Neutrosophic three-way concept lattice and its application in conflict analysis

2.1 Neutrosophic set

Smarandache presented the notion of neutrosophic set in1998. On the basis of neutrosophic set, Wang et al. presented the concept of single-valued neutrosophic set (SVNS) as follows.

Definition 1. [18] Let X be a set of objects. A single-valued neutrosophic set A in X is characterized by a truth-membership function T _A , an indeterminacy-membership function I _A and a falsity-membership function F _A , where T _A , I _A , F _A  : X → [0, 1]. For each x ∈ X, T _A  (x), I _A  (x) and F _A  (x) are the truth-membership degree, indeterminacy-membership degree and falsity-membership degree of x ∈ X to A respectively. The single-valued neutrosophic set A can be denoted by A = { [ x , ( T A ( x ) , I A ( x ) , F A ( x ) ) ] x ∈ X } . For convenience, call D = {(x₁, x₂, x₃) |x₁, x₂, x₃ ∈ [0, 1]} as the set of neutrosophic values. Obviously, (T _A  (x) , I _A  (x) , F _A  (x) ∈ D for each x ∈ X.

For single-valued neutrosophic sets, Smarandache [16, 38] presented an original definition of inclusion relation which is called type-1 inclusion relation. Wang [18] and Borzooei [39] presented another definition of inclusion relation which is called type-2 inclusion relation.

Definition 2. [16, 38] Let A, B be the two neutrosophic sets, A ⊆ ₁B if and only if ∀x ∈ X, T _A  (x) ⩽ T _B  (x), I _A  (x) ⩾ I _B  (x), F _A  (x) ⩾ F _B  (x).

Definition 3. [18, 39] Let A, B be the two neutrosophic sets, A ⊆ ₂B if and only if ∀x ∈ X, T _A  (x) ⩽ T _B  (x), I _A  (x) ⩽ I _B  (x), F _A  (x) ⩾ F _B  (x).

2.2 Pawlak’s conflict analysis model

The conflict analysis is proposed by Pawlak [28]. In Pawlak’s model, the relationship between agents and issues is described as an information system K = (X, A, R), where X is a nonempty and finite set of agents, A is a nonempty and finite set of issues, and R is a mapping, where R : X × A → {-1, 0, + 1}. R (x, a) = +1 means agent x agrees with issue a, R (x, a) = -1 means agent x objects to issue a, and R (x, a) = 0 means agent x is neutral towards a.

In the Pawlak’s model for conflict analysis, an auxiliary function based on K is presented.

Definition 4. [28] Let K = (X, A, R) be an information system, a ∈ A, for ∀x, y ∈ X, the auxiliary function is defined as: φ a ( x , y ) = { 1 , 0 , - 1 , if R ( x , a ) × R ( y , a ) = 1 or x = y , if R ( x , a ) × R ( y , a ) = 0 and x ≠ y , if R ( x , a ) × R ( y , a ) = - 1 .

In conflict analysis, φ _a  (x, y) = 1 indicating that x and y have the same attitude towards a, they are both agree with a or they are both object to a; if φ _a  (x, y) = 0, then at least one agent in x, y has a neutral attitude towards a; if φ _a  (x, y) = -1, it means that x and y have different attitudes about a.

Example 1. [37] The following Table 1 shows an example of conflict analysis. X is a set of four visitors and A is a set of five cities of China. Furthermore, a₁ is Guangzhou; a₂ refers to Chengdu; a₃ is Chongqing; a₄ denotes Tianjin; and a₅ stands for Jinan. As for whether to go to a₂(Chengdu), the attitude of x₁ and x₂ is positive, while x₃ does not want to go, and x₄ is neutral.

On the basis of the auxiliary function, Pawlak proposed the following distance function:

Definition 5. Let K = (X, A, R) be an information system, for ∀x, y ∈ X, the distance function is defined as: ρ ( x , y ) = ∑ a ∈ V 1 - φ a ( x , y ) 2 | A | .

Once the distance of two agents x, y is calculated, then they define the relationship of x and y into the following three categories:

Definition 6. [28] Let K = (X, A, R) be an information system. For ∀x, y ∈ X, x and y is:

(1) conflict if ρ (x, y) > 0.5;

(2) neutral if ρ (x, y) = 0.5;

(3) allied if ρ (x, y) < 0.5.

Formal concept analysis is a useful knowledge representation framework for describing and summarizing data in data analysis. There are some basic notions related to FCA.

Definition 7. [1] A formal context is a triple F = (G, M, I), where G ={ h₁, h₂, …, h _m  } is a nonempty and finite set of objects, M ={ o₁, o₂, …, o _n  } is a nonempty and finite set of attributes, I is the binary relationship between G and M, I ⊆ G × M. For any h ∈ G, o ∈ M, hIo or (h, o) ∈ I means that the object h has the attribute o.

For a formal context F = (G, M, I), H ⊆ G, O ⊆ M, Wille [1] defined the “↑, ↓” operators as follows: H ↑ = { o | o ∈ M , ∀ h ∈ H , ( h , o ) ∈ I } , O ↓ = { h | h ∈ G , ∀ o ∈ O , ( h , o ) ∈ I } .

Definition 8. [1] Let F = (G, M, I) be a formal context, H ⊆ G, O ⊆ M. If H^↑ = O and O^↓ = H, then we call (H, O) as a concept, where H and O are called the extent and intent of (H, O) respectively.

Theorem 1. [1] Let F = (G, M, I) be a formal context, L (G, M, I) = {(H, O) |H^↑ = O, O^↓ = H}. Then (L (G, M, I) , ⩽) is a complete lattice. It is called the concept lattice of the formal context F, where the partial order on L (G, M, I) is given by: ( H 1 , O 1 ) ≤ ( H 2 , O 2 ) ⇔ H 1 ⊆ H 2 ( ⇔ O 1 ⊇ O 2 ) and the infimum and supremum are respectively given by: ( H 1 , O 1 ) ∧ ( H 2 , O 2 ) = ( H 1 ∩ H 2 , ( O 1 ∪ O 2 ) ↓ ↑ ) ( H 1 , O 1 ) ∨ ( H 2 , O 2 ) = ( ( H 1 ∪ H 2 ) ↑ ↓ , O 1 ∩ O 2 )

Belohlavek [40] presented fuzzy formal concept analysis by combining fuzzy sets and formal concept analysis. A fuzzy formal context is a triple K = ( H , O , R ˜ ) , where H represents set of objects, O represents set of attributes, and R ˜ is the L-fuzzy relationship between H and O, R ˜ : H × O → L , L is a complete residuated lattice.

For any L-set A ∈ L ^H of objects, a L-set A ↑ ˜ ∈ L O of attributes can be defined using “ ↑ ˜ ” operator as follows [40]: for each o ∈ O, A ↑ ˜ ( o ) = ∧ h ∈ H ( A ( h ) → R ˜ ( h , o ) ) . Similarly, for any L-set B ∈ L ^O of attributes, a L-set B ↓ ˜ ∈ L H of objects set can be defined using “ ↓ ˜ ” operator as follows [40]: for each h ∈ H, B ↓ ˜ ( h ) = ∧ o ∈ O ( B ( o ) → R ˜ ( h , o ) ) .

If A ↑ ˜ = B and B ↓ ˜ = A , then (A, B) is called a L-fuzzy concept in K. Singh [24] proposed the three-way fuzzy concepts based on neutrosophic sets, where the extent and intent of a concept are neutrosophic sets on H and O respectively.

3.1 The object-induced neutrosophic three-way concept lattices

We present the object-induced neutrosophic three-way concept lattices by using three pairs of operators defined in Definition 11. And some related properties are presented in this subsection.

Definition 12. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, H ∈ 2 ^G , (O, P, Q) ∈2 ^M  × 2 ^M  × 2 ^M , μ : 2 ^G  → 2 ^M  × 2 ^M  × 2 ^M and ν : 2 ^M  × 2 ^M  × 2 ^M  → 2 ^G are defined as follows: H μ = ( H ↑ T , H ↑ I , H ↑ F ) , ( O , P , Q ) ν = O ↓ T ∩ P ↓ I ∩ Q ↓ F .

Definition 13. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, H ∈ 2 ^G , (O, P, Q) ∈2 ^M  × 2 ^M  × 2 ^M . If H ^μ  = (O, P, Q) and (O, P, Q)  ^ν  = H, then (H, (O, P, Q)) is called an object-induced neutrosophic three-way concept of F ̇ , where H and (O, P, Q) is called the extent and intent of (H, (O, P, Q)) respectively. The partial order between object-induced neutrosophic three-way concepts is defined as follows: ( H 1 , ( O 1 , P 1 , Q 1 ) ) ≤ ( H 2 , ( O 2 , P 2 , Q 2 ) ) ⇔ H 1 ⊆ H 2 ⇔ O 1 ⊇ O 2 , P 1 ⊇ P 2 , Q 1 ⊇ Q 2

Definition 14. Let M be a finite nonempty set, 2 ^M  × 2 ^M  × 2 ^M be the Cartesian product of 2 ^M . The partial order “≤” on 2 ^M  × 2 ^M  × 2 ^M is defined by: ( O 1 , P 1 , Q 1 ) ≤ ( O 2 , P 2 , Q 2 ) ⇔ O 1 ⊆ O 2 , P 1 ⊆ P 2 , Q 1 ⊆ Q 2 Additionally, the intersection “∩” and union “∪” on 2 ^M  × 2 ^M  × 2 ^M are presented by: ( O 1 , P 1 , Q 1 ) ∩ ( O 2 , P 2 , Q 2 ) = ( O 1 ∩ O 2 , P 1 ∩ P 2 , Q 1 ∩ Q 2 ) , ( O 1 , P 1 , Q 1 ) ∪ ( O 2 , P 2 , Q 2 ) = ( O 1 ∪ O 2 , P 1 ∪ P 2 , Q 1 ∪ Q 2 ) .

Based on these partial order relations, the following properties can be proved:

Theorem 2. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, H, H₁, H₂ ∈ 2 ^G , (O, P, Q), (O₁, P₁, Q₁), (O₂, P₂, Q₂) ∈ 2 ^M  × 2 ^M  × 2 ^M . The following properties hold:

(1) H 1 ⊆ H 2 ⇒ H 1 μ ≥ H 2 μ , (O₁, P₁, Q₁) ≤ (O₂, P₂, Q₂) ⇒ (O₁, P₁, Q₁)  ^ν  ⊇ (O₂, P₂, Q₂)  ^ν ;

(2) H ⊆ H ^μν , (O, P, Q) ≤ (O, P, Q)  ^νμ ;

(3) H ^μ  = H ^μνμ , (O, P, Q)  ^ν  = (O, P, Q)  ^νμν ;

(4) H ⊆ (O, P, Q)  ^ν  ⇔ (O, P, Q) ≤ X ^μ ;

(5) ( H 1 ∪ H 2 ) μ = H 1 μ ∩ H 2 μ , ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^ν  = (O₁, P₁, Q₁)  ^ν  ∩ (O₂, P₂, Q₂)  ^ν ;

(6) ( H 1 ∩ H 2 ) μ ⊇ H 1 μ ∪ H 2 μ , ((O₁, P₁, Q₁) ∩ (O₂, P₂, Q₂))  ^ν  ⊇ (O₁, P₁, Q₁)  ^ν  ∪ (O₂, P₂, Q₂)  ^ν .

Proof. (1) Assume that H₁ ⊆ H₂. For any o 1 ∈ H 2 ↑ T , o 2 ∈ H 2 ↑ I , o 3 ∈ H 2 ↑ F and h ∈ H₁, by h ∈ H₂ we have T _{o 1}  (h) ≥ δ, F _{o 1}  (h) < θ, T _{o 2}  (h) < θ, I _{o 2}  (h) ≥ σ, F _{o 2}  (h) < θ, T _{o 3}  (h) < θ and F _{o 3}  (h) ≥ δ. It follows that o 1 ∈ H 1 ↑ T , o 2 ∈ H 1 ↑ I , o 3 ∈ H 1 ↑ F and hence H 1 ↑ T ⊇ H 2 ↑ T , H 1 ↑ I ⊇ H 2 ↑ I , H 1 ↑ F ⊇ H 2 ↑ F . Consequently, we have H 1 μ ≥ H 2 μ .

The latter formula can be obtained in the same way.

(2) It can be directly proved by the definition.

(3) Let H ∈ 2 ^G . By (2) we have H ⊆ H ^μν and hence H ^μ  ≥ H ^μνμ by (1). Additionally, we have H ^μ  ≤ (H ^μ )  ^νμ  = H ^μνμ and consequently H ^μ  = H ^μνμ .

We can prove (O, P, Q)  ^ν  = (O, P, Q)  ^νμν similarly.

(4) H ⊆ (O, P, Q)  ^ν  ⇔ H ⊆ O^{↓ _T}  ∩ P^{↓ _I}  ∩ Q^{↓ _F}  ⇔ H ⊆ O^{↓ _T} , H ⊆ P^{↓ _I} , H ⊆ Q^{↓ _F}  ⇔ O ⊆ H^{↑ _T} , P ⊆ H^{↑ _I} , Q ⊆ H^{↑ _F}  ⇔ (O, P, Q) ≤ (H^{↑ _T} , H^{↑ _I} , H^{↑ _F} ) ⇔ (O, P, Q) ≤ H ^μ .

(5) We prove that ( H 1 ∪ H 2 ) μ = H 1 μ ∩ H 2 μ . By (1) we have ( H 1 ∪ H 2 ) ↑ T ⊆ H 1 ↑ T , ( H 1 ∪ H 2 ) ↑ T ⊆ H 2 ↑ T and hence ( H 1 ∪ H 2 ) ↑ T ⊆ H 1 ↑ T ∩ H 2 ↑ T . On the other hand, assume that o ∈ H 1 ↑ T ∩ H 2 ↑ T . It follows that o ∈ H 1 ↑ T and o ∈ H 2 ↑ T . For each h ∈ H₁ ∪ H₂, we have h ∈ H₁ or h ∈ H₂ and hence T _o  (h) ≥ δ, F _o  (h) < θ. Consequently, H 1 ↑ T ∩ H 2 ↑ T ⊆ ( H 1 ∪ H 2 ) ↑ T .

((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^ν  = (O₁, P₁, Q₁)  ^ν  ∩ (O₂, P₂, Q₂)  ^ν can be proved similarly.

(6) It can be proved directly by (1). □

Theorem 3. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context. For any the object-induced neutrosophic three-way concepts (H₁, (O₁, P₁, Q₁)) and (H₂, (O₂, P₂, Q₂)), the infimum and supremum of (H₁, (O₁, P₁, Q₁)) and (H₂, (O₂, P₂, Q₂)) are given by: ( H 1 , ( O 1 , P 1 , Q 1 ) ) ∨ ( H 2 , ( O 2 , P 2 , Q 2 ) ) = ( ( H 1 ∪ H 2 ) μ ν , ( O 1 , P 1 , Q 1 ) ∩ ( O 2 , P 2 , Q 2 ) ) , ( H 1 , ( O 1 , P 1 , Q 1 ) ) ∧ ( H 2 , ( O 2 , P 2 , Q 2 ) ) = ( H 1 ∩ H 2 , ( ( O 1 , P 1 , Q 1 ) ∪ ( O 2 , P 2 , Q 2 ) ) ν μ ) .

Proof. We prove the infimum formula firstly. By ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^ν  = (O₁, P₁, Q₁)  ^ν  ∩ (O₂, P₂, Q₂)  ^ν  = H₁ ∩ H₂ it follows that (H₁ ∩ H₂)  ^μ  = ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμ . Additionally, ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμν  = ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^ν  = H₁ ∩ H₂. We conclude that (H₁ ∩ H₂, ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμ ) is an object-induced neutrosophic three-way concept. By H₁ ∩ H₂ ⊆ H₁ and H₁ ∩ H₂ ⊆ H₂ we have (H₁ ∩ H₂, ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμ ) ≤ (H₁, (O₁, P₁, Q₁)) and (H₁ ∩ X₂, ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμ ) ≤ (H₂, (O₂, P₂, Q₂)). Assume that let (H₃, (O₃, P₃, Q₃) is an object-induced neutrosophic three-way concept with (H₃, (O₃, P₃, Q₃)) ≤ (H₁, (O₁, P₁, Q₁)) and (H₃, (O₃, P₃, Q₃)) ≤ (H₂, (O₂, P₂, Q₂)). It follows that H₃ ⊆ H₁, H₃ ⊆ H₂ and hence H₃ ⊆ H₁ ∩ H₂. We conclude that (H₃, (O₃, P₃, Q₃)) ≤ (H₁ ∩ H₂, ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμ ). Therefore, (H₁ ∩ H₂, ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂))  ^νμ ) is the infimum of (H₁, (O₁, P₁, Q₁)) and (H₂, (O₂, P₂, Q₂)).

The supremum formula can be proved in the same way. □

Let ONTWL ( F ̇ ) be the set of all object-induced neutrosophic three-way concepts of F ̇ . By Theorem 3, ONTWL ( F ̇ ) forms a complete lattice with respect to the partial order defined in Definition 13, it is called the object-induced neutrosophic three-way concept lattice of F ̇ .

Based on these theorems, we propose Algorithm 1 to compute the object-induced neutrosophic three-way concept lattice. The algorithm is described as follow:

Example 2. We consider the neutrosophic formal context F ̇ = ( G , M , I ̇ ) as shown in Table 2. Let δ = θ = σ = 0.5.

Based on Algorithm 1, we construct the object-induced neutrosophic three-way concept lattice of F ̇ which is depicted in Fig. 1. We denote it as ONTWL ( F ̇ ) . For example, let H₁ = {h₃, h₅}. Based on Algorithm 1 and Definition 11, we get that O = H 1 ↑ T = { ∅ } , P = H 1 ↑ O = { o 2 } , and Q = H 1 ↑ F = { o 3 } . Moreover, O^{↓ _T}  = G, P^{↓ _O}  = {h₃, h₅} and Q^{↓ _F}  = {h₂, h₃, h₄, h₅, h₆}. Then we derive that H₁ = O^{↓ _T}  ⋂ P^{↓ _O}  ⋂ Q^{↓ _F}  = {h₃, h₅}. So we can conclude that (h₃h₅, ∅ , o₂, o₃)) is an object-induced neutrosophic three-way concept. When writing concepts, we have omitted brackets and braces for convenience. For example, we use (h₃h₅, ∅ , o₂, o₃)) instead of ({h₃, h₅} , (∅ , {o₂} , {o₃})).

OPEN IN VIEWER

Fig. 1

The Object-induced Neutrosophic Three-way Concept Lattices ONTWL ( F ̇ ) .

Similar to the subsection 3.1, we present the attribute-induced neutrosophic three-way concept lattices.

Definition 15. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, (H, J, S) ∈ 2 ^G  × 2 ^G  × 2 ^G , O ∈ 2 ^M , f : 2 ^G  × 2 ^G  × 2 ^G  → 2 ^M and g : 2 ^M  → 2 ^G  × 2 ^G  × 2 ^G are defined as follows: ( H , J , S ) f = H ↑ T ∩ J ↑ I ∩ S ↑ F , O g = ( O ↓ T , O ↓ I , O ↓ F ) .

Definition 16. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, (H, J, S) ∈ 2 ^G  × 2 ^G  × 2 ^G , O ∈ 2 ^M . If (H, J, S)  ^f  = O and O ^g  = (H, J, S), then ((H, J, S) , O) is called an attribute-induced neutrosophic three-way concept of F ̇ , where (H, J, S) and O is called the extent and intent of ((H, J, S) , O) respectively. The partial order between attribute-induced neutrosophic three-way concepts is defined as follows: ( ( H 1 , J 1 , S 1 ) , O 1 ) ≤ ( ( H 2 , J 2 , S 2 ) , O 2 ) ⇔ H 1 ⊆ H 2 , J 1 ⊆ J 2 , S 1 ⊆ S 2 ⇔ O 1 ⊇ O 2 .

Definition 17. Let G be a finite nonempty set, 2 ^G  × 2 ^G  × 2 ^G be the Cartesian product of 2 ^G . The partial order “≤” on 2 ^G  × 2 ^G  × 2 ^G is defined by: ( H 1 , J 1 , S 1 ) ≤ ( H 2 , J 2 , S 2 ) ⇔ H 1 ⊆ H 2 , J 1 ⊆ J 2 , S 1 ⊆ S 2 Additionally, the intersection “∩” and union “∪” on 2 ^G  × 2 ^G  × 2 ^G are presented by: ( H 1 , J 1 , S 1 ) ∩ ( H 2 , J 2 , S 2 ) = ( H 1 ∩ H 2 , J 1 ∩ J 2 , S 1 ∩ S 2 ) ( H 1 , J 1 , S 1 ) ∪ ( H 2 , J 2 , S 2 ) = ( H 1 ∪ H 2 , J 1 ∪ J 2 , S 1 ∪ S 2 )

Based on these partial order relations, the following properties can be proved:

Theorem 4. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, (H, J, S) , (H₁, J₁, S₁), (H₂, J₂, S₂) ∈2 ^G  × 2 ^G  × 2 ^G , O, O₁, O₂ ∈ 2 ^M . The following properties hold:

(1) (H₁, J₁, S₁) ≤ (H₂, J₂, S₂) ⇒ (H₁, J₁, S₁)  ^f  ⊇ (H₂, J₂, S₂)  ^f , O 1 ⊆ O 2 ⇒ O 1 g ≥ O 2 g ;

(2) O ⊆ O ^gf , (H, J, S) ≤ (H, J, S)  ^fg ;

(3) O = O ^gfg , (H, J, S) = (H, J, S)  ^fgf ;

(4) (H, J, S) ≤ O ^g  ⇔ O ⊆ (H, J, S)  ^f ;

(5) ( O 1 ∪ O 2 ) g = O 1 g ∩ O 2 g , ((H₁, J₁, S₁) ∪ (H₂, J₂, S₂))  ^f  = (H₁, J₁, S₁)  ^f  ∩ (H₂, J₂, S₂)  ^f ;

(6) ( O 1 ∩ O 2 ) g ⊇ O 1 g ∪ O 2 g , ((H₁, J₁, S₁) ∩ (H₂, J₂, S₂))  ^f  ⊇ (H₁, J₁, S₁)  ^f  ∪ (H₂, J₂, S₂)  ^f .

Proof. The proof is similar to Theorem 2. □

Theorem 5. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context. For any the attribute-induced neutrosophic three-way concepts ((H₁, J₁, S₁) , O₁) and ((H₂, J₂, S₂) , O₂), the infimum and supremum of ((H₁, J₁, S₁) , O₁) and ((H₂, J₂, S₂) , O₂) are given by: ( ( H 1 , J 1 , S 1 ) , O 1 ) ∨ ( ( H 2 , J 2 , S 2 ) , O 2 ) = ( ( ( H 1 , J 1 , S 1 ) ∪ ( H 2 , J 2 , S 2 ) ) fg , O 1 ∩ O 2 ) , ( ( H 1 , J 1 , S 1 ) , O 1 ) ∧ ( ( H 2 , J 2 , S 2 ) , O 2 ) = ( ( H 1 , J 1 , S 1 ) ∩ ( H 2 , J 2 , S 2 ) , ( O 1 ∪ O 2 ) gf ) .

Proof. The proof is similar to Theorem 3. □

Let ANTWL ( F ̇ ) be the set of all attribute-induced neutrosophic three-way concepts of F ̇ . By Theorem 5, ANTWL ( F ̇ ) forms a complete lattice with respect to the partial order defined in Definition 16, it is called the attribute-induced neutrosophic three-way concept lattice of F ̇ .

Based on the above discussion, we also can give the following Algorithm 2 for computing attribute-induced neutrosophic three-way concept lattice.

Example 3. We continue considering the neutrosophic formal context F ̇ = ( G , M , I ̇ ) in the Example 2. Based on Algorithm 2, we construct the attribute-induced neutrosophic three-way concept lattice of F ̇ which is depicted in Fig. 2, and we denote it as ANTWL ( F ̇ ) .

OPEN IN VIEWER

Fig. 2

The Attribute-induced Neutrosophic Three-way Concept Lattice ANTWL ( F ̇ ) .

Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, the positive formal context F ̇ T = ( G , M , I ̇ T ) , the neutral formal context F ̇ I = ( G , M , I ̇ I ) and the negative formal context F ̇ F = ( G , M , I ̇ F ) are generated by F ̇ = ( G , M , I ̇ ) . L ( F ̇ T ) , L ( F ̇ I ) and L ( F ̇ F ) are respectively the concept lattices of F ̇ T , F ̇ I and F ̇ F . For any ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , we define “+ _OB ” as (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)). (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) is called an alternate object-induced neutrosophic three-way concept, and the set of all the alternate object-induced neutrosophic three-way concepts are denoted as OBAL ( F ̇ ) .

Theorem 6. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, and F ̇ T = ( G , M , I ̇ T ) the positive formal context is generated by F ̇ = ( G , M , I ̇ ) . For any ( H 1 , O ) ∈ L ( F ̇ T ) , we have H 1 ↑ T = H 1 ↑ + , O^{↓ _T}  = O^↓₊.

Proof. By considering Definition 11 and Definition 18, the proof is straightforward. □

Similarly, the following two Theorems can be obtained.

Theorem 7. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, and F ̇ I = ( G , M , I ̇ I ) the neutral formal context is generated by F ̇ = ( G , M , I ̇ ) . For any ( H 2 , P ) ∈ L ( F ̇ I ) , we have H 2 ↑ I = H 2 ↑ 0 , P^{↓ _T}  = P^↓₀.

Proof. By considering Definition 11 and Definition 19, the proof is straightforward. □

Theorem 8. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context, and F ̇ F = ( G , M , I ̇ F ) the negative formal context is generated by F ̇ = ( G , M , I ̇ ) . For any ( H 3 , Q ) ∈ L ( F ̇ F ) , we have H 3 ↑ F = H 3 ↑ - , Q^{↓ _F}  = Q^↓_-.

Proof. By considering Definition 11 and Definition 20, the proof is straightforward. □

Theorem 9. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context. For any ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , if H₁ = H₂ = H₃, then (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁, (O, P, Q)) is an object-induced neutrosophic three-way concept.

Proof. Firstly, we prove that H 1 μ = ( O , P , Q ) . By H 1 μ = ( H 1 ↑ T , H 1 ↑ I , H 1 ↑ F ) and ( H 1 , O ) ∈ L ( F ̇ T ) , it follows that H 1 ↑ T = H 1 ↑ + = O . Similarly, by ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , we have H 2 ↑ I = H 2 ↑ 0 = P and H 3 ↑ F = H 3 ↑ - = Q . Thus, it follows that H 1 μ = ( H 1 ↑ T , H 1 ↑ I , H 1 ↑ F ) = ( H 1 ↑ T , H 2 ↑ I , H 3 ↑ F ) = ( O , P , Q ) by H₁ = H₂ = H₃. Additionally, by (O, P, Q)  ^ν  = O^{↓ _T}  ∩ P^{↓ _I}  ∩ Q^{↓ _F} , O^{↓ _T}  = O^↓₊ = H₁, P^{↓ _I}  = P^↓₀ = H₂ = H₁ and Q^{↓ _F}  = Q^↓_- = H₃ = H₁, we have (O, P, Q)  ^ν  = H₁ ∩ H₁ ∩ H₁ = H₁. Therefore, (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁, (O, P, Q)) is an object-induced neutrosophic three-way concept. □

Theorem 10. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context. For any ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) satisfies the following properties:

(1) (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ ⊇ O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ ⊇ P, (H₁ ⋂ H₂ ⋂ H₃) ^↑_- ⊇ Q.

(2) H₁ ⋂ H₂ ⋂ H₃ = O^↓₊ ⋂ P^↓₀ ⋂ Q^↓_-.

Proof. (1) By the definition of H^↑₊, we have (H₁ ⋂ H₂ ⋂ H 3 ) ↑ + ⊇ ( H 1 ↑ + ⋃ H 2 ↑ + ⋃ H 3 ↑ + ) ⊇ H 1 ↑ + = O . Similarly, we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ ⊇ P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- ⊇ Q.

(2) By ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , we have O^↓₊ = H₁, O^↓₀ = H₂, O^↓_- = H₃ and hence H₁ ⋂ H₂ ⋂ H₃ = O^↓₊ ⋂ P^↓₀ ⋂Q^↓_-. □

Theorem 11. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context. For any ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , if (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ = O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ = P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- = Q, then (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) is an object-induced neutrosophic three-way concept.

Proof. By Theorem 10(2), we have H₁ ⋂ H₂ ⋂ H₃ = O^↓₊ ⋂ P^↓₀ ⋂ Q^↓_-, and (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ = O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ = P, (H₁ ⋂ H₂ ⋂ H₃) ^↑_- = Q. Therefore, (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) is an object-induced neutrosophic three-way concept. □

Definition 21. Let F ̇ = ( G , M , I ̇ ) be a neutrosophic formal context. For any ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , if (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ ⊃ O or (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ ⊃ P or (H₁ ⋂ H₂ ⋂ H₃) ^↑_- ⊃ Q, then (H₁, O) +  _OB  (H₂, P) +  _OB  (H₃, Q) = (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) is called a redundant object-induced neutrosophic three-way concept. The set of all the redundant object-induced neutrosophic three-way concepts are denoted as OBRE ( F ̇ ) .

By this definition, a redundant object-induced neutrosophic three-way concept is clearly not an object-induced neutrosophic three-way concept. The following Theorem presents a method for constructing object-induced neutrosophic three-way concept lattices based on the set of alternate object-induced neutrosophic three-way concept concepts and the set of redundant object-induced neutrosophic three-way concept concepts of a neutrosophic formal context.

Theorem 12. ONTWL ( F ̇ ) = OBAL ( F ̇ ) - OBRE ( F ̇ ) .

Proof. Firstly, we prove that ONTWL ( F ̇ ) ⊆ OBAL ( F ̇ ) - OBRE ( F ̇ ) . For each ( H , ( O , P , Q ) ) ∈ ONTWL ( F ̇ ) , we have H ^μ  = (H^{↑ _T} , H^{↑ _I} , H^{↑ _F} ) = (O, P, Q) and hence ( H 1 , O ) ∈ L ( F ̇ T ) , ( H 2 , P ) ∈ L ( F ̇ I ) and ( H 3 , Q ) ∈ L ( F ̇ F ) , where H₁ = H^{↑ _T ↓ _T} , H₂ = H^{↑ _I ↓ _I} and H₃ = H^{↑ _F ↓ _F} . Consequently, we have H₁ ⋂ H₂ ⋂ H₃ = H and (H, (O, P, Q)) = (H₁⋂ H₂ ⋂ H 3 , ( O , P , Q ) ) ∈ OBAL ( F ̇ ) . Additionally, we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ = O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ = P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- = Q. It follows that ( H 1 ⋂ H 2 ⋂ H 3 , ( O , P , Q ) ) ∉ OBRE ( F ̇ ) . Therefore, ONTWL ( F ̇ ) ⊆ OBAL ( F ̇ ) - OBRE ( F ̇ ) .

Secondly, we prove that ONTWL ( F ̇ ) ⊇ OBAL ( F ̇ ) - OBRE ( F ̇ ) . For any ( H 1 ⋂ H 2 ⋂ H 3 , ( O , P , Q ) ) ∈ OBAL ( F ̇ ) - OBRE ( F ̇ ) , by (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) ∈ OBAL ( F ̇ ) we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ ⊇ O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ ⊇ P, (H₁ ⋂ H₂ ⋂ H₃) ^↑_- ⊇ Q and H₁ ⋂ H₂ ⋂ H₃ = O^↓₊ ⋂ P^↓₀ ⋂ Q^↓_-. Since ( H 1 ⋂ H 2 ⋂ H 3 , ( O , P , Q ) ) ∉ OBRE ( F ̇ ) , we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ = O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ = P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- = Q. It follows that ( H 1 ⋂ H 2 ⋂ H 3 , ( O , P , Q ) ) ∈ ONTWL ( F ̇ ) . Therefore, ONTWL ( F ̇ ) = OBAL ( F ̇ ) - OBRE ( F ̇ ) is proved. □