# On Intuitionistic Fuzzy Filters of Intuitionistic Fuzzy Coframes

Authors: Rajesh K. Thumbakara
Journal: Journal of Mathematics
 (2013)

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## Abstract

Frame theory is the study of topology based on its open set lattice, and it was studied extensively by various authors. In this paper, we study  quotients of intuitionistic fuzzy filters of an intuitionistic fuzzy coframe. The quotients of intuitionistic fuzzy filters are shown to be filters of the given intuitionistic fuzzy coframe. It is shown that the collection of all intuitionistic fuzzy filters of a coframe and the collection of all intutionistic fuzzy quotient filters of an intuitionistic fuzzy filter are coframes.


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## Body

## 1. Introduction

Frame theory is topology seen through notions of lattice theory; here one takes the lattice of open sets as the basic notion. The concept of Frames has been studied by many mathematicians including Banaschewski, Dowker, and Johnstone. For details one can refer to [1–3]. In 1965 Zadeh [4] introduced the concept of fuzzy sets as the generalization of ordinary subsets. The concept of ideals of a fuzzy subring was studied by Mordeson and Malik in [5] and Prajapati in [6]. In 1983, Atanassov [7] proposed a generalization of the notion of fuzzy set, known as intuitionistic fuzzy sets. In our earlier paper [8, 9], we introduced the concept of intuitionistic fuzzy frames and intuitionistic fuzzy filters of a frame. In this paper, intuitionistic fuzzy filters of a coframe and intuitionistic fuzzy filters of an intuitionistic fuzzy coframe are studied and examined. Subcoframe of the collection of all intuitionistic fuzzy filters of the coframe F is obtained.


## 2. Preliminaries

In this section we will review some fundamental definitions.Definition 1.
LetF be any set; then for any arbitrary S⊆F, ∧S=inf{x∣x∈S} and ∨S=sup{x∣x∈S}.Definition 2 (see [1, 2]).
A frame is a complete latticeF satisfying the infinite distributive law a∧(∨S)=∨{a∧x∣x∈S} for any a∈F and S⊆F.Definition 3 (see [1, 2]).
A coframe is a complete latticeF satisfying the infinite distributive law a∨(∧S)=∧{a∨x∣x∈S} for any a∈F and S⊆F.Definition 4 (see [1, 2]).
A subsetM of a coframe F closed under arbitrary meets, finite joins and having unit (top) element eF and zero (bottom) element oF of the coframe F is called a sub coframe of F.Definition 5 (see [1, 2]).
For any coframeF, a subset P of  F is a filter if x,y∈P⇒x∧y∈P and x∈P,  y≥x⇒y∈P.Definition 6 (see [7]).
LetX be a nonempty set. An intuitionistic fuzzy set (IFS) A of X is an object of the form A={(x,μA(x),γA(x))∣x∈F}, where μA:X→[0,1] and  γA:X→[0,1] define, respectively, the degree  of membership and the degree of nonmembership of the element x∈X and 0≤μA(x)+γA(x)≤1, for all x∈X.Definition 7 (see [8]).
LetF be a coframe then an IFS A={(x,μA(x),γA(x))∣x∈F} in F is called an intuitionistic fuzzy coframe (IFCF) of F if it satisfies the following conditions:(i)
μA(x∨y)≥min{μA(x),μA(y)}γA(x∨y)≤max{γA(x),γA(y)} for all x,y∈F,(ii)
μA(∧S)≥inf{μA(x)∣x∈S}γA(∧S)≤sup{γA(x)∣x∈S} for arbitrary S⊆F,(iii)
μA(eF)=μA(oF)≥μA(x)γA(eF)=γA(oF)≤γA(x) for all x∈F, where eF and oF are, respectively, the unit and zero element of the coframe F.Definition 8 (see [7]).
If{Aα∣α∈Λ} is a family of IFS of F where Aα={(x,μAα(x),γAα(x))∣x∈F}, then
(1)⋂α∈ΛAα={(x,μ⋂α∈ΛAα(x),γ⋂α∈ΛAα(x))∣x∈F}={(x,inf{μAα(x)},sup{γAα(x)})∣x∈F},⋃α∈ΛAα={(x,μ⋃α∈ΛAα(x),γ⋃α∈ΛAα(x))∣x∈F}={(x,sup{μAα(x)},inf{γAα(x)})∣x∈F}.Definition 9 (see [9]).
The operation of meet∧ and join ∨ on a coframe F can be extended to operations ∧~ and ∨~ on  the set of all intuitionistic fuzzy set IFS of F as follows.
ForA,B∈IFS  one has
(2)A∧~B={(x,μA∧~B(x),γA∧~B(x))∣x∈F},
where  μA∧~B(x)=supu∧v=x{μA(u)∧μB(v)} and γA∧~B(x)=infu∧v=x{γA(u)∨γB(v)}(3)A∨~B={(x,μA∨~B(x),γA∨~B(x))∣x∈F},
where μA∨~B(x)=supu∨v=x{μA(u)∧μB(v)} and γA∨~B(x)=infu∨v=x{γA(u)∨γB(v)}.The original operation∧ and ∨ on a coframe F can be retrieved from ∧~ and ∨~ by embedding F into IFS as the set of all intuitionistic fuzzy singletons each of which is an IFS,
(4)1x={(y,μx(y),γx(y))∣y∈F},
where
(5)μx(y)={1ify=x,0otherwise,γx(y)={0ify=x,1otherwise.Lemma 10 (see [9]).
LetA,B,C∈IFS;  then (i)
A∨~(B∧~C)⊆(A∨~B)∧~(A∨~C)(ii)
A⊆B⇒A∨~C⊆B∨~C.


## 3. Intuitionistic Fuzzy Filter of a Coframe

In this section, we show that the collection of all intuitionistic fuzzy filters of the coframeF is a coframe. We use IFF(F) to denote intuitionistic fuzzy filter of the coframe F.Definition 11 (see [9]).
LetF be a coframe; then an IFS A on F is said to be an IFF(F) if   (F1)
μA(x∧y)≥min{μA(x),μA(y)}, for all x,y∈FγA(x∧y)≤max{γA(x),γA(y)}, for all x,y∈F,(F2)
μA(x∨y)≥max{μA(x),μA(y)}, for all x,y∈FγA(x∨y)≤min{γA(x),γA(y)}, for all x,y∈F,(F3)
μA(eF)=1,γA(eF)=0 where eF is the unit element of F.Lemma 12 (see [9]).
IfA and B are any two  IFF(F), then, A∩B is an  IFF(F).Result 1 (see [9]).
Union of any twoIFF(F) needs not be an IFF(F).Definition 13.
LetA be an IFS. Let 〈A〉=⋂{B∣A⊆B,BanIFF(F)} where A⊆B means μA(x)≤μB(x) and γA(x)≥γB(x) for all x∈F.  Then 〈A〉 is called the IFF(F) generated by A.Theorem 14.
LetA,B be two IFF(F); then A∧~B is an IFF(F) and A∧~B=〈A∪B〉.Proof.
Letx∈F; then
(6)(A∧~B)(x)={∫(x,supu∧v=x{μA(u)∧μB(v)},infu∧v=x{γA(u)∨γB(v)})∣x∈F∫}⊇(x,μA(x)∧μB(eF),γA(x)∨γB(eF))=(x,μA(x),γA(x))=A.
Hence A∧~B⊇A. Similarly, we have A∧~B⊇B. Thus, A∧~B⊇A∪B. Also,(F1)
(7)μA∧~B(x∧y)=supu∧v=x∧y{μA(u)∧μB(v)}≥⋁{∫μA(u1∧v1)∧μB(u2∧v2)∣u1∧u2=x,v1∧v2=y;u1,v1,u2,v2∈F∫}≥⋁{∫(μA(u1)∧μA(v1))∧(μB(u2)∧μB(v2))∣u1∧u2=x,v1∧v2=y;u1,v1,u2,v2∈F∫}=⋁{∫(μA(u1)∧μB(u2))∧(μA(v1)∧μB(v2))∣u1∧u2=x,v1∧v2=y;u1,v1,u2,v2∈F∫}=⋁{μA(u1)∧μB(u2)∣u1∧u2=x;u1,u2∈F}∧⋁{μA(v1)∧μB(v2)∣v1∧v2=y;v1,v2∈F}=μA∧~B(x)∧μA∧~B(y)∀x,y∈F,γA∧~B(x∧y)=infu∧v=x∧y{γA(u)∨γB(v)}≤⋀{∫γA(u1∧v1)∨γB(u2∧v2)∣u1∧u2=x,v1∧v2=y;u1,v1,u2,v2∈F∫}≤⋀{∫(γA(u1)∨γA(v1))∨(γB(u2)∨γB(v2))∣u1∧u2=x,v1∧v2=y;u1,v1,u2,v2∈F∫}=⋀{∫(γA(u1)∨γB(u2))∨(γA(v1)∨γB(v2))∣u1∧u2=x,v1∧v2=y;u1,v1,u2,v2∈F∫}=⋀{γA(u1)∨γB(u2)∣u1∧u2=x;u1,u2∈F}∨⋀{γA(v1)∨γB(v2)∣v1∧v2=y;v1,v2∈F}=γA∧~B(x)∨γA∧~B(y)∀x,y∈F,(F2)
(8)μA∧~B(x∨y)=supu∧v=x∨y{μA(u)∧μB(v)}≥⋁{μA(y∨u)∧μB(y∨v)∣u∧v=x;u,v∈F}[∵(y∨u)∧(y∨v)=x∨y]≥⋁{μA(u)∧μB(v)∣u∧v=x;u,v∈F}=μA∧~B(x)∀x,y∈F.Similarly, we haveμA∧~B(x∨y)≥μA∧~B(y) for all x,y∈F.Therefore,μA∧~B(x∨y)≥μA∧~B(x)∨μA∧~B(y), for all x,y∈F(9)γA∧~B(x∨y)=infu∧v=x∨y{γA(u)∨γB(v)}≤⋀{γA(y∨u)∨γB(y∨v)∣u∧v=x;u,v∈F}[∵(y∨u)∧(y∨v)=x∨y]≤⋀{γA(u)∨γB(v)∣u∧v=x;u,v∈F}=γA∧~B(x)∀x,y∈F.Similarly, we haveγA∧~B(x∨y)≤γA∧~B(y) for all x,y∈F.Therefore,γA∧~B(x∨y)≤γA∧~B(x)∧γA∧~B(y), for all x,y∈F.(F3)
SinceμA(eF)=μB(eF)=1, clearly μA∧~B(eF)=supu∧v=eF{μA(u)∧μB(v)}≥μA(eF)∧μB(eF)=1 and since γA(eF)=γB(eF)=0, we have γA∧~B(eF)=infu∧v=eF{γA(u)∧γB(v)}≤γA(eF)∧γB(eF)=0.Thus ifA,  B are IFF(F), then A∧~B is an IFF(F).Now letC be any IFF(F) such that A∪B⊆C, then,
(10)μA∧~B(x)=supu∧v=x{μA(u)∧μB(v)}≤supu∧v=x{μC(u)∧μC(v)}≤supu∧v=x{μC(u∧v)}=μC(x),γA∧~B(x)=infu∧v=x{γA(u)∨γB(v)}≥infu∧v=x{γC(u)∨γC(v)}≥infu∧v=x{γC(u∧v)}=γC(x).
Hence A∧~B⊆C. Thus A∧~B is the smallest filter of F such that A∪B⊆A∧~B.HenceA∧~B=〈A∪B〉.Proposition 15.
LetA,  B be two IFF(F); then A∨~B=A∩B.Proof.
We have, forx∈F,  μA∨~B(x)=supu∨v=x{μA(u)∧μB(v)}≥μA(x)∧μB(x)=μA∩B(x) and γA∨~B(x)=infu∨v=x{γA(u)∨γB(v)}≤γA(x)∨γB(x)=γA∩B(x). Hence A∩B⊆A∨~B.
Now sinceμi(u∨v)=μi(x)≥μi(u)∨μi(v) and hence μi(x)≥μi(u) and μi(x)≥μi(v) for i=A,B(11)μA∨~B(x)=supu∨v=x{μA(u)∧μB(v)}≤supu∨v=x{μA(x)∧μB(x)}=μA(x)∧μB(x)=μA∩B(x),γA∨~B(x)=infu∨v=x{γA(u)∨γB(v)}≥infu∨v=x{γA(x)∨γB(x)}=γA(x)∨γB(x)=γA∩B(x).
Hence A∨~B⊆A∩B. Therefore A∨~B=A∩B.Theorem 16.
Given an arbitrary collection(Aα)α∈∧ and A of IFF(F), then
(12)A∩(⋀α∈∧~Aα)=⋀α∈∧~(A∩Aα).Proof.
We have forw∈F,
(13)μA∩(∧~α∈∧Aα)(w)=μA(w)∧(⋀α∈∧~μAα)(w)=μA(w)∧sup∧α∈∧aα=w{⋀α∈∧μAα(aα)}=sup∧α∈∧aα=w{⋀α∈∧(μA(w)∧μAα(aα))}≤sup∧α∈∧aα=w{⋀α∈∧(μA(aα∨w)∧μAα(aα))}=sup∧α∈∧aα=w{⋀α∈∧(μA(aα)∧μAα(aα))}=sup∧α∈∧aα=w{⋀α∈∧(μA∩μAα)(aα)}=μ∧~α∈∧(A∩Aα)(w),γA∩(∧~α∈∧Aα)(w)=γA(w)∧(⋀α∈∧~γAα)(w)=γA(w)∧inf∧α∈∧aα=w{⋁α∈∧γAα(aα)}=inf∧α∈∧aα=w{⋁α∈∧(γA(w)∧γAα(aα))}≥inf∧α∈∧aα=w{⋁α∈∧(γA(aα∨w)∧γAα(aα))}=inf∧α∈∧aα=w{⋁α∈∧(γA(aα)∧γAα(aα))}=inf∧α∈∧aα=w{⋁α∈∧(γA∩γAα)(aα)}=γ∧~α∈∧(A∩Aα)(w).
Therefore,
(14)A∩(⋀α∈∧~Aα)⊆⋀α∈∧~(A∩Aα).
Also A∩Aα⊆Aα and A∩Aα⊆A for all α∈∧; hence  ⋀~α∈∧(A∩Aα)⊆⋀~α∈∧Aα and ⋀~α∈∧(A∩Aα)⊆A.  Therefore,
(15)⋀α∈∧~(A∩Aα)⊆A∩(⋀α∈∧~Aα).
From (14) and (15) we have ⋀~α∈∧(A∩Aα)=A∩(⋀~α∈∧Aα).Remark 17.
If we interchange the roles of∧~ and ∩ in Theorem 16, only one-sided inequality A∧~(⋂α∈∧Aα)⊆⋂α∈∧(A∧~Aα) holds.Theorem 18.
The setIFF of all intuitionistic fuzzy filters of the coframe F is a coframe.Proof.
I
F
F is a complete lattice, which is bounded above by AχF={(x,μAχF(x),γAχF(x))∣x∈F} where μAχF(x)=1,  γAχF(x)=0 for all x∈F and bounded below by AeF={(x,μeF(x),γeF(x))∣x∈F} where
(16)μeF(x)={1x=eF,0otherwise,γeF(x)={0x=eF1otherwise∀x∈F.
Also ⋀~α∈∧(A∨~Aα)=A∨~(⋀~α∈∧Aα) for arbitrary collection (Aα)α∈∧ and A of IFF(F) by Proposition 15 and Theorem 16. Hence IFF is a coframe.


## 4. Intuitionistic Fuzzy Filter of an Intuitionistic Fuzzy Coframe

Definition 19.
LetA be an intuitionistic fuzzy coframe (IFCF) of F and B an IFS with B⊆A. Then B is called an intuitionistic fuzzy filter of A(IFF(A)) if(i)
μB(x∧y)≥min{μB(x),μB(y)}γB(x∧y)≤max{γB(x),γB(y)}.(ii)
μB(x∨y)≥max{μA(x)∧μB(y),μB(x)∧μA(y)} for all x,y∈F.γB(x∨y)≤min{γA(x)∨γB(y),γB(x)∨γA(y)} for all x,y∈F.(iii)
μB(eF)=μA(eF) and γB(eF)=γA(eF) where eF is the unit element of F.If B is  an  IFF(A), then we write B⊲A.Theorem 20.
LetA be an IFCF of F and B an IFF(A). Then A∩B  is an IFF(A).Proof.
ObviouslyA∩B⊆A and(i)
(17)μA∩B(x∧y)=min{μA(x∧y),μB(x∧y)}≥min{min(μA(x),μA(y)),min(μB(x),μB(y))}≥min{min(μA(x),μB(x)),min(μA(y),μB(y))}=μA∩B(x)∧μA∩B(y),∀x,y∈F,γA∩B(x∧y)=max{γA(x∧y),γB(x∧y)}≤max{max(γA(x),γA(y)),max(γB(x),γB(y))}=max{max(γA(x),γB(x)),max(γA(y),γB(y))}=γA∩B(x)∨γA∩B(y),∀x,y∈F,(ii)
(18)μA∩B(x∨y)=min{μA(x∨y),μB(x∨y)}≥min{μA(x)∧μA(y),(μA(x)∧μB(y))∨(μB(x)∧μA(y))}=(μA(x)∧μA(y))∧[(μA(x)∧μB(y))∨(μB(x)∧μA(y))]=(μA(x)∧μA(y)∧μB(y))∨(μA(x)∧μB(x)∧μA(y))=(μA(x)∧μA∩B(y))∨(μA∩B(x)∧μA(y))∀x,y∈F,γA∩B(x∨y)=max{γA(x∨y),γB(x∨y)}≤max{γA(x)∨γA(y),(γA(x)∨γB(y))∧(γB(x)∨γA(y))}=(γA(x)∨γA(y))∨[(γA(x)∨γB(y))∧(γB(x)∨γA(y))]=(γA(x)∨γA(y)∨γB(y))∧(γA(x)∨γB(x)∨γA(y))=(γA(x)∨γA∩B(y))∧(γA∩B(x)∨γA(y))∀x,y∈F,(iii)
also clearlyμA∩B(eF)=μA(eF) and γA∩B(eF)=γA(eF) where eF is the unit element of F.Theorem 21.
LetA be an IFCF of F, and let B,  C be two IFF(A). Then B∩C is also an IFF(A).Proof.
ClearlyB∩C⊆A also(i)
(19)μB∩C(x∧y)≥μB∩C(x)∧μB∩C(y),γA∩B(x∧y)≤γA∩B(x)∨γA∩B(y)∀x,y∈F,(ii)
(20)μB∩C(x∨y)=min{μB(x∨y),μC(x∨y)}≥min{(μA(x)∧μB(y))∨(μB(x)∧μA(y)),(μA(x)∧μC(y))∨(μC(x)∧μA(y))}=[μA(x)∧μB(y)∨(μB(x)∧μA(y))]∧[(μA(x)∧μC(y))∨(μC(x)∧μA(y))]≥(μA(x)∧μB(y)∧μC(y))∨(μB(x)∧μC(x)∧μA(y))=[μA(x)∧μB∩C(y)]∨[μB∩C(x)∧μA(y)]∀x,y∈F,γB∩C(x∨y)=max{γB(x∨y),γC(x∨y)}≤max{(γA(x)∨γB(y))∧(γB(x)∨γA(y)),(γA(x)∨γC(y))∧(γC(x)∨γA(y))}=[(γA(x)∨γB(y))∧(γB(x)∨γA(y))]∨[(γA(x)∨γC(y))∧(γC(x)∨γA(y))]≤(γA(x)∨γA(y)∨γB(y))∧(γA(x)∨γB(x)∨γA(y))=(γA(x)∨γA∩B(y))∨(γA∩B(x)∨γA(y))∀x,y∈F,(iii)
alsoμB∩C(eF)=μA(eF) and γB∩C(eF)=γA(eF).Theorem 22.
LetA be an IFCF of F and B  an  IFS with B⊆A. Then B is an IFF(A) if and only if(i)
μB(x∧y)≥μB(x)∧μB(y);  γB(x∧y)≤γB(x)∨γB(y) for all x,y∈F,(ii)
A∨~B⊆B,(iii)
μB(eF)=μA(eF) and γB(eF)=γA(eF).Proof.
Suppose conditions (i), (ii), and (iii) holds to prove thatB is an IFF(A).
SinceA∨~B⊆B, one has
(21)μB(x∨y)≥μA∨~B(x∨y)=supu∨v=x∨y{μA(u)∧μB(v)}≥μA(x)∧μB(y)∀x,y∈F,γB(x∨y)≤γA∨~B(x∨y)=infu∨v=x∨y{γA(u)∨γB(v)}≤γA(x)∧γB(y)∀x,y∈F.
Also μB(x∨y)≥μA(y)∧μB(x) and γB(x∨y)≤γA(y)∧γB(x) for all x,y∈F.
Hence,(22)μB(x∨y)≥max{μA(x)∧μB(y),μB(x)∧μA(y)},γB(x∨y)≤min{γA(x)∨γB(y),γB(x)∨γA(y)}.
Therefore B is an IFF(A).
Conversely ifB  is  an  IFF(A), we have  μB(x∧y)≥min{μB(x),μB(y)},  γB(x∧y)≤max{γB(x),γB(y)},  and  μB(eF)=μA(eF),  γB(eF)=γA(eF) from Definition 19.
Now for allz∈F  with z=x∨y,  μB(z)≥supx∨y=z{μA(x)∧μB(y)}=μA∨~B(z) and  γB(z)≤infx∨y=z{γA(x)∨γB(y)}=γA∨~B(z). Hence A∨~B⊆B.Theorem 23.
LetA be an  IFCF of F, and let B,  C be two IFF(A); then B∧~C is an IFF(A) and B⊆B∧~C,  C⊆B∧~C.Proof.
(i) We haveμB∧~C(x∧y)≥μB∧~C(x)∧μB∧~C(y) and γB∧~C(x∧y)≤γB∧~C(x)∧γB∧~C(y) from the proof of Theorem 14.
(ii) Now by Theorem22 since B,  C are IFF(A), we have A∨~B⊆B  and A∨~C⊆C.
HenceA∨~(B∧~C)⊆(A∨~B)∧~(A∨~C)⊆B∧~C by Lemma 10.
(iii) Also clearlyμB∧~C(eF)=μA(eF) and γB∧~C(eF)=γA(eF).
HenceB∧~C is an IFF(A) by Theorem 22.
AgainμB∧~C(x)=supu∧v=x{μB(u)∧μC(v)}≥μB(x)∧μC(eF)=μB(x) and γB∧~C(x)=infu∧v=x{γB(u)∨γC(v)}≤γB(x)∨γC(eF)=γB(x) for every x∈F.
HenceB⊆B∧~C. Similarly C⊆B∧~C.


## 5. Quotient of Intuitionistic Fuzzy Filters of the Intuitionistic Fuzzy Coframe

Definition 24.
LetA be an IFCF of F, and let B,  C be IFF(A). Then the quotient of B by C is denoted as [B:C] and is defined as
(23)[B:C]=⋃{D∣D⊲A,D∨~C⊆B}.
The collection of all quotients of any intuitionistic fuzzy filter B of A is denoted by [B]A.Theorem 25.
LetA be an IFCF of F, and let B,C be IFF(A). Then [B:C] is an IFF(A). Also B⊆[B:C]⊆A.Proof.
LetA~={D∣D⊲AandD∨~C⊆B}. Suppose D,D′∈A~, then D and D′ are IFF(A) such that D∨~C⊆B and D′∨~C⊆B. Now by Theorem 23  D∧~D′ is an IFF(A).
Now(D∧~D′)∨~C⊆(D∨~C)∧~(D′∨~C)⊆B∧~B⊆B by Lemma 10 and since
(24)μB∧~B(x)=supu∧v=x{μB(u)∧μB(v)}≤supu∧v=xμB(u∧v)=μB(x),γB∧~B(x)=infu∧v=x{γB(u)∨γB(v)}≥γB(u∧v)=γB(x).
Hence D∧~D′∈A~.
Now(25)μ[B:C](x)∧μ[B:C](y)=(⋁D∈A~μD(x))∧(⋁D′∈A~μD′(y))=⋁D,D′∈A~{μD(x)∧μD′(y)}≤⋁D,D′∈A~μD∧~D′(x∧y)≤μ[B:C](x∧y),
also
(26)γ[B:C](x)∨γ[B:C](y)=(⋀D∈A~γD(x))∨(⋀D′∈A~γD′(y))=⋀D,D′∈A~{γD(x)∨γD′(y)}≥⋀D,D′∈A~γD∧~D′(x∧y)≥γ[B:C](x∧y),
since D∧~D′∈A~.
Hence(27)μ[B:C](x∧y)≥min{μ[B:C](x),μ[B:C](y)},γ[B:C](x∧y)≤max{γ[B:C](x),γ[B:C](y)}.
Now
(28)μ[B:C](x∨y)=⋁D∈A~μD(x∨y)≥⋁D∈A~{(μA(x)∧μD(y))∨(μD(x)∧μA(y))}≥⋁D∈A~(μA(x)∧μD(y))=μA(x)∧{⋁D∈A~(μD(y))}=μA(x)∧μ[B:C](y).
Similarly μ[B:C](x∨y)≥μ[B:C](x)∧μA(y).
Thus(29)μ[B:C](x∨y)≥max{μA(x)∧μ[B:C](y),μ[B:C](x)∧μA(y)}∀x,y∈F.
Now
(30)γ[B:C](x∨y)=⋀D∈A~γD(x∨y)≤⋀D∈A~{(γA(x)∨γD(y))∧(γD(x)∨γA(y))}≤⋀D∈A~(γA(x)∨γD(y))=γA(x)∨{⋀D∈A~γD(y)}=γA(x)∨γ[B:C](y).
Similarly γ[B:C](x∨y)≤γ[B:C](x)∨γA(y).
Hence(31)γ[B:C](x∨y)≤min{γA(x)∨γ[B:C](y),γ[B:C](x)∨γA(y)}∀x,y∈F.
Also
(32)μ[B:C](eF)=⋁D∈A~μD(eF)=μA(eF),γ[B:C](eF)=⋀D∈A~γD(eF)=γA(eF).
Now from (27), (29), (31), and (32), [B:C] is an IFF(A).
Also clearly[B:C]⊆A. Since B is an IFF(A), we have by Theorem 22, A∨~B⊆B.
Now by Lemma10 since C⊆A, we have C∨~B⊆A∨~B⊆B. Hence B∈A~ and so B⊆[B:C]. Thus B⊆[B:C]⊆A.Theorem 26.
LetA be an IFCF of F, and let B,  C,  E be IFF(A). Then the following holds:(1)
ifB⊆C; then [B:E]⊆[C:E] and [E:C]⊆[E:B],(2)
ifB⊆C; then [C:B]=A,(3)
[B:B]=A,(4)
[B:C]=[B:B∧~C].Proof.
(
1
) Let B⊆C.  Let A~={D∣D⊲AandD∨~E⊆B} and A~′={D∣D⊲AandD∨~E⊆C}.
IfD∈A~, then D⊲AandD∨~E⊆B⊆C. Therefore D∈A~′ and hence A~⊆A~′.
So[B:E]=⋃D∈A~D⊆⋃D∈A~′D=[C:E] and hence [B:E]⊆[C:E].
Similarly it can be shown that[E:C]⊆[E:B].
(
2
) Let A~={D∣D⊲AandD∨~B⊆C}. We have A⊲A. Also B⊲A implies that A∨~B⊆B⊆C.
HenceA∈A~, and so A⊆⋃D∈A~D=[C:B]. Also since [C:B] is an IFF(A),  [C:B]⊆A.
Therefore,[C:B]=A.
(
3
) We have B⊆B. So from Theorem 26(2) we have [B:B]=A.
(
4
) We have by Theorem 23  B∧~C is an IFF(A) and C⊆B∧~C.
Hence[B:B∧~C]⊆[B:C] from   Theorem 26(1).
LetA~={D∣D⊲AandD∨~C⊆B} and A~′={D∣D⊲AandD∨~(B∧~C)⊆B}.
IfD∈A~, then D⊲AandD∨~C⊆B. Now since D⊆A, D∨~B⊆A∨~B by Lemma 10.
Also by Theorem22, B⊲A implies A∨~B⊆B. Hence D∨~B⊆B.
Therefore, by Lemma10, D∨~(B∧~C)⊆(D∨~B)∧~(D∨~C)⊆B∧~B=B and hence D∈A~′.
So[B:C]=⋃D∈A~D⊆⋃D∈A~′D=[B:B∧~C]. Thus [B:B∧~C]=[B:C].Corollary 27.
LetA be an IFCF of F, and let B,  C be IFF(A). Then(1)
[B:B∩C]=A,(2)
[B∧~C:B∩C]=A,(3)
[[B:B]:C]=A.Proof.
(
1
) Since B∩C is an IFF(A) by Theorem 20 and B∩C⊆B, we have [B:B∩C]=A by Theorem 26.
(
2
) Since B∩C⊆B∧~C, we have [B∧~C:B∩C]=A by Theorem 26.
(
3
) By Theorem 26, [B:B]=A also C⊆A. Hence [[B:B]:C]=A by Theorem 26.Theorem 28.
LetA be an IFCF of F, and let B,  C,  E be IFF(A). Then [B:C∧~E]=[B:C]∩[B:E].Proof.
We haveC∧~E is an IFF(A) such that C⊆C∧~E and E⊆C∧~E by Theorem 23.
Hence[B:C∧~E]⊆[B:C] and [B:C∧~E]⊆[B:E] by Theorem 26.
Hence(33)[B:C∧~E]⊆[B:C]∩[B:E].
Let A~={D∣D⊲AandD∨~C⊆B}, A~′={D∣D⊲AandD∨~E⊆B} and  A~′′={D∣D⊲AandD∨~(C∧~E)⊆B}.
For everyx∈F,
(34)μ[B:C]∩[B:E](x)=μ[B:C](x)∧μ[B:E](x)=(⋁D∈A~μD(x))∧(⋁D′∈A~′μD′(x))=⋁D′∈A~′((⋁D∈A~μD(x))∧μD′(x))=⋁D′∈A~′(⋁D∈A~(μD(x)∧μD′(x)))=⋁D,D′{μD(x)∧μD′(x)∣D∈A~,D′∈A~′},
also
(35)γ[B:C]∩[B:E](x)=γ[B:C](x)∨γ[B:E](x)=(⋀D∈A~γD(x))∨(⋀D′∈A~′γD′(x))=⋀D′∈A~′((⋀D∈A~γD(x))∨γD′(x))=⋀D′∈A~′(⋀D∈A~(γD(x)∨γD′(x)))=⋀D,D′{γD(x)∨γD′(x)∣D∈A~,D′∈A~′}.
Let D∈A~,  D′∈A~′; then D,  D′ are IFF(A) and D∨~C⊆B,  D′∨~E⊆B.
Now by Theorem21  D∩D′ is an IFF(A), and also
(36)(D∩D′)∨~(C∧~E)⊆((D∩D′)∨~C)∧~((D∩D′)∨~E)⊆(D∨~C)∧~(D′∨~E)⊆B∧~B=B.
Hence D∩D′∈A~′′. Thus
(37)[B:C∧~E]=⋃D∈A′′D⊇⋃D,D′{D∩D′∣D∈A~,D′∈A~′}.
Now
(38)μ[B:C∧~E](x)≥⋁D,D′{μD∩D′(x)∣D∈A~,D′∈A~′}=⋁D,D′{μD(x)∧μD′(x)∣D∈A~,D′∈A~′}=μ[B:C]∩[B:E](x).
Also
(39)γ[B:C∧~E](x)≤⋀D,D′{γD∩D′(x)∣D∈A~,D′∈A~′}=⋀D,D′{γD(x)∨γD′(x)∣D∈A~,D′∈A~′}=γ[B:C]∩[B:E](x)
from (35).
Thus(40)[B:C∧~E]⊇[B:C]∩[B:E].
Consequently [B:C∧~E]=[B:C]∩[B:E] from (33) and (40).Theorem 29.
LetA be an IFCF of F, and let the arbitrary collection (Aα)α∈∧,  B, and C be IFF(A); then for the arbitrary collection [B:Aα]α∈∧ and [B:C] of quotients of B,
(41)[B:C]∩(⋀α∈∧~[B:Aα])=⋀α∈∧~([B:C]∩[B:Aα]).Proof.
We have(42)⋀α∈∧~([B:C]∩[B:Aα])=⋀α∈∧~[B:C∧~Aα].
By Theorem 28.
Now from Theorem23, C⊆C∧~Aα and Aα⊆C∧~Aα.
Therefore, by Theorem26, [B:C∧~Aα]⊆[B:C] and [B:C∧~Aα]⊆[B:Aα].
Hence(43)⋀α∈∧~[B:C∧~Aα]⊆[B:C],⋀α∈∧~[B:C∧~Aα]⊆⋀α∈∧~[B:Aα].
So,
(44)⋀α∈∧~[B:C∧~Aα]⊆[B:C]∩(⋀α∈∧~[B:Aα]).
Hence,
(45)⋀α∈∧~([B:C]∩[B:Aα])⊆[B:C]∩(⋀α∈∧~[B:Aα])
by (42).
Also forx∈F,
(46)μ[B:C]∩(∧~α∈∧[B:Aα])(x)=μ[B:C](x)∧μ∧~α∈∧[B:Aα](x)=μ[B:C](x)∧sup{⋀α∈∧μ[B:Aα](aα)∣aα∈F,⋀α∈∧aα=x}=sup{⋀α∈∧(μ[B:C](x)∧μ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}=sup{⋀α∈∧(μ[B:C](x)∧[μA(aα)∧μ[B:Aα](aα)])∣aα∈F,⋀α∈∧aα=x}(∵[B:Aα]⊆A)=sup{⋀α∈∧([μ[B:C](x)∧μA(aα)]∧μ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}≤sup{⋀α∈∧(μ[B:C](aα∨x)∧μ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}(∵[B:C]isanIFF(F))=sup{⋀α∈∧(μ[B:C](aα)∧μ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}=sup{⋀α∈∧(μ[B:C]∩[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}=μ∧~α∈∧([B:C]∩[B:Aα])(x),γ[B:C]∩(∧~α∈∧[B:Aα])(x)=γ[B:C](x)∨γ∧~α∈∧[B:Aα](x)=γ[B:C](x)∨inf{⋁α∈∧γ[B:Aα](aα)∣aα∈F,⋀α∈∧aα=x}=inf{⋁α∈∧(γ[B:C](x)∨γ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}=inf{⋁α∈∧((γ[B:C](x)∨γA(aα))∨γ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}≥inf{⋁α∈∧(γ[B:C](aα∨x)∨γ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}(∵[B:C]isanIFF(F))=inf{⋁α∈∧(γ[B:C](aα)∨γ[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}=sup{⋁α∈∧(μ[B:C]∩[B:Aα](aα))∣aα∈F,⋀α∈∧aα=x}=μ∧~α∈∧([B:C]∩[B:Aα])(x).
Hence,
(47)[B:C]∩(⋀α∈∧~[B:Aα])⊆⋀α∈∧~([B:C]∩[B:Aα]).
Therefore [B:C]∩(⋀~α∈∧[B:Aα])=⋀~α∈∧([B:C]∩[B:Aα]) from (45) and (47).Remark 30.
If we interchange the roles of⋀~ and ⋂ in Theorem 29, only one-sided inequality [B:C]∧~(⋂α∈∧[B:Aα])⊆⋂α∈∧([B:C]∧~[B:Aα]) holds.Theorem 31.
LetA be an IFCF of F; then the collection of all intuitionistic fuzzy quotient filter of any intuitionistic fuzzy filter B of A, [B]A is a coframe.Proof.
[
B
]
A is a complete lattice, which is bounded above by A and below by Ae={(x,μAe(x),γAe(x))∣x∈F} where
(48)μAe(x)={μA(eF),x=eF,0,otherwise,γAe(x)={γA(eF),x=eF,0,otherwise.
Also [B:C]∨~(∧~α∈∧[B:Aα])=∧~α∈∧([B:C]∨~[B:Aα]) for arbitrary collection (Aα)α∈∧, B and C of  intuitionistic fuzzy filters of A from Proposition 15 and Theorem 29.
Hence[B]A is a coframe.Theorem 32.
The collection of all intuitionistic fuzzy quotient filters of any intuitionistic fuzzy filterB of the intuitionistic fuzzy coframe AχF of F, [B]AχF is the sub coframe of the collection of all intuitionistic fuzzy filters IFF of  F, where AχF={(x,μAχF(x),γAχF(x))∣x∈F},  μAχF(x)=1,  γAχF(x)=0 for all x∈F.Proof.
Follows from Theorems18 and 31 as [B]AχF⊆IFF. Also [B:B]=AχF and Ae=AeF are, respectively, top and bottom elements of [B]AχF.



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