T-Rough Sets Based on the Lattices

Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran

http://cjms.journals.umz.ac.ir

ISSN: 1735-0611 CJMS.2(1)(2013), 39-53

T-Rough Sets Based on the Lattices

S.B. Hosseini1 and E. Hosseinpour 2

1 _{Department of Mathematics, Sari Branch, Islamic Azad University,} Sari, Iran.

2 _{Department of Mathematics, Sari Branch, Islamic Azad University,} Sari, Iran.

Abstract.The aim of this paper is to introduce and study set-valued homomorphism on lattices andT-rough lattice with respect to a sublattice. This paper deals with T-rough set approach on the lattice theory. The result of this study contributes to,T-rough fuzzy set and approximation theory and proved in several papers. Keywords: approximation space; lattice; prime ideal; rough ideal;

T-rough set; set-valued homomorphism;T-rough fuzzy ideal

1. INTRODUCTION and PRELIMINARIES

Lattice theory plays an important role in the rough set theory and fuzzy set theory. Various uncertainties in real world applications can bring difficulties in determining the crisp membership functions of fuzzy sets. They involve not only vagueness (lack of sharp class boundaries), but also ambiguity (lack of information). Hence many extensions have been developed to represent these uncertainties in membership values, such as interval-valued fuzzy sets. The notion of rough sets has been in-troduced by Pawlak in his papers [19],. . . ,[26] and Pawlak and Skowron

1_{Corresponding author: sbhosseini@iausari.ac.ir}

[27], [28], [29]. It soon invoked a natural question concerning about pos-sible connection between rough sets and algebraic systems. The alge-braic approach to rough sets have been given and studied by Bonikowaski in [6], Iwinski in [14], Rosenfeild in [30] and W. Zhang, W. Wu in [34]. Banerjee and Pal in [2],Biswas in [3, 4], Biswas and Nanda in [5], Nanda in [18], introduced the notion of rough set and rough subgroups. Kuroki in[16] introduced the notion of rough ideals in a semigroups. Davvaz [8] introduced the notion of rough subring with respect to an ideal of a ring. Dubois and Prade [9] combined fuzzy sets and rough sets in a fruitful way by defining rough fuzzy sets and fuzzy rough sets. Qi-Mei Xiao and Zhen-Liang Zhang in[31, 34] discussed the lower and the upper approximations of prime ideals and of fuzzy prime ideals in a semigroup with details. Davvaz in [7] introducedT-rough set and T-rough homo-morphism in a group. Rough set theory 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. The lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and upper approximation is the union of all the equivalence classes which have a non-empty intersection with the set. It is common that a partition induces an equivalence relation on a set and vice versa. The properties of rough sets can thus be examined via either partition or equivalence classes. Rough sets are a suitable mathematical model of vague concepts, i.e., concepts without sharp boundaries. Hosseini et al.[12, 13] studied some properties ofT-rough set in semigroup and com-mutative rings. In this paper,T-rough set and ideal based on lattice is defined and some properties are given. We attempt to conduct a further study along this line. In particular, We prove some more general and fundamental properties of the generalized rough sets. We discuss the relations between the upper and lowerT-rough prime ideals on lattices and the upper and lower approximations of their homomorphism images and generalize some theorems have been proved.

The following definitions and preliminaries are required in the se-quel of our work and hence presented in brief. Some of them were in [8, 19, 20]. Suppose that U is a non-empty set. A partition or classi-fication of U is a family Θ of non-empty subsets of U such that each element of U is contained in exactly one element of Θ. Recall that an equivalence relationθon a setU is a reflexive, symmetric and transitive binary relation on U. Each partition Θ induces an equivalence relation θon U by setting

Conversely, each equivalence relationθon U induces a partition Θ ofU whose classes have the form

[x]θ={y∈U |xθy}.

Definition 1.1. A pair (U, θ) where U 6= ∅ and θ is an equivalence relation on U is called an approximation space.

Definition 1.2. For an approximation space (U, θ) by a rough approx-imation in (U, θ) we mean a mapping Apr : P(U) −→ P(U)×P(U) defined by for everyX ∈P(U), Apr(X) = (Apr(X), Apr(X)), where

Apr(X) ={x∈U |[x]θ ⊆X}, Apr(X) ={x∈U |[x]θ

\

X 6=∅}. Apr(X) is called a lower rough approximation of X in (U, θ) whereas Apr(X) is called upper rough approximation of X in (U, θ).

Definition 1.3. Given an approximation space (U, θ) a pair (A, B) in P(U)×P(U) is called a rough set in (U, θ) if (A, B) = (Apr(X), Apr(X)) for someX∈P(U).

Definition 1.4. A subset X of U is called definable if Apr(X) = Apr(X). If X⊆U is given by a predicate P and x∈U, then

(1)x∈Apr(X) means thatx certainly has property P, (2)x∈Apr(X) means thatx possibly has property P,

(3)x∈U \Apr(X) means thatx definitely does not have property P.

Proposition 1.5. [14, 18] Let U be a nonempty set and θ be an equiv-alence relation on U. For any subsetsA, B ⊆U, we have

(i) Apr(A)⊆A⊆Apr(A);

(ii) If A⊆B, then Apr(A)⊆Apr(A) and Apr(A)⊆Apr(B); (iii) Apr(A∩B) =Apr(A)∩Apr(B);

(iv) Apr(A)∪Apr(B) =Apr(A∪B).

2. T-ROUGH PRIME IDEAL OF A LATTICE

In this section,we introduce lattices and define the concept of set-valued homomorphism on lattices. some basic properties of generalized lower and upper approximation sets in a lattice are investigated.

Definition 2.1. [11] An order (L,≤) is a lattice if sup{a,b}andinf{a,b}

exist for alla, b∈ L.

Definition 2.3. [7] Let (L,≤) and (K,≤) be two lattices and A ∈

P∗(K) whereP∗(K) denotes the set of all non-empty subsets ofK. Let T : L → P∗(K) be a set-valued mapping. The upper inverse and the lower inverse ofA under T are defined by

T−1(A) ={x∈L|T(x)\A6=∅}; T+(A) ={x∈L|T(x)⊆A}.

Definition 2.4. The pair (T+(A), T−1(A)) is referred to as the gener-alized rough set with respect to A, induced by T or T- rough set with respect toA.

Example 2.5. Let (L, θ) be an approximation space andT :L→P∗(L) be a set-valued mapping where T(x) = [x]θ for all x ∈L, then for any A⊆L, T+(A) =Apr(A) and T−1(A) =Apr(A).

Proposition 2.6. [7] Let L and K be two lattices and A, B ∈P∗(K). Let T :L→P∗(K) be a set-valued mapping. Then the following points hold:

(i)] T−1(AS

B) =T−1(A)S

T−1(B); (ii) T+(AT

B) =T+(A)T

T+(B);

(iii) A⊆B implies T+(A)⊆T+(B) and T−1(A)⊆T−1(B); (iv) T+(A)S

T+(B)⊆T+(AS B); (v) T−1(AT

B)⊆T−1(A)T

T−1(B).

Definition 2.7. [11] (i) A non-empty subset K of L is a sublattice of the lattice (L,W

,V

) if a∨b, a∧b∈K for alla, b∈K.

(ii) Let L be a lattice and I be a nonempty subset ofL. I is called an ideal of L, ifx∧a, a∨b∈I for all x∈L anda, b∈I.

(iii) A proper idealP ofLis called a prime ideal, ifa, b∈Landa∧b∈P implies thata∈P orb∈P.

Definition 2.8. [1] If A and B are non-empty subsets of L, we define AV

B and AW

B as follows:

A^B ={a∧b|a∈A, b∈B}; A_B ={a∨b|a∈A, b∈B}.

Definition 2.9. Let Land K be two lattices andT :L→P∗(K) be a set-valued mapping. T is called a set-valued homomorphism if

(i) T(x∧y) =T(x)V T(y); (ii) T(x∨y) =T(x)W

T(y); for all x, y∈L.

Proof. Let x, y ∈ T+(S) , by Definition 2.3 ,T(x), T(y) ⊆S. Since S is a sublattice of K, we have

T(x∨y) =T(x)_T(y)⊆S and T(x∧y) =T(x)^T(y)⊆S. It shows thatx∨y, x∧y∈T+(S).

Moreover, let x, y ∈ T−1_{(S), by Definition 2.3, T(x)}T

S 6= ∅ and T(y)T

S 6= ∅. Suppose a ∈ T(x)T

S and b ∈ T(y)T

S. Since S is a sublattice ofK, we havea∨b∈S anda∨b∈T(x)W

T(y) =T(x∨y). It implies that a∨b∈T(x∨y)T

S. HenceT(x∨y)T

S 6=∅. It means that x∨y ∈ T−1(S). Again, a∧b ∈ S and a∧b ∈ T(x)V

T(y). So that a∧b∈ T(x∧y)T

S. ThereforeT(x∧y)T

S 6=∅. It means that x∧y∈T−1(S).

Corollary 2.11. Let L and K be two lattices and T :L→P∗(K) be a set-valued homomorphism. If S is a sublattice of K and T+(S) 6=∅ 6= T−1(S), then (T+(S), T−1(S))is a T-rough sublattice ofL.

Proposition 2.12. Let L andK be two lattices andT :L→P∗(K) be a set-valued homomorphism. If A, B be non-empty subsets ofK , then (1) T+(A)W

T+(B)⊆T+(AW B); (2) T+(A)V

T+(B)⊆T+(AV B).

Proof. (1). Suppose z be any element of T+(A)W

T+(B). Then z=a∨b for somea∈T+(A) and b∈T+(B). By definition, T(a)⊆A and T(b)⊆B. Since

T(a∨b) =T(a)_T(b) ={x∨y|x∈T(a), y ∈T(b)} ⊆ {x∨y|x∈A, y∈B}=A_B, we imply thata∨b∈T+(AW

B) and soz∈T+(AW B). (2).The proof is similar to the proof of (1).

The following examples show that the converse of above proposition is not true.

Example 2.13. (1). Let L = {0,1,2, ...,8}. Let a∨b = max{a, b}

and a∧ b = min{a, b} for all a, b ∈ L . Then (L,∨,∧) is a lat-tice. If we consider equivalence classes [0] = {0,1,2}, [3] = {3,4}, [5] ={5,6,7,8}andT :L→P∗(L) be a set-valued homomorphism with T(x) = [x] for all x∈L. LetA={3,4,5,7},B ={0,1,2,3,6,8}. Then AW

B ={3,4,5,6,7,8},T+(AW

B) ={3,4,5,6,7,8},T+(A) ={3,4}, T+(B) ={0,1,2} and T+(A)W

T+(B) = {3,4}. And soT+(AW B)* T+(A)W

T+(B).

withT(x) = [0, x] for all x∈L. And letA={0,1₂},B ={1 3,

1

2}. Then T+(A) = {0}, T+(B) = ∅, T+(AV

B) = {0}, T+(A)V

T+(B) = ∅. ThereforeT+(AV

B)*T+(A)VT+(B).

Proposition 2.14. Let L andK be two lattices andT :L→P∗(K) be a set-valued homomorphism. If A, B be non-empty subsets ofK, then (1) T−1(A)W

T−1(B)⊆T−1(AW B); (2) T−1(A)V

T−1(B)⊆T−1(AV B).

Proof. (1). Let z ∈ T−1(A)W

T−1(B). Then z = a∨b for some a∈T−1(A) andb∈T−1(B). HenceT(a)T

A6=∅andT(b)T

B 6=∅and so there existx∈T(a)T

A andy∈T(b)T

B. Thereforex∨y∈AW B and x∨y ∈T(a)W

T(b) = T(a∨b). Thus x∨y ∈ T(a∨b)T (AW

B) which implies that T(a∨b)T

(AW

B)6=∅. Soz=a∨b∈T−1(AW B). (2). The proof is similar to the proof of (1).

The following examples show that the converse of above noted proposi-tion is not true.

Example 2.15. (i) LetL={0, x1, x2, x3, x4, x5,1}be the following lat-tice and T : L → P∗(L) be a set-valued homomorphism with T(x) =

{x5} for all x ∈ L. And let A = {x2, x3} , B = {x1, x2}. Then T−1(A) = ∅, T−1(B) = ∅, T−1(AW

B) = L, T−1(A)W

T−1(B) = ∅. Therefore

T−1(AW

B)*T−1(A)WT−1(B).

@ @ @ I 6 0

x1 @x2 _x3

@ @ I

6 6

x4 x5

@ @ @ I 1

(ii) Let L be the above noted lattice and T : L → P∗(L) be a set-valued homomorphism with T(x) = {x2} for all x ∈ L. And let A =

{x4} , B = {x5}. Then T−1(A) = ∅, T−1(B) = ∅, T−1(AV

B) = L, T−1(A)V

T−1(B) =∅. Therefore T−1(AV

B)*T−1(A)VT−1(B).

Lemma 2.16. Let L and K be two lattices and T : L → P∗(K) be a set-valued homomorphism. If I is an ideal of K and T+(I) 6= ∅, then T+(I) is an ideal of L.

T(x)W

T(y)⊆I.Thereforex∨y∈T+(I) andT(r∧x) =T(r)V

T(x)⊆

I.Thus r∧x∈T+(I). Hence T+(I) is an ideal ofL.

Lemma 2.17. Let L and K be two lattices and T : L → P∗(K) be a set-valued homomorphism. If I is an ideal of K and T−1_{(I) 6=∅, then} T−1(I) is an ideal ofL.

Proof. Let x, y ∈T−1(I). By Definition 2.3 provided, T(x)T I 6=∅

and T(y)T

I 6= ∅. Let a ∈ T(x)T

I and b ∈ T(y)T

I. Since I is an ideal ofK, thereforea∨b∈I anda∨b∈T(x)W

T(y) =T(x∨y). Thus a∨b∈T(x∨y)T

I. HenceT(x∨y)T

I 6=∅ and so x∨y∈T−1(I). Now letr∈Landx∈T−1(I). By Definition 2.3,we haveT(x)T

I 6=∅. Leta∈T(x)T

I. Since I is an ideal of K, thereforeT(r)V

a⊆I and T(r)^a⊆T(r)^T(x) =T(r∧x).

Thus T(r∧x)T

I 6=∅ and so r∧x ∈ T−1(I). Therefore T−1(I) is an ideal of L.

Corollary 2.18. Let L and K be two lattices and T : L → P∗(K) be a set-valued homomorphism. If I is an ideal of K and T+(I) 6=∅ and T−1(I)6=∅, then (T+(I), T−1(I)) is aT-rough ideal of L.

Theorem 2.19. Let L and K be two lattices and T :L→ P∗(K) be a set-valued homomorphism. IfP is a prime ideal ofK andL6=T+(P)6=

∅, then T+(P) is a prime ideal of L.

Proof. Let x, y∈L and x∧y ∈T+(P). By Definition 2.3,we have T(x)V

T(y) =T(x∧y) ⊆ P. And so, for anya ∈ T(x) and b ∈T(y), a∧b∈P. Now we show thatT(x)⊆P orT(y)⊆P. SupposeT(x)*P and T(y) * P. Then there is a ∈ T(x) such that a /∈ P and there is b∈T(y) such thatb /∈P. SinceP is a prime ideal ofK, we deduce that a∧b /∈P as which is a contradiction. Hence T(x)⊆P orT(y)⊆P. It means thatx∈T+(P) or y∈T+(P).

Theorem 2.20. Let L and K be two lattices and T : L → P∗(K) be a set-valued homomorphism. If P is a prime ideal of K and L 6= T−1(P)6=∅ , then T−1(P) is a prime ideal of L .

Proof. Letx, y∈Land x∧y∈T−1(P), then T(x∧y)T

P 6=∅. On the other hand, {a∧b |a ∈T(x), b ∈ T(y)} = T(x∧y). So there are a∈T(x), b∈T(y) such that a∧b∈P. Since P is a prime ideal of K, we havea∈P orb∈P. Hencea∈T(x)T

P orb∈T(y)T

P. It means thatT(x)T

P 6=∅ orT(y)T

P 6=∅. Hence x∈T−1(P) or y ∈T−1(P). ThereforeT−1(P) is a prime ideal ofL.

∅andL6=T−1(P)6=∅, Then(T+(P), T−1(P))is aT-rough prime ideal of L.

3. T-ROUGH QUOTIENT IDEAL IN LATTICES

In this section, we defineT-rough quotient sets with respect to a set-valued homomorphism and investigate some their properties. LetLand K be two lattices and T :L→P∗(K) be a set-valued homomorphism . Let us set _TL ={T(x)|x∈L}. It is clear that L_T is a lattice.

Definition 3.1. Let Land K be two lattices andT :L→P∗(K) be a set-valued homomorphism. The lower T-rough quotient and the upper T-rough quotient set with respect toA∈P∗(K) are

T+(A)

T ={T(x)|T(x)⊆A};

T−1(A)

T ={T(x)|T(x)∩A6=∅}.

Lemma 3.2. Let Land K be two lattices and T :L→P∗(K) be a set-valued homomorphism. IfA∈P∗(K) be an ideal of K and T+_T(A) 6=∅ , then T+_T(A) is an ideal of L_T.

Proof. Let T(x), T(y) ∈ T+_T(A). Since A ∈ P∗(K) is an ideal of K, soT(x∨y) =T(x)W

T(y) ⊆A. Therefore T(x)W

T(y) ∈ T+_T(A). Now supposeT(r)∈ _TLandT(x)∈ T+_T(A), By Definition 3.1,T(x)⊆A. Since A is an ideal of K, thus T(x∧r) = T(x)V

T(r) ⊆ A. It means that T(x)V

T(r)∈ T+_T(A). Therefore T+_T(A) is an ideal of L_T.

Lemma 3.3. Let Land K be two lattices and T :L→P∗(K) be a set-valued homomorphism. IfA∈P∗(K) be an ideal of K and T−_T1(A) 6=∅, then T−_T1(A) is an ideal of L_T.

Proof. Suppose T(x), T(y) ∈ T−_T1(A). By Definition 3.1, we have T(x)T

A 6=∅ and T(y)T

A 6= ∅. Let a∈ T(x)T

A and b∈ T(y)T A. SinceA∈P∗(K) is an ideal ofK, soa∨b∈T(x)W

T(y) =T(x∨y) and a∨b∈A. Thereforea∨b∈T(x∨y)T

A. It means thatT(x)W

T(y)∈

T−1_(A)

T . Now suppose T(r) ∈ L

T and T(x) ∈

T−1_(A)

T , By Definition 3.1 provided, T(x)T

A 6= ∅. Let a ∈ T(x)T

A. Since A is an ideal of K, T(r)V

a⊂Aand

T(r)^a⊆T(r)^T(x) =T(x∧r). ThereforeT(x∧r)T

A6=∅. It meas thatT(x)V

T(r)∈ T−1_T(A).

Corollary 3.4. Let L and K be two lattices and T :L→ P∗(K) be a set-valued homomorphism. IfA∈P∗(K)is an ideal ofKand T+_T(A) 6=∅

Proposition 3.5. Let L and K be two lattices and T :L→ P∗(K) be a set-valued homomorphism. If P ∈ P∗(K) be a T-lower rough prime ideal ofK and L_T 6= T+_T(P) 6=∅, Then T+_T(P) is a lower T- rough quotient prime ideal of _TL.

Proof. By Lemma 3.2 provided, we have T+_T(P) is an ideal of L_T. Therefore we assume that T(x), T(y) ∈ L_T such that T(x)V

T(y) ∈

T+_(P)

T . Thus T(x∧y) ∈ T+_(P)

T . Hence T(x∧y) = T(x)

V

T(y) ⊆ P. It follows thatx∧y ∈T+(P). Since P is a lowerT- rough prime ideal of K,x∈T+(P) or y ∈T+(P). It means that T(x) ⊆P orT(y) ⊆P. Thus T(x) ∈ T+_T(P) orT(y) ∈ T+_T(P). Hence T+_T(P) is a lower T- rough quotient prime ideal of _TL.

Proposition 3.6. Let L and K be two lattices and T :L→ P∗(K) be a set-valued homomorphism. IfP ⊆K be an upperT-rough prime ideal of K and L_T 6= T−_T1(P) 6=∅ , Then T−1_T(P) is an upper T-rough quotient prime ideal of _TL.

Proof. By Lemma 3.3 provided, we have T−1_T(P) is an ideal of L_T. Then we assume thatT(x), T(y)∈ L

T such thatT(x∧y) =T(x)

V

T(y)∈

T−1_(P)

T . Hence T(x∧y)

T

P 6=∅. It shows that x∧y∈ T−1_{(P). Since} P is an upper T-rough prime ideal of K, thus we have x ∈ T−1(P) or y ∈ T−1(P). It means that T(x)T

P 6= ∅ or T(y)T

P 6= ∅. Thus T(x)∈ T−1_T(P) or T(y)∈ T−_T1(P). Therefore T−_T1(P) is an upperT-rough quotient prime ideal of _TL.

Corollary 3.7. Let L and K be two lattices and T :L→ P∗(K) be a set-valued homomorphism. If(T+(P), T−1(P))be aT-rough prime ideal of K and L_T 6= T+_T(P) =6 ∅ and L_T 6= T−_T1(P) 6=∅, Then T+_T(P) and T−1_T(P) are the lower T- rough and the upper T- rough quotient prime ideal of

L

T,respectively.

4. SET-VALUED HOMOMORPHISM INDUCED BY A

HOMOMORPHISM

In this section, by using a homomorphism on a lattice and a set-valued homomorphism, we define a set-set-valued homomorphism and check out some relations between of them.

Definition 4.1. [11] LetLandKbe two lattices. A mappingf :L→K is called a homomorphism if

(i) f(x∧y) =f(x)V f(y); (ii) f(x∨y) =f(x)W

A surjection(onto) homomorphism is called an epimorphism and an injection(one to one) homomorphism is a monomorphism. An isomor-phism is a bijection homomorisomor-phism.

Lemma 4.2. Let LandK be two lattices andf :L→K be an isomor-phism and let T2 : K → P∗(K) be a set-valued homomorphism. Then T1(x) ={u∈L|f(u) ∈T2(f(x))} is a set-valued homomorphism from L toP∗(L).

Proof. First, we show that T1 is a well-defined mapping. Suppose x1, x2 ∈L, and x1 =x2, then

y1∈T1(x1)⇔f(y1)∈T2(f(x1)) =T2(f(x2))

⇔y1 ∈T1(x2).

So,T1(x1) =T1(x2). Now we show thatT1(x∨y) =T1(x)WT1(y).From the definition ofT1, we have

T1(x∨y) ={u∈L|f(u)∈T2(f(x∨y))}

={u∈L|f(u)∈T2(f(x)∨f(y))}.

Suppose f(u) ∈ T2(f(x)∨f(y)) = {c∨d |c ∈ T2(f(x)), d∈ T2(f(y)}. Since f is onto, then there are s, t∈L such that f(s) =c, f(t) = d. It follows that

{f(s)∨f(t)|f(s)∈T2(f(x)), f(t)∈T2(f(y))}=

{f(s∨t)|s∈T1(x), t∈T1(y)}. So, f(u) = f(w∨z) for some w ∈ T1(x), z ∈ T1(y). Since f is one to

one, we deduce thatu=w∨z. Hence T1(x∨y) =T1(x)W T1(y). Also to similar reason, we have T1(x∧y) =T1(x)VT1(y). Hence T1 is a set-valued homomorphism from LtoP∗(L).

Theorem 4.3. Let L and K be two lattices and f : L → K be an isomorphism and let T2 :K → P∗(K) be a set-valued homomorphism. If T1(x) = {u ∈ L | f(u) ∈ T2(f(x))} and A is a nonempty subset of K, then

(1) f(T₁+(A)) =T₂+(f(A)); (2)f(T₁−1(A)) =T₂−1(f(A)).

So,

f(z)∈T2(f(x))⇒z∈T1(x)⇒z∈A,

⇒f(z)∈f(A)⇒w∈f(A),

⇒T2(f(x))⊆f(A)⇒f(x)∈T₂+(f(A)), =⇒y∈T₂+(f(A)).

Thereforef(T₁+(A))⊆T₂+(f(A)).

Conversely, ify ∈T₂+(f(A)), thenT2(y)⊆f(A).On the other hand, f is onto, then there is x ∈ L such that y = f(x). Hence, we have T2(f(x))⊆f(A).

Letu∈T1(x), thenf(u)∈f(A),therefore there exists a∈A such that f(u) =f(a).Butf is one to one, sou=a. ThenT1(x)⊂A. Therefore, x∈T₁+(A). It implies thaty∈f(T₁+(A)).

So,T₂+(f(A))⊆f(T₁+(A)).

(2).Ify∈f(T₁−1(A)), then there exists x∈T₁−1(A) such thaty=f(x). But ifx∈T₁−1(A), then T1(x)∩A6=∅.Let a∈T1(x)∩A, therefore

f(a)∈T2(f(x))∩f(A)⇒T2(f(x))∩f(A)6=∅,

⇒f(x)∈T₂−1(f(A)),

⇒y∈T₂−1(f(A)). It means thatf(T₁−1(A))⊆T₂−1(f(A)).

Conversely, ify∈T₂−1(f(A)),sincef is onto , then there existsx∈L such thaty=f(x),andT2(y)∩f(A)=6 ∅.So, we haveT2(f(x))∩f(A)6= ∅. Hence there is z ∈ T2(f(x))∩f(A). It means that there exists a ∈ A such that z = f(a) ∈ T2(f(x)). Then a ∈ T1(x)∩A 6= ∅. It obtains that x∈ T₁−1(A). Then y =f(x)∈ f(T₁−1(A)). It follows that T₂−1(f(A))⊆f(T₁−1(A)).

5. T-ROUGH FUZZY PRIME IDEAL WITH RESPECT TO

A FUZZY PRIME IDEAL

Definition 5.1. Let (L, θ) be an approximation space. A subset fuzzy is a mappingµfrom L to [0,1].For every x∈L, we define ,

Apr(µ)(x) = ^ a∈[x]θ

µ(a); Apr(µ)(x) = _ a∈[x]θ

µ(a).

They are called , respectively, the lower and upper approximation of the fuzzy subset µ. Apr(µ) = (Apr(µ), Apr(µ)) is called a rough fuzzy set with respect to θ if Apr(µ) 6= Apr(µ)). Let µ be a fuzzy subset of L, λ∈[0,1]. Then the sets

µλ={x∈L|µ(x)≥λ}; µsλ ={x∈L|µ(x)> λ}

are called, respectively, λ-levelest and λ-strong levelest of the fuzzy set µ.

Definition 5.2. [28] A fuzzy subset µ of a lattice L is called a fuzzy ideal if

(i) µ(x∨y)≥µ(x)∧µ(y) for all x, y∈L; (ii) µ(x∧y)≥µ(x)∧µ(y) for allx, y∈L.

It is a fuzzy prime ideal if µ(x∧y) = µ(x) or µ(x∧y) = µ(y) for all x, y∈L.

Definition 5.3. Let L and K be two lattices and T :L → P∗(K) be a set-valued homomorphism. Let µ be a fuzzy ideal of K . For every x∈L, we define

T+(µ)(x) = ^ a∈T(x)

µ(a) ; T−1(µ)(x) = _ a∈T(x)

µ(a).

T+_{(µ) and T}−1_{(µ) are called , respectively, the lower T-rough and} the upper T-rough fuzzy subsets of L. (T+(µ), T−1(µ)) is said to be T-rough fuzzy set of L. If T+(µ) and T−1(µ) are fuzzy prime ideals, (T+(µ), T−1(µ)) is said to beT-rough fuzzy prime ideal of L.

The following theorem and lemma have been proved in [7, 13]:

Theorem 5.4. Let µbe a fuzzy subset of a latticeL. Thenµ is a fuzzy ideal( fuzzy prime ideal) of L iff µλ, µsλ are, if they are nonempty, ideals [prime ideals] ofL for every λ∈[0,1].

Lemma 5.5. Let Land K be two lattices and T :L→P∗(K) be a set-valued homomorphism. If µis a fuzzy ideal ofK, then for all λ∈[0,1], (1) (T+(µ))λ =T+(µλ);

(2) (T−1(µ))λ=T−1(µλ); (3) (T+(µ))s_λ =T+(µs_λ); (4) (T−1(µ))s_λ=T−1(µs_λ).

Theorem 5.6. Let L and K be two lattices and T :L → P∗(K) be a set-valued homomorphism. If µ is a fuzzy ideal of K, then T+(µ) is a fuzzy ideal ofL.

Theorem 5.7. Let L and K be two lattices and T :L → P∗(K) be a set-valued homomorphism. Ifµis a fuzzy ideal of K, thenT−1(µ) is an fuzzy ideal ofL.

If θ is a complete congruence relation(that is, equivalence relation and [x]θW[y]θ= [x∨y]θ and [x]θV[y]θ = [x∧y]θ forx, y∈L) onL and defineT :L→P∗(L) whereT(x) = [x]θ for everyx∈L, we generalized theorems proved in [8].

Theorem 5.8. If µis a fuzzy prime ideal of K and T :L→P∗(K) be a set-valued homomorphism, thenT+_{(µ), T}−1_{(µ) are fuzzy prime ideals} of L.

Definition 5.9. Letf be a mapping from a setXto a setY.Letµbe a fuzzy set ofX andλbe a fuzzy set ofY.Then the inverse imagef−1(λ) ofλdefined by

f−1(λ)(x) =λ(f(x))f or all x∈X. The image f(µ) ofµis the fuzzy set inY defined by

f(µ)(y) =

W

x∈f−1_(y){µ(x)}, iff−1(y)6=∅

0, iff−1(y) =∅. for all y∈Y.

Theorem 5.10. Let L and K be two lattices and f : L → K be an isomorphism and T2 :K →P∗(K) be a set-valued homomorphism. Let µ be a fuzzy subset of L. IfT1(x) ={u∈L|f(u)∈T2(f(x))}, then (1)T₁+(µ)is a fuzzy(prime) ideal ofL if and only if T₂+(f(µ))is a fuzzy (prime)ideal of K;

(2) T₁−1(µ) is a fuzzy (prime)ideal of L if and only if T₂−1(f(µ)) is a fuzzy (prime)ideal of K.

6. CONCLUSION

presented by Davvaz [7, 8]. Further, we studied and investigated some of their interesting properties of a set- valued homomorphism induced by a lattice homomorphism.

ACKNOWLEDGMENT The authors are grateful to reviewers for their valuable comments and suggestions for improving the paper.

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