Binary relations applied to the fuzzy substructures of quantales under rough environment

Definition 2.1

[1] A quantale 𝒬 is a complete lattice equipped with an associative binary operation ⨀ distributing over arbitrary joins. In other words, for any m, x ∈ 𝒬 and {m _j }, {x _j } ⊆ 𝒬 (j ∈ I), it holds

1. m⨀( ⋁ j ∈ I x j ) = ⋁ j ∈ I (m⨀ x j );

2. ( ⋁ j ∈ I m j )⨀x = ⋁ j ∈ I ( m j ⨀x).

Definition 2.2

[1] Let 𝒬 be a quantale and ∅ ≠ S ⊆ 𝒬 . Then, S is said to be the subquantale ( SQ ) of 𝒬 if

1. ⋁ j ∈ I x j ∈ S ,

2. x ⨀ y ∈ S , ∀ x , y , x j ∈ S .

Definition 2.3

[4] Let 𝒬 be a quantale and ∅ ≠ S ⊆ 𝒬 . Then, S is said to be an ideal ( Id ) if the following hold

1. ∀ a , b ∈ S then a ⋁ b ∈ S ;

2. ∀ a , b ∈ 𝒬 and b ∈ S such that a ≤ b ⇒ a ∈ S ;

3. ∀ a ∈ 𝒬 and b ∈ S ⇒ a ⨀ b ∈ S and b ⨀ a ∈ S .

Definition 2.4

[7] Suppose a non-empty finite set 𝒬 and λ be an equivalence relation on 𝒬 . Let [ x ] λ denote the equivalence class of the relation λ containing x ∈ 𝒬 . If a subset of 𝒬 can be expressed as a union of equivalence classes of 𝒬 , then it is said to be definable set in 𝒬 . Let a subset S of 𝒬 cannot be expressed as a union of equivalence classes of 𝒬 . Then, we say it undefinable set. However, we can approximate that undefinable set by two definable sets in 𝒬 . The first one is called λ -upper approximation of S and the second is called λ -lower approximation of S . They are defined as follows:

Definition 2.5

[3] Let 𝒬 be a quantale. Then, the function λ : 𝒬 ⟶ [ 0 , 1 ] is known as fuzzy subset (FSST) of 𝒬 . Moreover,

• A FSST λ : 𝒬 ⟶ [ 0 , 1 ] is non-empty if λ is not a zero map. Let F ( 𝒬 ) be the collection of all FSSTs of 𝒬 .

• Let λ and φ be two FSSTs of 𝒬 . Then, λ ⊆ φ if and only if λ ( c ) ≤ φ ( c ) ∀ c ∈ 𝒬 . Clearly, λ = φ ⇔ λ ⊆ φ and φ ⊆ λ .

• Let λ and φ be two FSSTs of 𝒬 Then, the union and intersection of λ and φ are ( λ ⋃ φ ) ( k ) = max { λ ( k ) , φ ( k ) } and ( λ ⋂ φ ) ( k ) = min { λ ( k ) , φ ( k ) } ∀ k ∈ 𝒬 , respectively.

Definition 2.6

[33] Let 𝒬 be a quantale and λ ≠ ∅ be a FSST of 𝒬 and α ∈ [ 0 , 1 ] . Then,

Definition 2.7

Let 𝒬 be a quantale and λ ≠ ∅ be a FSST of 𝒬 . Then, λ is a fuzzy subquantale ( FSQ ) of 𝒬 if the following conditions hold

1. λ ( ⋁ j ∈ I x j ) ≥ min j ∈ I λ ( x j ) ;

2. λ ( x ⨀ y ) ≥ min { λ ( x ) , λ ( y ) } ∀ x , y ∈ 𝒬 , { x j } ⊆ 𝒬 .

Definition 2.8

[5] Let 𝒬 be a quantale and λ ≠ ∅ be a FSST of 𝒬 . Then, λ is called fuzzy ideal (Fid) of 𝒬 , if the following conditions hold

1. x ≤ y ⟹ λ ( x ) ≥ λ ( y ) ;

2. min { λ ( x ) , λ ( y ) } ≤ λ ( x ⋁ y ) ;

3. max { λ ( x ) , λ ( y ) } ≤ λ ( x ⨀ y ) ∀ x , y ∈ 𝒬 .

Definition 2.9

[5] Let λ be a non-constant FId of 𝒬 . Then, λ is FPId of 𝒬 if λ ( x ⨀ y ) = λ ( x ) or λ ( x ⨀ y ) = λ ( y ) , ∀ x , y ∈ 𝒬 .

Definition 2.10

[5] Let λ be a non-constant FId 𝒬 . Then, λ is FPId of 𝒬 if λ ( x ⨀ x ) = λ ( x ) for all x ∈ 𝒬 .

Definition 2.11

[8] Fuzzy rough sets were introduced by Dubois and Prade [8]. Let 𝒬 be a non-empty finite set called the universe set and λ be an equivalence relation on 𝒬 . Then, ( 𝒬 , λ ) is called an approximation space. Let η be a FSST. If x ∈ 𝒬 such that

Definition 2.12

Let 𝒬 be a quantale. An equivalence relation T on 𝒬 is called a congruence on 𝒬 if we have a T b , c T d ⟹ ( a ⨀ 1 b , c ⨀ 2 d ) ∈ T and a i T b i ⟹ ⋁ i ∈ I a i T ⋁ i ∈ I b i ∀ a , b , c , d , a i , b i ∈ 𝒬 .

Definition 2.13

Let Ɽ from 𝒬 1 to 𝒬 2 be a CPR. Then,

1. Ɽ is called ⋁ -complete w.r.t. aftersets if r Ɽ ⋁ s Ɽ = ( r ⋁ s ) Ɽ ∀ r , s ∈ 𝒬 1 .

2. Ɽ is called ⨀ -complete w.r.t. aftersets if r Ɽ ⨀ s Ɽ = ( r ⨀ s ) Ɽ ∀ r , s ∈ 𝒬 1 .

3. Ɽ is called ⋁ -complete w.r.t. foresets if Ɽ l ⋁ Ɽ m = Ɽ ( l ⋁ m ) ∀ l , m ∈ 𝒬 2 .

4. Ɽ is called ⨀ -complete w.r.t. foresets if Ɽ l ⨀ Ɽ m = Ɽ ( l ⨀ m ) ∀ l , m ∈ 𝒬 2 .

5. A CPR Ɽ from 𝒬 1 to 𝒬 2 is called complete relation (CMR) w.r.t. aftersets/foresets if it is both ⋁ -complete and ⨀ -complete.

Definition 3.1

Let L ⊆ 𝒬 1 × 𝒬 2 . Then, L : 𝒬 1 ⟶ 𝒬 2 is a BIR. For a FSST λ of 𝒬 2 , the upper approximation ( L ¯ λ ) and the lower approximation ( L ¯ λ ) of λ w.r.t. aftersets are the two FSSTs over 𝒬 1 and can be defined as:

Theorem 3.2

Let L : 𝒬 1 ⟶ 𝒬 2 and I : 𝒬 1 ⟶ 𝒬 2 be the two BIRs and λ 1 and λ 2 be the two non-empty FSSTs of 𝒬 2 . Then, we have

1. λ 1 ≤ λ 2 ⇒ L ¯ λ 1 ≤ L ¯ λ 2 ,

2. λ 1 ≤ λ 2 ⇒ L ¯ λ 1 ≤ L ¯ λ 2 ,

3. L ¯ λ 1 ⋂ L ¯ λ 2 ⊇ L ¯ λ 1 ⋂ λ 2 ,

4. L ¯ λ 1 ⋂ L ¯ λ 2 = L ¯ λ 1 ⋂ λ 2 ,

5. L ¯ λ 1 ⋃ L ¯ λ 2 = L ¯ λ 1 ⋃ λ 2 ,

6. L ¯ λ 1 ⋃ L ¯ λ 1 ⊆ L ¯ λ 1 ⋃ λ 2 ,

7. L ⊆ I ⇒ L ¯ λ 1 ⊆ I ¯ λ 1 ,

8. L ⊆ I ⇒ L ¯ λ 1 ⊇ L ¯ λ 2 .

Proof

Proofs are simple and can be seen from [24].□

Theorem 3.3

Let L : 𝒬 1 ⟶ 𝒬 2 and I : 𝒬 1 ⟶ 𝒬 2 be the two BIRs and λ 1 and λ 2 be the two non-empty FSSTs of 𝒬 1 . Then, we have

1. φ 1 ≤ φ 2 ⇒ L ¯ φ 1 ≤ L ¯ φ 2 ,

2. φ 1 ≤ φ 2 ⇒ L ¯ φ 1 ≤ L ¯ φ 2 ,

3. L ¯ φ 1 ⋂ L ¯ φ 2 ⊇ L ¯ , φ 1 ⋂ φ 2

4. L ¯ φ 1 ⋂ L ¯ φ 2 = L ¯ φ 1 ⋂ φ 2 ,

5. L ¯ φ 1 ⋃ L ¯ φ 2 ⊇ L ¯ φ 1 ⋃ φ 2 ,

6. L ¯ φ 1 ⋃ L ¯ φ 2 = L ¯ φ 1 ⋃ φ 2 ,

7. L ⊆ I ⇒ L ¯ φ 1 ⊆ I ¯ φ 1 ,

8. L ⊆ I ⇒ L ¯ φ 1 ⊇ I ¯ φ 1 .

Proof

Proofs are similar to the proofs of Theorem 3.2.□

Theorem 3.4

Let L : 𝒬 1 ⟶ 𝒬 2 and I : 𝒬 1 ⟶ 𝒬 2 be two BIRs. If λ is a FSST of 𝒬 2 , then:

1. ( L ⋂ I ) — λ ⊆ L ¯ λ ⋂ I ¯ λ ,

2. ( L ⋃ I ) — λ ⊇ L ¯ λ ⋃ I ¯ λ .

Proof

The proof is simple and obtained by parts ( 4 ) and ( 5 ) of Theorem 3.2.□

Example 3.5

Let 𝒬 1 = { 0 , x , 1 } and 𝒬 2 = { 0 ′ , c , d , 1 ′ } be two complete lattices shown in Figures 1 and 2 with the binary operations shown in Tables 1 and 2, respectively. Then, 𝒬 1 and 𝒬 2 are two quantales. Let L ⊆ 𝒬 1 × 𝒬 2 and I ⊆ 𝒬 1 × 𝒬 2 defined as:

Theorem 3.6

Let L : 𝒬 1 ⟶ 𝒬 2 and I : 𝒬 1 ⟶ 𝒬 2 be two BIRs. If φ is a FSST of 𝒬 1 , then we have

1. ( L ⋂ I ) — φ ⊆ L ¯ φ ⋂ I ¯ φ ,

2. ( L ⋂ I ) — φ ⊇ L ¯ φ ⋃ I ¯ . φ

Proof

The proof is straightforward.□

Example 3.7

Consider the quantales in Example 3.5. Let L ⊆ 𝒬 1 × 𝒬 2 and I ⊆ 𝒬 1 × 𝒬 2 be the BIRs defined by:

Definition 4.1

Let L be a BIR from 𝒬 1 to 𝒬 2 and ∅ ≠ λ be a FSST of 𝒬 2 . Then, λ is called generalized upper (lower) rough fuzzy substructure of 𝒬 1 w.r.t. aftersets, if the upper (lower) approximation [ L ¯ λ ( L ¯ λ ) ] is a fuzzy substructure of 𝒬 1 . Similarly, let ∅ ≠ φ be a FSST of 𝒬 1 and L be a BIR. Then, φ is called generalized upper (lower) rough fuzzy substructure of 𝒬 1 w.r.t. foresets, if the upper (lower) approximation [ L ¯ φ ( L ¯ ) φ ] is a fuzzy substructure of 𝒬 2 .

Theorem 4.2

Let L be CPR and ∅ ≠ λ be a FSQ of 𝒬 2 . Then, λ is a generalized upper rough fuzzy subquantale ( GU r RFSQ) of 𝒬 1 w.r.t. to aftersets.

Proof

Let L be CPR and ∅ ≠ λ is a FSQ of 𝒬 2 . Then, λ will satisfy the following properties:

1. λ ( ⋁ j ∈ I x j ) ≥ min j ∈ I λ ( x j ) ,

2. λ ( x ⨀ 2 y ) ≥ min { λ ( x ) , λ ( y ) } ∀ x , y , x j ∈ 𝒬 2 .