# Generalized Rough Fuzzy Ideals in Quantales

Authors: Saqib Mazher Qurashi; Muhammad Shabir
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The paper examines the generalized rough fuzzy ideals of quantales. There are some intrinsic relations between fuzzy prime (primary) ideals of quantales and generalized rough fuzzy prime (primary) ideals of quantales. Homomorphic images of “generalized rough ideals, generalized rough prime (primary) ideals, and generalized rough fuzzy prime (primary) ideals” which are incited by quantale homomorphism are examined.


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## Body

## 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]. Kazancı 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∗-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 Section2, 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]).
LetU,σ be an approximation space, where U is a nonempty set, and let σ be an equivalence relation on U. For x∈U, the equivalence class of x, containing x, is denoted by xσ. For A⊆U, the upper and lower approximations of A are, respectively, defined as σ¯A=x∈U∣xσ∩A≠∅, σ_A=x∈U∣xσ⊆A. It is easy to verify that σ_A⊆A⊆σ¯A for all A⊆U.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 Qt and Qt′ for quantales, unless stated otherwise.Definition 2 (see [47]).
A complete latticeQt having associative binary operation ∗ is called a quantale if it satisfiesy∗⋁i∈Izi=⋁i∈Iy∗zi; ⋁i∈Iyi∗z=⋁i∈Iyi∗z,for ally,z,yi,zi∈Qti∈I.We will represent the top element ofQt by 1 and the bottom element by 0 throughout the paper. Let A,B⊆Qt, and we define A∗B by the set x∗y∣x∈A,y∈B, A∨B by x∨y∣x∈A,y∈B and ⋁i∈IAi=⋁i∈Ixi∣xi∈Ai.Definition 3 (see [44]).
LetQt be a quantale. A nonempty subset A of Qt is said to be an ideal of Qt if the following conditions hold:(1)
For allz1,z2∈A, z1∨z2∈A is implied.(2)
Ifz∈Qt, z′∈A and z⩽z′ imply z∈A.(3)
For allz∈Qt and z′∈A, then z∗z′∈A and z′∗z∈A.An idealA is said to be a prime ideal if z∗z′∈A implies z∈A or z′∈A for all z,z′∈Qt.An idealA is said to be a semiprime ideal if z∗z∈A implies z∈A for all z∈Qt.Primary ideal is an idealA of Qt if for all x,z∈Qt,  x∗z∈A and x∉A imply zn∈A for some positive integer n, where zn=z∗⋯∗z︸n.As it is well known in the fuzzy theory established by Zadeh [23], a fuzzy subset g of Qt is defined as a map from Qt to the unit interval 0,1. The symbols ∧ and ∨ will denote the respective infimum and supremum.Definition 4 (see [24]).
Let(W,σ) be an approximation space. A fuzzy subset g is a mapping from W to 0,1, then for x∈W, one defines(1)σ_gz=⋀p∈zσgp;σ¯gz=⋁p∈zσgp.
They are called the lower and upper approximations ofg, respectively. If σ_(g)≠σ¯(g), then σ(g)=(σ_(g),σ¯(g)) is called a rough fuzzy set with respect to σ. For α∈0,1, the sets(2)gα=x∈W∣gx≥α;gα+=x∈W∣gx>αare called α-cut and strong α-cut of the fuzzy set g, respectively.Definition 5 (see [46]).
A nonempty fuzzy subsetg of Qt is called a fuzzy ideal of Qt, if the following conditions are satisfied:(1)
Ifc≤d, then g(d)≤g(c).(2)
g(c)∧g(d)≤g(c∨d).(3)
g(c)∨g(d)≤g(c∗d).From(1) and (2) in Definition 5, it is observed that g(c∨d)=g(c)∧g(d) for all c,d∈Qt. Thus, a fuzzy set g is a fuzzy ideal of Qt if and only if g(c∨d)=g(c)∧g(d) and g(c∗d)≥g(c)∨g(d) for all c,d∈Qt.Definition 6 (see [46]).
A nonconstant fuzzy idealg of a quantale Qt is called a fuzzy prime ideal of Qt if for all c,d∈Qt,g(c∗d)=g(c) or g(c∗d)=gd.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]).
Letg be a fuzzy subset of a quantale Qt. Then g is a fuzzy (prime, semiprime, primary) ideal of Qt if and only if for each α∈[0,1], gα (resp., gα+) is either empty or (prime, semiprime, primary) ideal of Qt.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(Qt) to denote the set of all fuzzy subsets of Qt.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]).
LetU and W be two nonempty universes. Let H be a set-valued mapping given by H:U→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 σH={(x,y)∣y∈H(x)}. Obviously, if σ is an arbitrary relation from U to W, then a set-valued mapping Hσ:U→P(W) can be defined by Hσ(x)={y∈W∣(x,y)∈σ}, where x∈U. For any set A⊆W, the lower and upper approximations represented by H_(A) and H¯(A), respectively, are defined as(3)H_A=z∈U∣Hz⊆A,H¯A=z∈U∣Hz∩A≠∅.
We call the pair(H_(A),H¯(A)) generalized rough set, and H_, H¯ are termed as lower and upper generalized approximation operators, respectively.IfW=U and σH={(x,y)∣y∈H(x)} is an equivalence relation on U, then the pair (U, σH) 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(Qt,∗1) and (Qt′,∗2) be two quantales. A set-valued mapping H:Qt→P∗(Qt′), where P∗(Qt′) represents the collection of all nonempty subsets of Qt′, is called a set-valued homomorphism if, for all ai,a,b∈Qt(i∈I),(1)
H(a)∗2H(b)⊆H(a∗1b),(2)
⋁i∈IH(ai)⊆H(⋁i∈Iai).A set-valued mappingH:Qt→P∗(Qt′) 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. BesidesH will mean the map H:Qt→P∗(Qt′), 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 Definition4 and generalized it in the following.Definition 10.
Let(Qt,∗1) and (Qt′,∗2) be two quantales and let H be a SV-Hom. Let g be any fuzzy subset of Qt′. Then for every z∈Qt, one defines(4)H_gz=infa∈Hzga;H¯gz=supa∈Hzga.
HereH_(g) is the generalized lower approximation and H¯(g) is the generalized upper approximation of the fuzzy subset g. The pair (H_(g),H¯(g)) is called generalized rough fuzzy set of Qt if H_(g)≠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.
LetH be a SV-Hom. Then for every collection gii∈I⊆FQt′,(1)
H_infi∈I⁡gi=infi∈I⁡H_(gi);(2)
H¯supi∈I⁡gi=supi∈I⁡H¯(gi).Proof.
(
1
) For x∈Qt, we have(5)H_infi∈I⁡gix=infa∈Hxinfi∈I⁡gia=infi∈I⁡infa∈Hx⁡gia=infi∈I⁡H_gix.
The other item has the similar proof.Proposition 12.
Let(Qt,∗1) and (Qt′,∗2) be two quantales and let H be a SV-Hom. Let g be a fuzzy subset of Qt′. Then for each α∈0,1, one has the following:(1)
(H_(g))α=H_(gα);(2)
(H¯(g))α=H¯(gα);(3)
(H_(g))α+=H_(gα+);(4)
(H¯(g))α+=H¯(gα+).Proof.
(
1
) Let(6)z∈H_gα⟺H_gz≥α⟺infa∈Hz⁡ga≥α⟺ga≥α∀a∈Hz;⟺Hz⊆gα⟺z∈H_gα.
Axioms(2), (3), and (4) are similar to the proof of (1).Definition 13.
LetH be a SV-Hom. A fuzzy subset g of the quantale Qt′ is called a lower [anupper] GRF ideal of Qt′ if H_(g)[H¯(g)] is a f-ideal of Qt. A fuzzy subset g of Qt′, which is both an upper and a lower GRF ideal of Qt′, is called GRF ideal of Qt′.Now, lower approximations and upper approximations off-ideals of quantales are being studied in the following.Theorem 14.
LetH be a SSV-Hom and let g be a f-ideal of Qt′. Then H_(g) is a f-ideal of Qt.Proof.
Sinceg is a f-ideal of Qt′, by Definition 5, we have g(a∨b)=g(a)∧g(b) and g(a∗b)≥g(a)∨g(b)∀a,b∈Qt′. As H is a SSV-Hom, so H(z1∨z2)=H(z1)∨H(z2),  ∀z1,z2∈Qt.
Therefore,(7)H_gz1∨z2=infe∈Hz1∨z2⁡ge=infe∈Hz1∨Hz2⁡ge.
Sincee∈H(z1)∨H(z2), there exist c1∈H(z1) and c2∈H(z2) such that e=c1∨c2.
Hence,(8)H_gz1∨z2=infc1∨c2∈Hz1∨Hz2⁡gc1∨c2=infc1∈Hz1,c2∈Hz2⁡gc1∧gc2=infinfc1∈Hz1⁡gc1,infc2∈Hz2⁡gc2=H_gz1∧H_gz2.
Hence,(9)H_gz1∨z2=H_gz1∧H_gz2∀z1,z2∈Qt.
Again sinceH is a SSV-Hom, hence H(z1∗1z2)=H(z1)∗2H(z2)∀z1,z2∈Qt.
Thus we have(10)H_gz1∗1z2=infe∈Hz1∗1z2⁡ge=infe∈Hz1∗2Hz2⁡ge.
Now sincee∈H(z1)∗2H(z2), there exist c1∈H(z1), c2∈H(z2) such that e=c1∗2c2.
Thus,(11)H_gz1∗1z2=infc1∗2c2∈Hz1∗2Hz2⁡gc1∗2c2≥infc1∗2c2∈Hz1∗2Hz2⁡gc1∨gc2=infc1∈Hz1,c2∈Hz2⁡gc1∨gc2=sup⁡infc1∈Hz1⁡gc1,infc2∈Hz2⁡gc2=H_gz1∨H_gz2.
Hence,(12)H_gz1∗1z2≥H_gz1∨H_gz2∀z1,z2∈Qt.
Thus, by (9) and (12), H_(g) is a f-ideal of Qt.Theorem 15.
LetH be a SSV-Hom and let g be a f-ideal of Qt′. Then H¯(g) is a f-ideal of Qt.Proof.
SinceH is a SSV-Hom, therefore H(z1∨z2)=H(z1)∨H(z2)∀z1,z2∈Qt. Also g is f-ideal of Qt′; hence g(a∨b)=g(a)∧g(b)∀a,b∈Qt′.
Consider(13)H¯gz1∨z2=supc∈Hz1∨z2⁡gc=supc∈Hz1∨Hz2⁡gc.
Forc∈H(z1)∨H(z2), we have a∈H(z1) and b∈H(z2) such that c=a∨b.
Hence,(14)H¯gz1∨z2=supa∨b∈Hz1∨Hz2⁡ga∨b=supa∈Hz1,b∈Hz2⁡ga∧gb=inf⁡supa∈Hz1⁡ga,supb∈Hz2⁡gb=H¯gz1∧H¯gz2.
Thus,(15)H¯gz1∨z2=H¯gz1∧H¯gz2∀z1,z2∈Qt.
Now,(16)H¯gz1∗1z2=supc∈Hz1∗2z2⁡gc=supc∈Hz1∗2Hz2⁡gc.
Forc∈H(z1)∗2H(z2), there exist a∈H(z1) and b∈H(z2) such that c=a∗2b.
Hence,(17)H¯gz1∗1z2=supa∗2b∈Hz1∗2Hz2⁡ga∗2b≥supa∈Hz1,b∈Hz2⁡ga∨gb=sup⁡supa∈Hz1⁡ga,supb∈Hz2⁡gb=H¯gz1∨H¯gz2.
Thus,(18)H¯gz1∗1z2≥H¯gz1∨H¯gz2∀z1,z2∈Qt.
Hence by (15) and (18), we have H¯(g) is a f-ideal of Qt.By the above two theorems, we have immediately the following corollary.Corollary 16.
LetH be a SSV-Hom and let g be a f-ideal of Qt′. Then g is a GRF ideal of Qt′.Proposition 17.
LetH be a SSV-Hom. Let gii∈I be a family of f-ideals of Qt′. Then H_(infi∈I(gi)) is a f-ideal of Qt.Proof.
Since everygi is a f-ideal for i∈I, therefore ∀x,y∈Qt,(19)H_infi∈I⁡gix∨y=infi∈I⁡H_gix∨y=infi∈I⁡H_gix∨y=infi∈I⁡inf⁡H_gix,H_giy=inf⁡infi∈I⁡H_gix,infi∈I⁡H_giy=inf⁡H_infi∈I⁡gix,H_infi∈I⁡giy=H_infi∈I⁡gix∧H_infi∈I⁡giy.
Hence,(20)H_infi∈I⁡gix∨y=H_infi∈I⁡gix∧H_infi∈I⁡giy∀x,y∈Qt,H_infi∈I⁡gix∗1y=infi∈I⁡H_gix∗1y=infi∈I⁡H_gix∗1y≥infi∈I⁡sup⁡H_gix,H_giy=sup⁡infi∈I⁡H_gix,infi∈I⁡H_giy=sup⁡H_infi∈I⁡gix,H_infi∈I⁡giy=H_infi∈I⁡gix∨H_infi∈I⁡giy.
Hence,(21)H_infi∈I⁡gix∗1y≥H_infi∈I⁡gix∨H_infi∈I⁡giy∀x,y∈Qt.
Therefore,H_(infi∈I⁡gi) is a f-ideal of Qt.Theorem 18.
LetH be a SSV-Hom and let g be a f-ideal of Qt′. Then H_(g) (respectively, H¯(g)) is a f-ideal of Qt if and only if for each α∈0,1, H_(gα) (respectively, H¯(gα)), where gα≠∅, is an ideal of Qt.Proof.
SupposeH_(g) is a f-ideal of Qt. We need to show that H_(gα) is an ideal of Qt. Let z1,z2∈H_(gα). Then H_(g)(z1)≥α, H_(g)(z2)≥α. But since H_(g) is a f-ideal, so H_(g)(z1∨z2)=H_(g)(z1)∧H_(g)(z2)≥α. Hence z1∨z2∈H_(gα). Let y∈H_(gα), z∈Qt, and z≤y. Then H_(g)(z)≥H_(g)(y)≥α. Thus z∈H_(gα). Suppose y∈H_(gα) and ∀z∈Qt, then H_(g)(z∗1y)≥H_(g)(z)∨H_(g)(y)=H_(g)(y)≥α, and we get z∗1y∈H_(gα). Similarly, y∗1z∈H_(gα). Hence, H_(gα) is an ideal of Qt.
Conversely, assumeH_(gα) is an ideal of Qt. We will show that H_(g) is a f-ideal of Qt. For any z1,z2∈Qt, let α=H_(g)(z1)∧H_(g)(z2)∈ rang(H_(g)). Then H_(g)(z1)≥α and H_(g)(z2)≥α; that is, z1∈H_(gα) and z2∈H_(gα). Hence, z1∨z2∈H_(gα).
Consider(22)H_gz1∨z2=infc∈Hz1∨z2⁡gc=infc∈Hz1∨Hz2⁡gc.
SinceH is a SSV-Hom, for c∈H(z1)∨H(z2), there exist a1∈H(z1) and a2∈H(z2) such that c=a1∨a2.
Hence we obtain(23)H_gz1∨z2=infa1∨a2∈Hz1∨Hz2⁡ga1∨a2=infa1∨a2∈Hz1∨Hz2⁡ga1∧ga2=infa1∈Hz1,a2∈Hz2⁡ga1∧ga2=inf⁡infa1∈Hz1ga1,infa2∈Hz2⁡ga2=H_gz1∧H_gz2.
SoH_(g)(z1∨z2)=H_(g)(z1)∧H_(g)(z2)∀z1,z2∈H_(gα).
Now forz∈H_(gα) and ∀y∈Qt, we obtain z∗1y∈H_(gα) and y∗1z∈H_(gα). Hence, H_(g)(z∗1y)≥α and H_(g)(z)≥α. If either H_(g)(y)≥α or H_(g)(y)<α, in both the cases, H_(g)(x)∨H_(g)(y)≥α. We suppose H_(g)(x)∨H_(g)(y)=α. So H_(g)(x∗1y)≥H_(g)(x)∨H_(g)(y). Hence, H_(g) is a f-ideal of Qt.Example 19.
Let(Qt,∗1) and (Qt′,∗2) be two quantales, where Qt and Qt′ are depicted in Figures 1 and 2 and the binary operations ∗1 and ∗2 on both the quantales are the same as the meet operation in the lattices Qt and Qt′ as shown in Tables 1 and 2.Table 1
Binary operation∗1 subject to Qt.
∗
1
0
a
1
0
0
0
0
a
0
a
a
1
0
a
1Table 2
Binary operation∗2 subject to Qt′.
∗
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
′Figure 1
Illustration ofQt.Figure 2
Illustration ofQt′.LetH be a SSV-Hom as defined by H(0)=0′, H(a)=i,j, H(1)=1′. Let g be a f-ideal of Qt′ 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 Qt′ are as follows: H_(g)=0.9/0+0.6/a+0.6/1 and H¯(g)=0.9/0+0.7/a+0.6/1. It is easily verified that H_(g) and H¯(g) are f-ideals of Qt.ConsiderH:Qt′→P∗(Qt′) defined by H(0′)=H(i)=H(j)={0′} and H(1′)=Qt′. Then H is a SV-Hom.Letμ be a fuzzy subset of Qt′ defined by μ(x)={1,  ifx=0′;0.7,  ifx≠0′}∀x∈Qt′. Then μ is a f-ideal of Qt′. Hence GLA and GUA of f-ideal μ of Qt′ are H_(μ)=1/0′+1/i+1/j+0.7/1′ and H¯(μ)=1/0′+1/i+1/j+1/1′. It is observed that H_(μ) is not a f-ideal of Qt′ and H¯(μ) is a constant f-ideal. Hence it is important to take SSV-Hom.Definition 20.
LetH be a SV-Hom and let g be a fuzzy subset of a quantale Qt′. Then g is called an upper [a lower] GRF prime ideal of Qt′ if H¯(g)[H_(g)] is a f-prime ideal of Qt. A fuzzy subset g of Qt′, which is both an upper and a lower GRF prime ideal, is called GRF prime ideal of Qt′.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.
LetH be a SSV-Hom. If g is a f-prime ideal of Qt′, then H_(g) is a f-prime ideal of Qt.Proof.
Asg is a f-prime ideal of Qt′, therefore g(c∗2b)=g(c) or g(c∗2b)=g(b)∀c,b∈Qt′ and hence, g is a f-ideal of Qt′, so by Theorem 14, H_(g) is a f-ideal of Qt.
Consider(24)H_gx1∗1y1=infe∈Hx1∗1y1⁡ge=infe∈Hx1∗2Hy1⁡ge.
SinceH is a SSV-Hom, therefore for e∈H(x1)∗2H(y1) there exist c∈H(x1) and b∈H(y1) such that e=c∗2b.
Hence,(25)H_gx1∗1y1=infc∗2b∈Hx1∗2Hy1⁡gc∗2b=infc∈Hx1,b∈Hy1⁡gc∗2b=infc∈Hx1,b∈Hy1⁡gcorgb=infc∈Hx1⁡gcorinfb∈Hy1⁡gb=H_gx1orH_gy1.
Thus,H_(g)(x1∗1y1)=H_(g)(x1) or H_(g)(x1∗1y1)=H_(g)(y1)∀x1,y1∈Qt. Hence H_(g) is a f-prime ideal of Qt.Proposition 22.
LetH be a SSV-Hom. If g is a f-prime ideal of Qt′, then H¯(g) is a f-prime ideal of Qt.Proof.
The proof is similar as reported in Proposition21.By the above two theorems, we have immediately the following corollary.Corollary 23.
LetH be a SSV-Hom and let g be a f-prime ideal of Qt′. Then g is a GRF prime ideal of Qt′.Theorem 24.
LetH be a SSV-Hom and let H_(g) be a f-ideal of Qt. Then H_(g) is a f-prime ideal of Qt if and only if H_(g)(x∗1y)=H_(g)(x)∨H_(g)(y)∀x,y∈Qt.Proof.
LetH_(g) be a f-prime ideal of Qt. Then H_(g)(x∗1y)=H_(g)(x) or H_(g)(x∗1y)=H_(g)(y).
This implies that(26)H_gx∗1y≤H_gx∨H_gy.
AsH_(g) is a f-ideal of Qt, hence by definition of f-ideal, we have(27)H_gx∗1y≥H_gx∨H_gy.
By (26) and (27), we obtain H_(g)(x∗1y)=H_(g)(x)∨H_(g)(y). Conversely, suppose that H_(g)(x∗1y)=H_(g)(x)∨H_(g)(y)∀x,y∈Qt. We have to show that H_(g) is a f-prime ideal. As [0,1] is totally ordered, so H_(g)(x)∨H_(g)(y)=H_(g)(x) or H_(g)(x)∨H_(g)(y)=H_(g)(y). Hence H_(g)(x∗1y)=H_(g)(x) or H_(g)(x∗1y)=H_(g)(y)∀x,y∈Qt. This shows that H_(g) is a f-prime ideal of Qt.Theorem 25.
LetH be a SSV-Hom and let g be a f-prime ideal of Qt′. Then H_(g) (respectively, H¯(g)) is a f-prime ideal of Qt if and only if, for each α∈0,1, H_(gα) (respectively, H¯(gα)), where gα≠∅, is a prime ideal of Qt.Proof.
Asg is a f-prime ideal of Qt′, therefore g(a∗2c)=g(a) or g(a∗2c)=g(c)∀a,c∈Qt′. Suppose H_(g) is a f-prime ideal of Qt, then H_(g) is a f-ideal of Qt. By Theorem 18, H_(gα) is an ideal of Qt. In order to show that H_(gα) is a prime ideal for all α∈[0,1], we have to show that for a∗1c∈H_(gα) implies that a∈H_(gα) or c∈H_(gα). Let a∗1c∈H_(gα). Then H_(g)(a)=H_(g)(a∗1c)≥α or H_(g)(a∗1c)=H_(g)(c)≥α. Thus, a∈H_(gα) or c∈H_(gα). Hence H_(gα) is a prime ideal of Qt.
Conversely, suppose thatH_(gα) is a prime ideal of Qt, then H_(gα) is an ideal of Qt. By Theorem 18, H_(g) is a f-ideal of Qt.
Consider(28)H_gx2∗1y2=infd∈Hx2∗1y2⁡gd=infd∈Hx2∗2Hy2⁡gd.
SinceH is a SSV-Hom, we have a∈H(x), c∈H(y) such that d=a∗2c.
Hence,(29)H_gx2∗1y2=infa∗2c∈Hx2∗2Hy2⁡ga∗2c=infa∈Hx2,c∈Hy2⁡ga∗2c=infa∈Hx2,c∈Hy2⁡gaorgc=infa∈Hx2⁡gaorinfc∈Hy2⁡gc=H_gx2orH_gy2.
ThereforeH_(g)(x2∗1y2)=H_(g)(x2) or H_(g)(x2∗1y2)=H_(g)(y2)∀x2,y2∈Qt. Hence H_(g) is a f-prime ideal of QtTheorem 26.
LetH be a SSV-Hom and let g be a f-semiprime ideal of Qt′. Then H_(g) is a f-semiprime ideal of Qt.Proof.
Asg is a f-semiprime ideal of Qt′, therefore g(d2)=g(d)∀d∈Qt′ and g is a f-ideal of Qt′, so by Theorem 14, H_(g) is a f-ideal of Qt.
Hence consider(30)H_gy=infd∈Hy⁡gd=infd∈Hy⁡gd2=infd∗2d∈Hy∗2Hy⁡gd2=infd∗2d∈Hy∗1y⁡gd2=infd2∈Hy2⁡gd2=H_gy2.
ThusH_(g)(y)=H_(g)(y2)∀y∈Qt. Therefore H_(g) is a f-semiprime ideal of Qt.Theorem 27.
LetH be a SSV-Hom and let g be a f-semiprime ideal of Qt′. Then H¯(g) is a f-semiprime ideal of Qt.Proof.
Proof is similar as reported in Theorem26.Corollary 28.
LetH be a SSV-Hom and let g be a f-semiprime ideal of Qt′. Then g is a GRF semiprime ideal of Qt′.Theorem 29.
Letg be a f-semiprime ideal of Qt′ and let H be a SSV-Hom. Then H_(g) (respectively, H¯(g)) is a f-semiprime ideal of Qt if and only if, for each α∈0,1, H_(gα) (respectively, H¯(gα)), where gα≠∅, is a semiprime ideal of Qt.Proof.
SupposeH_(g) is a f-semiprime ideal of Qt, then H_(g) is a f-ideal of Qt. By Theorem 18, H_(gα) is an ideal of Qt. In order to show that H_(gα) is a semiprime ideal ∀α∈0,1, we have to show that for a∗1a∈H_(gα) implies a∈H_(gα). Let a∗1a∈H_(gα). Since H_(g) is a f-semiprime ideal, we have H_(g)(a)=H_(g)(a∗1a)≥α. Thus, we have a∈H_(gα). Hence H_(gα) is a semiprime ideal of Qt.
Conversely, suppose thatH_(gα) is a semiprime ideal of Qt. Then H_(gα) is an ideal of Qt. By Theorem 18, H_(g) is a f-ideal.
ForH_(g) to be a f-semiprime ideal, we have to show that H_(g)(z∗1z)=H_(g)(z)∀z∈Qt. As H is a SSV-Hom and g is a f-semiprime ideal of Qt′, consider(31)H_gz=infc∈Hz⁡gc=infc∈Hz⁡gc2=infc∗2c∈Hz∗2Hz⁡gc2=infc∗2c∈Hz∗1z⁡gc2=infc2∈Hz2⁡gc2=H_gz2.
ThusH_(g)(z)=H_(g)(z2)∀z∈Qt. Hence H_(g) is a f-semiprime ideal of Qt.Example 30.
Let(Qt,∗1) and Qt′,∗2 be two quantales, where Qt and Qt′ are depicted in Figures 1 and 2 and the binary operations ∗1 and ∗2 on both the quantales are the same as the meet operation in the lattices Qt and Qt′ as shown in Tables 1 and 2.LetH:Qt→P∗(Qt′) be a SSV-Hom as defined in Example 19.Letλ be a fuzzy subset of Qt′ defined by λ=0.9/0′+0.6/i+0.9/j+0.6/1′. Then one can verify that λ is a f-prime ideal of Qt′.Hence GUA and GLA of thef-prime ideal λ are H¯(λ)=0.9/0+0.9/a+0.6/1 and H_(λ)=0.9/0+0.6/a+0.6/1. It is observed that H¯(λ) and H_(λ) are nonconstant f-prime ideals of Qt.Letg be a fuzzy subset of Qt′ defined by g(x)={1,  ifx=0′;0.6,  ifx≠0′}∀x∈Qt′. Then g is a f-semiprime ideal of Qt′. Hence GLA and GUA of f-semiprime ideal g are as follows: H_(g)=1/0+0.6/a+0.6/1 and H¯(g)=1/0+0.6/a+0.6/1. It is clear that H¯(g) and H_(g) are f-semiprime ideals of Qt.Theorem 31.
Letg be a f-primary ideal of Qt′ and let H be a SSV-Hom. Then H_(g) is a f-primary ideal of Qt.Proof.
Asg is a f-primary ideal of Qt′, therefore g(a∗2b)=g(a) or g(a∗2b)=g(bn)∀a,b∈Qt′ and hence, g is a f-ideal of Qt′, so by Theorem 14, H_(g) is f-ideal of Qt. Since H is given as SSV-Hom, consider(32)H_gz∗1y=infd∈Hz∗1y⁡gd=infa∗2b∈Hz∗2Hy⁡ga∗2b=infa∈Hz,b∈Hy⁡ga∗2b=infa∈Hz,b∈Hy⁡gaorgbn=infa∈Hz⁡gaorinfb∈Hy⁡gbn=infa∈Hz⁡gaorinfbn∈Hyn⁡gbn=H_gzorH_gyn.
Herebn=b∗2b∗2,…,∗2b∈H(y)∗2H(y)∗2,…, ∗2H(y)=H(y∗1y∗1y∗1,…,∗1y)=H(yn) up to n times for some positive integer n. Thus H_(g)(z∗1y)=H_(g)(z) or H_(g)(z∗1y)=H_(g)(yn)∀z,y∈Qt. Therefore H_(g) is a f-primary ideal of Qt.Theorem 32.
Letg be a f-primary ideal of Qt′ and let H be a SSV-Hom. Then H¯(g) is a f-primary ideal of Qt.Proof.
The proof is similar to the proof of Theorem31.Theorem 33.
LetH be a SSV-Hom and let g be a nonconstant f-primary ideal of Qt′. Then H_(g) (respectively, H¯(g)) is a f-primary ideal of Qt if and only if for each α∈[0,1], H_(gα) (respectively, H¯(gα)), where gα≠∅, is a primary ideal of Qt.


## 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(Qt,∗1) and (Qt′,∗2) be two quantales. A map f:Qt→Qt′ is called a quantale homomorphism if(1)
f(a∗1b)=f(a)∗2f(b);(2)
f(⋁i∈Iai)=⋁i∈If(ai)∀a,b,ai∈Qt(i∈I).
A quantale homomorphismf:Qt→Qt′ is called anepimorphism if f is onto  Qt′ and f is called amonomorphism if f is one-one. If f is bijective, then it is called anisomorphism.
It is clear that ifx≤y, then f(x)≤f(y); that is, f is order-preserving.Proposition 35.
Let(Qt,∗1) and (Qt′,∗2) be two quantales, let f:Qt→Qt′ be an epimorphism, and let H2:Qt′→P∗(Qt′) be a SV-Hom. Then one has the following:(1) Iff is one to one and H1(x)=y∈Qt∣f(y)∈H2(f(x))∀x∈Qt, then H1 is a SV-Hom from Qt to P∗Qt.(2) IfH2 is a SSV-Hom, then H1 is a SSV-Hom.Proof.
(
1
) First of all, we show that H1 is a well-defined mapping. Suppose x1=x2, then we have y1∈H1(x1)⇔f(y1)∈H2(f(x1))=H2(f(x2))⇔y1∈H1(x2). Thus we have H1(x1)=H1(x2). Now we show that H1 is SV-Hom. Suppose y∈H1(x1)∗1H1(x2), then there exist a∈H1(x1) and b∈H1(x2) such that y=a∗1b. Since H2 is a SV-Hom and f is a quantale homomorphism, then fa∗2fb∈H2fx1∗2H2fx2⊆H2(f(x1)∗2f(x2))=H2(f(x1∗1x2)). Therefore, f(a∗1b)=f(a)∗2f(b)∈H2(f(x1∗1x2)). Hence y=a∗1b∈H1(x1∗1x2). Thus, we have H1(x1)∗1H1(x2)⊆H1(x1∗1x2). Now we show that ⋁i∈IH1(xi)⊆H1(⋁i∈Ixi)∀xi∈Qt(i∈I). Let y∈⋁i∈IH1(xi), then there exists ai∈H1(xi)∀i∈I such that y=⋁i∈Iai. Hence f(y)=f(⋁i∈Iai)=⋁i∈If(ai)∈⋁i∈IH2(f(xi))⊆H2(⋁i∈If(xi))=H2(f(⋁i∈Ixi)). Finally, y=⋁i∈Iai∈H1(⋁i∈Ixi). Hence ⋁i∈IH1(xi)⊆H1(⋁i∈Ixi). So, H1 is a SV-Hom from Qt to P∗(Qt).
(
2
) It is similar to part(1).Theorem 36.
Letf:Qt→Qt′ be a quantale isomorphism and let H2:Qt′→P∗(Qt′) be a SV-Hom. Set H1(m)=z∈Qt∣f(z)∈H2(f(m))∀m∈Qt and ∀∅≠A⊆Qt′, then(1)
f(H¯1(A))=H¯2(f(A));(2)
f(H_1(A))=H_2(f(A));(3)
f(x)∈f(H¯1(A))⇔x∈H¯1(A).Proof.
(
1
) Let z∈f(H¯1(A)). Then there exists x∈H¯1(A) such that f(x)=z. Since x∈H¯1(A), H1(x)∩A≠∅. Suppose z′∈H1(x)∩A, then f(z′)∈f(A), and by the definition of H1(x), we obtain f(z′)∈H2(f(x)). Thus, H2(f(x))∩f(A)≠∅, and hence z=f(x)∈H¯2(f(A)). Thus, we obtain f(H¯1A)⊆H¯2(f(A)).
Now we takey∈H¯2(f(A)), then there is m∈Qt such that f(m)=y. Hence H2(f(m))∩f(A)≠∅. So there is z1∈A such that f(z1)∈f(A) and f(z1)∈H2(f(m)). By the definition of H1(m), we have z1∈H1(m). Thus H1(m)∩A≠∅. This gives m∈H¯1(A). Hence, y=f(m)∈f(H¯1(A)). Thus H¯2(f(A))⊆f(H¯1A). Finally, we obtain f(H¯1A)=H¯2(f(A)).
(
2
) Suppose z∈f(H_1(A)), then there exists m∈H_1(A) such that f(m)=z and H1(m)⊆A. Suppose z′∈H2(f(m)), then there is n′∈Qt such that f(n′)=z′; hence f(n′)∈H2(f(m)). Thus n′∈H1(m)⊆A, so z′=f(n′)∈f(A). Hence, H2(f(m))⊆f(A). Thus z=f(m)∈H_2(f(A)), so we have f(H_1(A))⊆H_2(f(A)).
Now lety∈H_2(f(A)). Then there exists n∈Qt such that f(n)=y and H2(f(n))⊆f(A). Suppose n′∈H1(n), then f(n′)∈H2(f(n))⊆f(A) and hence n′∈A. Thus H1(n)⊆A and we obtain n∈H_1(A). Hence f(n)=y∈f(H_1(A)) and thus, H_2(f(A))⊆f(H_1(A)). Hence finally, we have f(H_1(A))=H_2(f(A)).
(
3
) Let x∈H¯1(A). Then f(x)∈f(H¯1(A)). Conversely, suppose that f(x)∈f(H¯1(A)), then there is y∈H¯1(A) such that f(x)=f(y). Since f is ono-one, hence x=y∈H¯1(A).Remark 37.
From Theorem36(3), it is easily obtained that f(x)∈f(H_1(A))⇔x∈H_1(A)∀∅≠A⊆Qt′.Theorem 38.
Letf:Qt→Qt′ be a surjective quantale homomorphism and let H2:Qt′→P∗(Qt′) be a SV-Hom. Let H1(x)=y∈Qt∣f(y)∈H2(f(x))∀x∈Qt and ∀∅≠A⊆Qt′. Then,(1)
H¯1(A) is an ideal of Qt iff  H¯2(f(A)) is an ideal of Qt′;(2)
H¯1(A) is a prime ideal of Qt iff H¯2(f(A)) is a prime ideal of Qt′;(3)
H¯1(A) is a semiprime ideal of Qt iff H¯2(f(A)) is a semiprime ideal of Qt′;(4)
H¯1(A) is a primary ideal of Qt iff H¯2(f(A)) is a primary ideal of Qt′.Proof.
(
1
) Suppose H¯1(A) is an ideal of Qt. We show that H¯2(f(A)) is an ideal of Qt′, where f(H¯1(A))=H¯2(f(A)) by Theorem 36(1).
(i) Letx,z∈f(H¯1(A)). Then there exists x1,z1∈H¯1(A) such that f(x1)=x and f(z1)=z. Since f is a surjective quantale homomorphism and H¯1(A) is an ideal of Qt, we have x∨z=f(x1)∨f(z1)=f(x1∨z1)∈f(H¯1(A)). Therefore x∨z∈f(H¯1(A))∀x,z∈f(H¯1(A)).
(ii) Letz≤x∈f(H¯1(A)). Then we obtain x1∈H¯1(A) and z1∈Qt such that f(x1)=x and f(z1)=z. Since f(z1)≤f(x1), we have f(x1∨z1)=f(x1)∨f(z1)=f(x1)∈f(H¯1(A)). But H¯1(A) is a lower set and z1≤x1∨z1, and we have z1∈H¯1(A). Thus z=f(z1)∈f(H¯1(A)).
(iii) Letx∈f(H¯1(A)) and z∈Qt′. Then there exist x1∈H¯1(A) and z1∈Qt such that f(x1)=x and f(z1)=z. Since H¯1(A) is an ideal and f is a quantale homomorphism, we have x1∗1z1∈H¯1(A). Hence x∗2z=f(x1)∗2f(z1)=f(x1∗1z1)∈f(H¯1(A)). In the same way, we have z∗2x∈f(H¯1(A)). Hence, f(H¯1(A)) is an ideal of Qt′. But f(H¯1(A))=H¯2(f(A)). So H¯2(f(A)) is an ideal of Qt′.
Conversely, supposef(H¯1(A))=H¯2(f(A)) is an ideal of Qt′.
(i) Letz1,z2∈H¯1(A). Then f(z1),f(z2)∈f(H¯1(A)). Since f(H¯1(A)) is directed, f(z1∨z2)=f(z1)∨f(z2)∈f(H¯1(A)). So by Theorem 36(3), we have z1∨z2∈H¯1(A). Hence H¯1(A) is directed.
(ii) Letz1≤z2∈H¯1(A). Then f(z1)≤f(z2)∈f(H¯1(A)). Since f(H¯1(A)) is a lower set, then f(z1)∈f(H¯1(A)). By Theorem 36(3), we obtain z1∈H¯1(A). So H¯1(A) is a lower set.
(iii) Supposey′∈Qt and y∈H¯1(A), then f(y′)∈Qt′ and f(y)∈f(H¯1(A)). But f(H¯1(A)) is an ideal of Qt′, and we have f(y∗1y′)=f(y)∗2f(y′)∈f(H¯1(A)). Thus by Theorem 36(3), we have y∗1y′∈H¯1(A). Similarly, y′∗1y∈H¯1(A). So, H¯1(A) is an ideal of Qt.
(
2
) Let H¯1(A) be a prime ideal of Qt. Then H¯1(A) is obviously an ideal of Qt and H¯1(A)≠Qt. By part (1), H¯2(f(A)) is an ideal of Qt′. We also have that H¯2(f(A))=f(H¯1(A))≠Qt′. Now suppose y1,y2∈Qt′ and y1∗2y2∈H¯2(f(A)). Since f is surjective, there are z1,z2∈Qt such that y1=f(z1), y2=f(z2). Then f(z1∗1z2)=f(z1)∗2f(z2)=y1∗2y2∈f(H¯1(A)). By Theorem 36(3), we obtain z1∗1z2∈H¯1(A). But H¯1(A) is prime, and we have z1∈H¯1(A) or z2∈H¯1(A). Thus y1∈f(H¯1(A))=H¯2(f(A)) or y2∈f(H¯1(A))=H¯2(f(A)). So H¯2(f(A)) is a prime ideal of Qt′.
Conversely, letH¯2(f(A)) be a prime ideal of Qt′. Then H¯2(f(A)) is an ideal of Qt′. Since f(H¯1A)=H¯2(fA)≠Qt′, thus H¯1(A)≠Qt. By part (1), H¯1(A) is an ideal of Qt. Now suppose z1,z2∈Qt and z1∗1z2∈H¯1(A). So f(z1)∗2f(z2)=f(z1∗1z2)∈f(H¯1(A)). Since f(H¯1(A))=H¯2(f(A)) is prime, we have f(z1)∈f(H¯1(A)) or f(z2)∈f(H¯1(A)). So by Theorem 36(3), we have z1∈H¯1(A) or z2∈H¯1(A). Thus H¯1(A) is a prime ideal of Qt.
The remaining parts(3) and (4) are similar to the proof (2).Theorem 39.
Letf:Qt→Qt′ be a surjective quantale homomorphism and let H2:Qt′→P∗(Qt′) be a SV-Hom. Set H1(x)=y∈Qt∣f(y)∈H2(f(x))∀x∈Qt and ∀∅≠B⊆Qt′. Then the following hold:(1)
H_1(B) is an ideal of Qt iff  H_2(f(B)) is an ideal of Qt′;(2)
H_1(B) is a prime ideal of Qt iff H_2(f(B)) is a prime ideal of Qt′;(3)
H_1(B) is a semiprime ideal of Qt iff H_2(f(B)) is a semiprime ideal of Qt′;(4)
H_1(B) is a primary ideal of Qt iff H_2(f(B)) is a primary ideal of Qt′.Proof.
The proofs of all the parts can be obtained by Theorem38.


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

Theorem 40.
Letf:Qt→Qt′ be a surjective quantale homomorphism, let H2:Qt′→P∗(Qt′) be a SV-Hom, and let λ be a fuzzy subset of Qt′. If H1(x)=y∈Qt∣f(y)∈H2(f(x))∀x∈Qt, then(1)
H¯1(λ) is a f-ideal of Qt iff H¯2(f(λ)) is a f-ideal of Qt′;(2)
H¯1(λ) is a f-prime ideal of Qt iff H¯2(f(λ)) is a f-prime ideal of Qt′;(3)
H¯1(λ) is a f-semiprime ideal of Qt iff H¯2(f(λ)) is a f-semiprime ideal of Qt′;(4)
H¯1(λ) is a f-primary ideal of Qt iff H¯2(f(λ)) is a f-primary ideal of Qt′.In the above,f(λ)(y)=⋁λ(x)∣f(x)=y,x∈Qt, y∈Qt′; that is, f(λ) is the standard Zadeh image of the fuzzy subset λ under the mapping f.Proof.
(
1
) We first point out that, for each α∈0,1, (f(λ))α+=f(λα+) and (H¯1(λ))α+≠∅ if and only if (H¯2(f(λ)))α+≠∅.
LetH¯1(λ) be a f-ideal of Qt. Then for all α∈(0,1], if (H¯2(fλ))α+≠∅, then (H¯1(λ))α+≠∅. By Theorem 18, we have (H¯1(λ))α+ is an ideal of Qt. Also by using Proposition 12, we obtain that H¯1(λα+) is an ideal of Qt. Now, by Theorem 38(1), we have that (H¯2(f(λ)))α+=H¯2(f(λ))α+=H¯2(f(λα+)) is an ideal of Qt′. Thus, by Theorem 18, we have that H¯2(f(λ)) is a f-ideal of Qt′.
Conversely, supposeH¯2(f(λ)) is a f-ideal of Qt′. We have that (H¯2(f(λ)))α+=H¯2(f(λ))α+=H¯2(f(λα+)) is an ideal of Qt′ by utilizing Theorem 18. It is obtained from Theorem 38(1) that H¯1(λα+) is an ideal of Qt. Hence by Theorem 18, H¯1(λ) is a f-ideal of Qt.
(
2
) Let H¯1(λ) be a f-prime ideal of Qt. Now for H¯2(f(λ))α+≠∅, then (H¯1(λ))α+≠∅ for any α∈0,1. Since H¯1(λ) is a f-prime ideal of Qt, then by Theorem 25, we have that (H¯1(λ))α+ is a prime ideal of Qt. It is also obtained from Proposition 12  that H¯1(λα+) is a prime ideal of Qt. Hence (H¯2(f(λ)))α+=H¯2(f(λ))α+=H¯2(f(λα+)) is a prime ideal of Qt′, by Theorem 38(2). Thus, by Theorem 25, we have that H¯2(f(λ)) is a f-prime ideal of Qt′.
Conversely, supposeH¯2(f(λ)) is a f-prime ideal of Qt′. By Theorem 25, we obtain that (H¯2(f(λ)))α+=H¯2(f(λ))α+=H¯2(f(λα+)) is a prime ideal of Qt′. Thus it is obtained, from Theorem 38(2), that H¯1(λα+) is a prime ideal of Qt. Hence H¯1(λ) is a f-prime ideal of Qt by Theorem 25.
Axioms(3) and (4) can be obtained in a similar way.Theorem 41.
Letf be a surjective quantale homomorphism from a quantale (Qt,∗1) onto a quantale (Qt′,∗2). Let H2:Qt′→P∗(Qt′) be a SV-Hom and let λ be a fuzzy subset of Qt′. If H1(x)=y∈Qt∣f(y)∈H2(f(x))∀x∈Qt, then(1)
H_1(λ) is a f-ideal of Qt iff H_2(f(λ)) is a f-ideal of Qt′;(2)
H_1(λ) is a f-prime ideal of Qt iff H_2(f(λ)) is a f-prime ideal of Qt′;(3)
H_1(λ) is a f-semiprime ideal of Qt iff H_2(f(λ)) is a f-semiprime ideal of Qt′;(4)
H_1(λ) is a f-primary ideal of Qt iff H_2(f(λ)) is a f-primary ideal of Qt′.Proof.
The proof is similar as reported in Theorem40.


## 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.



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