci. 100 (1997) 139–163. [13] N. Kuroki, J.N. Mordeson, Structure of rough sets and rough groups, J. Fuzzy Math. 5 (1) (1997) 183–191.
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[14] N. Kuroki, P.P. Wang, The lower and upper approximations in a fuzzy group, Inform. Sci. 90 (1996) 203–220. [15] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst. 8 (1982) 133–139. [16] J.N. Mordeson, M.S. Malik, Fuzzy Commutative Algebra, World Publishing, Singapore, 1998. [17] S. Nanda, S. Majumdar, Fuzzy rough sets, Fuzzy Sets Syst. 45 (1992) 157–160. [18] Z. Pawlak, Rough sets, Int. J. Inf. Comp. Sci. 11 (1982) 341–356. [19] Z. Pawlak, Rough Sets––Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishing, Dordrecht, 1991. [20] L. Polkowski, A. Skowron (Eds.), Rough Sets in Knowledge Discovery, 1. Methodology and Applications, Studies in Fuzziness and Soft Computing, vol. 18, Physical-Verlag, Heidelberg, 1998. [21] L. Polkowski, A. Skowron (Eds.), Rough Sets in Knowledge Discovery, 2. Applications, Studies in Fuzziness and Soft Computing, vol. 19, Physical-Verlag, Heidelberg, 1998. [22] J. Pomykala, J.A. Pomykala, The stone algebra of rough sets, Bull. Polish Acad. Sci. Math. 36 (1988) 495–508. [23] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. [24] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.
Roughness in rings B. Davvaz Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran Received 26 June 2003; received in revised form 29 September 2003; accepted 8 October 2003
Abstract In this paper basic notions of the rough set theory will be given. In fact, the paper concerns a relationship between rough sets and ring theory. We shall introduce the notion of rough subring (resp. ideal) with respect to an ideal of a ring which is an extended notion of a subring (resp. ideal) in a ring, and we shall give some properties of the lower and the upper approximations in a ring. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Rough set; Lower approximation; Upper approximation; Rough subring; Rough ideal
1. Introduction The concept of rough set was originally proposed by Pawlak [18,19] as a formal tool for modelling and processing incomplete information in information systems. Since then the subject has been investigated in many papers (see [11,20,21]). The theory of rough set is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. A key notion in Pawlak rough set model is an equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes which have a non-empty intersection with the set. Some authors, for example,
E-mail address: [email protected] (B. Davvaz). 0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.10.001
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Bonikowaski [2], Iwinski [9], and Pomykala and Pomykala [22] studied algebraic properties of rough sets. The lattice theoretical approach has been suggested by Iwinski [9]. Pomykala and Pomykala [22] showed that the set of rough sets forms a Stone algebra. Comer [3] presented an interesting discussion of rough sets and various algebras related to the study of algebraic logic, such as Stone algebras and relation algebras. It is a natural question to ask what does happen if we substitute an algebraic structure instead of the universe set. Biswas and Nanda [1] introduced the notion of rough subgroups. Kuroki in [12], introduced the notion of a rough ideal in a semigroup. Kuroki and Wang [14] gave some properties of the lower and upper approximations with respect to the normal subgroups. Also, Kuroki and Mordeson in [13] studied the structure of rough sets and rough groups. Jun applied the rough set theory to BCK-algebras [10]. In [4–6] the present author applied the concept of approximation spaces in the theory of algebraic hyperstructures, and in [7] investigated the similarity between rough membership functions and conditional probability. Dubois and Prade [8] began to investigate the problem of fuzzification of a rough set. Fuzzy rough sets are a notion introduced as a further extension of the idea of rough sets. Now, we propose a more general approach to this issue. In this paper basic notions of the rough set theory will be given. In fact, the paper concerns a relationship between rough sets and ring theory. We shall introduce the notion of rough subring (resp. ideal) with respect to an ideal of a ring which is an extended notion of a subring (resp. ideal) in a ring, and we shall give some properties of the lower and the upper approximations in a ring.
2. Rough sets Suppose that U is a non-empty set. A partition or classification of U is a family P of non-empty subsets of U such that each element of U is contained in exactly one element of P. Recall that an equivalence relation h on a set U is a reflexive, symmetric, and transitive binary relation on U . Each partition P induces an equivalence relation h on U by setting xhy () x and
y are in the same class of P:
Conversely, each equivalence relation h on U induces a partition P of U whose classes have the form ½xh ¼ fy 2 U jxhyg. The following notation will be used. Given a non-empty universe U , by PðU Þ we will denote the power-set on U . If h is an eqivalence relation on U then for every x 2 U , ½xh denotes the equivalence class of h determined by x. For any X U , we write X c to denote the complementation of X in U , that is the set U n X .
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Definition 2.1. A pair ðU ; hÞ where U 6¼ ; and h is an equivalence relation on U , is called an approximation space. Definition 2.2. For an approximation space ðU ; hÞ, by a rough approximation in ðU ; hÞ we mean a mapping Apr : PðU Þ ! PðU Þ PðU Þ defined by for every X 2 PðU Þ, AprðX Þ ¼ ðAprðX Þ; AprðX ÞÞ; where AprðX Þ ¼ fx 2 X j½xh X g;
AprðX Þ ¼ fx 2 X j½xh \ X 6¼ ;g:
AprðX Þ is called a lower rough approximation of X in ðU ; hÞ, where as AprðX Þ is called upper rough approximation of X in ðU ; hÞ. Definition 2.3. Given an approximation space ðU ; hÞ, a pair ðA; BÞ 2 PðU Þ PðU Þ is called a rough set in ðU ; hÞ iff ðA; BÞ ¼ AprðX Þ for some X 2 PðU Þ. For the sake of illustration, let ðU ; hÞ is an approximation space, where U ¼ fx1 ; x2 ; . . . ; x8 g and an equivalence relation h with the following equivalence classes: E1 ¼ fx1 ; x4 ; x5 g; E2 ¼ fx2 ; x5 ; x7 g; E3 ¼ fx3 g; E4 ¼ fx6 g: Let X ¼ fx3 ; x5 g, then AprðX Þ ¼ fx3 g and AprðX Þ ¼ fx2 ; x3 ; x5 ; x7 g and so ðfx3 g; fx2 ; x3 ; x5 ; x7 gÞ ¼ AprðX Þ is a rough set. The reader will find in [13,18–23] a deep study of rough set theory. Definition 2.4. A subset X of U is called definable if AprðX Þ ¼ AprðX Þ. If X U is given by a predicate P and x 2 U , then 1. x 2 AprðX Þ means that x certainly has property P , 2. x 2 AprðX Þ means that x possibly has property P , 3. x 2 U n AprðX Þ means that x definitely does not have property P .
Definition 2.5. Let AprðAÞ ¼ ðAprðAÞ, AprðAÞÞ and AprðBÞ ¼ ðAprðBÞ, AprðBÞÞ be any two rough sets in the approximation space ðU ; hÞ. Then
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ii(i) AprðAÞ t AprðBÞ ¼ ðAprðAÞ [ AprðBÞ; AprðAÞ [ AprðBÞÞ, i(ii) AprðAÞ u AprðBÞ ¼ ðAprðAÞ \ AprðBÞ; AprðAÞ \ AprðBÞÞ, (iii) AprðAÞ v AprðBÞ () AprðAÞ u AprðBÞ ¼ AprðAÞ. When AprðAÞ v AprðBÞ, we say that AprðAÞ is a rough subset of AprðBÞ. Thus in the case of rough sets AprðAÞ and AprðBÞ, AprðAÞ v AprðBÞ if and only if AprðAÞ AprðBÞ and AprðAÞ AprðBÞ: This property of rough inclusion has all the properties of set inclusion. The rough complement of AprðAÞ denoted by AprC ðAÞ is defined by AprC ðAÞ ¼ ðU n AprðAÞ; U n AprðAÞÞ: Also, we can define AprðAÞ n AprðBÞ as follows: AprðAÞ n AprðBÞ ¼ AprðAÞ u AprC ðBÞ ¼ ðAprðAÞ n AprðBÞ; AprðAÞ n AprðBÞÞ:
3. Rough ideals Throughout this paper R is a ring. Let I be an ideal of R and X be a nonempty subset of R. Then the sets AprI ðX Þ ¼ fx 2 Rjx þ I X g;
AprI ðX Þ ¼ fx 2 Rjðx þ IÞ \ X 6¼ ;g
are called, respectively, lower and upper approximations of the set X with respect to the ideal I. Let I be an ideal of R; for a, b 2 R we say a is congruent to b mod I, written as a bðmod IÞ if a b 2 I. It is easy to see that the relation a bðmod IÞ is an equivalence relation. Therefore, when U ¼ R and h is the above equivalence relation, then we use the pair ðR; IÞ instead of the approximation space ðU ; hÞ. Also, in this case we use the symbols AprI ðX Þ and AprI ðX Þ instead of AprðX Þ and AprðX Þ. Proposition 3.1. For every approximation space ðR; IÞ and every subsets A, B R, we have: (1) (2) (3) (4)
AprI ðAÞ A AprI ðAÞ; AprI ð;Þ ¼ ; ¼ AprI ð;Þ; AprI ðRÞ ¼ R ¼ AprI ðRÞ; If A B, then AprI ðAÞ AprI ðBÞ and AprI ðAÞ AprI ðBÞ;
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(5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
151
AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAprI ðAÞÞ ¼ AprI ðAÞ; AprI ðAÞ ¼ ðAprI ðAC ÞÞC ; AprI ðAÞ ¼ ðAprI ðAC ÞÞC ; AprI ðA \ BÞ ¼ AprI ðAÞ \ AprI ðBÞ; AprI ðA \ BÞ AprI ðAÞ \ AprI ðBÞ; AprI ðA [ BÞ AprI ðAÞ [ AprI ðBÞ; AprI ðA [ BÞ ¼ AprI ðAÞ [ AprI ðBÞ; AprI ðx þ IÞ ¼ AprI ðx þ IÞ for all x 2 R.
Proof. The proof is similar to the [12, Theorem 2.1] and [4, Proposition 4.1]. h The following example shows that the converse of 12 and 13 in Proposition 3.1 is not true. Example 3.2. Let R ¼ Z15 , I ¼ f0; 5; 10g and A ¼ f1; 2; 3; 6; 7; 9; 11; 13g and B ¼ f1; 6; 7; 8; 10; 11g. Then AprI ðAÞ ¼ f1; 6; 11g, AprI ðBÞ ¼ f1; 6; 11g, AprI ðA [ BÞ ¼ f1; 3; 6; 8; 11; 13g, AprI ðAÞ ¼ f1; 2; 3; 4; 6; 7; 8; 9; 11; 12; 13; 14g, AprI ðBÞ ¼ f0; 1; 2; 3; 5; 6; 7; 8; 10; 11; 12; 13g, AprI ðA \ BÞ ¼ f1; 2; 6; 7; 11; 12g, and so AprI ðA [ BÞ 6 AprI ðAÞ [ AprI ðBÞ and AprI ðAÞ \ AprI ðBÞ 6 AprI ðA \ BÞ: By Proposition 3.1 (parts 5, 6, 7, 8 and 15) we immediately get: Corollary 3.3. For every approximation space ðR; IÞ, i(i) for every A R, AprI ðAÞ and AprI ðAÞ are definable sets, (ii) for every x 2 R, x þ I is definable set.
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If A and B are non-empty subsets of R, let AB denote the set of all finite sums fa1 b1 þ a2 b2 þ þ an bn jn 2 N; ai 2 A; bi 2 Bg. Proposition 3.4. Let I be an ideal of R, and A, B non-empty subsets of R, then AprI ðAÞ AprI ðBÞ ¼ AprI ðA BÞ: Pn Proof. Suppose x be any element of AprI ðAÞ AprI ðBÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðAÞ and bi 2 AprI ðBÞ. Hence ðai þ IÞ \ A 6¼ ; and ðbi þ IÞ \ B 6¼ ;, and so there exist xi 2 ðai þ IÞ \ A and yi 2 ðbi þ IÞ \ B for 1 6 i 6 n. Pn Pn Pn Therefore i¼1 xi yi 2 AB and i¼1 xi yi 2 i¼1 ai bi þ I. Thus ðx þ IÞ \ AB 6¼ ; which yields that x 2 AprI ðABÞ, and so AprI ðAÞ AprI ðBÞ AprI ðA BÞ. Conversely, let x 2 AprI ðABÞ,Pthen ðx þ IÞ \ AB 6¼ ;. Hence there exists n y 2 x þ I and y 2 AB, and so y ¼ i¼1 ai bi for some ai 2 A and bi 2 B. Now, we have x2yþI ¼
n X i¼1
ai bi þ I ¼
n X
ðai þ IÞðbi þ IÞ:
i¼1
P Then there exist xi 2 ai þ I and yi 2 bi þ I such that x ¼ ni¼1 xi yi . Since ai 2 ðxi þ IÞ \ A and bi 2 ðyi þ IÞ \ B, we get xi 2 AprI ðAÞ and yi 2 AprI ðBÞ, which yields that x 2 AprI ðAÞ AprI ðBÞ, and so AprI ðABÞ AprI ðAÞ AprI ðBÞ. h Proposition 3.5. Let I be an ideal of R, and A; B non-empty subsets of R, then AprI ðAÞ þ AprI ðBÞ ¼ AprI ðA þ BÞ: Proof. The proof is similar to the proof of Proposition 3.4, by considering the suitable modification by using the definition of A þ B. h Proposition 3.6. Let I be an ideal of R, and A; B non-empty subsets of R, then AprI ðAÞ AprI ðBÞ AprI ðA BÞ: Pn Proof. Suppose x be any element of AprI ðAÞ AprI ðBÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðAÞ and bi 2 AprI ðBÞ. Hence ai þ I A and bi þ I B for Pn Pn 1 6 i 6 n. Since i¼1 ðai þ IÞðbi þ IÞ AB, we get i¼1 ai bi þ I ¼ x þ I AB, and so x 2 AprI ðABÞ.
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The following example shows that the converse of Proposition 3.6 is not true. Example 3.7. Let R ¼ Z12 , I ¼ f0; 4; 8g, A ¼ f0; 2; 4; 6; 10g and B ¼ f0; 1; 4; 8g. Then AprI ðAÞ ¼ f2; 6; 10g, AprI ðBÞ ¼ f0; 4; 8g, AprI ðAÞ AprI ðBÞ ¼ f0; 4; 8g, A B ¼ f0; 2; 4; 6; 8; 10g, AprI ðA BÞ ¼ f0; 2; 4; 6; 8; 10g. Proposition 3.8. Let I be an ideal of R, and A, B non-empty subsets of R, then AprI ðAÞ þ AprI ðBÞ AprI ðA þ BÞ: Proof. The proof is similar to the proof of Proposition 3.6, by considering the suitable modification by using the definition of A þ B. h The following example shows that AprI ðA þ BÞ AprI ðAÞ þ AprI ðBÞ does not hold in general. Example 3.9. Let R ¼ Z12 , I ¼ f0; 6g, A ¼ f0; 1; 2; 5; 6; 8g and B ¼ f0; 3; 4; 6; 9g. Then AprI ðAÞ ¼ f0; 2; 6; 8g, AprI ðBÞ ¼ f0; 3; 6; 9g, AprI ðAÞ þ AprI ðBÞ ¼ f0; 2; 3; 5; 6; 8; 9; 11g, AprI ðA þ BÞ ¼ f0; 2; 3; 4; 5; 6; 8; 9; 10; 11g. Lemma 3.10. Let I, J be two ideals of R such that I J and let A be a non-empty subset of R, then i(i) AprJ ðAÞ AprI ðAÞ, (ii) AprI ðAÞ AprJ ðAÞ. Proof. It is straightforward.
h
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The following corollary follows from Lemma 3.10. Corollary 3.11. Let I, J be two ideals of R and A a non-empty subset of R, then i(i) AprI ðAÞ \ AprJ ðAÞ AprðI\J Þ ðAÞ, (ii) AprðI\J Þ ðAÞ AprI ðAÞ \ AprJ ðAÞ. Proposition 3.12. Let I, J be two ideals of R, then AprI ðJ Þ is an ideal of R. Proof. Suppose a; b 2 AprI ðJ Þ and r 2 R, then ða þ IÞ \ J 6¼ ; and ðb þ IÞ \ J 6¼ ;. So there exist x 2 ða þ IÞ \ J and y 2 ðb þ IÞ \ J . Since J is an ideal of R, we have x y 2 J and x y 2 ða þ IÞ ðb þ IÞ ¼ a b þ I. Hence ða b þ IÞ \ J 6¼ ;, which implies a b 2 AprI ðJ Þ. Also, we have rx 2 J and rx 2 rða þ IÞ ¼ ra þ I. So ðra þ IÞ \ J 6¼ ;, which implies ra 2 AprI ðJ Þ. Therefore AprI ðJ Þ is an ideal of R. h Similarly, if I is an ideal and J is a subring of R, then AprI ðJ Þ is a subring of R. Proposition 3.13. Let I, J be two ideals of R, then AprI ðJ Þ is an ideal of R. Proof. Suppose a; b 2 AprI ðJ Þ and r 2 R, then a þ I J and b þ I J . It is easy to see that ða b þ IÞ J and ðra þ IÞ J . Hence a b 2 AprI ðJ Þ and ra 2 AprI ðJ Þ. h Similarly, if I is an ideal and J is a subring of R, then AprI ðJ Þ is a subring of R. Definition 3.14. Let I be an ideal of R and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a rough set in the approximation space ðR; IÞ. If AprI ðAÞ and AprI ðAÞ are ideals (resp. subrings) of R, then we call AprI ðAÞ a rough ideal (resp. subring). Note that a rough subring also is called a rough ring. Corollary 3.15 i(i) Let I, J be two ideals of R, then AprI ðJ Þ and AprJ ðIÞ are rough ideals. (ii) Let I be an ideal and J is a subring of R, then AprI ðJ Þ is a rough ring. Proposition 3.16. Let I, J be two ideals of R and K a subring of R. Then i(i) AprI ðKÞ AprJ ðKÞ AprðIþJ Þ ðKÞ, (ii) AprI ðKÞ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ.
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Pn
Proof. (i) Suppose x be any element of AprI ðKÞ AprJ ðKÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðKÞ and bi 2 AprJ ðKÞ. Hence ðai þ IÞ \ K 6¼ ; and ðbi þ J Þ \ K 6¼ ;, and so there exist elements xi ; yi 2 R such that xi 2 ðai þ IÞ \ K and yi 2 ðbi þ J Þ \ K. Since K is a subring of R, we have P n i¼1 xi yi 2 K. On the other hand we have n n n X X X xi y i 2 ðai þ IÞðbi þ J Þ ¼ ai bi þ I þ J : i¼1
i¼1
Therefore n X
i¼1
! ai bi þ I þ J
\ K 6¼ ;;
i¼1
Pn which implies i¼1 ai bi 2 AprðIþJ Þ ðKÞ. Pn (ii) Let x be any element of AprI ðKÞ AprJ ðKÞ. Then x ¼ i¼1 ai bi for some ai 2 AprI ðKÞ and bi 2 AprJ ðKÞ. Therefore ai þ I K and bi þ J K. Now, we have n n X X ðai þ IÞðbi þ J Þ K or ai bi þ I þ J K; i¼1
Pn
i¼1
which yields i¼1 ai bi 2 AprðIþJ Þ ðKÞ. Conversely, since I I þ J and J I þ J , then by Lemma 3.10, we have AprðIþJ Þ ðKÞ AprI ðKÞandAprðIþJ Þ ðKÞ AprJ ðKÞ: Thus AprðIþJ Þ ðKÞ AprðIþJ Þ ðKÞ AprI ðKÞ AprJ ðKÞ. Since AprðIþJ Þ ðKÞ is a subring of R, we obtain AprðIþJ Þ ðKÞ AprI ðKÞ AprJ ðKÞ:
The inclusion symbol in Proposition 3.16 (i) may not be replaced by an equals sign, as the next example shows. Example 3.17. Let R ¼ Z15 , I ¼ f0g, J ¼ f0; 5; 10g and K ¼ f0; 3; 6; 9; 12g. Then AprI ðKÞ ¼ f0; 3; 6; 9; 12g ¼ K
and
AprJ ðKÞ ¼ Z15 :
So AprI ðKÞ AprJ ðKÞ ¼ K Z15 ¼ K, whereas AprðIþJ Þ ðKÞ ¼ Z15 . This shows that AprI ðKÞ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ is not true in general. Proposition 3.18. Let I, J be two ideals of R and K a subring of R. Then AprI ðKÞ þ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ:
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Proof. Since I I þ J and J I þ J , by Lemma 3.10, we get AprðIþJ Þ ðKÞ AprI ðKÞ and AprðIþJ Þ ðKÞ AprJ ðKÞ and so AprðIþJ Þ ðKÞ AprI ðKÞþ AprJ ðKÞ. Now, let x 2 AprI ðKÞ þ AprJ ðKÞ, then x ¼ a þ b for some a 2 AprI ðKÞ and b 2 AprJ ðKÞ. Hence a þ I K and b þ J K. So x þ I þ J ¼ a þ b þ I þ J ¼ a þ I þ b þ J K þ K ¼ K; which yields x 2 AprðIþJ Þ ðKÞ.
h
Proposition 3.19. Let I, J be two ideals of R and K a subring of R. Then AprI ðKÞ þ AprJ ðKÞ ¼ AprðIþJ Þ ðKÞ: Proof. Since I I þ J and J I þ J , then by Lemma 3.10, we have AprI ðKÞ AprðIþJ Þ ðKÞ
and
AprJ ðKÞ AprðIþJ Þ ðKÞ:
Therefore AprI ðKÞ þ AprJ ðKÞ AprIþJ ðKÞ. Conversely, let x be an arbitrary element of AprIþJ ðKÞ, then ðx þ I þ J Þ \ K 6¼ ;. So there exists a 2 I such that ðx þ a þ J Þ \ K 6¼ ;, which yields x þ a 2 AprJ ðKÞ. On the other hand, since a 2 I, then ða þ IÞ \ K ¼ I \ K 6¼ ; which implies that a 2 AprI ðKÞ. Now, we have x ¼ a þ ðx þ aÞ 2 AprI ðKÞ þ AprJ ðKÞ:
Now, let R and R0 be two rings and f : R ! R0 a homomorphism from R to R . It is well known, ker f is an ideal of R. 0
Theorem 3.20. Let R and R0 be two rings and f a homomorphism from R to R0 . If A is a non-empty subset of R, then f ðAprker f ðAÞÞ ¼ f ðAÞ: Proof. Since A Aprker f ðAÞ, it follows that f ðAÞ f ðAprker f ðAÞÞ. Conversely, let y 2 f ðAprker f ðAÞÞ. Then there exists an element x 2 Aprker f ðAÞ such that f ðxÞ ¼ y, so we have ðx þ ker f Þ \ A 6¼ ;. Thus there exists an element a 2 ðx þ ker f Þ \ A. Then a ¼ x þ b for some b 2 ker f , that is, x ¼ a b. Then we have y ¼ f ðxÞ ¼ f ða bÞ ¼ f ðaÞ f ðbÞ ¼ f ðaÞ 2 f ðAÞ; and so f ðAprker f ðAÞÞ f ðAÞ. h
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Let f : R ! R0 be a homomorphism and A be a subset of R. Clearly we have f ðAprker f ðAÞÞ f ðAÞ. But, the following example shows that, in general, f ðAprker f ðAÞÞ 6¼ f ðAÞ. Example 3.21. The map f : Z12 ! Z4 given by f ð1Þ ¼ 3 defines a homomorphism. We obtain kerf ¼ f0; 4; 8g. Suppose A ¼ f1; 5; 6; 9g, then f ðAÞ ¼ f2; 3g, Aprker f ðAÞ ¼ f1; 5; 9g and f ðAprker f ðAÞÞ ¼ f3g. The lower and upper approximations can be presented in an equivalent form as follows: Let I be an ideal of R, and A a non-empty subset of R. Then Apr ðAÞ ¼ fa þ I 2 R=Ija þ I Ag; I
AprI ðAÞ ¼ fa þ I 2 R=Ijða þ IÞ \ A 6¼ ;g: Proposition 3.22. Let I, J be two ideals of R, then AprI ðJ Þ is an ideal of R=I. Proof. Assume that a þ I, b þ I 2 AprI ðJ Þ and r þ I 2 R=I. Then ða þ IÞ \ J 6¼ ; and ðb þ IÞ \ J 6¼ ;, so there exist x 2 ða þ IÞ \ J and y 2 ðb þ IÞ \ J . Since J is an ideal of R, we have x y 2 J and rx 2 J . Also, we have x y 2 ða þ IÞ ðb þ IÞ ¼ a b þ I;
rx 2 rða þ IÞ ¼ ra þ I:
Therefore ða b þ IÞ \ J 6¼ ; and ðra þ IÞ \ J 6¼ ;, which imply ða þ IÞ ðb þ IÞ 2 AprI ðJ Þ and ðr þ IÞða þ IÞ 2 AprI ðJ Þ. Therefore AprI ðJ Þ is an ideal of R=I. h Proposition 3.23. Let I, J be two ideals of R, then Apr ðJ Þ is an ideal of R=I. I
Proof. It is straightforward.
h
Similarly, if I is an ideal and J is a subring of R, then Apr ðJ Þ and AprI ðJ Þ I are subrings of R=I. 4. Fuzzy sets and fuzzy rough sets Zadeh in [24] introduced the notion of a fuzzy subset A of a non-empty set U as a membership function lA : U ! ½0; 1 which associates with each point x 2 U its ‘‘degree of membership’’ lA ðxÞ 2 ½0; 1. Let A and B be fuzzy subsets in U . Then (1) A ¼ B iff lA ðxÞ ¼ lB ðxÞ for all x 2 U ; (2) A B iff lA ðxÞ 6 lB ðxÞ for all x 2 U ; (3) C ¼ A [ B iff lC ðxÞ ¼ maxflA ðxÞ; lB ðxÞg for all x 2 U ;
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(4) D ¼ A \ B iff lD ðxÞ ¼ minflA ðxÞ; lB ðxÞg for all x 2 U ; (5) The complement of A denoted by AC , is defined by lAC ðxÞ ¼ 1 lA ðxÞ for all x 2 U . In 1971, Rosenfeld [23] introduced fuzzy sets in the realm of group theory and formulated the concept of a fuzzy subgroup of a group. Since then many researchers are engaged in extending the concepts of abstract algebra to the broader framework of the fuzzy setting. In 1982, Liu [15] defined and studied fuzzy subrings and fuzzy ideals of a ring. A fuzzy subset A of a ring R is called a fuzzy subring of R if, for all x; y 2 R i(i) lA ðx yÞ P minflA ðxÞ; lA ðyÞg, (ii) lA ðxyÞ P minflA ðxÞ; lA ðyÞg. If the condition (ii) is replaced by lA ðxyÞ P maxflA ðxÞ; lA ðyÞg; then A is called a fuzzy ideal of R. The reader will find in [11,16] some basic definitions and results about the fuzzy algebra. Let ðU ; hÞ is an approximation space and AprðX Þ a rough set in ðU ; hÞ. A fuzzy rough set AprðAÞ ¼ ðAprðAÞ; AprðAÞÞ in AprðX Þ is characterised by a pair of maps lAprðAÞ : AprðX Þ ! ½0; 1 and
lAprðAÞ : AprðX Þ ! ½0; 1
with the property that lAprðAÞ ðxÞ 6 lAprðAÞ ðxÞ
for all x 2 AprðX Þ:
Dubois and Prade [8] introduced the problem of fuzzification of a rough set. Also, Nanda and Majumdar in [17] investigated and discussed the concept of fuzzy rough sets. For two fuzzy rough sets AprðAÞ ¼ ðAprðAÞ; AprðAÞÞ and
AprðBÞ ¼ ðAprðBÞ; AprðBÞÞ
in AprðX Þ we define (1) AprðAÞ ¼ AprðBÞ iff lAprðAÞ ðxÞ ¼ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
lAprðAÞ ðxÞ ¼ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
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(2) AprðAÞ AprðBÞ iff lAprðAÞ ðxÞ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
lAprðAÞ ðxÞ lAprðBÞ ðxÞ
for all x 2 AprðX Þ;
(3) AprðCÞ ¼ AprðAÞ t AprðBÞ iff lAprðCÞ ðxÞ ¼ maxflAprðAÞ ðxÞ; lAprðBÞ ðxÞg
for all x 2 AprðX Þ;
lAprðCÞ ðxÞ ¼ maxflAprðAÞ ðxÞ; lAprðBÞ ðxÞg
for all x 2 AprðX Þ;
(4) AprðDÞ ¼ AprðAÞ u AprðBÞ iff lAprðDÞ ðxÞ ¼ minflAprðAÞ ðxÞ; lAprðBÞ ðxÞg for all x 2 AprðX Þ; lAprðDÞ ðxÞ ¼ minflAprðAÞ ðxÞ; lAprðBÞ ðxÞg
for all x 2 AprðX Þ;
(5) we define the complement AprC ðAÞ of AprðAÞ by the ordered pair C ðAprC ðAÞ; Apr ðAÞÞ of membership functions where lAprC ðAÞ ðxÞ ¼ 1 lAprðAÞ ðxÞ
for all x 2 AprðX Þ;
lAprC ðAÞ ðxÞ ¼ 1 lAprðAÞ ðxÞ
for all x 2 AprðX Þ:
5. Fuzzy rough ideals Let I be an ideal of R and AprI ðX Þ ¼ ðAprI ðX Þ; AprI ðX ÞÞ a rough ring. The d ðX Þ ¼ Apr ðX Þ n Apr ðX Þ is called the boundary region of X . difference Apr I I I Let AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ be a fuzzy rough set of AprI ðX Þ. We define lApr ðAÞ : AprI ðX Þ ! ½0; 1 as follows: I ( lApr ðAÞ ðxÞ if x 2 AprI ðX Þ; I lApr ðAÞ ðxÞ ¼ I d ðX Þ: 0 if Apr I
Definition 5.1. Let AprI ðX Þ is a rough ring. An interval-valued fuzzy subset A is given by A ¼ fðx; ½lApr ðAÞ ðxÞ; lAprI ðAÞ ðxÞ j x 2 AprI ðX Þg; I
where ðAprI ðAÞ; AprI ðAÞÞ is a fuzzy rough set of AprI ðX Þ. Denote lf A ðxÞ ¼ ½lApr ðAÞ ðxÞ; lAprI ðAÞ ðxÞ for all x 2 AprI ðX Þ, and Dð½0; 1Þ I denotes the family of all closed subintervals of ½0; 1. If lApr ðAÞ ðxÞ ¼ I lAprI ðAÞ ðxÞ ¼ c where 0 6 c 6 1, then we have lf A ðxÞ ¼ ½c; c which we also assume, for the sake of convenience, to belong to Dð½0; 1Þ. Thus lf A ðxÞ 2 Dð½0; 1Þ for all x 2 AprI ðX Þ.
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Definition 5.2. Let D1 ¼ ½a1 ; b1 , D2 ¼ ½a2 ; b2 be elements of Dð½0; 1Þ then we define rmaxðD1 ; D2 Þ ¼ ½a1 _ a2 ; b1 _ b2 ; rminðD1 ; D2 Þ ¼ ½a1 ^ a2 ; b1 ^ b2 : We call D2 6 D1 if and only if a2 6 a1 and b2 6 b1 . We write D1 P D2 if D2 6 D1 . Definition 5.3. Let AprI ðX Þ is a rough ring. A fuzzy rough set AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ in AprI ðX Þ is called a fuzzy rough subring if for all x; y 2 AprI ðX Þ, the following hold: f f i(i) lf A ðx þ yÞ P rminf l A ðxÞ; l A ðyÞg, f f (ii) lf A ðxyÞ P rminf l A ðxÞ; l A ðyÞg. If the condition (ii) is replaced by f f lf A ðxyÞ P rmaxf l A ðxÞ; l A ðyÞg; then A is called a fuzzy rough ideal. In Definition 5.3, if X be a definable set, i.e., AprI ðX Þ ¼ AprI ðX Þ ¼ X , then X is a ring. Suppose lf A ðxÞ ¼ ½lApr ðAÞ ðxÞ; lAprI ðAÞ ðxÞ. If lApr ðAÞ ðxÞ ¼ I I lAprI ðAÞ ðxÞ ¼ c where 0 6 c 6 1, then we have lf A ðxÞ ¼ ½c; c ¼ c. Therefore if we consider AprI ðAÞ ¼ AprI ðAÞ, then Definition 5.3 is the ordinary definition of fuzzy subring and fuzzy ideal, and so a non-empty subset A of X is a subring (ideal) of X if and only if the characteristic function of A is a fuzzy subring (ideal) of X . Lemma 5.4. Let AprI ðX Þ be a rough ring. If AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ and AprI ðBÞ ¼ ðAprI ðBÞ; AprI ðBÞÞ are two fuzzy rough subrings (resp. ideals) of AprI ðX Þ then A \ B is a fuzzy rough subring (resp. ideal) of AprI ðX Þ. Proof. The proof is straightforward.
h
Definition 5.5. Let AprðX Þ be a rough ring and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a fuzzy rough set of AprðX Þ. Then we define At ¼ fx 2 AprI ðX ÞjlApr ðAÞ ðxÞ P tg; I
At ¼ fx 2 AprI ðX ÞjlAprI ðAÞ ðxÞ P tg: ðAt ; At Þ is called a level rough set.
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Theorem 5.6. Let AprI ðX Þ be a rough ring and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a fuzzy rough set of AprI ðX Þ. Then AprI ðAÞ is a fuzzy rough subring of AprI ðX Þ if and only if for every t 2 Im lApr ðAÞ \ Im lAprI ðAÞ , ðAt ; At Þ is a rough subring of I AprI ðX Þ. Proof. Suppose for every 0 6 t 6 1, ðAt ; At Þ is a rough subring of AprI ðX Þ. For every x; y 2 AprI ðX Þ we must prove all of the conditions in Definition 5.3. We f put rminf lf A ðxÞ; l A ðyÞg ¼ ½t0 ; t1 , then minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg ¼ t0 ; I
minflAprI ðAÞ ðxÞ; lAprI ðAÞ ðyÞg ¼ t1 :
I
We have x; y 2 At1 then x y 2 At1 and xy 2 At1 . On the other hand if d ðX Þ or y 2 Apr d ðX Þ, then t0 ¼ 0 and so l x 2 Apr Apr ðAÞ ðx yÞ P 0 ¼ t0 and I I I d d l ðxyÞ P 0 ¼ t0 . If x 62 Apr ðX Þ or y 62 Apr ðX Þ then l ðxÞ ¼ Apr ðAÞ
I
I
Apr ðAÞ
I
I
lApr ðAÞ ðxÞ and lApr ðAÞ ðyÞ ¼ lApr ðAÞ ðyÞ. So t0 ¼ minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg. I
I
I
I
I
Therefore x 2 At0 , y 2 At0 , and so x y 2 At0 and xy 2 At0 which imply lApr ðAÞ ðx yÞ P t0 and lApr ðAÞ ðxyÞ P t0 . Therefore I
I
lf A ðx yÞ ¼ ½lApr ðAÞ ðx yÞ; lAprI ðAÞ ðx yÞ P ½t0 ; t1 ; I
lf A ðxyÞ ¼ ½lApr ðAÞ ðxyÞ; lAprI ðAÞ ðxyÞ P ½t0 ; t1 ; I
thus f f lf A ðx yÞ P rminf l A ðxÞ; l A ðyÞg; f f lf A ðxyÞ P rminf l A ðxÞ; l A ðyÞg: Therefore AprI ðAÞ is a fuzzy rough subring of AprI ðX Þ. Conversely, assume that AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ is a fuzzy rough subring of AprI ðX Þ. We show that for every 0 6 t 6 1, At and At are subrings. For every x; y 2 At , we have lApr ðAÞ ðxÞ P t and lApr ðAÞ ðyÞ P t, so I
I
f rminf lf A ðxÞ; l A ðyÞg P ½t; minflAprI ðAÞ ðxÞ; lAprI ðAÞ ðyÞg P ½t; minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg: I
I
Therefore lf A ðx yÞ P ½t; minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg; I
I
lf A ðxyÞ P ½t; minflApr ðAÞ ðxÞ; lApr ðAÞ ðyÞg: I
I
Since x; y 2 AprI ðX Þ we have x y 2 AprI ðX Þ and xy 2 AprI ðX Þ and so lApr ðAÞ ðx yÞ P t I
and
lApr ðAÞ ðxyÞ P t: I
Therefore x y 2 At and xy 2 At . Now, let x; y 2 At , then we have
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B. Davvaz / Information Sciences 164 (2004) 147–163
lAprI ðAÞ ðxÞ P t
and
lAprI ðAÞ ðyÞ P t;
f then rminf lf A ðxÞ; l A ðyÞg P ½0; t, so and lf A ðx yÞg P ½0; t
lf A ðxyÞg P ½0; t:
Hence lAprI ðAÞ ðx yÞ P t
and
lAprI ðAÞ ðxyÞ P t:
Therefore x y 2 At and xy 2 At .
h
Theorem 5.7. Let AprI ðX Þ is a rough ring and AprI ðAÞ ¼ ðAprI ðAÞ; AprI ðAÞÞ a fuzzy rough set of AprI ðX Þ. Then AprI ðAÞ is a fuzzy rough ideal of AprI ðX Þ if and only if for every 0 6 t 6 1, ðAt ; At Þ is a rough ideal of AprI ðX Þ. Proof. The proof is similar to the proof of Theorem 5.6.
h
Acknowledgements The author would like to thank the anonymous referees for their very constructive comments.
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