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Generalized Rough Fuzzy Ideals in Quantales.

1. Introduction

The idea of "theory of rough sets" proposed by Pawlak [1, 2] to manage uncertainty and granularity in the information system has attracted the concern and attention of scientists and experts in different fields of science and technology. Late years have seen its wide applications in algebraic systems, knowledge discovery, data mining, expert systems, pattern recognition, granular computing, graph theory, machine learning, partially ordered sets, and so forth [3-15]. It is noted that the significant concepts in the classical theory of rough set are the lower and upper approximations obtained from equivalence relation on a universal set. In many cases, as pointed out by numerous researchers, the implementation of theory of rough set becomes restrictive if we use the condition of the equivalence relation in the model of Pawlak rough set. To get control of this issue, several authors generalized the classical rough set theory by using more general binary relations [16-20] or by employing coverings [21,22]. Besides, theory of rough sets can also be generalized to the fuzzy environment by employing the notion of fuzzy sets of Zadeh [23], and the resulting notions are called fuzzy rough sets [24-27].

Recently, researchers have connected the ideas and techniques for rough set hypothesis to different algebraic structures. Biswas and Nanda [4] took a group as a ground set and presented the notions of rough groups and rough subgroups. Kuroki and Mordeson [28] discussed the structure of rough sets and rough groups. At that point in [29], Kuroki presented the thought of rough ideals in semigroup. Rough prime ideals and rough fuzzy prime ideals in semigroups were proposed by Xiao and Zhang [17]. Davvaz [30] gave the concepts of rough ideals in rings. He also wrote a short note on algebraic T-rough sets [31]. Kazanci and Davvaz in [32] gave rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings. To overcome the confinement of equivalence relations in the process of establishing rough sets in a ring, Yamak et al. [19] introduced the concept of set-valued mappings as the basis of the generalized upper and lower approximations of a ring with the help of an ideal. Roughness in modules was researched by Davvaz and Mahdavipour [33]. Rasouli and Davvaz [14] presented the notion of rough ideals in MV-algebra. In BCK-algebras, rough ideals were defined by Jun [34]. Classical approximation theory has also been applied to some partially ordered structures. For instance, in [10], Estaji et al. investigated the concepts of upper and lower rough ideals in a lattice by introducing the relationships between lattice theory and rough sets. Zhan et al. discussed "a new rough set theory: rough soft hemirings" in [35]. Ma et al. gave the "applications of a kind of novel Z-soft fuzzy rough ideals to hemirings" and investigated "a survey of decision making methods based on certain hybrid soft set models" [36, 37].

The combination of rough set theory to soft set theory is very important. Feng et al. proposed rough soft sets by combining Pawlak rough sets and soft sets. In particular, Feng et al. put forth a novel concept of soft rough fuzzy sets by combining rough sets, soft sets, and fuzzy sets and we call it Feng-soft rough fuzzy set [38]. And in 2011, Meng et al. further discussed the Feng-soft rough fuzzy sets and put forward another kind of soft rough fuzzy sets, which is called Meng-soft rough fuzzy set [39]. These sets are limited and have a rigorous restrictive condition. Based on the above reason, Zhan and Zhu provided a novel concept of soft rough fuzzy sets, which is called a Z-soft rough fuzzy set [40]. As reported in [41, 42], characterizations of two kinds of hemirings based on probability spaces and reviews on decision making methods based on (fuzzy) soft sets and rough soft sets are discussed, respectively.

The structure of quantale was proposed by Mulvey [43] to study the spectrum of [C.sup.*]-algebras. The idea of ideals (prime, primary) of quantale was given by Wang and Zhao [44, 45]. Xiao and Li [16] generalized the ideals of quantale by means of set-valued mappings. The start of theory of rough sets for applying in algebraic structures, for example, semigroups, rings, modules, and groups, has been focused on a congruence relation. However, we obtain the restricted applications by using the congruence relation. To take care of this issue, Davvaz [19, 31] introduced the idea of a set-valued homomorphism for rings and groups. In this paper, we intend to generalize the results which have been proved in [46].

The arrangement of the paper is as per the following. In Section 2, we review some principal properties of rough sets, rough fuzzy sets, and ideals of quantale. In Section 3, we have introduced "generalized rough fuzzy ideal" and "generalized rough fuzzy prime (semiprime, primary) ideals" of quantales and give a few properties of such ideals. In Section 4, we will describe the images of generalized rough ideals and discuss how they are related. We will explain the relation between lower (upper) generalized rough and lower (upper) generalized approximations of their homomorphic images by using quantale homomorphism and set-valued homomorphism of quantale. In Section 5, we will discuss generalized rough fuzzy prime (primary) ideals based on quantale homomorphism. At last, the conclusion is given in Section 6.

2. Preliminaries

Here, we review a few ideas and results which will be vital in the following.

Definition 1 (see [2]). Let (U, [sigma]) be an approximation space, where U is a nonempty set, and let [sigma] be an equivalence relation on U. For x [member of] U, the equivalence class of x, containing x, is denoted by [[x].sub.[sigma]]. For A [subset or equal to] U, the upper and lower approximations of A are, respectively, defined as [mathematical expression not reproducible].

For more details on rough sets, rough fuzzy sets, and fuzzy rough sets, we refer to [2, 24, 26, 27]. Throughout this paper, we shall use [Q.sub.t] and [Q'.sub.t] for quantales, unless stated otherwise.

Definition 2 (see [47]). A complete lattice [Q.sub.t] having associative binary operation * is called a quantale if it satisfies

[mathematical expression not reproducible].

We will represent the top element of [Q.sub.t] by 1 and the bottom element by 0 throughout the paper. Let A,B [subset or equal to] [Q.sub.t], and we define A * B by the set {x * y | x [member of] A, y [member of] B}, A [disjunction] B by [x [disjunction] y | x [member of] A, y [member of] B} and [[disjunction].sub.i[member of]I][A.sub.i] = {[[disjunction].sub.i[member of]I][A.sub.i] } [x.sub.i] [member of] [A.sub.i]}.

Definition 3 (see [44]). Let [Q.sub.t] be a quantale. A nonempty subset A of [Q.sub.t] is said to be an ideal of [Q.sub.t] if the following conditions hold:

(1) For all [z.sub.1], [z.sub.2] [member of] A, [z.sub.1] [disjunction] [z.sub.2] [member of] A is implied.

(2) If z [member of] [Q.sub.t], z' [member of] A and z [less than or equal to] z' imply z [member of] A.

(3) For all z [member of] [Q.sub.t] and z' [member of] A, then z * z' [member of] A and z' * z [member of] A.

An ideal A is said to be a prime ideal if z * z' [member of] A implies z [member of] A or z [member of] A for all z, z [member of] [Q.sub.t].

An ideal A is said to be a semiprime ideal if z * z [member of] A implies z [member of] A for all z [member of] [Q.sub.t].

Primary ideal is an ideal A of [Q.sub.t] if for all x, z [member of] [Q.sub.t], x * z [member of] A and x [not member of] A imply [z.sup.n] [member of] A for some positive integer n, where [mathematical expression not reproducible].

As it is well known in the fuzzy theory established by Zadeh [23], a fuzzy subset g of [Q.sub.t] is defined as a map from [Q.sub.t] to the unit interval [0,1]. The symbols [conjunction] and [disjunction] will denote the respective infimum and supremum.

Definition 4 (see [24]). Let (W, [sigma]) be an approximation space. A fuzzy subset g is a mapping from W to [0, 1], then for x [member of] W, one defines

[mathematical expression not reproducible]. (1)

They are called the lower and upper approximations of g, respectively. If [[sigma].bar](g) [not equal to] [bar.[sigma]] (g), then [sigma](g) = ([[sigma].bar](g), [bar.[sigma]](g)) is called a rough fuzzy set with respect to [sigma]. For [alpha] [member of] [0, 1], the sets

[mathematical expression not reproducible] (2)

are called [alpha]-cut and strong [alpha]-cut of the fuzzy set g, respectively.

Definition 5 (see [46]). A nonempty fuzzy subset g of [Q.sub.t] is called a fuzzy ideal of [Q.sub.t], if the following conditions are satisfied:

(1) If c [less than or equal to] d, then g(d) [less than or equal to] g(c).

(2) g(c) [conjunction] g(d) [less than or equal to] g(c [disjunction] d).

(3) g(c) [disjunction] g(d) [less than or equal to] g(c * d).

From (1) and (2) in Definition 5, it is observed that g(c [disjunction] d) = g(c) [conjunction] g(d) for all c,d [member of] [Q.sub.t]. Thus, a fuzzy set g is a fuzzy ideal of [Q.sub.t] if and only if g(c [disjunction] d) = g(c) [conjunction] g(d) and g(c * d) [greater than or equal to] g(c) [conjunction] g(d) for all c,d [member of] [Q.sub.t].

Definition 6 (see [46]). A nonconstant fuzzy ideal g of a quantale [Q.sub.t] is called a fuzzy prime ideal of [Q.sub.t] if for all c,d [member of] [Q.sub.t],

g(c * d) = g(c) or g(c * d) = g(d).

Note that we require a fuzzy prime ideal of a quantale to be a nonconstant in order to keep consistent with the definition of prime ideals of quantales [45]. Therefore, throughout this paper, a fuzzy ideal of a quantale is always assumed to be nonconstant. For fuzzy semiprime and fuzzy primary ideals, see [46].

Proposition 7 (see [46]). Let g be a fuzzy subset of a quantale [Q.sub.t]. Then g is a fuzzy (prime, semiprime, primary) ideal of [Q.sub.t] if and only if for each [mathematical expression not reproducible] is either empty or (prime, semiprime, primary) ideal of [Q.sub.t].

Throughout this paper, f-ideal, f-prime, f-semiprime, and f-primary ideals will denote fuzzy ideal, fuzzy prime, fuzzy semiprime, and fuzzy primary ideals of quantales, unless stated otherwise. We use F([Q.sub.t]) to denote the set of all fuzzy subsets of [Q.sub.t].

The concept of generalized rough sets is a generalization of Pawlak's rough set. In rough set theory, an equivalence relation is the basic requirement for lower and upper approximations. Sometimes it is difficult to find such an equivalence relation among the elements of the set under investigation. In such situations, generalized rough set approach can be useful.

Definition 8 (see [19]). Let U and W be two nonempty universes. Let H be a set-valued mapping given by H :U [right arrow] P(W), where P(W) is the power set of W. Then the triple (U, W, H) is referred to as a generalized approximation space or generalized rough set. Any set-valued function from U to P(W) defines a binary relation from U to W by setting [[sigma].sub.H] = {(x, y) | y [member of] H(x)}. Obviously, if [sigma] is an arbitrary relation from U to W, then a set-valued mapping [H.sub.[sigma]] : U [right arrow] P(W) can be defined by [H.sub.[sigma]](x) = {y [member of] W | (x, y) [member of] [sigma]}, where x [member of] U. For any set A [subset or equal to] W, the lower and upper approximations represented by [H.bar](A) and [bar.H](A), respectively, are defined as

[H.bar](A) = {z [member of] U | H(z) [subset or equal to] A}, [H.bar](A) = {z [member of] U | H(z) [intersection] A [not equal to] 0}. (3)

We call the pair ([H.bar.](A), [bar.H](A)) generalized rough set, and [H.bar.], [bar.H] are termed as lower and upper generalized approximation operators, respectively.

If W = U and [[sigma].sub.H] = {(x, y) | y [member of] H(x)} is an equivalence relation on U, then the pair (U, [[sigma].sub.H]) is the Pawlak approximation space. Therefore, a generalized rough set is an extended notion of Pawlak's rough set [16].

Definition 9 (see [16]). Let ([Q.sub.t], [*.sub.1]) and ([Q'.sub.t], [*.sub.2]) be two quantales. A set-valued mapping H : [Q.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]), where [P.sup.*]([Q.sub.t]) represents the collection of all nonempty subsets of [Q'.sub.t], is called a set-valued homomorphism if, for all [a.sub.i], a, b [member of] [Q.sub.t] (i [member of] I),

(1) H(a) [*.sub.2] H(b) [subset or equal to] H(a[*.sub.1] b),

(2) [[disjunction].sub.i[member of]I]H([a.sub.i]) [subset or equal to] H([[disjunction].sub.i[member of]I] [a.sub.i]).

A set-valued mapping H : [Q.sub.t] [right arrow] [P.sup.*] ([Q'.sub.t]) is called a strong set-valued homomorphism if we replace inclusion by equality in (1) and (2).

From here onwards by SV-Hom, we will mean the set-valued homomorphism. For strong set-valued homomorphism, we will use SSV-Hom. Besides H will mean the map H : [Q.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]), unless stated otherwise.

3. Generalized Rough Fuzzy Prime (Primary) Ideals in Quantale

In this section, we will introduce the generalized rough fuzzy ideal in quantales and resulting properties of such ideals are presented. Now we use the concept from Definition 4 and generalized it in the following.

Definition 10. Let ([Q.sub.t], [*.sub.1]) and ([Q'.sub.t], [*.sub.2]) be two quantales and let H be a SV-Hom. Let g be any fuzzy subset of [Q'.sub.t]. Then for every z [member of] [Q.sub.t], one defines

[mathematical expression not reproducible]. (4)

Here [H.bar](g) is the generalized lower approximation and [bar.H] (g) is the generalized upper approximation of the fuzzy subset g. The pair ([H.bar] (g), [bar.H](g)) is called generalized rough fuzzy set of [Q.sub.t] if [H.bar] (g) [not equal to] [bar.H] (g).

From here onward by GLA, GUA, and GRF, we will mean generalized lower approximation, generalized upper approximation, and generalized rough fuzzy set, respectively.

Lemma 11. Let H be a SV-Hom. Then for every collection [{[g.sub.i]}.sub.i[member of]I] [subset or equal to] F([Q'.sub.t]),

(1) [mathematical expression not reproducible];

(2) [mathematical expression not reproducible].

Proof. (1) For x [member of] [Q.sub.t], we have

[mathematical expression not reproducible]. (5)

The other item has the similar proo Proposition 12. Let ([Q.sub.t], and ([Q'.sub.t], [*.sub.2]) be two quantales and let H be a SV-Hom. Let g be a fuzzy subset of [Q'.sub.t]. Then for each [alpha] [member of] [0,1], one has the following:

(1) [([H.bar](g)).sub.[alpha]] = [H.bar]([g.sub.[alpha]]);

(2) [([bar.H](g)).sub.[alpha]] = [bar.H]([g.sub.[alpha]]);

(3) [mathematical expression not reproducible];

(4) [mathematical expression not reproducible].

Proof. (1) Let

[mathematical expression not reproducible]. (6)

Axioms (2), (3), and (4) are similar to the proof of (1).

Definition 13. Let H be a SV-Hom. A fuzzy subset g of the quantale [Q'.sub.t] is called a lower [an upper] GRF ideal of [Q'.sub.t] if [H.bar](g) [[bar.H](g)] is a f-ideal of [Q.sub.t]. A fuzzy subset g of [Q'.sub.t], which is both an upper and a lower GRF ideal of [Q'.sub.t], is called GRF ideal of [Q'.sub.t].

Now, lower approximations and upper approximations of f-ideals of quantales are being studied in the following.

Theorem 14. Let H be a SSV-Hom and let g be a f-ideal of [Q'.sub.t]. Then [H.bar](g) is a f-ideal of [Q.sub.t].

Proof. Since g is a f-ideal of [Q'.sub.t], by Definition 5, we have g(a[disjunction] b) = g(a) [conjunction] g(b) and g(a * b) [greater than or equal to] g(a) [disjunction] g(b) [for all]a, b [member of] [Q'.sub.t]. As H is a SSV-Hom, so H([z.sub.1] [disjunction] [z.sub.2]) = H([z.sub.1]) [disjunction] H([z.sub.2]), V[z.sub.1],[z.sub.2] [member of] [Q.sub.t].

Therefore,

[mathematical expression not reproducible]. (7)

Since [epsilon] [member of] H([z.sub.1]) [disjunction] H([z.sub.2]), there exist [c.sub.1] [member of] H([z.sub.1]) and [c.sub.2] [member of] H([z.sub.2]) such that e = [c.sub.1] v [c.sub.2].

Hence,

[mathematical expression not reproducible]. (8)

Hence,

[mathematical expression not reproducible]. (9)

Again since H is a SSV-Hom, hence H([z.sub.1], [*.sub.1] [z.sub.2]) = H([z.sub.1])[*.sub.2] H([z.sub.2]) [for all][z.sub.1],[z.sub.2] [member of] [Q.sub.t].

Thus we have

[mathematical expression not reproducible]. (10)

Now since [mathematical expression not reproducible].

Thus,

[mathematical expression not reproducible]. (11)

Hence,

[mathematical expression not reproducible]. (12)

Thus, by (9) and (12), [H.bar](g) is a f-ideal of [Q.sub.t].

Theorem 15. Let H be a SSV-Hom and let g be a f-ideal of [Q'.sub.t]. Then [bar.H](g) is a f-ideal of [Q.sub.t].

Proof. Since H is a SSV-Hom, therefore [mathematical expression not reproducible]. Consider

[mathematical expression not reproducible]. (13)

For c [member of] H([z.sub.1]) [disjunction] H([z.sub.2]), we have a [member of] H([z.sub.1]) and b [member of] H([z.sub.2]) such that c = a [disjunction] b. Hence,

[mathematical expression not reproducible]. (14)

Thus,

[mathematical expression not reproducible]. (15)

Now,

[mathematical expression not reproducible]. (16)

For c [member of] H([z.sub.1])[*.sub.2] H([z.sub.2]), there exist a [member of] H([z.sub.1]) and b [member of] H([z.sub.2]) such that c = a [*.sub.2] b.

Hence,

[mathematical expression not reproducible]. (17)

Thus,

[mathematical expression not reproducible]. (18)

Hence by (15) and (18), we have [bar.H](g) is a f-ideal of [Q.sub.t].

By the above two theorems, we have immediately the following corollary.

Corollary 16. Let H be a SSV-Hom and let g be a f-ideal of [Q'.sub.t]. Then g is a GRF ideal of [Q'.sub.t].

Proposition 17. Let H be a SSV-Hom. Let [{[g.sub.i]}.sub.i[member of]I] be a family of f-ideals of [Q'.sub.t]. Then [H.bar]([inf.sub.i[member of]I]([g.sub.i])) is a f-ideal of [Q.sub.t].

Proof. Since every [g.sub.i] is a f-ideal for i [member of] I, therefore [for all]x, y [member of] [Q.sub.t],

[mathematical expression not reproducible]. (19)

Hence,

[mathematical expression not reproducible]. (20)

Hence,

[mathematical expression not reproducible]. (21)

Therefore, [H.bar]([inf.sub.i[member of]I]([g.sub.i])) is a f-ideal of [Q.sub.t].

Theorem 18. Let H be a SSV-Hom and let g be a f-ideal of [Q'.sub.t]. Then [H.bar](g) (respectively, [H.bar](g)) is a f-ideal of [Q.sub.t] if and only if for each [alpha] [member of] [0,1], [H.bar] ([g.sub.[alpha]]) (respectively, [H.bar] ([g.sub.[alpha]])), where [g.sub.[alpha]] [not equal to] 0, is an ideal of [Q.sub.t].

Proof. Suppose [H.bar](g) is a f-ideal of [Q.sub.t]. We need to show that [mathematical expression not reproducible].

Conversely, assume [H.bar]([g.sub.[alpha]]) is an ideal of [Q.sub.t]. We will show [mathematical expression not reproducible].

Consider

[mathematical expression not reproducible]. (22)

Since H is a SSV-Hom, for c [member of] H([z.sub.1]) [disjunction] H([z.sub.2]), there exist [a.sub.1] [member of] H([z.sub.1]) and [a.sub.2] [member of] H([z.sub.2]) such that c = [a.sub.1] [disjunction] [a.sub.2].

Hence we obtain

[mathematical expression not reproducible]. (23)

So [mathematical expression not reproducible].

Now for [mathematical expression not reproducible].

Example 19. Let ([Q.sub.t], [*.sub.1]) and ([Q'.sub.t], [*.sub.2]) be two quantales, where [Q.sub.t] and [Q'.sub.t] are depicted in Figures 1 and 2 and the binary operations [*.sub.1] and [*.sub.2] on both the quantales are the same as the meet operation in the lattices [Q.sub.t] and [Q'.sub.t] as shown in Tables 1 and 2.

Let H be a SSV-Hom as defined by H(0) = [0'}, H(a) = [i, j}, H(1) = {1'}. Let g be a f-ideal of [Q'.sub.t] defined by g = 0.9/0' + 0.6/i + 0.7/j + 0.6/1'. Then GLA and GUA of the f-ideal g of [Q'.sub.t] are as follows: [H.bar](g) = 0.9/0 + 0.6/a + 0.6/1 and [bar.H](g) = 0.9/0 + 0.7/a + 0.6/1. It is easily verified that [H.bar](g) and [bar.H](g) are f-ideals of [Q.sub.t].

Consider H : [Q'.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) defined by H(0') = H(i) = H(j) = [0'} and H(1') = [Q'.sub.t]. Then H is a SV-Hom.

Let [mu] be a fuzzy subset of defined by [mathematical expression not reproducible]. It is observed that [H.bar](p) is not a f-ideal of [Q'.sub.t] and [bar.H]([mu]) is a constant f-ideal. Hence it is important to take SSV-Hom.

Definition 20. Let H be a SV-Hom and let g be a fuzzy subset of a quantale [Q'.sub.t]. Then g is called an upper [a lower] GRF prime ideal of [Q'.sub.t] if [ba.rH](g) [[H.bar](g)] is a f-prime ideal of [Q.sub.t]. A fuzzy subset g of [Q'.sub.t], which is both an upper and a lower GRF prime ideal, is called GRF prime ideal of [Q'.sub.t].

Similarly, we can define upper [lower] GRF semiprime (primary) ideals of quantale. Thus the concept of generalized rough fuzzy ideals of quantales extends the notion of rough fuzzy ideals.

Proposition 21. Let H be a SSV-Hom. If g is a f -prime ideal of [Q'.sub.t] then [H.bar](g) is a f-prime ideal of [Q.sub.t].

Proof. As g isa f-prime ideal of [Q'.sub.t], therefore g(c[*.sub.2] b) = g(c) or g(c[*.sub.2] b) = g(b) [for all]c, b [member of] [Q'.sub.t] and hence, g is a f-ideal of [Q'.sub.t], so by Theorem 14, [H.bar](g) is a f-ideal of [Q.sub.t]. Consider

[mathematical expression not reproducible]. (24)

Since H is a SSV-Hom, therefore for [member of] e H([x.sub.1])[*.sub.2] H([y.sub.1]) there exist c [member of] H([x.sub.1]) and b [member of] H([y.sub.1]) such that [member of] = c[*.sub.2] b.

Hence,

[mathematical expression not reproducible]. (25)

Thus, [H.bar](g)([x.sub.1] [*.sub.1] [y.sub.1]) = [H.bar](g)([x.sub.1]) or [H.bar](g)([x.sub.1][*.sub.1] [y.sub.1]) = [H.bar](g)([y.sub.1]) [for all][x.sub.1],[y.sub.1] [member of] [Q.sub.t]. Hence [H.bar](g) is a f-prime ideal of [Q.sub.t].

Proposition 22. Let H be a SSV-Hom. If g is a f -prime ideal of [Q'.sub.t], then [bar.H](g) is a f-prime ideal of [Q.sub.t].

Proof. The proof is similar as reported in Proposition 21.

By the above two theorems, we have immediately the following corollary.

Corollary 23. Let H be a SSV-Hom and let g be a f-prime ideal of [Q'.sub.t]. Then g is a GRF prime ideal of [Q'.sub.t].

Theorem 24. Let H be a SSV-Hom and let [H.bar](g) be a f-ideal of [Q.sub.t]. Then [H.bar](g) is a f -prime ideal of [Q.sub.t] if and only if [H.bar](g)(x[*.sub.1] y) = [H.bar](g)(x) [H.bar](g)(y) [for all]x, y [member of] [Q.sub.t].

Proof. Let [H.bar](g) be a f-prime ideal of [Q.sub.t]. Then g([d.sup.2]) = [H.bar](g)(x[*.sub.1] y) = [H.bar](g)(x) or [H.bar](g)(x[*.sub.1] y) = [H.bar](g)(y).

This implies that

[H.bar](g)(x[*.sub.1] y) [less than or equla to] [H.bar](g)(x) [disjunction] [H.bar](g)(y). (26)

As [H.bar] (g) is a f-ideal of [Q.sub.t], hence by definition of f-ideal, we have

[H.bar](g)(x[*.sub.1] y) [greater than or equal to] [H.bar](g)(x) [disjunction] [H.bar](g)(y). (27)

By (26) and (27), we obtain [mathematical expression not reproducible].

Theorem 25. Let H be a SSV-Hom and let g be a f-prime ideal of [Q'.sub.t]. Then [H.bar](g) (respectively, [bar.H](g)) is a f-prime ideal of [Q.sub.t] if and only if, for each [alpha] [member of] [0,1], [H.bar]([g.sub.[alpha]]) (respectively, [bar.H]([g.sub.[alpha]])), where [g.sub.[alpha]] [not equal to] 0, is a prime ideal of [Q.sub.t].

Proof. As g isa f-prime ideal of [Q'.sub.t], therefore g(a[*.sub.2] c) = g(a) or g(a[*.sub.2] c) = g(c) [for all]a,c [member of] [Q'.sub.t]. Suppose [H.bar](g) is a f-prime ideal of [Q.sub.t], then [H.bar](g) is a f-ideal of [Q.sub.t]. By Theorem 18, [H.bar]([g.sub.[alpha]]) is an ideal of [Q.sub.t]. In order to show that [mathematical expression not reproducible].

Conversely, suppose that [H.sub.bar]([g.sub.[alpha]]) is a prime ideal of [Q.sub.t], then [H.sub.bar]([g.sub.[alpha]]) is an ideal of [Q.sub.t]. By Theorem 18, [H.sub.bar](g) is a f-ideal of [Q.sub.t].

Consider

[mathematical expression not reproducible]. (28)

Since H is a SSV-Hom, we have a [member of] H(x), c [member of] H(y) such that d = a[*.sub.2] c.

Hence,

[mathematical expression not reproducible]. (29)

Therefore [mathematical expression not reproducible].

Theorem 26. Let H be a SSV-Hom and let g be a f-semiprime ideal of [Q'.sub.t]. Then [H.bar](g) is a f-semiprime ideal of [Q.sub.t].

Proof. As g is a f-semiprime ideal of [Q'.sub.t], therefore g([d.sup.2]) = g(d) [for all]d [member of] [Q'.sub.t] and g is a f-ideal of [Q'.sub.t], so by Theorem 14, [H.bar](g) is a f-ideal of [Q.sub.t].

Hence consider

[mathematical expression not reproducible]. (30)

Thus [H.bar](g)(y) = [H.bar] (g)([y.sup.2]) [for all]y [member of] [Q.sub.t]. Therefore H(g) is a f-semiprime ideal of [Q.sub.t].

Theorem 27. Let H be a SSV-Hom and let g be a f-semiprime ideal of [Q'.sub.t]. Then [bar.H](g) is a f-semiprime ideal of [Q.sub.t].

Proof. Proof is similar as reported in Theorem 26.

Corollary 28. Let H be a SSV-Hom and let g be a f-semiprime ideal of [Q'.sub.t]. Then g is a GRF semiprime ideal of [Q'.sub.t].

Theorem 29. Let g be a f-semiprime ideal of [Q'.sub.t] and let H be a SSV-Hom. Then [H.bar](g) (respectively, [bar.H](g))is a f-semiprime ideal of [Q.sub.t] if and only if, for each a [member of] [0,1], [H.sub.bar]([g.sub.[alpha]]) (respectively, [bar.H]([g.sub.[alpha]])), where [g.sub.[alpha]] [not equal to] 0, is a semiprime ideal of [Q.sub.t].

Proof. Suppose [H.sub.bar](g) is a f-semiprime ideal of [Q.sub.t], then [H.bar](g) is a f-ideal of [Q.sub.t]. By Theorem 18, [H.sub.bar]([g.sub.[alpha]]) is an ideal of [Q.sub.t]. In order to show that [H.sub.bar]([g.sub.[alpha]]) is a semiprime ideal [for all][alpha] [member of] [0,1], we have to show that for a[*.sub.1] a [member of] [H.sub.bar]([g.sub.[alpha]]) implies a [member of] [H.bar]([g.sub.[alpha]]). Let a[*.sub.1] a [member of] [H.sub.bar]([g.sub.[alpha]]). Since [H.sub.bar](g) is a f-semiprime ideal, we have [H.bar](g)(a) = H(g)(a[*.sub.1] a) [greater than or equal to] [alpha]. Thus, we have a [member of] H([g.sub.[alpha]]). Hence [H.sub.bar]([g.sub.[alpha]]) is a semiprime ideal of [Q.sub.t].

Conversely, suppose that [H.bar]([g.sub.[alpha]]) is a semiprime ideal of [Q.sub.t]. Then [H.sub.bar]([g.sub.[alpha]]) is an ideal of [Q.sub.t]. By Theorem 18, [H.bar](g) is a f-ideal.

For [H.bar](g) to be a f-semiprime ideal, we have to show that [H.bar](g)(z[*.sub.1] z) = [H.bar](g)(z) [for all]z [member of] [Q.sub.t]. As H is a SSV-Hom and g is a f-semiprime ideal of [Q'.sub.t], consider

[mathematical expression not reproducible]. (31)

Thus [H.bar](g)(z) = [H.bar] (g)([z.sub.2]) [for all]z [member of] [Q.sub.t]. Hence [H.bar] (g) is a f-semiprime ideal of [Q.sub.t].

Example 30. Let ([Q.sub.t],[*.sub.1]) and ([Q'.sub.t],[*.sub.2]) be two quantales, where [Q.sub.t] and are depicted in Figures 1 and 2 and the binary operations [*.sub.1] and [*.sub.2] on both the quantales are the same as the meet operation in the lattices [Q.sub.t] and [Q'.sub.t] as shown in Tables 1 and 2.

Let H : [Q.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) be a SSV-Hom as defined in Example 19.

Let [lambda] be a fuzzy subset of [Q'.sub.t] defined by [lambda] = 0.9/0' + 0.6/i+ 0.9/j + 0.6/1'. Then one can verify that [lambda] is a f-prime ideal of [Q'.sub.t].

Hence GUA and GLA of the f-prime ideal [lambda] are [H.bar]([lambda]) = 0.9/0 + 0.9/a + 0.6/1 and [H.bar]([lambda]) = 0.9/0 + 0.6/a + 0.6/1. It is observed that [H.bar]([lambda]) and [H.sub.bar]([lambda]) are nonconstant f-prime ideals of [Q.sub.t].

Let g be a fuzzy subset of [Q.sub.t] defined by g(x) = [1, if x = 0'; 0.6, if x = 0'} [for all]x [member of] [Q'.sub.t]. Then g is a f-semiprime ideal of [Q'.sub.t]. Hence GLA and GUA of f-semiprime ideal g are as follows: [H.bar](g) = 1/0+ 0.6/a + 0.6/1 and [bar.H](g) = 1/0 +0.6/a + 0.6/1. It is clear that [bar.H] (g) and [H.sub.bar](g) are f-semiprime ideals of [Q.sub.t].

Theorem 31. Let g be a f-primary ideal of [Q'.sub.t] and let H be a SSV-Hom. Then [H.sub.bar](g) is a f-primary ideal of [Q.sub.t].

Proof. As g is a f-primary ideal of [Q'.sub.t], therefore g(a[*.sub.2] b) = g(a) or g(a[*.sub.2] b) = g([b.sup.n]) [for all]a, b [member of] [Q'.sub.t] and hence, g is a f-ideal of [Q'.sub.t], so by Theorem 14, [H.sub.bar](g) is f-ideal of [Q.sub.t]. Since H is given as SSV-Hom, consider

[mathematical expression not reproducible]. (32)

Here [b.sup.n] = b[*.sub.2] b[*.sub.2], ..., [*.sub.2] b [member of] H(y)[*.sub.2] H(y)[*.sub.2], ..., [*.sub.2] H(y) = H(y[*.sub.1] y[*.sub.1] y[*.sub.1], ..., [*.sub.1] y) = H([y.sup.n]) up to n times for some positive integer n. Thus [H.sub.bar](g)(z[*.sub.1] y) = H(g)(z) or [H.bar](g)(z[*.sub.1] y) = H(g)([y.sup.n]) [for all]z, y [member of] [Q.sub.t]. Therefore [H.bar](g) is a f-primary ideal of [Q.sub.t].

Theorem 32. Let g be a f-primary ideal of [Q'.sub.t] and let H be a SSV-Hom. Then [bar.H](g) is a f-primary ideal of [Q.sub.t].

Proof. The proof is similar to the proof of Theorem 31.

Theorem 33. Let H be a SSV-Hom and let g be a nonconstant f-primary ideal of [Q'.sub.t]. Then [H.bar](g) (respectively, [bar.H](g)) is a f-primary ideal of [Q.sub.t] if and only if for each [alpha] [member of] [0,1], [H.sub.bar]([g.sub.[alpha]]) (respectively, [bar.H]([g.sub.[alpha]])), where [g.sub.[alpha] [not equal to] 0, is a primary ideal of [Q.sub.t].

4. Homomorphic Images of Generalized Rough Ideals Based on Quantale Homomorphism

In this section, we will describe the images of generalized lower and upper approximations by using quantale homomorphism and set-valued homomorphism of quantales.

Definition 34 (see [47]). Let ([Q.sub.t], [*.sub.1]) and ([Q'.sub.t],[*.sub.2]) be two quantales. A map f : [Q.sub.t] [right arrow] [Q'.sub.t] is called a quantale homomorphism if

(1) f(a[*.sub.1] b) = f(a)[*.sub.2] f(b);

(2) f([[disjunction].sub.i[member of]I] [a.sub.i]) = [[disjunction].sub.i[member of]I] ([a.sub.i]) [for all]a,b,a, [member of] [Q.sub.t] (i [member of] I).

A quantale homomorphism f : [Q.sub.t] [right arrow] [Q'.sub.t] is called an epimorphism if f is onto [Q'.sub.t] and f is called a monomorphism if f is one-one. If f is bijective, then it is called an isomorphism.

It is clear that if x [less than or equal to] y, then f(x) [less than or equal to] f(y); that is, f is order-preserving.

Proposition 35. Let ([Q.sub.t], [*.sub.1]) and ([Q'.sub.t], [*.sub.2]) be two quantales, let f : [Q.sub.t] [right arrow] [Q'.sub.t] be an epimorphism, and let [H.sub.2] : [Q'.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) be a SV-Hom. Then one has the following:

(1) If f is one to one and [H.sub.1](x) = [y [member of] [Q.sub.t] | f(y) [member of] [H.sub.2](f(x))} [for all]x [member of] [Q.sub.t], then [H.sub.1] is a SV-Hom from [Q.sub.t] to [P.sup.*] ([Q.sub.t]).

(2) If [H.sub.2] is a SSV-Hom, then [H.sub.1] is a SSV-Hom.

Proof. (1) First of all, we show that [H.sub.1] is a well-defined mapping. Suppose [x.sub.1] = [x.sub.2], then we have [mathematical expression not reproducible].

(2) It is similar to part(1).

Theorem 36. Let f : [Q.sub.t] [right arrow] [Q'.sub.t] be a quantale isomorphism and let [H.sub.2] : [Q'.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) be a SV-Hom. Set [H.sub.1](m) = [z e [Q.sub.t] | f(z) [member of] [H.sub.2](f(m))} [for all]m [member of] [Q.sub.t] and [for all]0 [not equal to] A [subset or equal to] [Q'.sub.t], then

(1) f([[bar.H].sub.1] (A)) = [[bar.H].sub.2](f(A));

(2) f([[H.bar].sub.1] (A)) = [[H.bar].sub.2](f(A));

(3) f(x) [member of] f)[[bar.H].sub.1](A)) [??] x [member of] [[bar.H].sub.1] (A).

Proof. (1) Let [mathematical expression not reproducible].

Now we Take [mathematical expression not reproducible].

(2) Suppose Z [member of] (f ([H.bar].sub.1](A)), then there exists [mathematical expression not reproducible].

Now let y [member of] (f ([H.bar].sub.1](A)). Then there exists [mathematical expression not reproducible]

(3) Let x [member of] [[bar.H].sub.1](A). Then f(x) [member of] f([[bar.H.sub.1](A)). Conversely, Suppose that f(x) [member of] f([[bar.H].sub.1], (A)), then there is y [member of] y [member of] f([[bar.H].sub.1](A) such that f(x) = f(y). Since f is one-one, hence x = y [member of] [[bar.H].sub.1] (A).

Remark 37. From Theorem 36(3), it is easily obtained that f(x) [member of] f([[H.bar].sub.1](A)) [??] x [member of][[H.bar].sub.1](A) [for all]0 [not equal to] A [subset or equal to] [Q'.sub.t].

Theorem 38. Let f : [Q.sub.t] [right arrow] [Q'.sub.t] be a surjective quantale homomorphism and let [H.sub.2] : [Q'.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) be a SV-Hom. Let [H.sub.1](x) = [y [member of] [Q.sub.t] | f(y) [member of] [H.sub.2](f(x))}[for all]x [member of] [Q.sub.t] and [for all]0 [not equal to] A [subset or equal to] [Q'.sub.t]. Then,

(1) [[bar.H].sub.1](A) is an ideal of [Q.sub.t] iff [[bar.H].sub.2](f(A)) is an ideal of [Q'.sub.t];

(2) [[bar.H].sub.1](A) is a prime ideal of [Q.sub.t] iff [[bar.H].sub.2](f(A)) is a prime ideal of [Q'.sub.t];

(3) [[bar.H].sub.1](A) is a semiprime ideal of [Q.sub.t] iff [[bar.H].sub.2](f(A)) is a semiprime ideal of [Q'.sub.t];

(4) [[bar.H].sub.1](A) is a primary ideal of [Q.sub.t] iff [[bar.H].sub.2](f(A)) is a primary ideal of [Q'.sub.t].

Proof. (1) Supposed [[bar.H].sub.1](A) is an ideal of [Q.sub.t]. We show that [[bar.H].sub.2] f(A)) is an ideal of [Q'.sup.t], where f ([[bar.H].sub.1](A)) = [[bar.H].sub.2] f(A)) by Theorem 36(1).

(i) Let x, z [member of] f([[bar.H].sub.1] (A)). Then there exists [x.sub.1], [z.sub.1] [member of] [[bar.H].sub.1] (A) such that f([x.sub.1]) = x and f([z.sub.1]) = z. Since f is a surjective quantale homomorphism and [[bar.H].sub.1] (A) is an ideal of [Q.sub.t], we have xvz = f([x.sub.1]) [disjunction] f([z.sub.1]) = f([x.sub.1] [disjunction] [z.sub.1]) [member of]f([H.sub.1] (A)). Therefore x [disjunction] [member of] f([[bar.H].sub.1] (A)) [for all]x,z [member of] f([[bar.H].sub.1] (A)).

(ii) Let z [less than or equal to] x [member of]f([[bar.H].sub.1] (A)). Then we obtain [mathematical expression not reproducible].

(iii) Let x [member of] f([[bar.H].sub.1] (A)) and z [member of] [Q'.sub.t]. Then there exist [x.sub.1] [member of] [[bar.H].sub.1] (A) and [z.sub.1] [member of] [Q.sub.t] such that f([x.sub.1] ) = x and f([z.sub.1] ) = z. Since [[bar.H].sub.1] (A) is an ideal and f is a quantale homomorphism, we have [mathematical expression not reproducible].

Conversely, suppose f([[bar.H].sub.1] (A)) = [[bar.H].sub.2] (f(A)) is an ideal of [Q'.sub.t].

(1) Let [mathematical expression not reproducible]. So by Theorem 36(3), we have [z.sub.1] [disjunction] [z.sub.2] [member of] [[bar.H].sub.1] (A). Hence [H.sub.1] (A) is directed.

(ii) Let [mathematical expression not reproducible]. By Theorem 36(3), we obtain [z.sub.1] [member of] [[bar.H].sub.1] (A). So [[bar.H].sub.1] (A) is a lower set.

(iii) Suppose [mathematical expression not reproducible].

(2) Let [mathematical expression not reproducible].

Conversely, let [mathematical expression not reproducible].

The remaining parts (3) and (4) are similar to the proof (2).

Theorem 39. Let f : [Q.sub.t] [right arrow] [Q'.sub.t] be a surjective quantale homomorphism and let [H.sub.2] : [Q'.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) be a SV-Hom. Set [H.sub.1](x) = [y [member of] [Q.sub.t] | f(y) [member of] [H.sub.2](f(x))} [for all] x [member of][Q.sub.t] and [for all]0 [not equal to] B [subset or equal to] [Q'.sub.t]. Then the following hold:

(1) [[H.bar].sub.1](B) is an ideal of [Q.sub.t] iff [[H.bar].sub.2] (f(B)) is an ideal of [Q'.sub.t];

(2) [[H.bar].sub.1](B) is a prime ideal of [Q.sub.t] iff [[H.bar].sub.2] (f(B)) is a prime ideal of [Q'.sub.t];

(3) [[H.bar].sub.1] (B) is a semiprime ideal of [Q.sub.t] iff [[H.bar].sub.2] (f(B)) is a semiprime ideal of [Q'.sub.t];

(4) [[H.bar].sub.1] (B) is a primary ideal of [Q.sub.t] iff [[H.bar].sub.2] (f(B)) is a primary ideal of [Q'.sub.t].

Proof. The proofs of all the parts can be obtained by Theorem 38.

5. Generalized Rough Fuzzy Prime (Primary) Ideals Induced by Quantale Homomorphism

Theorem 40. Let f : [Q.sub.t] [right arrow] [Q'.sub.t] be a surjective quantale homomorphism and let [H.sub.2] : [Q'.sub.t] [right arrow] [P.sup.*]([Q'.sub.t]) be a SV-Hom, and let [lambda] be a fuzzy subset of [mathematical expression not reproducible], then

(1) [[bar.H].sub.1] ([lambda]) is a f-ideal of [Q.sub.t] iff [[bar.H].sub.2] (f([lambda])) is a f-ideal of [Q'.sub.t];

(2) [[bar.H].sub.1] ([lambda]) is a f-prime ideal of [Q.sub.t] iff [[bar.H].sub.2] (f([lambda])) is a f prime ideal of [Q'.sub.t];

(3) [[bar.H].sub.1] ([lambda]) is a f-semiprime ideal of [Q.sub.t] iff [[bar.H].sub.2] (f([lambda])) is a f-semiprime ideal of [Q'.sub.t];

(4) [[bar.H].sub.1] (X) is a f-primary ideal of [Q.sub.t] iff [[bar.H].sub.2] (f(X)) is a f-primary ideal of [Q'.sub.t].

In the above, f([lambda])(y) = [disjunction] {[lambda](x) | f (x) = y, x [member of] [Q.sub.t]}, y [member of] [Q'.sub.t]; that is, f([lambda]) is the standard Zadeh image of the fuzzy subset [lambda] under the mapping [lambda].

Proof. (1) We first point out that, for each [mathematical expression not reproducible].

Let [[bar.H].sub.1] ([lambda]) be a f-ideal of [Q.sub.t]. Then for all [alpha] [member of] (0,1], if [mathematical expression not reproducible]. By Theorem 18, we have [mathematical expression not reproducible] is an ideal of [Q.sub.t]. Also by using Proposition 12, we obtain that [mathematical expression not reproducible] is an ideal of [Q.sub.t]. Now, by Theorem 38(1), we have that [mathematical expression not reproducible] is an ideal of [Q'.sub.t]. Thus, by Theorem 18, we have that [[bar.H].sub.2] (f([lambda])) is a f-ideal of [Q'.sub.t].

Conversely, suppose [mathematical expression not reproducible] is an ideal of [Q.sub.t] by utilizing Theorem 18. It is obtained from Theorem 38(1) that [mathematical expression not reproducible] is an ideal of [Q.sub.t]. Hence by Theorem 18, Hi (X) is a f-ideal of [Q.sub.t].

(2) Let [[bar.H].sub.1] ([lambda]) be a f-prime ideal of [Q.sub.t]. Now for [mathematical expression not reproducible]. Since [[bar.H].sub.1] ([lambda]) is a f-prime ideal of [Q.sub.t], then by Theorem 25, we have that [mathematical expression not reproducible] is a prime ideal of [Q.sub.t]. It is also obtained from Proposition 12 that [mathematical expression not reproducible] is a prime ideal of [Q.sub.t]. Hence [mathematical expression not reproducible] is a prime ideal of [Q.sub.t], by Theorem 38(2). Thus, by Theorem 25, we have that [[bar.H].sub.2] (f([lambda])) is a f-prime ideal of [Q'.sub.t].

Conversely, suppose [[bar.H].sub.2] (f(X)) is a f-prime ideal of [Q'.sub.t]. By Theorem 25, we obtain that [mathematical expression not reproducible] is a prime ideal of [Q'.sub.t]. Thus it is obtained, from Theorem 38(2), that [mathematical expression not reproducible] is a prime ideal of [Q.sub.t]. Hence [[bar.H].sub.1]([lambda]) is a f-prime ideal of [Q.sub.t] by Theorem 25.

Axioms (3) and (4) can be obtained in a similar way.

Theorem 41. Let f be a surjective quantale homomorphism from a quantale [mathematical expression not reproducible], then

(1) [[H.bar].sub.1] ([lambda]) is a f-ideal of [Q.sub.t] iff [[H.bar].sub.2] (f ([lambda])) is a f-ideal of [Q'.sub.t];

(2) [[H.bar].sub.1] ([lambda]) is a f-prime ideal of [Q.sub.t] iff [[H.bar].sub.2] (f([lambda])) is a f-prime ideal of [Q'.sub.t];

(3) [[H.bar].sub.1] ([lambda]) is a f-semiprime ideal of [Q.sub.t] iff [[H.bar].sub.1] (f([lambda])) is a f-semiprime ideal of [Q'.sub.t];

(4) [[H.bar].sub.1] ([lambda]) is a f-primary ideal of [Q.sub.t] iff [[H.bar].sub.2] (f(X)) is a f-primary ideal of [Q'.sub.t].

Proof. The proof is similar as reported in Theorem 40.

6. Conclusion

Pure and applied mathematics are two important branches of mathematics and rough set theory has its own importance in both the branches. When we combine rough set theory with algebraic structures, we obtain new interesting results and research topics. These research topics are attracted by computer scientists and mathematicians. Researchers apply roughness into the algebraic system and find interesting algebraic properties of them. The combination of fuzzy set and rough set theory leads to various models. The relations between fuzzy sets, rough sets, and quantale theory have been already considered in [46]. We have examined the generalized rough fuzzy set theory and its properties in quantale.

In the present paper, we substituted a universe set by a quantale and introduced the notions of generalized rough fuzzy prime (semiprime, primary) ideals in quantale. We see that the lower and upper approximations of fuzzy ideals, using SSV-Hom, are fuzzy ideals, respectively. It is also seen that the approximations of fuzzy prime (semiprime, primary) ideals using SSV-Hom are fuzzy prime (semiprime, primary) ideals, respectively. We have discussed the relation between upper (lower) generalized rough fuzzy (prime, semiprime, primary) ideals and upper (lower) generalized rough fuzzy approximations of their homomorphic images.

We believe that in the near future the idea of generalized roughness will be extended to other algebraic structures.

https://doi.org/10.1155/2018/1085201

Conflicts of Interest

There are no conflicts of interest related to this paper.

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Saqib Mazher Qurashi (iD) (1,2) and Muhammad Shabir (2)

(1) Department of Mathematics, Government College University, Faisalabad, Pakistan

(2) Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Correspondence should be addressed to Saqib Mazher Qurashi; saqibmazhar@gcuf.edu.pk

Received 7 September 2017; Revised 19 November 2017; Accepted 10 December 2017; Published 16 January 2018

Academic Editor: Rigoberto Medina

Caption: Figure 1: Illustration of [Q.sub.t].

Caption: Figure 2: Illustration of [Q'.sub.t].
Table 1: Binary operation [*.sub.1] subject to [Q.sub.t].

[*.sub.1]   0   a  1

0           0   0  0
a           0   a  a
1           0   a  1

Table 2: Binary operation [*.sub.2] subject to [Q'.sub.t].

[*.sub.2]   0'   i    j    1'

0'          0'   0'   0'   0'
i           0'   i    0'   i
j           0'   0'   j    j
1'          0'   i    j    1'
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Title Annotation:Research Article
Author:Qurashi, Saqib Mazher; Shabir, Muhammad
Publication:Discrete Dynamics in Nature and Society
Date:Jan 1, 2018
Words:9270
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