A result on single valued neutrosophic refined rough approximation operators 1

1 Introduction

In order to deal with imprecise information and knowledge. Smarandache [14, 15] first introduced the notion of neutrosophic set by fusing the non-standard analysis and a tri-component set. A neutrosophic set consists of three membership functions (truth-membership function, indeterminacy membership function and falsity-membership function), where every function value is a real standard or non-standard subset of the nonstandard unit interval ] 0^-, 1⁺ [. Since then, many authors have been studied various aspects of neutrosophic sets from different point of view, for example, in order to apply the neutrosophic idea to logics. Rivieccio [11] proposed neutrosophic logics which is a generalization of fuzzy logics and studied some basic properties. Guo and Cheng [4] and Guo and Sengur [5] obtained a good applications in image processing and cluster analysis by using neutrosophic sets. Salama and Broumi [13] and Broumi and Smarandache [3] first given a new hybrid mathematical structure called rough neutrosophic sets, handling incomplete and indeterminate information, and studied some operations and their properties.

Wang et al. [16] proposed single valued neutrosophic sets by simplifying neutrosophic sets. single valued neutrosophic sets can also be looked as an extension of intuitionistic fuzzy sets (Atanassov [1]), in which three membership functions are unrelated and their function values belong to the unit closed interval. Single valued neutrosophic sets results in a new hot research issue. Ye [19–21] proposed decision making based on correlation coefficients and weighted correlation coefficient of single valued neutrosophic sets, and illustrated the application of proposed methods. Majumdar and Samant [9] studied distance, similarity and entropy of single valued neutrosophic sets from a theoretical aspect.

Şahin and Küçük [12] proposed a subsethood measure of single valued neutrosophic sets based on distance and showed its effectiveness by an example. We known that there’s a certain connection between fuzzy relations and fuzzy rough approximation operators (resp., fuzzy topologies, information systems [6–8]). Hence, Yang et al. [17] firstly proposed single valued neutrosophic relations and studied some kinds of kernels and closures of single valued neutrosophic relations, then they proposed single valued neutrosophic rough sets [18] by fusing single valued neutrosophic sets and rough sets (Pawlak, [10]), and explored a general framework of the study of single valued neutrosophic rough sets. Concretely, they studied the hybrid model by using constructive and axiomatic approaches. Recently, Bao and Yang [2] introduced n-dimension single valued neutrosophic refined rough sets by combining single valued neutrosophic refined sets with rough sets and further studied the hybrid model from two perspectives– constructive viewpoint and axiomatic viewpoint. However, the supremum and infimum of n-dimension single valued neutrosophic refined rough approximation operators were not given. Along this line, in the present paper, let X be a set, H _n (X) and L _n (X) denote the family of all n-dimension single valued neutrosophic refined upper and lower approximation operators in X, respectively. We can define appropriate order relation ≦ on H _n (X) (resp., L _n (X)) such that both (H _n (X), ≦) and (L _n (X), ≦) are complete lattices. In particular, both (H, ≦) and (L, ≦) are complete lattices, where H and L denote the family of single valued neutrosophic upper and lower approximation operators in X, respectively.

2 Preliminaries

In this section, we briefly recall some basic definitions which will be used in the paper.

3. The lattice structure of n-dimension single valued neutrosophic refined rough approximation operators

In this section, we take SVNRRn(X) to represent the family of all n-dimension single valued neutrosophic refined relations in X.

Let H n ( X ) = { ℛ 1 ¯ | ℛ ∈ S V N R R n ( X ) } and L n ( X ) = { ℛ 1 ¯ | ℛ ∈ S V N R R n ( X ) } be the family of n-dimension single valued neutrosophic refined upper and lower approximation operators in X, respectively. We will study the lattice structure of n-dimension single valued neutrosophic refined rough approximation operators.

Theorem 3.1.

(1) Defined a relation ⊑ on SVNRR_n(X) as fol¬lows: 0130 if and only if for any (a, b) ∈ X × X, T i ℛ 1 ( a , b ) ≤ T i ℛ 2 ( a , b ) , I i ℛ 2 ( a , b ) ≤ I i ℛ 1 ( a , b ) and F i ℛ 2 ( a , b ) ≤ F i ℛ 1 ( a , b ) for all i = 1, 2,…,n. Then (SVNRR_n(X), ⊑) is a poset.

(2) Defined a relation ≦ on H_n(X) as follows: ℛ 1 ¯ ≦ ℛ 2 ¯ if and only if R 1 ¯ ( A ) ℛ 2 ¯ ( A ) for each A ∈ SVNRS_n(X). Then (H_n(X), ≦) is a poset.

(3) Defined a relation ≦ on L_n(X) as follows: ℛ 1 ¯ ≦ ℛ 2 ¯ if and only if ℛ 2 ¯ ( A ) ⊑ ℛ 1 ¯ ( A ) for each A ∈ SVNRSn(X). Then (Ln(X), ≦) is a poset.

Proof. The proof is straightforward and we omit it.

∀ { ℛ i } i ∈ I ⊆ S V N R R n ( X ) and I be a index set, we can define union and intersection of ℛ i as follows: T ⊔ i ∈ I ℛ i ( a , b ) = ∨ i ∈ I ˜ T ℛ i ( a , b ) , I ⊔ i ∈ I ℛ i ( a , b ) = ∧ i ∈ I ˜ I ℛ i ( a , b ) , F ⊔ i ∈ I ℛ i ( a , b ) = ∧ i ∈ I ˜ F ℛ i ( a , b ) , and T ⊓ i ∈ I ℛ i ( a , b ) = ∧ i ∈ I ˜ T ℛ i ( a , b ) , I ⊓ i ∈ I ℛ i ( a , b ) = ∧ i ∈ I ˜ I ℛ i ( a , b ) , F ⊓ i ∈ I ℛ i ( a , b ) = ∨ i ∈ I F ℛ i ( a , b ) , Then ⊔ i ∈ I ℛ i and ⊓ i ∈ I ℛ are two n-dimension single valued neutrosophic refined relations, and we easily show that ⊔ i ∈ I ℛ i and ⊓ i ∈ I ℛ i are infimum and supremum of { ℛ i } i ∈ I , respectively. Hence we can easily obtain the following theorem:

Theorem 3.2. (SVNRR_n(X), ⊑, ⊔, ⊓) is a complete lattice, X_n and ø _n are its top element and bottom element, respectively, where X_n and ø _n are two n-dimension single valued neutrosophic refined relations in X and defined as follows: ∀(a, b) ∈ X × X, T X n ( a , b ) = ( 1 , 1 , … , 1 ) , I X n ( a , b ) = ( 0 , 0 , … , 0 ) , F X n ( a , b ) = ( 0 , 0 , … , 0 ) , and T ∅ n ( a , b ) = ( 0 , 0 , … , 0 ) , I ∅ n ( a , b ) = ( 1 , 1 , … , 1 ) . F ∅ n ( a , b ) = ( 1 , 1 , … , 1 ) .

Remark 3.3. By Definition 2.9, ∀ A ∈ S V N R S n ( X ) , ∀ a ∈ X , we have

I ∅ n _ ( A ) ( a ) = ⋁ b ∈ X ˜ [ ( I ∅ n ( a , b ) ) ∧ I A ( b ) ] = ( 0 , 0 , ⋯ , 0 ) ,

F ∅ n _ ( A ) ( a ) X [ T ∅ n ( a , b ) ∧ F A ( b ) ] = ( 0 , 0 , ⋯ , 0 ) ,

and

T ∅ n ¯ ( A ) ( a ) = ⋁ b ∈ X ˜ [ T ∅ n ( a , b ) ∧ T A ( b ) ] = ( 0 , 0 , ⋯ , 0 ) ,

I ∅ n ¯ ( A ) ( a ) X [ I ∅ n ( a , b ) ∨ I A ( b ) ] = ( 1 , 1 , ⋯ , 1 ) ,

F ∅ n ¯ ( A ) ( a ) = ⋀ b ∈ X ˜ [ F ∅ n ( a , b ) ∨ F A ( b ) ] = ( 1 , 1 , ⋯ , 1 ) . Thus ∀ R ∈ SVNRR n ( X ) , ∅ n ¯ ( A ) ⊑ R ¯ ( A ) and R _ ( A ) ⊑ ∅ n _ ( A ) . This shows ∅ n ¯ ≦ R ¯ and ∅ n _ ≦ R _ . Hence, ∅ n ¯ is the bottom element of (H _n (X), ≦) and ∅ n _ is the bottom element of (L _n (X), ≦).

According to the result of Remark 3.3, we have the following two theorems:

Theorem 3.4. ∀ { R i ¯ } i ∈ I ⊆ (H _n (X), ≦) and I be a index set, we can define union and intersection of R ¯ i as follows: ∨ i ∈ I ^ ℛ i ¯ = ⊔ i ∈ I ℛ ¯ i , ∧ i ∈ I ^ ℛ i ¯ = [ ⊓ i ∈ I ℛ i ¯ ] , where [ ⊓ i ∈ I ℛ i ] = ⊔ { ℛ ∈ S V N R R n ( X ) | ∀ A ∈ S V N R S n ( X ) , ℛ ¯ i ( A ) } . Then ∨ i ∈ I ^ ℛ i ¯ and ∧ i ∈ I ^ ℛ i ¯ are supremum and infimum of { ℛ i ¯ } i ∈ I , respectively.

Proof.

Let ℛ = ⊔ i ∈ I ℛ i then ℛ i ⊑ ℛ for each. By Corollary 2.11 and Theorem 3.1, we have ℛ ¯ i ≦ ℛ ¯ . If ℛ ′ is a n dimension single valued neutrosophic refined relation such that ℛ ¯ i ≦ ℛ ′ ¯ for each i ∈ I, then ∀ A ∈ S V N R S n ( X ) , ℛ ¯ i ( A ) ⊑ ℛ ′ ¯ ( A ) . Hence, ℛ ¯ ( A ) = ⊔ i ∈ I ℛ ¯ i ( A ) = ⊔ i ∈ I ℛ ¯ i ( A ) ⊑ ℛ ′ ¯ ( A ) . Thus ℛ ¯ ≦ ℛ ′ ¯ . So ∨ i ∈ I ^ ℛ ¯ i = ℛ ¯ = ⊔ i ∈ I ℛ i ¯ is the supremum of { ℛ ¯ i } i ∈ I .

Let ℛ * = [ ⊔ i ∈ I ℛ i ] , then ∀B ∈ SVNRS_n(X), we have ∀ A ∈ S V N R S n ( X ) , ℛ ′ ¯ ( A ) ⊑ ℛ ¯ i ( A ) By Theorem 3.1, we have ℛ ★ ¯ ≦ ℛ ¯ i for each i ∈ i. If ℛ ′ is a n-dimension single valued neutrosophic refined relation such that ℛ ′ ¯ ≦ ℛ ¯ i for each i ∈ i, then ∀ A ∈ S V N R S n ( X ) , ℛ ′ ¯ ( A ) ⊑ ℛ ¯ i ( A ) .. Thus ℛ ′ ¯ ( A ) ⊑ ⊓ i ∈ I ℛ ¯ i ( A ) . By the construction of [ ⊓ i ∈ I ℛ j ] , we can easily obtain ℛ ′ ⊑ [ ⊓ i ∈ I ℛ i ] Hence, ℛ ′ ¯ ≦ [ ⊓ i ∈ I ℛ i ] ¯ = ℛ * ¯ . So ∧ i ∈ I ^ ℛ i ¯ = ℛ * ¯ = [ ⊓ i ∈ I ℛ i ] ¯ is the infimum of { ℛ i ¯ } i ∈ I .Theorem 3.5. ∀ { ℛ j ¯ } j ∈ j ⊆ ( L n ( X ) , ≦ ) and J be a index set, we can define union and intersection of ℛ j ¯ as follows: ∨ j ∈ J ^ ℛ j ¯ = ⊔ j ∈ J ℛ j ¯ , ∧ j ∈ J ^ ℛ j ¯ = < ⊔ j ∈ J ℛ j > , ¯ where < ⊔ j ∈ J ℛ j > = ⊔ { ℛ ∈ S V N R R n ( X ) | ∀ A ∈ S V N R S n ( X ) , ⊔ j ∈ J ℛ j ¯ ( A ) ⊑ ℛ ¯ ( A ) } .Then ∨ j ∈ J ^ ℛ j ¯ and ∧ j ∈ J ^ ℛ j ¯ are supremum and infimum of { ℛ j ¯ } j ∈ J , respectively.Proof. Let ℛ = ⊔ j ∈ J ℛ j , then ℛ j ⊑ ℛ for each j ∈ J.By Corollary 2.11 and Theorem 3.1, we have ℛ j ¯ ≦ ℛ ¯ . If ℛ ′ is a n-dimension single valued neutrosophic refined relation such that ℛ j ¯ ≦ ℛ ′ ¯ for each j ∈ J, then ∀ A ∈ S V N R S n ( X ) , ℛ ′ ¯ ( A ) ⊑ ℛ j ¯ ( A ) .. Hence, ℛ ¯ ( A ) = ⊔ j ∈ J ℛ j ¯ ( A ) = ⊓ j ∈ J ℛ j ¯ ( A ) ⊒ ℛ ′ ¯ ( A ) .. Thus ℛ j ¯ ≦ ℛ ′ ¯ . So ∨ j ∈ J ^ ℛ j ¯ = ℛ ¯ = ⊔ j ∈ J ℛ j ¯ is the supremum of { ℛ j ¯ } j ∈ J . Let 0305, then ∀B ∈ SVNRSn(X), we have ℛ ★ ¯ ( B ) = < ⊔ j ∈ J ℛ j ¯ > ( B ) ⊒ ⊔ j ∈ J ℛ j ¯ ( B ) ⊒ ℛ j ¯ ( B ) . By Theorem 3.1, we have ℛ ★ ¯ ≦ ℛ j ¯ for each j ∈ J. If ℛ ′ is a n-dimension single valued neutrosophic refined relation such that ℛ ′ ¯ ≦ ℛ j ¯ for each j ∈ J, then ∀ A ∈ S V N R S n ( X ) , ℛ j ¯ ( A ) ⊑ ℛ ′ ¯ ( A ) . Thus ⊔ j ∈ J ℛ j ( A ) ⊑ ℛ ′ ¯ ( A ) . By the construction of < ⊔ j ∈ J ℛ j > , we can easily obtain < ⊔ j ∈ J ℛ j > ⊒ ℛ ′ . By Corollary 2.11 and Theorem 3.1, we have ℛ ′ ≦ < ⊔ j ∈ J ℛ j > ¯ = ℛ ★ ¯ . So ∧ j ∈ J ^ ℛ j ¯ = ℛ ★ ¯ = < ⊔ j ∈ J ℛ j > ¯ is the infimum of { ℛ j ¯ } j ∈ J .

By Remark 3.3 and Theorems 3.4 and 3.5, we have the following theorem:

Theorem 3.6. Both ( H n ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) and (L _n (X), ≦ , ∨ ˆ , ∧ ˆ ) are complete lattices.

Obviously, when n = 1, both ( H 1 ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) and ( L 1 ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) are complete lattices. Notice that, when n = 1, the n-dimension single valued neutrosophic refined relation will degenerate into a single valued neutrosophic relation. Hence we have the following corollary:

Corollary 3.7. If H (resp., L) denote the family of single valued neutrosophic upper (resp., lower) approximation operators in X, then both (H, ≦) and (L, ≦) are all complete lattices.

4 Conclusion

Following the notion of single valued neutrosophic refined rough approximation operators as introduced by Bao and Yang(2017), we proved that both ( H n ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) and ( L n ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) are complete lattices. In particular, both (H, ≦) and (L, ≦) are all complete lattices. To study the category of single valued neutrosophic refined rough sets in our forthcoming research and there are still exists a problem to be solved as follows: What is the relationship between three complete lattices (SVNRR _n (X), ⊑, ⊔, ⊓), ( H n ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) and ( L n ( X ) , ≦ , ∨ ˆ , ∧ ˆ ) .