On Intuitionistic Fuzzy Filters of Intuitionistic Fuzzy Coframes

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.

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. Let 𝐹 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 lattice 𝐹 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 lattice 𝐹 satisfying the infinite distributive law 𝑎∨(∧𝑆)=∧{𝑎∨𝑥∣𝑥∈𝑆} for any 𝑎∈𝐹 and 𝑆⊆𝐹.

Definition 4 (see [1, 2]). A subset 𝑀 of a coframe 𝐹 closed under arbitrary meets, finite joins and having unit (top) element 𝑒_𝐹 and zero (bottom) element 𝑜_𝐹 of the coframe 𝐹 is called a sub coframe of 𝐹.

Definition 5 (see [1, 2]). For any coframe 𝐹, 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]). Let 𝑋 be a nonempty set. An intuitionistic fuzzy set (IFS) 𝐴 of 𝑋 is an object of the form 𝐴={(𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥))∣𝑥∈𝐹}, where 𝜇_𝐴∶𝑋→[0,1] and 𝛾_𝐴∶𝑋→[0,1] define, respectively, the degree of membership and the degree of nonmembership of the element 𝑥∈𝑋 and 0≤𝜇_𝐴(𝑥)+𝛾_𝐴(𝑥)≤1, for all 𝑥∈𝑋.

Definition 7 (see [8]). Let 𝐹 be a coframe then an IFS 𝐴={(𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥))∣𝑥∈𝐹} in 𝐹 is called an intuitionistic fuzzy coframe (IFCF) of 𝐹 if it satisfies the following conditions:(i)𝜇_𝐴(𝑥∨𝑦)≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)} 𝛾_𝐴(𝑥∨𝑦)≤max{𝛾_𝐴(𝑥),𝛾_𝐴(𝑦)} for all 𝑥,𝑦∈𝐹,(ii)𝜇_𝐴(∧𝑆)≥inf{𝜇_𝐴(𝑥)∣𝑥∈𝑆} 𝛾_𝐴(∧𝑆)≤sup{𝛾_𝐴(𝑥)∣𝑥∈𝑆} for arbitrary 𝑆⊆𝐹,(iii)𝜇_𝐴(𝑒_𝐹)=𝜇_𝐴(𝑜_𝐹)≥𝜇_𝐴(𝑥) 𝛾_𝐴(𝑒_𝐹)=𝛾_𝐴(𝑜_𝐹)≤𝛾_𝐴(𝑥) for all 𝑥∈𝐹, where 𝑒_𝐹 and 𝑜_𝐹 are, respectively, the unit and zero element of the coframe 𝐹.

Definition 8 (see [7]). If {𝐴_𝛼∣𝛼∈Λ} is a family of IFS of 𝐹 where 𝐴_𝛼={(𝑥,𝜇_𝐴𝛼(𝑥),𝛾_𝐴𝛼(𝑥))∣𝑥∈𝐹}, then _𝛼∈Λ𝐴_𝛼=𝑥,𝜇_{⋂𝛼∈Λ𝐴𝛼}(𝑥),𝛾_{⋂𝛼∈Λ𝐴𝛼}(𝑥)=𝜇∣𝑥∈𝐹𝑥,inf_𝐴𝛼(𝑥),sup𝛾_𝐴𝛼(𝑥)∣𝑥∈𝐹_𝛼∈Λ,𝐴_𝛼=𝑥,𝜇_{⋃𝛼∈Λ𝐴𝛼}(𝑥),𝛾_{⋃𝛼∈Λ𝐴𝛼}(𝑥)=𝜇∣𝑥∈𝐹𝑥,sup_𝐴𝛼(𝑥),inf𝛾_𝐴𝛼(𝑥)∣𝑥∈𝐹. (1)

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.  
For  𝐴,𝐵∈IFS  one has 𝐴∧𝐵=𝑥,𝜇_𝐴∧_𝐵(𝑥),𝛾_𝐴∧_𝐵(𝑥)∣𝑥∈𝐹, (2) where 𝜇_𝐴∧_𝐵(𝑥)=sup_{𝑢∧𝑣=𝑥}{𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)} and 𝛾_𝐴∧_𝐵(𝑥)=inf_{𝑢∧𝑣=𝑥}{𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣)}𝐴∨𝐵=𝑥,𝜇_𝐴∨_𝐵(𝑥),𝛾_𝐴∨_𝐵(𝑥)∣𝑥∈𝐹, (3) where 𝜇_𝐴∨_𝐵(𝑥)=sup_{𝑢∨𝑣=𝑥}{𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)} and 𝛾_𝐴∨_𝐵(𝑥)=inf_{𝑢∨𝑣=𝑥}{𝛾_𝐴(𝑢)∨𝛾_𝐵(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, 1_𝑥=𝑦,𝜇_𝑥𝑦,𝛾_𝑥𝑦∣𝑦∈𝐹, (4) where 𝜇_𝑥𝑦=𝛾1if𝑦=𝑥,0otherwise,_𝑥𝑦=0if𝑦=𝑥,1otherwise. (5)

Lemma 10 (see [9]). Let 𝐴,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 coframe 𝐹 is a coframe. We use IFF(𝐹) to denote intuitionistic fuzzy filter of the coframe 𝐹.

Definition 11 (see [9]). Let 𝐹 be a coframe; then an IFS 𝐴 on 𝐹 is said to be an IFF(𝐹) if (F1)𝜇_𝐴(𝑥∧𝑦)≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)}, for all 𝑥,𝑦∈𝐹  𝛾_𝐴(𝑥∧𝑦)≤max{𝛾_𝐴(𝑥),𝛾_𝐴(𝑦)}, for all 𝑥,𝑦∈𝐹,(F2)𝜇_A(𝑥∨𝑦)≥max{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)}, for all 𝑥,𝑦∈𝐹  𝛾_𝐴(𝑥∨𝑦)≤min{𝛾_𝐴(𝑥),𝛾_𝐴(𝑦)}, for all 𝑥,𝑦∈𝐹,(F3)𝜇_𝐴(𝑒_𝐹)=1,𝛾_𝐴(𝑒_𝐹)=0 where 𝑒_𝐹 is the unit element of 𝐹.

Lemma 12 (see [9]). If 𝐴 and 𝐵 are any two  IFF(𝐹), then, 𝐴∩𝐵 is an  IFF(𝐹).

Result 1 (see [9]). Union of any two IFF(𝐹) needs not be an IFF(𝐹).

Definition 13. Let 𝐴 be an IFS. Let ⟨𝐴⟩=⋂{𝐵∣𝐴⊆𝐵,𝐵anIFF(𝐹)} where 𝐴⊆𝐵 means 𝜇_𝐴(𝑥)≤𝜇_𝐵(𝑥) and 𝛾_𝐴(𝑥)≥𝛾_𝐵(𝑥) for all 𝑥∈𝐹. Then ⟨𝐴⟩ is called the IFF(𝐹) generated by 𝐴.

Theorem 14. Let 𝐴,𝐵 be two IFF(𝐹); then 𝐴∧𝐵 is an IFF(𝐹) and 𝐴∧𝐵=⟨𝐴∪𝐵⟩.

Proof. Let 𝑥∈𝐹; then 𝐴(𝑥)=∧𝐵𝑥,sup_{𝑢∧𝑣=𝑥}𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣),inf_{𝑢∧𝑣=𝑥}𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣∣⊇)𝑥∈𝐹𝑥,𝜇_𝐴(𝑥)∧𝜇_𝐵𝑒_𝐹,𝛾_𝐴(𝑥)∨𝛾_𝐵𝑒_𝐹=𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)=𝐴. (6) Hence 𝐴∧𝐵⊇𝐴. Similarly, we have 𝐴∧𝐵⊇B. Thus, 𝐴∧𝐵⊇𝐴∪𝐵. Also,(F1)𝜇_𝐴∧_𝐵𝑥∧𝑦=sup_{𝑢∧𝑣=𝑥∧𝑦}𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣≥𝜇_𝐴)𝑢₁∧𝑣₁∧𝜇_𝐵𝑢₂∧𝑣₂∣𝑢₁∧𝑢₂=𝑥,𝑣₁∧𝑣₂=𝑦;𝑢₁,𝑣₁,𝑢₂,𝑣₂≥𝜇∈𝐹_𝐴𝑢₁∧𝜇_𝐴𝑣₁∧𝜇_𝐵𝑢₂∧𝜇_𝐵𝑣₂∣𝑢₁∧𝑢₂=𝑥,𝑣₁∧𝑣₂=𝑦;𝑢₁,𝑣₁,𝑢₂,𝑣₂=𝜇∈𝐹_𝐴𝑢₁∧𝜇_𝐵𝑢₂∧𝜇_𝐴𝑣₁∧𝜇_𝐵𝑣₂∣𝑢₁∧𝑢₂=𝑥,𝑣₁∧𝑣₂=𝑦;𝑢₁,𝑣₁,𝑢₂,𝑣₂=𝜇∈𝐹_𝐴𝑢₁∧𝜇_𝐵𝑢₂∣𝑢₁∧𝑢₂=𝑥;𝑢₁,𝑢₂∧𝜇∈𝐹_𝐴𝑣₁∧𝜇_𝐵𝑣₂∣𝑣₁∧𝑣₂=𝑦;𝑣₁,𝑣₂∈𝐹=𝜇_𝐴∧_𝐵(𝑥)∧𝜇_𝐴∧_𝐵𝑦𝛾∀𝑥,𝑦∈𝐹,_𝐴∧_𝐵𝑥∧𝑦=inf_{𝑢∧𝑣=𝑥∧𝑦}𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣≤𝛾_𝐴)𝑢₁∧𝑣₁∨𝛾_𝐵𝑢₂∧𝑣₂∣𝑢₁∧𝑢₂=𝑥,𝑣₁∧𝑣₂=𝑦;𝑢₁,𝑣₁,𝑢₂,𝑣₂≤𝛾∈𝐹_𝐴𝑢₁∨𝛾_𝐴𝑣₁∨𝛾_𝐵𝑢₂∨𝛾_𝐵𝑣₂∣𝑢₁∧𝑢₂=𝑥,𝑣₁∧𝑣₂=𝑦;𝑢₁,𝑣₁,𝑢₂,𝑣₂=𝛾∈𝐹_𝐴𝑢₁∨𝛾_𝐵𝑢₂∨𝛾_𝐴𝑣₁∨𝛾_𝐵𝑣₂∣𝑢₁∧𝑢₂=𝑥,𝑣₁∧𝑣₂=𝑦;𝑢₁,𝑣₁,𝑢₂,𝑣₂=𝛾∈𝐹_𝐴𝑢₁∨𝛾_𝐵𝑢₂∣𝑢₁∧𝑢₂=𝑥;𝑢₁,𝑢₂∨𝛾∈𝐹_𝐴𝑣₁∨𝛾_𝐵𝑣₂∣𝑣₁∧𝑣₂=𝑦;𝑣₁,𝑣₂∈𝐹=𝛾_𝐴∧_𝐵(𝑥)∨𝛾_𝐴∧_𝐵𝑦∀𝑥,𝑦∈𝐹, (7)(F2)𝜇_𝐴∧_𝐵𝑥∨𝑦=sup_{𝑢∧𝑣=𝑥∨𝑦}𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣≥𝜇_𝐴)𝑦∨𝑢∧𝜇_𝐵∵∧≥𝜇𝑦∨𝑣∣𝑢∧𝑣=𝑥;𝑢,𝑣∈𝐹𝑦∨𝑢𝑦∨𝑣=𝑥∨𝑦_𝐴(𝑢)∧𝜇_𝐵(𝑣)∣𝑢∧𝑣=𝑥;𝑢,𝑣∈𝐹=𝜇_𝐴∧_𝐵(𝑥)∀𝑥,𝑦∈𝐹. (8)

Similarly, we have 𝜇_𝐴∧_𝐵(𝑥∨𝑦)≥𝜇_𝐴∧_𝐵(𝑦) for all 𝑥,𝑦∈𝐹.

Therefore, 𝜇_𝐴∧_𝐵(𝑥∨𝑦)≥𝜇_𝐴∧_𝐵(𝑥)∨𝜇_𝐴∧_𝐵(𝑦), for all 𝑥,𝑦∈𝐹𝛾_𝐴∧_𝐵𝑥∨𝑦=inf_{𝑢∧𝑣=𝑥∨𝑦}𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣≤𝛾_𝐴)𝑦∨𝑢∨𝛾_𝐵∵∧≤𝛾𝑦∨𝑣∣𝑢∧𝑣=𝑥;𝑢,𝑣∈𝐹𝑦∨𝑢𝑦∨𝑣=𝑥∨𝑦_𝐴(𝑢)∨𝛾_𝐵(𝑣)∣𝑢∧𝑣=𝑥;𝑢,𝑣∈𝐹=𝛾_𝐴∧_𝐵(𝑥)∀𝑥,𝑦∈𝐹. (9)

Similarly, we have 𝛾_𝐴∧_𝐵(𝑥∨𝑦)≤𝛾_𝐴∧_𝐵(𝑦) for all 𝑥,𝑦∈𝐹.

Therefore, 𝛾_𝐴∧_𝐵(𝑥∨𝑦)≤𝛾_𝐴∧_𝐵(𝑥)∧𝛾_𝐴∧_𝐵(𝑦), for all 𝑥,𝑦∈𝐹.(F3) Since 𝜇_𝐴(𝑒_𝐹)=𝜇_𝐵(𝑒_𝐹)=1, clearly 𝜇_𝐴∧_𝐵(𝑒_𝐹)=sup_{𝑢∧𝑣=𝑒𝐹}{𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)}≥𝜇_𝐴(𝑒_𝐹)∧𝜇_𝐵(𝑒_𝐹)=1 and since 𝛾_𝐴(𝑒_𝐹)=𝛾_𝐵(𝑒_𝐹)=0, we have 𝛾_𝐴∧_𝐵(𝑒_𝐹)=inf_{𝑢∧𝑣=𝑒𝐹}{𝛾_𝐴(𝑢)∧𝛾_𝐵(𝑣)}≤𝛾_𝐴(𝑒_𝐹)∧𝛾_𝐵(𝑒_𝐹)=0.

Thus if 𝐴,  𝐵 are IFF(𝐹), then 𝐴∧𝐵 is an IFF(𝐹).

Now let 𝐶 be any IFF(𝐹) such that 𝐴∪𝐵⊆𝐶, then, 𝜇_𝐴∧_𝐵(𝑥)=sup_{𝑢∧𝑣=𝑥}𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)≤sup_{𝑢∧𝑣=𝑥}𝜇_𝐶(𝑢)∧𝜇_𝐶(𝑣)≤sup_{𝑢∧𝑣=𝑥}𝜇_𝐶(𝑢∧𝑣)=𝜇_𝐶(𝑥),𝛾_𝐴∧_𝐵(𝑥)=inf_{𝑢∧𝑣=𝑥}𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣)≥inf_{𝑢∧𝑣=𝑥}𝛾_𝐶(𝑢)∨𝛾_𝐶(𝑣)≥inf_{𝑢∧𝑣=𝑥}𝛾_𝐶(𝑢∧𝑣)=𝛾_𝐶(𝑥). (10) Hence 𝐴∧𝐵⊆𝐶. Thus 𝐴∧𝐵 is the smallest filter of 𝐹 such that 𝐴∪𝐵⊆𝐴∧𝐵.

Hence 𝐴∧𝐵=⟨𝐴∪𝐵⟩.

Proposition 15. Let 𝐴,  𝐵 be two IFF(𝐹); then 𝐴∨𝐵=𝐴∩𝐵.

Proof. We have, for 𝑥∈𝐹,  𝜇_𝐴∨_𝐵(𝑥)=sup_{𝑢∨𝑣=𝑥}{𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)}≥𝜇_𝐴(𝑥)∧𝜇_𝐵(𝑥)=𝜇_𝐴∩𝐵(𝑥) and 𝛾_𝐴∨_𝐵(𝑥)=inf_{𝑢∨𝑣=𝑥}{𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣)}≤𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑥)=𝛾_𝐴∩𝐵(𝑥). Hence 𝐴∩𝐵⊆𝐴∨𝐵.
Now since 𝜇_𝑖(𝑢∨𝑣)=𝜇_𝑖(𝑥)≥𝜇_𝑖(𝑢)∨𝜇_𝑖(𝑣) and hence 𝜇_𝑖(𝑥)≥𝜇_𝑖(𝑢) and 𝜇_𝑖(𝑥)≥𝜇_𝑖(𝑣) for 𝑖=𝐴,𝐵𝜇_𝐴∨_𝐵(𝑥)=sup_{𝑢∨𝑣=𝑥}𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)≤sup_{𝑢∨𝑣=𝑥}𝜇_𝐴(𝑥)∧𝜇_𝐵(𝑥)=𝜇_𝐴(𝑥)∧𝜇_𝐵(𝑥)=𝜇_𝐴∩𝐵(𝑥),𝛾_𝐴∨_𝐵(𝑥)=inf_{𝑢∨𝑣=𝑥}𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣)≥inf_{𝑢∨𝑣=𝑥}𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑥)=𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑥)=𝛾_𝐴∩𝐵(𝑥). (11) Hence 𝐴∨𝐵⊆𝐴∩𝐵. Therefore 𝐴∨𝐵=𝐴∩𝐵.

Theorem 16. Given an arbitrary collection (𝐴_𝛼)_𝛼∈∧ and 𝐴 of IFF(𝐹), then 𝐴∩^∼_𝛼∈∧𝐴_𝛼=^∼_𝛼∈∧𝐴∩𝐴_𝛼. (12)

Proof. We have for 𝑤∈𝐹, 𝜇_𝐴∩(∧_{𝛼∈∧𝐴𝛼)}(𝑤)=𝜇_𝐴(𝑤)∧^∼_𝛼∈∧𝜇_𝐴𝛼(𝑤)=𝜇_𝐴(𝑤)∧sup_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝜇_𝐴𝛼𝑎_𝛼=sup_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝜇_𝐴(𝑤)∧𝜇_𝐴𝛼𝑎_𝛼≤sup_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝜇_𝐴𝑎_𝛼∨𝑤∧𝜇_𝐴𝛼𝑎_𝛼=sup_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝜇_𝐴𝑎_𝛼∧𝜇_𝐴𝛼𝑎_𝛼=sup_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝜇_𝐴∩𝜇_𝐴𝛼𝑎_𝛼=𝜇∧_{𝛼∈∧(𝐴∩𝐴𝛼)}(𝑤),𝛾_𝐴∩(∧_{𝛼∈∧𝐴𝛼)}(𝑤)=𝛾_𝐴(𝑤)∧^∼_𝛼∈∧𝛾_𝐴𝛼(𝑤)=𝛾_𝐴(𝑤)∧inf_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝛾_𝐴𝛼𝑎_𝛼=inf_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝛾_𝐴(𝑤)∧𝛾_𝐴𝛼𝑎_𝛼≥inf_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝛾_𝐴𝑎_𝛼∨𝑤∧𝛾_𝐴𝛼𝑎_𝛼=inf_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝛾_𝐴𝑎_𝛼∧𝛾_𝐴𝛼𝑎_𝛼=inf_{∧𝛼∈∧𝑎𝛼=𝑤𝛼∈∧}𝛾_𝐴∩𝛾_𝐴𝛼𝑎_𝛼=𝛾∧_{𝛼∈∧(𝐴∩𝐴𝛼)}(𝑤). (13) Therefore, 𝐴∩^∼_𝛼∈∧𝐴_𝛼⊆^∼_𝛼∈∧𝐴∩𝐴_𝛼. (14) Also 𝐴∩𝐴_𝛼⊆𝐴_𝛼 and 𝐴∩𝐴_𝛼⊆𝐴 for all 𝛼∈∧; hence ⋀_𝛼∈∧(𝐴∩𝐴_𝛼)⊆⋀_𝛼∈∧𝐴_𝛼 and ⋀_𝛼∈∧(𝐴∩𝐴_𝛼)⊆𝐴.  Therefore, ^∼_𝛼∈∧𝐴∩𝐴_𝛼⊆𝐴∩^∼_𝛼∈∧𝐴_𝛼. (15) From (14) and (15) we have ⋀_𝛼∈∧(𝐴∩𝐴_𝛼)=𝐴∩(⋀_𝛼∈∧𝐴_𝛼).

Remark 17. If we interchange the roles of ∧ and ∩ in Theorem 16, only one-sided inequality 𝐴∧(⋂_𝛼∈∧𝐴_𝛼)⊆⋂_𝛼∈∧(𝐴∧𝐴_𝛼) holds.

Theorem 18. The set 𝐼_𝐹𝐹 of all intuitionistic fuzzy filters of the coframe 𝐹 is a coframe.

Proof. 𝐼_𝐹𝐹 is a complete lattice, which is bounded above by 𝐴_𝜒𝐹={(𝑥,𝜇_𝐴𝜒𝐹(𝑥),𝛾_𝐴𝜒𝐹(𝑥))∣𝑥∈𝐹} where 𝜇_𝐴𝜒𝐹(𝑥)=1,  𝛾_𝐴𝜒𝐹(𝑥)=0 for all 𝑥∈𝐹 and bounded below by 𝐴_𝑒𝐹={(𝑥,𝜇_𝑒𝐹(𝑥),𝛾_𝑒𝐹(𝑥))∣𝑥∈𝐹} where 𝜇_𝑒𝐹(𝑥)=1𝑥=𝑒_𝐹,𝛾0otherwise,_𝑒𝐹(𝑥)=0𝑥=𝑒_𝐹1otherwise∀𝑥∈𝐹. (16) Also ⋀_𝛼∈∧(𝐴∨𝐴_𝛼)=𝐴∨(⋀_𝛼∈∧𝐴_𝛼) for arbitrary collection (𝐴_𝛼)_𝛼∈∧ and 𝐴 of IFF(𝐹) by Proposition 15 and Theorem 16. Hence 𝐼_𝐹𝐹 is a coframe.

4. Intuitionistic Fuzzy Filter of an Intuitionistic Fuzzy Coframe

Definition 19. Let 𝐴 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)𝜇_𝐵(𝑥∧𝑦)≥min{𝜇_𝐵(𝑥),𝜇_𝐵(𝑦)} 𝛾_𝐵(𝑥∧𝑦)≤max{𝛾_𝐵(𝑥),𝛾_𝐵(𝑦)}.(ii)𝜇_𝐵(𝑥∨𝑦)≥max{𝜇_𝐴(𝑥)∧𝜇_𝐵(𝑦),𝜇_𝐵(𝑥)∧𝜇_𝐴(𝑦)} for all 𝑥,𝑦∈𝐹. 𝛾_𝐵(𝑥∨𝑦)≤min{𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑦),𝛾_𝐵(𝑥)∨𝛾_𝐴(𝑦)} for all 𝑥,𝑦∈𝐹.(iii)𝜇_𝐵(𝑒_𝐹)=𝜇_𝐴(𝑒_𝐹) and 𝛾_𝐵(𝑒_𝐹)=𝛾_𝐴(𝑒_𝐹) where 𝑒_𝐹 is the unit element of 𝐹.If 𝐵 is an IFF(𝐴), then we write 𝐵⊲𝐴.

Theorem 20. Let 𝐴 be an IFCF of 𝐹 and 𝐵 an IFF(𝐴). Then 𝐴∩𝐵 is an IFF(𝐴).

Proof. Obviously 𝐴∩𝐵⊆𝐴 and(i)𝜇_𝐴∩𝐵𝜇𝑥∧𝑦=min_𝐴𝑥∧𝑦,𝜇_𝐵𝜇𝑥∧𝑦≥minmin_𝐴(𝑥),𝜇_𝐴𝑦,min𝜇_𝐵(𝑥),𝜇_𝐵𝑦≥minmin𝜇_𝐴(𝑥),𝜇_𝐵(𝑥),min𝜇_𝐴𝑦,𝜇_𝐵𝑦=𝜇_𝐴∩𝐵(𝑥)∧𝜇_𝐴∩𝐵𝑦𝛾,∀𝑥,𝑦∈𝐹,_𝐴∩𝐵𝛾𝑥∧𝑦=max_𝐴𝑥∧𝑦,𝛾_𝐵𝛾𝑥∧𝑦≤maxmax_𝐴(𝑥),𝛾_𝐴𝑦,max𝛾_𝐵(𝑥),𝛾_𝐵𝑦=maxmax𝛾_𝐴(𝑥),𝛾_𝐵(𝑥),max𝛾_𝐴𝑦,𝛾_𝐵𝑦=𝛾_𝐴∩𝐵(𝑥)∨𝛾_𝐴∩𝐵𝑦,∀𝑥,𝑦∈𝐹, (17)(ii)𝜇_𝐴∩𝐵𝜇𝑥∨𝑦=min_𝐴𝑥∨𝑦,𝜇_𝐵𝜇𝑥∨𝑦≥min_𝐴(𝑥)∧𝜇_𝐴𝑦,𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦∨𝜇_𝐵(𝑥)∧𝜇_𝐴𝑦=𝜇_𝐴(𝑥)∧𝜇_𝐴𝑦∧𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦∨𝜇_𝐵(𝑥)∧𝜇_𝐴𝑦=𝜇_𝐴(𝑥)∧𝜇_𝐴𝑦∧𝜇_𝐵𝑦∨𝜇_𝐴(𝑥)∧𝜇_𝐵(𝑥)∧𝜇_𝐴𝑦=𝜇_𝐴(𝑥)∧𝜇_𝐴∩𝐵𝑦∨𝜇_𝐴∩𝐵(𝑥)∧𝜇_𝐴𝑦𝛾∀𝑥,𝑦∈𝐹,_𝐴∩𝐵𝛾𝑥∨𝑦=max_𝐴𝑥∨𝑦,𝛾_𝐵𝛾𝑥∨𝑦≤max_𝐴(𝑥)∨𝛾_𝐴𝑦,𝛾_𝐴(𝑥)∨𝛾_𝐵𝑦∧𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦=𝛾_𝐴(𝑥)∨𝛾_𝐴𝑦∨𝛾_𝐴(𝑥)∨𝛾_𝐵𝑦∧𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦=𝛾_𝐴(𝑥)∨𝛾_𝐴𝑦∨𝛾_𝐵𝑦∧𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦=𝛾_𝐴(𝑥)∨𝛾_𝐴∩𝐵𝑦∧𝛾_𝐴∩𝐵(𝑥)∨𝛾_𝐴𝑦∀𝑥,𝑦∈𝐹, (18)(iii) also clearly 𝜇_𝐴∩𝐵(𝑒_𝐹)=𝜇_𝐴(𝑒_𝐹) and 𝛾_𝐴∩𝐵(𝑒_𝐹)=𝛾_𝐴(𝑒_𝐹) where 𝑒_𝐹 is the unit element of 𝐹.

Theorem 21. Let 𝐴 be an IFCF of 𝐹, and let 𝐵,  𝐶 be two IFF(𝐴). Then 𝐵∩𝐶 is also an IFF(𝐴).

Proof. Clearly 𝐵∩𝐶⊆𝐴 also(i)𝜇_𝐵∩𝐶𝑥∧𝑦≥𝜇_𝐵∩𝐶(𝑥)∧𝜇_𝐵∩𝐶𝑦,𝛾_𝐴∩𝐵𝑥∧𝑦≤𝛾_𝐴∩𝐵(𝑥)∨𝛾_𝐴∩𝐵𝑦∀𝑥,𝑦∈𝐹, (19)(ii)𝜇_𝐵∩𝐶𝜇𝑥∨𝑦=min_𝐵𝑥∨𝑦,𝜇_𝐶𝜇𝑥∨𝑦≥min_𝐴(𝑥)∧𝜇_𝐵𝑦∨𝜇_𝐵(𝑥)∧𝜇_𝐴𝑦,𝜇_𝐴(𝑥)∧𝜇_𝐶𝑦∨𝜇_𝐶(𝑥)∧𝜇_𝐴𝑦=𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦∨𝜇_𝐵(𝑥)∧𝜇_𝐴𝑦∧𝜇_𝐴(𝑥)∧𝜇_𝐶𝑦∨𝜇_𝐶(𝑥)∧𝜇_𝐴𝑦≥𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦∧𝜇_𝐶𝑦∨𝜇_𝐵(𝑥)∧𝜇_𝐶(𝑥)∧𝜇_𝐴𝑦=𝜇_𝐴(𝑥)∧𝜇_𝐵∩𝐶𝑦∨𝜇_𝐵∩𝐶(𝑥)∧𝜇_𝐴𝑦𝛾∀𝑥,𝑦∈𝐹,_𝐵∩𝐶𝛾𝑥∨𝑦=max_𝐵𝑥∨𝑦,𝛾_𝐶𝛾𝑥∨𝑦≤max_𝐴(𝑥)∨𝛾_𝐵𝑦∧𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦,𝛾_𝐴(𝑥)∨𝛾_𝐶𝑦∧𝛾_𝐶(𝑥)∨𝛾_𝐴𝑦=𝛾_𝐴(𝑥)∨𝛾_𝐵𝑦∧𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦∨𝛾_𝐴(𝑥)∨𝛾_𝐶𝑦∧𝛾_𝐶(𝑥)∨𝛾_𝐴𝑦≤𝛾_𝐴(𝑥)∨𝛾_𝐴𝑦∨𝛾_𝐵𝑦∧𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦=𝛾_𝐴(𝑥)∨𝛾_𝐴∩𝐵𝑦∨𝛾_𝐴∩𝐵(𝑥)∨𝛾_𝐴𝑦∀𝑥,𝑦∈𝐹, (20)(iii) also 𝜇_𝐵∩𝐶(𝑒_𝐹)=𝜇_𝐴(𝑒_𝐹) and 𝛾_𝐵∩𝐶(𝑒_𝐹)=𝛾_𝐴(𝑒_𝐹).

Theorem 22. Let 𝐴 be an IFCF of 𝐹 and 𝐵  an  IFS with 𝐵⊆𝐴. Then 𝐵 is an IFF(𝐴) if and only if(i)𝜇_𝐵(𝑥∧𝑦)≥𝜇_𝐵(𝑥)∧𝜇_𝐵(𝑦);  𝛾_𝐵(𝑥∧𝑦)≤𝛾_𝐵(𝑥)∨𝛾_𝐵(𝑦) for all 𝑥,𝑦∈𝐹,(ii)𝐴∨𝐵⊆𝐵,(iii)𝜇_𝐵(𝑒_𝐹)=𝜇_𝐴(𝑒_𝐹) and 𝛾_𝐵(𝑒_𝐹)=𝛾_𝐴(𝑒_𝐹).

Proof. Suppose conditions (i), (ii), and (iii) holds to prove that 𝐵 is an IFF(𝐴).
Since 𝐴∨𝐵⊆𝐵, one has 𝜇_𝐵𝑥∨𝑦≥𝜇_𝐴∨_𝐵𝑥∨𝑦=sup_{𝑢∨𝑣=𝑥∨𝑦}𝜇_𝐴(𝑢)∧𝜇_𝐵(𝑣)≥𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦𝛾∀𝑥,𝑦∈𝐹,_𝐵𝑥∨𝑦≤𝛾_𝐴∨_𝐵𝑥∨𝑦=inf_{𝑢∨𝑣=𝑥∨𝑦}𝛾_𝐴(𝑢)∨𝛾_𝐵(𝑣)≤𝛾_𝐴(𝑥)∧𝛾_𝐵𝑦∀𝑥,𝑦∈𝐹. (21) Also 𝜇_𝐵(𝑥∨𝑦)≥𝜇_𝐴(𝑦)∧𝜇_𝐵(𝑥) and 𝛾_𝐵(𝑥∨𝑦)≤𝛾_𝐴(𝑦)∧𝛾_𝐵(𝑥) for all 𝑥,𝑦∈𝐹.
Hence, 𝜇_𝐵𝑥∨𝑦≥max𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦,𝜇_𝐵(𝑥)∧𝜇_𝐴𝑦,𝛾_𝐵𝑥∨𝑦≤min𝛾_𝐴(𝑥)∨𝛾_𝐵𝑦,𝛾_𝐵(𝑥)∨𝛾_𝐴𝑦. (22) Therefore 𝐵 is an IFF(𝐴).
Conversely if 𝐵  is  an  IFF(𝐴), we have 𝜇_𝐵(𝑥∧𝑦)≥min{𝜇_𝐵(𝑥),𝜇_𝐵(𝑦)},  𝛾_𝐵(𝑥∧𝑦)≤max{𝛾_𝐵(𝑥),𝛾_𝐵(𝑦)},  and  𝜇_𝐵(𝑒_𝐹)=𝜇_𝐴(𝑒_𝐹),  𝛾_𝐵(𝑒_𝐹)=𝛾_𝐴(𝑒_𝐹) from Definition 19.
Now for all 𝑧∈𝐹 with 𝑧=𝑥∨𝑦,  𝜇_𝐵(𝑧)≥sup_{𝑥∨𝑦=𝑧}{𝜇_𝐴(𝑥)∧𝜇_𝐵(𝑦)}=𝜇_𝐴∨_𝐵(𝑧) and  𝛾_𝐵(𝑧)≤inf_{𝑥∨𝑦=𝑧}{𝛾_𝐴(𝑥)∨𝛾_𝐵(𝑦)}=𝛾_𝐴∨_𝐵(𝑧). Hence 𝐴∨𝐵⊆𝐵.

Theorem 23. Let 𝐴 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 have 𝜇_𝐵∧_𝐶(𝑥∧𝑦)≥𝜇_𝐵∧_𝐶(𝑥)∧𝜇_𝐵∧_𝐶(𝑦) and 𝛾_𝐵∧_𝐶(𝑥∧𝑦)≤𝛾_𝐵∧_𝐶(𝑥)∧𝛾_𝐵∧_𝐶(𝑦) from the proof of Theorem 14.
(ii) Now by Theorem 22 since 𝐵,  𝐶 are IFF(𝐴), we have 𝐴∨𝐵⊆𝐵  and 𝐴∨𝐶⊆𝐶.
Hence 𝐴∨(𝐵∧𝐶)⊆(𝐴∨𝐵)∧(𝐴∨𝐶)⊆𝐵∧𝐶 by Lemma 10.
(iii) Also clearly 𝜇_𝐵∧_𝐶(𝑒_𝐹)=𝜇_𝐴(𝑒_𝐹) and 𝛾_𝐵∧_𝐶(𝑒_𝐹)=𝛾_𝐴(𝑒_𝐹).
Hence 𝐵∧𝐶 is an IFF(𝐴) by Theorem 22.
Again 𝜇_𝐵∧_𝐶(𝑥)=sup_{𝑢∧𝑣=𝑥}{𝜇_𝐵(𝑢)∧𝜇_𝐶(𝑣)}≥𝜇_𝐵(𝑥)∧𝜇_𝐶(𝑒_𝐹)=𝜇_𝐵(𝑥) and 𝛾_𝐵∧_𝐶(𝑥)=inf_{𝑢∧𝑣=𝑥}{𝛾_𝐵(𝑢)∨𝛾_𝐶(𝑣)}≤𝛾_𝐵(𝑥)∨𝛾_𝐶(𝑒_𝐹)=𝛾_𝐵(𝑥) for every 𝑥∈𝐹.
Hence 𝐵⊆𝐵∧𝐶. Similarly 𝐶⊆𝐵∧𝐶.

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

Definition 24. Let 𝐴 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 [𝐵∶𝐶]=𝐷∣𝐷⊲𝐴,𝐷∨𝐶⊆𝐵. (23) The collection of all quotients of any intuitionistic fuzzy filter 𝐵 of 𝐴 is denoted by [𝐵]_𝐴.

Theorem 25. Let 𝐴 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. Let 𝐴={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 𝜇_𝐵∧_𝐵(𝑥)=sup_{𝑢∧𝑣=𝑥}𝜇_𝐵(𝑢)∧𝜇_𝐵(𝑣)≤sup_{𝑢∧𝑣=𝑥}𝜇_𝐵(𝑢∧𝑣)=𝜇_𝐵(𝑥),𝛾_𝐵∧_𝐵(𝑥)=inf_{𝑢∧𝑣=𝑥}𝛾_𝐵(𝑢)∨𝛾_𝐵(𝑣)≥𝛾_𝐵(𝑢∧𝑣)=𝛾_𝐵(𝑥). (24) Hence 𝐷∧𝐷^′∈𝐴.
Now 𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_{[𝐵∶𝐶]}𝑦=_𝐷∈𝐴𝜇_𝐷(𝑥)∧_𝐷′∈𝐴𝜇_𝐷′𝑦=_{𝐷,𝐷′∈}𝐴𝜇_𝐷(𝑥)∧𝜇_𝐷′𝑦≤_{𝐷,𝐷′∈}𝐴𝜇_𝐷∧_𝐷^′𝑥∧𝑦≤𝜇_{[𝐵∶𝐶]}𝑥∧𝑦, (25) also 𝛾_{[𝐵∶𝐶]}(𝑥)∨𝛾_{[𝐵∶𝐶]}𝑦=_𝐷∈𝐴𝛾_𝐷(𝑥)∨_𝐷′∈𝐴𝛾_𝐷′𝑦=_{𝐷,𝐷′∈}𝐴𝛾_𝐷(𝑥)∨𝛾_𝐷′𝑦≥_{𝐷,𝐷′∈}𝐴𝛾_𝐷∧_𝐷^′𝑥∧𝑦≥𝛾_{[𝐵∶𝐶]}𝑥∧𝑦, (26) since 𝐷∧𝐷^′∈𝐴.
Hence 𝜇_{[𝐵∶𝐶]}𝑥∧𝑦≥min𝜇_{[𝐵∶𝐶]}(𝑥),𝜇_{[𝐵∶𝐶]}𝑦,𝛾_{[𝐵∶𝐶]}𝑥∧𝑦≤max𝛾_{[𝐵∶𝐶]}(𝑥),𝛾_{[𝐵∶𝐶]}𝑦. (27) Now 𝜇_{[𝐵∶𝐶]}𝑥∨𝑦=_𝐷∈𝐴𝜇_𝐷≥𝑥∨𝑦_𝐷∈𝐴𝜇_𝐴(𝑥)∧𝜇_𝐷𝑦∨𝜇_𝐷(𝑥)∧𝜇_𝐴𝑦≥_𝐷∈𝐴𝜇_𝐴(𝑥)∧𝜇_𝐷𝑦=𝜇_𝐴(𝑥)∧_𝐷∈𝐴𝜇_𝐷𝑦=𝜇_𝐴(𝑥)∧𝜇_{[𝐵∶𝐶]}𝑦. (28) Similarly 𝜇_{[𝐵∶𝐶]}(𝑥∨𝑦)≥𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_𝐴(𝑦).
Thus 𝜇_{[𝐵∶𝐶]}𝑥∨𝑦≥max𝜇_𝐴(𝑥)∧𝜇_{[𝐵∶𝐶]}𝑦,𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_𝐴𝑦∀𝑥,𝑦∈𝐹. (29) Now 𝛾_{[𝐵∶𝐶]}𝑥∨𝑦=_𝐷∈𝐴𝛾_𝐷≤𝑥∨𝑦_𝐷∈𝐴𝛾_𝐴(𝑥)∨𝛾_𝐷𝑦∧𝛾_𝐷(𝑥)∨𝛾_𝐴𝑦≤_𝐷∈𝐴𝛾_𝐴(𝑥)∨𝛾_𝐷𝑦=𝛾_𝐴(𝑥)∨_𝐷∈𝐴𝛾_𝐷𝑦=𝛾_𝐴(𝑥)∨𝛾_{[𝐵∶𝐶]}𝑦. (30) Similarly 𝛾_{[𝐵∶𝐶]}(𝑥∨𝑦)≤𝛾_{[𝐵∶𝐶]}(𝑥)∨𝛾_𝐴(𝑦).
Hence 𝛾_{[𝐵∶𝐶]}𝑥∨𝑦≤min𝛾_𝐴(𝑥)∨𝛾_{[𝐵∶𝐶]}𝑦,𝛾_{[𝐵∶𝐶]}(𝑥)∨𝛾_𝐴𝑦∀𝑥,𝑦∈𝐹. (31) Also 𝜇_{[𝐵∶𝐶]}𝑒_𝐹=_𝐷∈𝐴𝜇_𝐷𝑒_𝐹=𝜇_𝐴𝑒_𝐹,𝛾_{[𝐵∶𝐶]}𝑒_𝐹=_𝐷∈𝐴𝛾_𝐷𝑒_𝐹=𝛾_𝐴𝑒_𝐹. (32) 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 Lemma 10 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. Let 𝐴 be an IFCF of 𝐹, and let 𝐵,  𝐶,  𝐸 be IFF(𝐴). Then the following holds:(1)if 𝐵⊆𝐶; then [𝐵∶𝐸]⊆[𝐶∶𝐸] and [𝐸∶𝐶]⊆[𝐸∶𝐵],(2)if 𝐵⊆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}.
If 𝐷∈𝐴, then 𝐷⊲𝐴and𝐷∨𝐸⊆𝐵⊆𝐶. Therefore 𝐷∈𝐴^′ and hence 𝐴⊆𝐴^′.
So [𝐵∶𝐸]=⋃_𝐷∈𝐴𝐷⊆⋃_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.
Hence 𝐴∈𝐴, and so 𝐴⊆⋃_𝐷∈𝐴𝐷=[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).
Let 𝐴={𝐷∣𝐷⊲𝐴and𝐷∨𝐶⊆𝐵} and 𝐴^′={𝐷∣𝐷⊲𝐴and𝐷∨(𝐵∧𝐶)⊆𝐵}.
If 𝐷∈𝐴, then 𝐷⊲𝐴and𝐷∨𝐶⊆𝐵. Now since 𝐷⊆𝐴, 𝐷∨𝐵⊆𝐴∨𝐵 by Lemma 10.
Also by Theorem 22, 𝐵⊲𝐴 implies 𝐴∨𝐵⊆𝐵. Hence 𝐷∨𝐵⊆𝐵.
Therefore, by Lemma 10, 𝐷∨(𝐵∧𝐶)⊆(𝐷∨𝐵)∧(𝐷∨𝐶)⊆𝐵∧𝐵=𝐵 and hence 𝐷∈𝐴^′.
So [𝐵∶𝐶]=⋃_𝐷∈𝐴𝐷⊆⋃_𝐷∈𝐴^′𝐷=[𝐵∶𝐵∧𝐶]. Thus [𝐵∶𝐵∧𝐶]=[𝐵∶𝐶].

Corollary 27. Let 𝐴 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. Let 𝐴 be an IFCF of 𝐹, and let 𝐵,  𝐶,  𝐸 be IFF(𝐴). Then [𝐵∶𝐶∧𝐸]=[𝐵∶𝐶]∩[𝐵∶𝐸].

Proof. We have 𝐶∧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 𝐵∶𝐶∧𝐸⊆[𝐵∶𝐶]∩[𝐵∶𝐸]. (33) Let 𝐴={𝐷∣𝐷⊲𝐴and𝐷∨𝐶⊆𝐵}, 𝐴^′={𝐷∣𝐷⊲𝐴and𝐷∨𝐸⊆𝐵} and  𝐴^′′={𝐷∣𝐷⊲𝐴and𝐷∨(𝐶∧𝐸)⊆𝐵}.
For every 𝑥∈𝐹,  𝜇_{[𝐵∶𝐶]∩[𝐵∶𝐸]}(𝑥)=𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_{[𝐵∶𝐸]}(𝑥)=_𝐷∈𝐴𝜇_𝐷(𝑥)∧_𝐷′∈𝐴^′𝜇_𝐷′(𝑥)=_𝐷′∈𝐴^′𝐷∈𝐴𝜇_𝐷(𝑥)∧𝜇_𝐷′(𝑥)=_𝐷′∈𝐴^′𝐷∈𝐴𝜇_𝐷(𝑥)∧𝜇_𝐷′(𝑥=_𝐷,𝐷′)𝜇_𝐷(𝑥)∧𝜇_𝐷′(𝑥)∣𝐷∈𝐴,𝐷^′∈𝐴^′, (34) also 𝛾_{[𝐵∶𝐶]∩[𝐵∶𝐸]}(𝑥)=𝛾_{[𝐵∶𝐶]}(𝑥)∨𝛾_{[𝐵∶𝐸]}(𝑥)=_𝐷∈𝐴𝛾_𝐷(𝑥)∨_𝐷′∈𝐴^′𝛾_𝐷′(𝑥)=_𝐷′∈𝐴^′𝐷∈𝐴𝛾_𝐷(𝑥)∨𝛾_𝐷′(𝑥)=_𝐷′∈𝐴^′𝐷∈𝐴𝛾_𝐷(𝑥)∨𝛾_𝐷′(𝑥=_𝐷,𝐷′)𝛾_𝐷(𝑥)∨𝛾_𝐷′(𝑥)∣𝐷∈𝐴,𝐷^′∈𝐴^′. (35) Let 𝐷∈𝐴,  𝐷^′∈𝐴^′; then 𝐷,  𝐷^′ are IFF(𝐴) and 𝐷∨𝐶⊆𝐵,  𝐷^′∨𝐸⊆𝐵.
Now by Theorem 21  𝐷∩𝐷^′ is an IFF(𝐴), and also 𝐷∩𝐷^′∨𝐶⊆∧𝐸𝐷∩𝐷^′∨𝐶∧𝐷∩𝐷^′⊆𝐷∧𝐷∨𝐸∨𝐶^′∨𝐸⊆𝐵∧𝐵=𝐵. (36) Hence 𝐷∩𝐷^′∈𝐴^′′. Thus 𝐵∶𝐶∧𝐸=_{𝐷∈𝐴′′}𝐷⊇_𝐷,𝐷′𝐷∩𝐷^′∣𝐷∈𝐴,𝐷^′∈𝐴^′. (37) Now 𝜇_[𝐵∶𝐶∧_𝐸](𝑥)≥_𝐷,𝐷′𝜇_{𝐷∩𝐷′}(𝑥)∣𝐷∈𝐴,𝐷^′∈𝐴^′=_𝐷,𝐷′𝜇_𝐷(𝑥)∧𝜇_𝐷′(𝑥)∣𝐷∈𝐴,𝐷^′∈𝐴^′=𝜇_{[𝐵∶𝐶]∩[𝐵∶𝐸]}(𝑥). (38) Also 𝛾_[𝐵∶𝐶∧_𝐸](𝑥)≤_𝐷,𝐷′𝛾_{𝐷∩𝐷′}(𝑥)∣𝐷∈𝐴,𝐷^′∈𝐴^′=_𝐷,𝐷′𝛾_𝐷(𝑥)∨𝛾_𝐷′(𝑥)∣𝐷∈𝐴,𝐷^′∈𝐴^′=𝛾_{[𝐵∶𝐶]∩[𝐵∶𝐸]}(𝑥) (39) from (35).
Thus 𝐵∶𝐶∧𝐸⊇[𝐵∶𝐶]∩[𝐵∶𝐸]. (40) Consequently [𝐵∶𝐶∧𝐸]=[𝐵∶𝐶]∩[𝐵∶𝐸] from (33) and (40).

Theorem 29. Let 𝐴 be an IFCF of 𝐹, and let the arbitrary collection (𝐴_𝛼)_𝛼∈∧,  𝐵, and 𝐶 be IFF(𝐴); then for the arbitrary collection [𝐵∶𝐴_𝛼]_𝛼∈∧ and [𝐵∶𝐶] of quotients of 𝐵, [𝐵∶𝐶]∩^∼_𝛼∈∧𝐵∶𝐴_𝛼=^∼_𝛼∈∧[𝐵∶𝐶]∩𝐵∶𝐴_𝛼. (41)

Proof. We have ^∼_𝛼∈∧[𝐵∶𝐶]∩𝐵∶𝐴_𝛼=^∼_𝛼∈∧𝐵∶𝐶∧𝐴_𝛼. (42) By Theorem 28.
Now from Theorem 23, 𝐶⊆𝐶∧𝐴_𝛼 and 𝐴_𝛼⊆𝐶∧𝐴_𝛼.
Therefore, by Theorem 26, [𝐵∶𝐶∧𝐴_𝛼]⊆[𝐵∶𝐶] and [𝐵∶𝐶∧𝐴_𝛼]⊆[𝐵∶𝐴_𝛼].
Hence ^∼_𝛼∈∧𝐵∶𝐶∧𝐴_𝛼⊆[𝐵∶𝐶_{∼𝛼∈∧}],𝐵∶𝐶∧𝐴_𝛼⊆^∼_𝛼∈∧𝐵∶𝐴_𝛼. (43) So, ^∼_𝛼∈∧𝐵∶𝐶∧𝐴_𝛼⊆[𝐵∶𝐶]∩^∼_𝛼∈∧𝐵∶𝐴_𝛼. (44) Hence, ^∼_𝛼∈∧[𝐵∶𝐶]∩𝐵∶𝐴_𝛼⊆[𝐵∶𝐶]∩^∼_𝛼∈∧𝐵∶𝐴_𝛼 (45) by (42).
Also for 𝑥∈𝐹, 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]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥=sup_𝛼∈∧𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥=sup_𝛼∈∧𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_𝐴𝑎_𝛼∧𝜇_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥∵𝐵∶𝐴_𝛼⊆𝐴=sup_𝛼∈∧𝜇_{[𝐵∶𝐶]}(𝑥)∧𝜇_𝐴𝑎_𝛼∧𝜇_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥≤sup_𝛼∈∧𝜇_{[𝐵∶𝐶]}𝑎_𝛼∨𝑥∧𝜇_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥(∵[𝐵∶𝐶]isanIFF(𝐹))=sup_𝛼∈∧𝜇_{[𝐵∶𝐶]}𝑎_𝛼∧𝜇_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥=sup_𝛼∈∧𝜇_{[𝐵∶𝐶]∩[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼∧=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]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥=inf_𝛼∈∧𝛾_{[𝐵∶𝐶]}(𝑥)∨𝛾_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥=inf_𝛼∈∧𝛾_{[𝐵∶𝐶]}(𝑥)∨𝛾_𝐴𝑎_𝛼∨𝛾_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥≥inf_𝛼∈∧𝛾_{[𝐵∶𝐶]}𝑎_𝛼∨𝑥∨𝛾_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥(∵[𝐵∶𝐶]isanIFF(𝐹))=inf_𝛼∈∧𝛾_{[𝐵∶𝐶]}𝑎_𝛼∨𝛾_{[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼=𝑥=sup_𝛼∈∧𝜇_{[𝐵∶𝐶]∩[𝐵∶𝐴𝛼]}𝑎_𝛼∣𝑎_𝛼∈𝐹,_𝛼∈∧𝑎_𝛼∧=𝑥=𝜇_{𝛼∈∧([𝐵∶𝐶]∩[𝐵∶𝐴𝛼])}(𝑥). (46) Hence, [𝐵∶𝐶]∩^∼_𝛼∈∧𝐵∶𝐴_𝛼⊆^∼_𝛼∈∧[𝐵∶𝐶]∩𝐵∶𝐴_𝛼. (47) Therefore [𝐵∶𝐶]∩(⋀_𝛼∈∧[𝐵∶𝐴_𝛼])=⋀_𝛼∈∧([𝐵∶𝐶]∩[𝐵∶𝐴_𝛼]) from (45) and (47).

Remark 30. If we interchange the roles of ⋀ and ⋂ in Theorem 29, only one-sided inequality [𝐵∶𝐶]∧(⋂_𝛼∈∧[𝐵∶𝐴_𝛼])⊆⋂_𝛼∈∧([𝐵∶𝐶]∧[𝐵∶𝐴_𝛼]) holds.

Theorem 31. Let 𝐴 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 𝐴_𝑒={(𝑥,𝜇_𝐴𝑒(𝑥),𝛾_𝐴𝑒(𝑥))∣𝑥∈𝐹} where 𝜇_𝐴𝑒(𝑥)=𝜇_𝐴𝑒_𝐹,𝑥=𝑒_𝐹,𝛾0,otherwise,_𝐴𝑒(𝑥)=𝛾_𝐴𝑒_𝐹,𝑥=𝑒_𝐹,0,otherwise. (48) Also [𝐵∶𝐶]∨(∧_𝛼∈∧[𝐵∶𝐴_𝛼])=∧_𝛼∈∧([𝐵∶𝐶]∨[𝐵∶𝐴_𝛼]) for arbitrary collection (𝐴_𝛼)_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 filter 𝐵 of the intuitionistic fuzzy coframe 𝐴_𝜒𝐹 of 𝐹, [𝐵]_𝐴𝜒𝐹 is the sub coframe of the collection of all intuitionistic fuzzy filters 𝐼_𝐹𝐹 of 𝐹, where 𝐴_𝜒𝐹={(𝑥,𝜇_𝐴𝜒𝐹(𝑥),𝛾_𝐴𝜒𝐹(𝑥))∣𝑥∈𝐹},  𝜇_𝐴𝜒𝐹(𝑥)=1,  𝛾_𝐴𝜒𝐹(𝑥)=0 for all 𝑥∈𝐹.

Proof. Follows from Theorems 18 and 31 as [𝐵]_𝐴𝜒𝐹⊆𝐼_𝐹𝐹. Also [𝐵∶𝐵]=𝐴_𝜒𝐹 and 𝐴_𝑒=𝐴_𝑒𝐹 are, respectively, top and bottom elements of [𝐵]_𝐴𝜒𝐹.

Acknowledgments

The author is thankful to Professor T. Thrivikraman, Former Head, Department of Mathematics, Cochin University of Science and Technology, Cochin, India. The author also acknowledges the financial assistance received from University Grants Commission, New Delhi.