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

Abstract

The theory of three-way concept analysis has been developed into an effective tool for data analysis and knowledge discovery. In this paper, we propose neutrosophic three-way concept lattice by combining neutrosophic set with three-way concept analysis and present an approach for conflict analysis by using neutrosophic three-way concept lattice. Firstly, we propose the notion of neutrosophic formal context, in which the relationships between objects and attributes are expressed by neutrosophic numbers. Three pairs of concept derivation operators are proposed. The basic properties of object-induced and attribute-induced neutrosophic three-way concept lattices are examined. Secondly, we divide the neutrosophic formal context into three classical formal contexts and propose the notions of the candidate neutrosophic three-way concepts and the redundant neutrosophic three-way concepts. Two approaches of constructing object-induced (attribute-induced) neutrosophic three-way concept lattices are presented by using candidate, redundant and original neutrosophic three-way concepts respectively. Finally, we apply the neutrosophic formal concept analysis to the conflict analysis and put forward the corresponding optimal strategy and the calculation method of the alliance.

Keywords

Three-way concept analysis neutrosophic set neutrosophic three-way concept lattice conflict analysis

1 Introduction

In 1982, Wille proposed the formal concept analysis [1], also known as the concept lattice theory. The formal concept analysis is based on the formal context, and the formal concepts in the formal context are generated by some derivation operators. According to a specific order relationship, all formal concepts in a formal context form a complete lattice which is named as concept lattice, and it describes the hierarchical structure between concepts. Concept lattice theory has been widely used in knowledge engineering, machine learning, pattern recognition, expert systems, decision analysis, data mining and other fields.

There are two important extensions of formal concept which are named as the attribute-oriented concept and the object-oriented concept. On the basis of Wille’s concept lattice, Duntsch and Gediga [2] introduced a pair of operators to construct the attribute-oriented concept lattice. It provided a supplementary view of the data in a formal context. The notion of object-oriented concept lattices is introduced by Yao [3, 4]. In addition, the role of different concept lattices in data analysis is compared. Qi et al. [5] proposed three-way concepts in formal contexts by considering jointly possessed attributes and jointly not possessed attributes, extended the traditional two-way concept lattice to three-way. Li et al. [6] proposed two models to construct three-way approximate concepts in incomplete contexts, and then the equivalence of the two models is proved. After that, they discussed attribute reduction and attribute characteristics in the three-way approximate concept lattices. Yao [7] presented a common conceptual framework of the notions of interval sets and incomplete formal contexts for representing partially-known concepts. Qi et al. [8] presented the algorithms of constructing three-way concept lattices. Based on the formal context and its complement context, Yang [9] proposed a new method to construct object-induced (OE) three-way concept lattices and attribute-induced (AE) three-way concept lattices by using the candidate OE (AE) concepts and redundant OE (AE) concepts. Qian [10] presented a method to construct three-way concept lattices for attribute dual context. It is proved that in the attribute dual context, three-way concept lattices and concept lattices are isomorphic. Burusco and González [11] proposed the notion of L-fuzzy formal context and presented the method for calculating the L-fuzzy concept lattices. Xu [12] proposed the attribute-induced fuzzy three-way concepts and the object-induced fuzzy three-way concepts by using fuzzy three-way operator and its inverse operator. The concept-cognitive-learning approach to the fuzzy three-way concept is presented. Ji [13] introduced the theory of Pythagorean fuzzy set into the fuzzy three-way concept lattice and presented the construction method of the Pythagorean fuzzy three-way concept lattice in the Pythagorean fuzzy formal context. Yao [14] presented a common conceptual framework of the notions of interval sets and incomplete formal contexts for representing partially-known concepts. While a complete formal context is induced by a binary relation, an incomplete formal context is induced by an interval binary relation that is interpreted as a family of binary relations. Zhao et al. [15] discussed the relationship between different kinds of three-way concept lattices by reformulating and extending the properties of three-way operators.

Neutrosophic set is proposed by Smarandache [16]. The truth-membership function, the falsity-membership function, and the indeterminacy-membership function are used to characterize the neutrosophic set. Based on the neutrosophic set, Wang et al. [17, 18] proposed the notions of single-valued neutrosophic sets (SVNS) and interval neutrosophic sets (INS). Some set-theoretic operations on SVNS and INS are presented. Ye [19, 20] proposed the correlation coefficients, weighted correlation coefficients and cross-entropy for SVNS. The cross-entropy is applied to multi-attribute decision-making. Harish [21] presented some new kinds of operations named as neutrality addition and scalar multiplication for the pairs of single-valued neutrosophic numbers. Harish et al. [22] presented novel algorithms for solving the multiple attribute decision-making problems under the neutrosophic set environment. He [23] also proposed some distance measures to study discrimination between the SVNSs. Singh [24] combined neutrosophic set with three-way fuzzy concept lattice. He extended the attribute of fuzzy concepts to neutrosophic set, and extended the object set of fuzzy concepts to the covering neutrosophic set of attributes. In addition, Singh [25] proposed complex neutrosophic concept lattice and applied it to air quality analysis. To solve the problems of voting, Singh [26] introduced n-valued neutrosophic context and its graphical structure visualization for descriptive analysis. Mao et al. [27] applied the interval neutrosophic set theory to fuzzy formal concept and proposed a new interval neutrosophic fuzzy concept lattice.

Pawlak [28] initiated the study of conflict analysis by using rough sets. To describe conflict situations, the auxiliary functions and distance functions are presented. Lang [29, 30] presented the notions of the maximum positive alliance, central alliance, and negative alliance for conflict situations according to the two thresholds. The incremental algorithm for constructing probabilistic conflict, neutral and allied sets in dynamic information systems are surveyed. Based on Bayesian minimum risk theory, Lang [31] presented an approach to calculate the positive, neutral and negative alliances by using Pythagorean fuzzy loss function given by experts. Sun et al. [32] put forward a new rough set model for conflict analysis. Based on the rough set theory of two universes, a matrix method of conflict analysis is developed for analyzing and solving conflict situations. Fan et al. [33] put forward a conflict analysis model based on three-way decision. According to the conflict situation, two pairs of evaluation functions are defined by using inclusion degrees. In addition, the trisections of agent set and issue set are used to determine the sub-optimal feasible consensus strategies. Yao [34] suggested three levels of conflict consisting of strong conflict, weak conflict and non-conflict and clarified the importance of conceptual and semantics interpretations in conflict. Sun et al. [35] established the rules of acceptance, rejection and non-commitment in conflict decision-making information system respectively. Through the Bayesian risk decision-making theory over two universes rough sets, the methods of calculating the thresholds of alliance partition are given and a feasible consensus strategy is presented. Lang [36] presented a general model of three-way conflict analysis by using conditional alliance and conflict evaluations of two agents with respect to a subset of issues and made a comparative study with the existing models. Zhi [37] discussed conflict analysis under one-vote veto and described an incremental algorithm for conflict analysis in order to dealing with dynamic environment. In real life, many people have “selective phobia” in some cases. That is to say, when they are faced with some choices, they will be unconfident and lack of self-consciousness. They feel difficult to give own opinions clearly. The final choice of them is often “everything is fine”. It means they can either agree to the issue or reject the issue, or remain neutral. This is different from people who keep a neutral attitude. We note that a neutrosophic set is characterized independently by a truth membership function, a falsity membership function and an indeterminacy membership function. So it is suitable to use neutrosophic numbers to distinguish the two situations mentioned above.

According to all the above discussions, we apply neutrosophic set into formal concept analysis to solve the problems in conflict analysis. We note that the existing neutrosophic three-way concept lattices are mainly constructed by using a pair of derivation operators which are natural generalization of Wille’s concept forming operators. Considering the needed of trisections of agent set and issue set in conflict analysis, different from [24] and [27], we use three pairs of concept forming operators to construct neutrosophic three-way concepts. In addition, Singh [24] proposed three-way fuzzy concepts by using neutrosophic set and Gödel residuated lattice, the intent and extent of concepts are neutrosophic sets. The operator “→”žin Gödel residuated lattice is related to an inclusion relation. The inclusion relation for neutrosophic sets, proposed by Smarandache [16, 38], divides three membership functions into two groups and do not really take advantage of the three membership functions. Actually, this kind of inclusion relation is only suitable for some extreme cases. So we present three pairs of operators to distinguish the different importance of the three membership functions. The main contributions of this study are summarized as follows:

Firstly, we present three pairs of approximation operators, namely positive operators, neutral operators and negative operators, to construct neutrosophic three-way concept lattices.

Besides, the candidate neutrosophic three-way concepts, the redundant neutrosophic three-way concepts and the original neutrosophic three-way concepts are presented. Accordingly, two type of construction methods of the object-induced (attribute-induced) neutrosophic three-way concept lattices are examined.

Finally, we define auxiliary function in neutrosophic conflict information system by using the three pairs of operators in neutrosophic three-way concept lattices and put forward the corresponding optimal strategy and the calculation method of the alliance.

The remaining content of this paper is arranged as follows. In Section 2, we review some related properties of neutrosophic set and concept lattice. In Section 3, we introduce the neutrosophic formal contexts and propose the positive operator, neutral operator and negative operator. The object-induced neutrosophic three-way concept lattices and the attribute-induced neutrosophic three-way concept lattices are investigated. In Section 4, we divide the neutrosophic formal context into positive formal context, neutral formal context and negative formal context. The relationships between the concepts generated in positive formal context, neutral formal context, negative formal context and the concepts generated in the neutrosophic formal context are examined. In Section 5, we apply neutrosophic formal concepts to conflict analysis. The auxiliary function is constructed by the attribute-induced neutrosophic three-way concepts. According to the auxiliary function, the distance function, union classes, neutral classes and conflict classes are investigated.

2 Preliminaries

In this section, we recall some fundamental notions and properties related to neutrosophic set, conflict analysis and concept lattice.

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 $A = {[x, (T_{A} (x), I_{A} (x), F_{A} (x))] x \in X} .$ 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: $φ_{a} (x, y) = {\begin{matrix} 1, \\ 0, \\ - 1, \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} if R (x, a) \times R (y, a) = 1 or x = y, \\ if R (x, a) \times R (y, a) = 0 and x \neq y, \\ if R (x, a) \times R (y, a) = - 1 . \end{matrix}$

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.

Table 1
An information system of Example 1

a ₁ a ₂ a ₃ a ₄ a ₅

x ₁ +1 +1 –1 –1 +1

x ₂ –1 +1 +1 –1 +1

x ₃ +1 –1 –1 –1 0

x ₄ –1 0 –1 +1 +1

	a ₁	a ₂	a ₃	a ₄	a ₅
x ₁	+1	+1	–1	–1	+1
x ₂	–1	+1	+1	–1	+1
x ₃	+1	–1	–1	–1	0
x ₄	–1	0	–1	+1	+1

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: $ρ (x, y) = \frac{\sum_{a \in V} \frac{1 - φ_{a} (x, y)}{2}}{| A |} .$

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.

2.3 Formal concept analysis

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: $\begin{matrix} H^{↑} = {o | o \in M, \forall h \in H, (h, o) \in I}, \\ O^{↓} = {h | h \in G, \forall o \in O, (h, o) \in I} . \end{matrix}$

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: $(H_{1}, O_{1}) \leq (H_{2}, O_{2}) \Leftrightarrow H_{1} \subseteq H_{2} (\Leftrightarrow O_{1} \supseteq O_{2})$ and the infimum and supremum are respectively given by: $\begin{matrix} (H_{1}, O_{1}) \land (H_{2}, O_{2}) = (H_{1} \cap H_{2}, {(O_{1} \cup O_{2})}^{↓ ↑}) \\ (H_{1}, O_{1}) \lor (H_{2}, O_{2}) = ({(H_{1} \cup H_{2})}^{↑ ↓}, O_{1} \cap O_{2}) \end{matrix}$

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

For any L-set A ∈ L^H of objects, a L-set $A^{\tilde{↑}} \in L^{O}$ of attributes can be defined using “ $\tilde{↑}$ ” operator as follows [40]: for each o ∈ O, $A^{\tilde{↑}} (o) = \land_{h \in H} (A (h) \to \tilde{R} (h, o)) .$ Similarly, for any L-set B ∈ L^O of attributes, a L-set $B^{\tilde{↓}} \in L^{H}$ of objects set can be defined using “ $\tilde{↓}$ ” operator as follows [40]: for each h ∈ H, $B^{\tilde{↓}} (h) = \land_{o \in O} (B (o) \to \tilde{R} (h, o)) .$

If $A^{\tilde{↑}} = B$ and $B^{\tilde{↓}} = A$ , 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 Neutrosophic three-way concept lattice

In this section, we combine neutrosophic set and formal concept analysis to construct the neutrosophic three-way concept lattice. Different from the related situation presented in [24], the relationship between objects and attributes is characterized by neutrosophic sets.

Definition 9. A neutrosophic formal context is a triple $\dot{F} = (G, M, \dot{I})$ , 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, $\dot{I}$ is a neutrosophic set on G × M, that is, for ∀ (h, o) ∈ G × M, $\dot{I} (h, o)$ is a neutrosophic number. If $\dot{I} (h, o) = (T_{o} (h), I_{o} (h), F_{o} (h))$ , where (T_o (h) , I_o (h) , F_o (h)) ∈ D, it means the degree of which object h has attribute o is T_o (h), the degree of which object h is uncertain whether it has attribute o is I_o (h), and the degree of which object h does not have attribute o is F_o (h).

Definition 10. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, h ∈ G, o ∈ M. Three pairs of operators are defined as follows:

The positive operators: ↑_T : G → 2^M, ↓_T : M → 2^G: $\begin{matrix} h^{↑_{T}} = {o | o \in M, T_{o} (h) \geq δ, F_{o} (h) < θ}, \\ o^{↓_{T}} = {h | h \in G, T_{o} (h) \geq δ, F_{o} (h) < θ} . \end{matrix}$ The neutral operators: ↑_I : G → 2^M, ↓_I : M → 2^G: $\begin{matrix} h^{↑_{I}} = {o | o \in M, T_{o} (h) < θ, I_{o} (h) \geq σ, F_{o} (h) < θ}, \\ o^{↓_{I}} = {h | h \in G, T_{o} (h) < θ, I_{o} (h) \geq σ, F_{o} (h) < θ} . \end{matrix}$ And the negative operators: ↑_F : G → 2^M, ↓_F : M → 2^G: $\begin{matrix} h^{↑_{F}} = {o | o \in M, F_{o} (h) \geq δ, T_{o} (h) < θ}, \\ o^{↓_{F}} = {h | h \in G, F_{o} (h) \geq δ, T_{o} (h) < θ} . \end{matrix}$ Where δ, θ, σ are the thresholds with 0 ≤ δ, θ, σ ≤ 1 and δ ≥ θ.

Definition 11. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, H ∈ 2^G, O ∈ 2^M. Three pairs of operators are defined as follows:

The positive operators: ↑_T : 2^G → 2^M, ↓_T : 2^M → 2^G: $\begin{matrix} H^{↑_{T}} = {o | o \in M, \forall h \in H, T_{o} (h) \geq δ, F_{o} (h) < θ}, \\ O^{↓_{T}} = {h | h \in G, \forall o \in O, T_{o} (h) \geq δ, F_{o} (h) < θ} . \end{matrix}$ The neutral operators: ↑_I : 2^G → 2^M, ↓_I : 2^M → 2^G: $\begin{matrix} H^{↑_{I}} = {o | o \in M, \forall h \in H, T_{o} (h) < θ, I_{o} (h) \geq σ, \\ F_{o} (h) < θ}, \\ O^{↓_{I}} = {h | h \in G, \forall o \in O, T_{o} (h) < θ, I_{o} (h) \geq σ, \\ F_{o} (h) < θ} . \end{matrix}$ And the negative operators: ↑_F : 2^G → 2^M, ↓_F : 2^M → 2^G: $\begin{matrix} H^{↑_{F}} = {o | o \in M, \forall h \in H, F_{o} (h) \geq δ, T_{o} (h) < θ}, \\ O^{↓_{F}} = {h | h \in G, \forall o \in O, F_{o} (h) \geq δ, T_{o} (h) < θ} . \end{matrix}$ Where δ, θ, σ are the thresholds with 0 ≤ δ, θ, σ ≤ 1 and δ ≥ θ.

It is worth to note that: we think that only when T_o (h) is bigger than a certain threshold while F_o (h) is less than a certain threshold, it can be confirmed that the object h has the attribute o.

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 $\dot{F} = (G, M, \dot{I})$ 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: $\begin{matrix} H^{μ} = (H^{↑_{T}}, H^{↑_{I}}, H^{↑_{F}}), \\ {(O, P, Q)}^{ν} = O^{↓_{T}} \cap P^{↓_{I}} \cap Q^{↓_{F}} . \end{matrix}$

Definition 13. Let $\dot{F} = (G, M, \dot{I})$ 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 $\dot{F}$ , 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: $\begin{matrix} (H_{1}, (O_{1}, P_{1}, Q_{1})) \leq (H_{2}, (O_{2}, P_{2}, Q_{2})) \Leftrightarrow \\ H_{1} \subseteq H_{2} \Leftrightarrow O_{1} \supseteq O_{2}, P_{1} \supseteq P_{2}, Q_{1} \supseteq Q_{2} \end{matrix}$

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: $\begin{matrix} (O_{1}, P_{1}, Q_{1}) \leq (O_{2}, P_{2}, Q_{2}) \Leftrightarrow O_{1} \subseteq O_{2}, \\ P_{1} \subseteq P_{2}, Q_{1} \subseteq Q_{2} \end{matrix}$ Additionally, the intersection “∩” and union “∪” on 2^M × 2^M × 2^M are presented by: $\begin{matrix} (O_{1}, P_{1}, Q_{1}) \cap (O_{2}, P_{2}, Q_{2}) = (O_{1} \cap O_{2}, P_{1} \cap P_{2}, \\ Q_{1} \cap Q_{2}), \\ (O_{1}, P_{1}, Q_{1}) \cup (O_{2}, P_{2}, Q_{2}) = (O_{1} \cup O_{2}, P_{1} \cup P_{2}, \\ Q_{1} \cup Q_{2}) . \end{matrix}$

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

Theorem 2. Let $\dot{F} = (G, M, \dot{I})$ 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) $H_{1} \subseteq H_{2} \Rightarrow H_{1}^{μ} \geq H_{2}^{μ}$ , (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) $(H_{1} \cup H_{2})^{μ} = H_{1}^{μ} \cap H_{2}^{μ}$ , ((O₁, P₁, Q₁) ∪ (O₂, P₂, Q₂)) ^ν = (O₁, P₁, Q₁) ^ν ∩ (O₂, P₂, Q₂) ^ν;

(6) $(H_{1} \cap H_{2})^{μ} \supseteq H_{1}^{μ} \cup H_{2}^{μ}$ , ((O₁, P₁, Q₁) ∩ (O₂, P₂, Q₂)) ^ν ⊇ (O₁, P₁, Q₁) ^ν ∪ (O₂, P₂, Q₂) ^ν.

Proof. (1) Assume that H₁ ⊆ H₂. For any $o_{1} \in H_{2}^{↑_{T}}$ , $o_{2} \in H_{2}^{↑_{I}}$ , $o_{3} \in H_{2}^{↑_{F}}$ 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 $o_{1} \in H_{1}^{↑_{T}}$ , $o_{2} \in H_{1}^{↑_{I}}$ , $o_{3} \in H_{1}^{↑_{F}}$ and hence $H_{1}^{↑_{T}} \supseteq H_{2}^{↑_{T}}$ , $H_{1}^{↑_{I}} \supseteq H_{2}^{↑_{I}}$ , $H_{1}^{↑_{F}} \supseteq H_{2}^{↑_{F}}$ . Consequently, we have $H_{1}^{μ} \geq H_{2}^{μ}$ .

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 ${(H_{1} \cup H_{2})}^{μ} = H_{1}^{μ} \cap H_{2}^{μ}$ . By (1) we have ${(H_{1} \cup H_{2})}^{↑_{T}} \subseteq H_{1}^{↑_{T}}$ , ${(H_{1} \cup H_{2})}^{↑_{T}} \subseteq H_{2}^{↑_{T}}$ and hence ${(H_{1} \cup H_{2})}^{↑_{T}} \subseteq H_{1}^{↑_{T}} \cap H_{2}^{↑_{T}}$ . On the other hand, assume that $o \in H_{1}^{↑_{T}} \cap H_{2}^{↑_{T}}$ . It follows that $o \in H_{1}^{↑_{T}}$ and $o \in H_{2}^{↑_{T}}$ . For each h ∈ H₁ ∪ H₂, we have h ∈ H₁ or h ∈ H₂ and hence T_o (h) ≥ δ, F_o (h) < θ. Consequently, $H_{1}^{↑_{T}} \cap H_{2}^{↑_{T}} \subseteq {(H_{1} \cup H_{2})}^{↑_{T}}$ .

((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 $\dot{F} = (G, M, \dot{I})$ 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: $\begin{matrix} (H_{1}, (O_{1}, P_{1}, Q_{1})) \lor (H_{2}, (O_{2}, P_{2}, Q_{2})) = \\ ((H_{1} \cup H_{2})^{μ ν}, (O_{1}, P_{1}, Q_{1}) \cap (O_{2}, P_{2}, Q_{2})), \\ (H_{1}, (O_{1}, P_{1}, Q_{1})) \land (H_{2}, (O_{2}, P_{2}, Q_{2})) = \\ (H_{1} \cap H_{2}, ((O_{1}, P_{1}, Q_{1}) \cup (O_{2}, P_{2}, Q_{2}))^{ν μ}) . \end{matrix}$

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 $ONTWL (\dot{F})$ be the set of all object-induced neutrosophic three-way concepts of $\dot{F}$ . By Theorem 3, $ONTWL (\dot{F})$ 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 $\dot{F}$ .

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

Algorithm 1 Computing the object-induced neutrosophic three-way concept lattice
Input: A neutrosophic formal context $\dot{F} = (G, M, \dot{I})$
Output: The object-induced neutrosophic three-way concept (H, (O, P, Q))
1: Set thresholds δ, θ and σ.
2: for each H ⊆ Gdo
3: Calculate H^↑_T, H^↑_O, and H^↑_F according to Definition 11.
4: Let O = H^↑_T, P = H^↑_O, and Q = H^↑_F.
5: Calculate O^↓_T, P^↓_O and Q^↓_F according to Definition 11.
6: Let H = O^↓_T ⋂ P^↓_O ⋂ Q^↓_F.
7: end for
8: return The object-induced neutrosophic three-way concept (H, (O, P, Q))

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

Table 2
The Neutrosophic formal context $\dot{F} = (G, M, \dot{I})$

o ₁ o ₂ o ₃

h ₁ (1.0,0.2,0.3) (0.9,0.1,0.3) (0.1,0.7,0.3)

h ₂ (0.9,0.1,0.1) (0.9,0.9,0.9) (0.1,0.2,0.9)

h ₃ (0.1,0.2,0.9) (0.1,0.6,0.2) (0.2,0.3,0.8)

h ₄ (0.6,0.0,0.1) (0.1,0.1,0.9) (0.1,0.3,0.9)

h ₅ (0.9,0.1,0.2) (0.1,0.9,0.1) (0.1,0.1,0.9)

h ₆ (0.0,0.0,1.0) (0.9,0.1,0.1) (0.2,0.1,0.9)

	o ₁	o ₂	o ₃
h ₁	(1.0,0.2,0.3)	(0.9,0.1,0.3)	(0.1,0.7,0.3)
h ₂	(0.9,0.1,0.1)	(0.9,0.9,0.9)	(0.1,0.2,0.9)
h ₃	(0.1,0.2,0.9)	(0.1,0.6,0.2)	(0.2,0.3,0.8)
h ₄	(0.6,0.0,0.1)	(0.1,0.1,0.9)	(0.1,0.3,0.9)
h ₅	(0.9,0.1,0.2)	(0.1,0.9,0.1)	(0.1,0.1,0.9)
h ₆	(0.0,0.0,1.0)	(0.9,0.1,0.1)	(0.2,0.1,0.9)

Based on Algorithm 1, we construct the object-induced neutrosophic three-way concept lattice of $\dot{F}$ which is depicted in Fig. 1. We denote it as $ONTWL (\dot{F})$ . For example, let H₁ = {h₃, h₅}. Based on Algorithm 1 and Definition 11, we get that $O = H_{1}^{↑_{T}} = {\emptyset}$ , $P = H_{1}^{↑_{O}} = {o_{2}}$ , 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})).

Fig. 1

The Object-induced Neutrosophic Three-way Concept Lattices $ONTWL (\dot{F})$ .

3.2 The attribute-induced neutrosophic three-way concept lattices

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

Definition 15. Let $\dot{F} = (G, M, \dot{I})$ 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^Gare defined as follows: $\begin{matrix} {(H, J, S)}^{f} = H^{↑_{T}} \cap J^{↑_{I}} \cap S^{↑_{F}}, \\ O^{g} = (O^{↓_{T}}, O^{↓_{I}}, O^{↓_{F}}) . \end{matrix}$

Definition 16. Let $\dot{F} = (G, M, \dot{I})$ 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 $\dot{F}$ , 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: $\begin{matrix} ((H_{1}, J_{1}, S_{1}), O_{1}) \leq ((H_{2}, J_{2}, S_{2}), O_{2}) \Leftrightarrow \\ H_{1} \subseteq H_{2}, J_{1} \subseteq J_{2}, S_{1} \subseteq S_{2} \Leftrightarrow O_{1} \supseteq O_{2} . \end{matrix}$

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: $\begin{matrix} (H_{1}, J_{1}, S_{1}) \leq (H_{2}, J_{2}, S_{2}) \Leftrightarrow H_{1} \subseteq H_{2}, \\ J_{1} \subseteq J_{2}, S_{1} \subseteq S_{2} \end{matrix}$ Additionally, the intersection “∩” and union “∪” on 2^G × 2^G × 2^G are presented by: $\begin{matrix} (H_{1}, J_{1}, S_{1}) \cap (H_{2}, J_{2}, S_{2}) = (H_{1} \cap H_{2}, J_{1} \\ \cap J_{2}, S_{1} \cap S_{2}) \\ (H_{1}, J_{1}, S_{1}) \cup (H_{2}, J_{2}, S_{2}) = (H_{1} \cup H_{2}, J_{1} \\ \cup J_{2}, S_{1} \cup S_{2}) \end{matrix}$

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

Theorem 4. Let $\dot{F} = (G, M, \dot{I})$ 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, $O_{1} \subseteq O_{2} \Rightarrow O_{1}^{g} \geq O_{2}^{g}$ ;

(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) $(O_{1} \cup O_{2})^{g} = O_{1}^{g} \cap O_{2}^{g}$ , ((H₁, J₁, S₁) ∪ (H₂, J₂, S₂)) ^f = (H₁, J₁, S₁) ^f ∩ (H₂, J₂, S₂) ^f;

(6) $(O_{1} \cap O_{2})^{g} \supseteq O_{1}^{g} \cup O_{2}^{g}$ , ((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 $\dot{F} = (G, M, \dot{I})$ 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: $\begin{matrix} ((H_{1}, J_{1}, S_{1}), O_{1}) \lor ((H_{2}, J_{2}, S_{2}), O_{2}) = \\ (((H_{1}, J_{1}, S_{1}) \cup (H_{2}, J_{2}, S_{2}))^{fg}, O_{1} \cap O_{2}), \\ ((H_{1}, J_{1}, S_{1}), O_{1}) \land ((H_{2}, J_{2}, S_{2}), O_{2}) = \\ ((H_{1}, J_{1}, S_{1}) \cap (H_{2}, J_{2}, S_{2}), (O_{1} \cup O_{2})^{gf}) . \end{matrix}$

Proof. The proof is similar to Theorem 3. □

Let $ANTWL (\dot{F})$ be the set of all attribute-induced neutrosophic three-way concepts of $\dot{F}$ . By Theorem 5, $ANTWL (\dot{F})$ 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 $\dot{F}$ .

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

Algorithm 2 Computing the attribute-induced neutrosophic three-way concept lattice
Input: A neutrosophic formal context $\dot{F} = (G, M, \dot{I})$
Output: The attribute-induced neutrosophic three-way concept ((H, J, S) , O)
1: Set thresholds δ, θ and σ;
2: for each O ∈ Mdo
3: Calculate O^↓_T, O^↓_O and O^↓_F according to Definition 11.
4: Let H = O^↓_T, J = O^↓_O, and S = O^↓_F.
5: Calculate H^↑_T, J^↑_O, and S^↑_F according to Definition 11.
6: Let O = H^↑_T ⋂ J^↑_O ⋂ S^↑_F.
7: end for
8: return The attribute-induced neutrosophic three-way concept ((H, J, S) , O)

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

Fig. 2

The Attribute-induced Neutrosophic Three-way Concept Lattice $ANTWL (\dot{F})$ .

4 Constructing the neutrosophic three-way concept lattices

In the previous section, we proposed the notion of neutrosophic three-way concept lattices. In this section, we presented a method to constructing neutrosophic three-way concept lattices. Some definitions which provided in this section are positive concept lattice, neutral concept lattice and negative concept lattice, alternate object-induced (attribute-induced) neutrosophic three-way concepts, redundant object-induced (attribute-induced) neutrosophic three-way concepts. We transform the neutrosophic formal context into three classical formal contexts. Then the neutrosophic three-way concept lattices are constructed by using the related three classical formal concept lattices.

Definition 18. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. $\dot{F}$ induces a positive formal context ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$ , where ${\dot{I}}_{T} \subseteq G \times M$ , ${\dot{I}}_{T} = {(h, o) | T_{o} (h) \geq δ, F_{o} (h) < θ}$ .

Let $L ({\dot{F}}_{T})$ be the concept lattice of ${\dot{F}}_{T}$ . We call $L ({\dot{F}}_{T})$ as the positive concept lattice. In order to avoid confusion, the operators in the positive formal context are denoted by “↑₊, ↓ ₊”, i.e., for any H ⊆ G and O ⊆ M, we have: $\begin{matrix} H^{↑_{+}} = {o | o \in M, \forall h \in H, (h, o) \in {\dot{I}}_{T}}, \\ O^{↓_{+}} = {h | h \in G, \forall o \in O, (h, o) \in {\dot{I}}_{T}} . \end{matrix}$

Similarly, the definitions of neutral and negative formal context are presented as follows:

Definition 19. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. $\dot{F}$ induces a neutral formal context ${\dot{F}}_{I} = (G, M, {\dot{I}}_{I})$ , where ${\dot{I}}_{I} \subseteq G \times M$ , ${\dot{I}}_{I} = {(h, o) | T_{o} (h) < θ, I_{o} (h) \geq σ, F_{o} (h) < θ .}$ .

Let $L ({\dot{F}}_{I})$ be the concept lattice of ${\dot{F}}_{I}$ . We call $L ({\dot{F}}_{I})$ as the neutral concept lattice. In order to avoid confusion, the operators in the neutral formal context are denoted by “↑₀, ↓ ₀”, i.e., for any H ⊆ G and O ⊆ M, we have: $\begin{matrix} H^{↑_{0}} = {o | o \in M, \forall h \in H, (h, o) \in {\dot{I}}_{I}}, \\ O^{↓_{0}} = {h | h \in G, \forall o \in O, (h, o) \in {\dot{I}}_{I}} . \end{matrix}$

Definition 20. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. $\dot{F}$ induces a negative formal context ${\dot{F}}_{F} = (G, M, {\dot{I}}_{F})$ , where ${\dot{I}}_{F} \subseteq G \times M$ , ${\dot{I}}_{F} = {(h, o) | T_{o} (h) < θ, F_{o} (h) \geq δ}$ .

Let $L ({\dot{F}}_{F})$ be the concept lattice of ${\dot{F}}_{F}$ . We call $L ({\dot{F}}_{F})$ as the negative concept lattice. In order to avoid confusion, the operators in the negative formal context are denoted by “↑_-, ↓ _-”, i.e., for any H ⊆ G and O ⊆ M, we have: $\begin{matrix} H^{↑_{-}} = {o | o \in M, \forall h \in H, (h, o) \in {\dot{I}}_{F}}, \\ O^{↓_{-}} = {h | h \in G, \forall o \in O, (h, o) \in {\dot{I}}_{F}} . \end{matrix}$

Example 4. The neutrosophic formal context in Example 2 is transformed into the following positive formal context, neutral formal context and negative formal context:

The concept lattices $L ({\dot{F}}_{T})$ , $L ({\dot{F}}_{I})$ and $L ({\dot{F}}_{F})$ are depicted in Figs. 3, 4 and 5 respectively.

Fig. 3

Positive Concept Lattice $L ({\dot{F}}_{T})$ .

Fig. 4

Neutral Concept Lattice $L ({\dot{F}}_{I})$ .

Fig. 5

Negative Concept Lattice $L ({\dot{F}}_{F})$ .

4.1 Type-I constructing the object-induced neutrosophic three-way concept lattices

Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, the positive formal context ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$ , the neutral formal context ${\dot{F}}_{I} = (G, M, {\dot{I}}_{I})$ and the negative formal context ${\dot{F}}_{F} = (G, M, {\dot{I}}_{F})$ are generated by $\dot{F} = (G, M, \dot{I})$ . $L ({\dot{F}}_{T}), L ({\dot{F}}_{I})$ and $L ({\dot{F}}_{F})$ are respectively the concept lattices of ${\dot{F}}_{T}$ , ${\dot{F}}_{I}$ and ${\dot{F}}_{F}$ . For any $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 $OBAL (\dot{F})$ .

Theorem 6. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, and ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$ the positive formal context is generated by $\dot{F} = (G, M, \dot{I})$ . For any $(H_{1}, O) \in L ({\dot{F}}_{T})$ , 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 $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, and ${\dot{F}}_{I} = (G, M, {\dot{I}}_{I})$ the neutral formal context is generated by $\dot{F} = (G, M, \dot{I})$ . For any $(H_{2}, P) \in L ({\dot{F}}_{I})$ , 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 $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, and ${\dot{F}}_{F} = (G, M, {\dot{I}}_{F})$ the negative formal context is generated by $\dot{F} = (G, M, \dot{I})$ . For any $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 $(H_{1}, O) \in L ({\dot{F}}_{T})$ , it follows that $H_{1}^{↑_{T}} = H_{1}^{↑_{+}} = O$ . Similarly, by $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , (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 $⋂ H_{3})^{↑_{+}} \supseteq (H_{1}^{↑_{+}} ⋃ H_{2}^{↑_{+}} ⋃ H_{3}^{↑_{+}}) \supseteq H_{1}^{↑_{+}} = O$ . Similarly, we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ ⊇ P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- ⊇ Q.

(2) By $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , we have O^↓₊ = H₁, O^↓₀ = H₂, O^↓_- = H₃ and hence H₁ ⋂ H₂ ⋂ H₃ = O^↓₊ ⋂ P^↓₀ ⋂Q^↓_-. □

Theorem 11. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 $OBRE (\dot{F})$ .

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. $ONTWL (\dot{F}) = OBAL (\dot{F}) - OBRE (\dot{F}) .$

Proof. Firstly, we prove that $ONTWL (\dot{F}) \subseteq OBAL (\dot{F}) - OBRE (\dot{F})$ . For each $(H, (O, P, Q)) \in ONTWL (\dot{F})$ , we have H^μ = (H^↑_T, H^↑_I, H^↑_F) = (O, P, Q) and hence $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ and $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , 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 ⋂ $H_{3}, (O, P, Q)) \in OBAL (\dot{F})$ . Additionally, we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ = O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ = P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- = Q. It follows that $(H_{1} ⋂ H_{2} ⋂ H_{3}, (O, P, Q)) \notin OBRE (\dot{F})$ . Therefore, $ONTWL (\dot{F}) \subseteq OBAL (\dot{F}) - OBRE (\dot{F})$ .

Secondly, we prove that $ONTWL (\dot{F}) \supseteq OBAL (\dot{F}) - OBRE (\dot{F})$ . For any $(H_{1} ⋂ H_{2} ⋂ H_{3}, (O, P, Q)) \in OBAL (\dot{F}) - OBRE (\dot{F})$ , by (H₁ ⋂ H₂ ⋂ H₃, (O, P, Q)) $\in OBAL (\dot{F})$ we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ ⊇ O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ ⊇ P, (H₁ ⋂ H₂ ⋂ H₃) ^↑_- ⊇ Q and H₁ ⋂ H₂ ⋂ H₃ = O^↓₊ ⋂ P^↓₀ ⋂ Q^↓_-. Since $(H_{1} ⋂ H_{2} ⋂ H_{3}, (O, P, Q)) \notin OBRE (\dot{F})$ , we have (H₁ ⋂ H₂ ⋂ H₃) ^↑₊ = O, (H₁ ⋂ H₂ ⋂ H₃) ^↑₀ = P and (H₁ ⋂ H₂ ⋂ H₃) ^↑_- = Q. It follows that $(H_{1} ⋂ H_{2} ⋂ H_{3}, (O, P, Q)) \in ONTWL (\dot{F})$ . Therefore, $ONTWL (\dot{F}) = OBAL (\dot{F}) - OBRE (\dot{F})$ is proved. □

Example 5. Consider the neutrosophic formal context in Table 2, then we have:

$OBAL (\dot{F}) = {(\emptyset, (M, M, M)), (h_{1}, (o_{1} o_{2}, o_{3}, \emptyset))$ ,

(h₁, (o₁o₂, ∅ , ∅)) , (h₁, (o₁, o₃, ∅)) , (h₁, (o₂, o₃, ∅)),

(h₁, (∅ , o₃, ∅)) , (h₅, (o₁, o₂, o₃)) , (h₅, (o₁, o₂, ∅)) , (h₂

h₄h₅, (o₁, ∅ , o₃)) , (h₁h₂h₄h₅, (o₁, ∅ , ∅)) , (h₄, (o₁, ∅ ,

o₂o₃)) , (h₄, (∅ , ∅ , o₂o₃)) (h₆, (o₂, ∅ , o₁o₃)) , (h₆, (o₂, ∅ ,

o₃)) , (h₁h₆, (o₂, ∅ , ∅)) , (h₃h₅, (∅ , o₂, o₃)) , (h₃h₅, (∅ ,

o₂, ∅)) , (h₃, (∅ , o₂, o₁o₃)) , (h₂h₃h₄h₅h₆, (∅ , ∅ , o₃)) ,

(h₃h₆, (∅ , ∅ , o₁o₃)) , (G, (∅ , ∅ , ∅))}.

$OBRE (\dot{F}) = {(h_{1}, (o_{1} o_{2}, \emptyset, \emptyset)), (h_{1}, (o_{1}, o_{3}, \emptyset)),$

(h₁, (o₂, o₃, ∅)) , (h₁, (∅ , o₃, ∅)) , (h₅, (o₁, o₂, ∅)) , (h₄,

(∅ , ∅ , o₂o₃)) , (h₆, (o₂, ∅ , o₃)) , . (h₃h₅, (∅ , o₂, ∅))}.

$ONTWL (\dot{F}) = OBAL (\dot{F}) - OBRE (\dot{F}) = {(\emptyset, (M,$

M, M)) , (h₁, (o₁o₂, o₃, ∅)) , (h₅, (o₁, o₂, o₃)) , (h₂h₄h₅,

(o₁, ∅ , o₃)) , (h₁h₂h₄h₅, (o₁, ∅ , ∅)) , (h₄, (o₁, ∅ , o₂o₃)) ,

(h₆, (o₂, ∅ , o₁o₃)) , (h₁h₆, (o₂, ∅ , ∅)) , (h₃h₅, (∅ , o₂, o₃)) ,

(h₃, (∅ , o₂, o₁o₃)) , (h₂h₃h₄h₅h₆, (∅ , ∅ , o₃)) , (h₃h₆,

(∅ , ∅ , o₁o₃)) , (G, (∅ , ∅ , ∅))}.

4.2 Type-I constructing the attribute-induced neutrosophic three-way concept lattices

Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, the positive formal context ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$ , the neutral formal context ${\dot{F}}_{I} = (G, M, {\dot{I}}_{I})$ and the negative formal context ${\dot{F}}_{F} = (G, M, {\dot{I}}_{F})$ are generated by $\dot{F} = (G, M, \dot{I})$ . $L ({\dot{F}}_{T}), L ({\dot{F}}_{I})$ and $L ({\dot{F}}_{F})$ are respectively the concept lattices of ${\dot{F}}_{T}$ , ${\dot{F}}_{I}$ and ${\dot{F}}_{F}$ . For any $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(J, O_{2}) \in L ({\dot{F}}_{I})$ and $(S, O_{3}) \in L ({\dot{F}}_{F})$ , we define “+_AT” as (H, O₁) + _AT (J, O₂) + _AT (S, O₃) = ((H, J, S) , O₁ ⋂ O₂ ⋂ O₃). ((H, J, S) , O₁ ⋂ O₂ ⋂ O₃) is called an alternate attribute-induced neutrosophic three-way concept, and the set of all the alternate attribute-induced neutrosophic three-way concept is denoted as $ATAL (\dot{F})$ .

The proofs of the following conclusions on attribute-induced neutrosophic three-way concept lattices are similar to that of object-induced neutrosophic three-way concept lattices (Section 4.1). So we omit them.

Theorem 13. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(J, O_{2}) \in L ({\dot{F}}_{I})$ and $(S, O_{3}) \in L ({\dot{F}}_{F})$ , if O₁ = O₂ = O₃, then (H, O₁) + _AT (J, O₂) + _AT (S, O₃) = ((H, J, S) , O₁) is an attribute-induced neutrosophic three-way concept.

Theorem 14. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(J, O_{2}) \in L ({\dot{F}}_{I})$ and $(S, O_{3}) \in L ({\dot{F}}_{F})$ , (H, O₁) + _AT (J, O₂) + _AT (S, O₃) = ((H, J, S) , O₁ ⋂ O₂ ⋂ O₃) satisfies the following properties:

(1) (O₁ ⋂ O₂ ⋂ O₃) ^↓₊ ⊇ H, (O₁ ⋂ O₂ ⋂ O₃) ^↓₀ ⊇ J, (O₁ ⋂ O₂ ⋂ O₃) ^↓_- ⊇ Q.

(2) O₁ ⋂ O₂ ⋂ O₃ = H^↑₊ ⋂ J^↑₀ ⋂ S^↑_-.

Theorem 15. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context. For any $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(J, O_{2}) \in L ({\dot{F}}_{I})$ and $(S, O_{3}) \in L ({\dot{F}}_{F})$ , if (O₁ ⋂ O₂ ⋂ O₃) ^↓₊ = H, (O₁ ⋂ O₂ ⋂ O₃) ^↓₀ = J and (O₁ ⋂ O₂ ⋂ O₃) ^↓_- = S, then (H, O₁) + _AT (J, O₂) + _AT (S, O₃) = ((H, J, S) , O₁ ⋂ O₂ ⋂ O₃ is an attribute-induced neutrosophic three-way concept.

Definition 22. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic formal context, the positive formal context ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$ , the neutral formal context ${\dot{F}}_{I} = (G, M, {\dot{I}}_{I})$ and the negative formal context ${\dot{F}}_{F} = (G, M, {\dot{I}}_{F})$ are generated by $\dot{F} = (G, M, \dot{I})$ . $L ({\dot{F}}_{T})$ , $L ({\dot{F}}_{I})$ and $L ({\dot{F}}_{F})$ are respectively the concept lattices of ${\dot{F}}_{T}$ , ${\dot{F}}_{I}$ and ${\dot{F}}_{F}$ . For any $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(J, O_{2}) \in L ({\dot{F}}_{I})$ and $(S, O_{3}) \in L ({\dot{F}}_{F})$ , if (O₁⋂ O₂ ⋂ O₃) ^↓₊ ⊃ H or (O₁ ⋂ O₂ ⋂ O₃) ^↓₀ ⊃ J or (O₁⋂ O₂ ⋂ O₃) ^↓_- ⊃ S, then (H, O₁) + _AT (J, O₂) + _AT (S, O₃) = ((H, J, S) , O₁ ⋂ O₂ ⋂ O₃) is called a redundant attribute-induced neutrosophic three-way concept, and the set of all the redundant attribute-induced neutrosophic three-way concepts are denoted as $ATRE (\dot{F})$ .

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

Theorem 16. $ANTWL (\dot{F}) = ATAL (\dot{F}) - ATRE (\dot{F}) .$

Example 6. Consider the neutrosophic formal context in Table 2, then we have:

$ATAL (\dot{F}) = {((\emptyset, \emptyset, \emptyset), M) ., ((h_{1}, \emptyset, \emptyset), o_{1} o_{2}), ((\emptyset, \emptyset,$ h₃h₆) , o₁o₃) , ((∅ , ∅ , h₄) , o₂o₃) , ((h₁h₂h₄h₅, ∅ , h₃h₆) , o₁) , ((h₁, ∅ , h₃h₆) , o₁) , ((h₁h₂h₄h₅, ∅ , ∅) , o₁) , ((h₁h₆, h₃h₅, h₄) , o₂) , ((h₁, h₃h₅, ∅) , o₂) , ((h₁h₆, ∅ , ∅) , o₂) , ((h₁h₆, h₃h₅, ∅) , o₂) , ((∅ , h₃h₅, ∅) , o₂) , ((∅ , h₃h₅, h₄) , o₂) , ((h₁, ∅ , h₄) , o₂) , ((h₁, h₃h₅, h₄) , o₂) , ((h₁h₆, ∅ , h₄) , o₂) , ((∅ , h₁, h₂h₃h₄h₅h₆) , o₃) , ((∅ , ∅ , h₂h₃h₄h₅h₆) , o₃) , ((∅ , h₁, h₃h₆) , o₃) , ((∅ , h₁, ∅) , o₃) , ((∅ , h₁, h₄) , o₃) , ((G, G, G) , ∅)} .

$ATRE (\dot{F}) = {((h_{1}, \emptyset, h_{3} h_{6}), o_{1}), ((h_{1} h_{2} h_{4} h_{5}, \emptyset, \emptyset),$ o₁) , ((h₁, h₃h₅, ∅) , o₂) , ((h₁h₆, ∅ , ∅) , o₂) , ((h₁h₆, h₃h₅, ∅) , o₂) , ((∅ , h₃h₅, ∅) , o₂) , ((∅ , h₃h₅, h₄) , o₂) , ((h₁, ∅ , h₄) , o₂) , ((h₁, h₃h₅, h₄) , o₂) , ((h₁h₆, ∅ , h₄) , o₂) , ((∅ , ∅ , h₂h₃h₄h₅h₆) , o₃) , ((∅ , h₁, h₃h₆) , o₃) , ((∅ , h₁, ∅) , o₃) , ((∅ , h₁, h₄) , o₃)}.

$ANTWL (\dot{F}) = ATAL (\dot{F}) - ATRE (\dot{F}) = {((\emptyset, \emptyset, \emptyset),$ M) , ((h₁, ∅ , ∅) , o₁o₂) , ((∅ , ∅ , h₃h₆) , o₁o₃) , ((∅ , ∅ , h₄) , o₂o₃) , ((h₁h₂h₄h₅, ∅ , h₃h₆) , o₁) , ((h₁h₆, h₃h₅, h₄) , o₂) , ((∅ , h₁, h₂h₃h₄h₅h₆) , o₃) , ((G, G, G) , ∅)}.

4.3 Type-II 2 constructing the object-induced neutrosophic three-way concept lattices

In Section 4.1, we present type-I constructing the object-induced neutrosophic three-way concept lattices by $OBAL (\dot{F})$ and $OBRE (\dot{F})$ . In this section, we will propose another method to construct the object-induced neutrosophic three-way concept lattices, which named as type-II 2 constructing the object-induced neutrosophic three-way concept lattices. Compared with the type-I, the type-II 2 does not have redundant concepts, so the calculation is simpler.

Let F = (G, M, I) be a formal context, H ⊆ G, (H^↑↓, H^↑) ∈ L (G, M, I). For any (H′, O′) ∈ L (G, M, I), if H ⊆ H′, then we have O′ ⊆ H^↑. The following Theorems provide the Type-II 2 method for constructing object-induced neutrosophic three-way concept lattices based on the property.

Theorem 17. Let $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{2}, P) \in L ({\dot{F}}_{I})$ , H₁ ∩ H₂ = H. Then (H^↑_T↓_T ∩ H^↑_I↓_I, (H^↑_T, H^↑_I, $(H^{↑_{T} ↓_{T}} \cap H^{↑_{I} ↓_{I}})^{↑_{F}})) \in ONTWL (\dot{F})$ .

Proof. We prove (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_T = H^↑_T firstly. By (H^↑_T↓_T ∩ H^↑_I↓_I) ⊆ H^↑_T↓_T, we have (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_T ⊇ H^{↑_T↓_T↑_T} = H^↑_T. Then we prove that (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_T ⊆ H^↑_T. By H₁ ∩ H₂ = H ⊆ H^↑_T↓_T and H₁ ∩ H₂ = H ⊆ H^↑_I↓_I, we have H₁ ∩ H₂ ⊆ H^↑_T↓_T ∩ H^↑_I↓_I, so (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_T ⊆ (H₁ ∩ H₂) ^↑_T = H^↑_T. Therefore, (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_T = H^↑_T. Similarly, we have (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_I = H^↑_I and (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_F = (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_F. Conversely, (H^↑_T↓_T ∩ H^↑_I↓_I) ⊆ (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_F↓_F, so H^↑_T↓_T ∩ H^↑_I↓_I ∩ (H^↑_T↓_T ∩ H^↑_I↓_I) ^↑_F↓_F = H^↑_T↓_T ∩ H^↑_I↓_I. In summary, $(H_{1}^{'} \cap H_{2}^{'}, (O^{'}, P^{'}, Q^{'})) \in ONTWL (\dot{F})$ . □

Theorem 18. Let $(H_{1}, O) \in L ({\dot{F}}_{T})$ , $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , H₁ ∩ H₃ = H. Then (H^↑_T↓_T ∩ H^↑_F↓_F, (H^↑_T, $(H^{↑_{T} ↓_{T}} \cap H^{↑_{F} ↓_{F}})^{↑_{I}}, H^{↑_{F}})) \in ONTWL (\dot{F})$ .

Theorem 19. Let $(H_{2}, P) \in L ({\dot{F}}_{I})$ , $(H_{3}, Q) \in L ({\dot{F}}_{F})$ , H₂ ∩ H₃ = H. Then $(H^{↑_{I} ↓_{I}} \cap H^{↑_{F} ↓_{F}}, ((H^{↑_{I} ↓_{I}} \cap H^{↑_{F} ↓_{F}})^{↑_{T}}, H^{↑_{I}}, H^{↑_{F}})) \in ONTWL (\dot{F})$ .

We call the concepts constructed by Theorem 17–19 as original object-induced neutrosophic three-way concepts, and the set of all the original object-induced neutrosophic three-way concepts are denoted as $OBOR (\dot{F})$ .

Theorem 20. $ONTWL (\dot{F}) = OBOR (\dot{F})$ .

Proof. It is obvious that $OBOR (\dot{F}) \subseteq ONTWL (\dot{F})$ , then we prove that $ONTWL (\dot{F}) \subseteq OBOR (\dot{F})$ . If $(H, (O, P, Q)) \in ONTWL (\dot{F})$ , then we have H^μ = (H^↑_T, H^↑_I, H^↑_F) = (O, P, Q), hence $(H, O) \in L ({\dot{F}}_{T})$ , $(H, P) \in L ({\dot{F}}_{I})$ and $(H, Q) \in L ({\dot{F}}_{T})$ . So we can construct (H, (O, P, Q)) by Theorem 17 or 18 or 19. It follows that $(H, (O, P, Q)) \in OBOR (\dot{F})$ . Therefore, $ONTWL (\dot{F}) = OBOR (\dot{F})$ . □

Example 7. Consider the neutrosophic formal context in Table 2, then we have: $ONTWL (\dot{F}) = OBOR (\dot{F}) = {(\emptyset, (M, M, M)), (h_{1},$ (o₁o₂, o₃, ∅)) , (h₅, (o₁, o₂, o₃)) , (h₂h₄h₅, (o₁, ∅ , o₃)), (h₁h₂h₄h₅, (o₁, ∅ , ∅)) , (h₄, (o₁, ∅ , o₂o₃)) , (h₆, (o₂, ∅ , o₁o₃)) , (h₁h₆, (o₂, ∅ , ∅)) , (h₃h₅, (∅ , o₂, o₃)) , (h₃, (∅ , o₂, o₁o₃)) , (h₂h₃h₄h₅h₆, (∅ , ∅ , o₃)) , (h₃h₆, (∅ , ∅ , o₁o₃)) , (G, (∅ , ∅ , ∅))}.

4.4 Type-II 2 constructing the attribute-induced neutrosophic three-way concept lattices

Similar to Section 4.3, we can construct the attribute-induced neutrosophic three-way concept lattices by another way. The proofs of the following conclusions are similar to section 4.3. So we omit them.

Let F = (G, M, I) be a formal context, O ⊆ M, (O^↓, O^↓↑) ∈ L (G, M, I). For any (H′, O′) ∈ L (G, M, I), if O ⊆ O′, then we have H′ ⊆ O^↓. The following Theorems provide the Type-II 2 method for constructing attribute-induced neutrosophic three-way concept lattices based on the property.

Theorem 21. Let $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(J, O_{2}) \in L ({\dot{F}}_{I})$ , O₁ ∩ O₂ = O. Then ((O^↓_T, O^↓_I, (O^↓_T↑_T ∩ O^↓_I↑_I) ^↓_F) , O^↓_T↑_T ∩ O^↓_I↑_I).

Theorem 22. Let $(H, O_{1}) \in L ({\dot{F}}_{T})$ , $(S, O_{3}) \in L ({\dot{F}}_{F})$ , O₁ ∩ O₃ = O. Then ((O^↓_T, (O^↓_T↑_T ∩ O^↓_F↑_F) ^↓_I, O^↓_F) , O^↓_T↑_T ∩ O^↓_F↑_F).

Theorem 23. Let $(J, O_{2}) \in L ({\dot{F}}_{I})$ , $(S, O_{3}) \in L ({\dot{F}}_{F})$ , O₂ ∩ O₃ = O. Then (((O^↓_I↑_I ∩ O^↓_F↑_F) ^↓_T, O^↓_I, O^↓_F) , O^↓_I↑_I ∩ O^↓_F↑_F).

We call the concepts constructed by Theorem 21–23 as original attribute-induced neutrosophic three-way concepts, and the set of all the original attribute-induced neutrosophic three-way concepts are denoted as $ATOR (\dot{F})$ .

Theorem 24. $ANTWL (\dot{F}) = ATOR (\dot{F})$ .

Example 8. Consider the neutrosophic formal context in Table 2, then we have: $ANTWL (\dot{F}) = ATOR (\dot{F}) = {((\emptyset, \emptyset, \emptyset), M), ((h_{1},$ ∅, ∅) , o₁o₂) , ((∅ , ∅ , h₃h₆) , o₁o₃) , ((∅ , ∅ , h₄) , o₂o₃), ((h₁h₂h₄h₅, ∅ , h₃h₆) , o₁) , ((h₁h₆, h₃h₅, h₄t) , o₂) , ((∅ , h₁, h₂h₃h₄h₅h₆) , o₃) , ((G, G, G) , ∅)}.

5 Conflict analysis model based on neutrosophic three-way concept lattices

In this section, we study the conflict analysis model based on the neutrosophic three-way concept lattices. Some definitions which provided in this section are the neutrosophic conflict information system, the auxiliary function, the union class, the neutral class, the conflict class and so on.

Definition 23. A neutrosophic conflict information system is a triple $\dot{F} = (G, M, \dot{I})$ , where G = {h₁, h₂, …, h_n} is the set of all agents, M = {o₁, o₂, …, o_m} is the set of all issues, $\dot{I} : G \times M \to D$ , D is the set of neutrosophic values. For any h ∈ G and o ∈ M, $\dot{I} (h, o) = (T_{o} (h), I_{o} (h), F_{o} (h))$ . It indicates that the degree of agent h agrees with issue o is T_o (h), the degree of agent h objects to issue o is F_o (h) and the degree of agent h is neutral towards o is I_o (h).

Example 9. Table 6 is a neutrosophic conflict information system $\dot{F} = (G, M, \dot{I})$ , where G = {h₁, h₂, h₃, h₄, h₅, h₆}, M = {o₁, o₂, o₃}. G is the set of four visitors, M is the set of three cities of China. o₁ is Beijing, o₂ is Shanghai, o₃ is Chengdu. o₁ (h₁) = (1.0, 0.2, 0.3) represents the degree of h₁ wants to go to Beijing is 1.0, the degree of whether h₁ goes to Beijing or not is 0.2, the degree of h₁ does not want to go to Beijing is 0.3.

Table 3
Positive formal context ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$

o ₁ o ₂ o ₃

h ₁ 1 1 0

h ₂ 1 0 0

h ₃ 0 0 0

h ₄ 1 0 0

h ₅ 1 0 0

h ₆ 0 1 0

	o ₁	o ₂
h ₁	1	1
h ₂	1	0
h ₃	0	0
h ₄	1	0
h ₅	1	0
h ₆	0	1

Table 4

Neutral formal context ${\dot{F}}_{I} = (G, M, {\dot{I}}_{I})$

	o ₁	o ₂	o ₃
h ₁	0	0	1
h ₂	0	0	0
h ₃	0	1	0
h ₄	0	0	0
h ₅	0	1	0
h ₆	0	0	0

Table 5

Negative formal context ${\dot{F}}_{F} = (G, M, {\dot{I}}_{F})$

	o ₁	o ₂	o ₃
h ₁	0	0	0
h ₂	0	0	1
h ₃	1	0	1
h ₄	0	1	1
h ₅	0	0	1
h ₆	1	0	1

Table 6

Neutrosophic conflict infonnation system $\dot{F} = (G, M, \dot{I})$

	o ₁	o ₂	o ₃
h ₁	(1.0,0.2,0.3)	(0.9,0.1,0.3)	(0.1,0.7,0.3)
h ₂	(0.9,0.1,0.1)	(0.9,0.9,0.9)	(0.1,0.2,0.9)
h ₃	(0.1,0.2,0.9)	(0.1,0.6,0.2)	(0.2,0.3,0.8)
h ₄	(0.6,0.0,0.1)	(0.1,0.1,0.9)	(0.1,0.3,0.9)
h ₅	(0.9,0.1,0.2)	(0.1,0.9,0.1)	(0.1,0.1,0.9)
h ₆	(0.0,0.0,1.0)	(0.9,0.1,0.1)	(0.2,0.1,0.9)

Definition 24. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system, ∀h_i, h_j ∈ G, o ∈ M, the attribute-induced neutrosophic three-way concept constructed by o is ((o^↓_T, o^↓_I, o^↓_F) , o^gf), and we define the auxiliary function φ_o (h_i, h_j) as follows: $\begin{matrix} φ_{o} (h_{i}, h_{j}) = \\ {\begin{matrix} 1, \\ - 1, \\ 0, \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} if h_{i}, h_{j} \in o^{↓_{T}} or h_{i}, h_{j} \in o^{↓_{I}} or h_{i}, h_{j} \in o^{↓_{F}}, \\ if h_{i} \in o^{↓_{T}}, h_{j} \in o^{↓_{F}} or h_{i} \in o^{↓_{F}}, h_{j} \in o^{↓_{T}}, \\ else . \end{matrix} \end{matrix}$

Similarly, we can define the distance function as follows:

Definition 25. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system, ∀h_i, h_j ∈ G, the distance ζ is defined as: $ζ (h_{i}, h_{j}) = \frac{\sum_{o \in V} \frac{1 - φ_{o} (h_{i}, h_{j})}{2}}{| M |} .$

Similarly, for two given agents, we calculate their distance, then we can define their relationship as the following three kinds of categories:

Definition 26. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system, ∀h_i, h_j ∈ G, 0 ≤ t₁ < t₂ ≤ 1, t₁, t₂ are the thresholds given by experts, then h_i and h_j is:

(1) conflict if ζ (h_i, h_j) > t₂;

(2) neutral if t₁ ≤ ζ (h_i, h_j) ≤ t₂;

(3) allied if ζ (h_i, h_j) < t₁.

Definition 27. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system, ∀h_i, h_j ∈ G, 0 ≤ t₁ < t₂ ≤ 1, through the distance ζ, the union class [h_i] _POS, the neutral class [h_i] _NEU and the conflict class [h_i] _NEG are defined as follows:

[h_i] _POS ={ h_j ∈ G|ζ (h_i, h_j) < t₁ },

[h_i] _NEU ={ h_j ∈ G|t₁ ≤ ζ (h_i, h_j) ≤ t₂ },

[h_i] _NEG ={ h_j ∈ G|ζ (h_i, h_j) > t₂ }.

Definition 28. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system, ∀h_i, h_j ∈ G, 0 ≤ t₁ < t₂ ≤ 1, the positive region POS (h_i), neutral region NEU (h_i) and negative region NEG (h_i) are defined as:

(1) POS (h_i) is the maximum set that satisfies:

“POS (h_i) ⊆ [h_i] _POS, ∀h_j, h_m ∈ POS (h_i) (ζ (h_j, h_m) lessthant₁)”.

(2) NEU (h_i) is the maximum set that satisfies:

“NEU (h_i) ⊆ [h_i] _NEU, ∀h_j, h_m ∈ NEU (h_i) (t₁ ≤ ζ (h_j, h_m) ≤ t₂)”.

(3) NEG (h_i) is the maximum set that satisfies:

“NEG (h_i) ⊆ [h_i] _NEG, ∀h_j, h_m ∈ NEG (h_i) (ζ (h_j, h_m) greaterthant₂)”.

It is easy to conclude that the distance between any two agents in the positive region is less than t₁, the distance between any two agents in the neutral region is between t₁ and t₂, and the distance between any two agents in the negative region is more than t₂.

The above method is considered from the perspective of the attribute-induced neutrosophic three-way concepts, then we will consider from the perspective of the object-induced neutrosophic three-way concepts. For the convenience, we denote the set of union issues of h_i and h_j as α (h_i, h_j), the set of conflicting issues of h_i and h_j as β (h_i, h_j) and the set of neutral issues of h_i and h_j as γ (h_i, h_j).

Theorem 25. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system. ∀h_i, h_j ∈ G, o ∈ M, the object-induced neutrosophic three-way concept constructed by h_i, h_j are $(h_{i}^{μ ν}, (h_{i}^{↑_{T}}, h_{i}^{↑_{I}}, h_{i}^{↑_{F}}))$ , $(h_{j}^{μ ν}, (h_{j}^{↑_{T}}, h_{j}^{↑_{I}}, h_{j}^{↑_{F}}))$ respectively, then we have the following properties:

(1) $α (h_{i}, h_{j}) = {h_{i}^{↑_{T}} \cap h_{j}^{↑_{T}}} \cup {h_{i}^{↑_{I}} \cap h_{j}^{↑_{I}}} \cup {h_{i}^{↑_{F}} \cap h_{j}^{↑_{F}}}$ ,

(2) $β (h_{i}, h_{j}) = {h_{i}^{↑_{T}} \cap h_{j}^{↑_{F}}} \cup {h_{i}^{↑_{F}} \cap h_{j}^{↑_{T}}}$ ,

(3) $γ (h_{i}, h_{j}) = {h_{i}^{↑_{T}} \cap h_{j}^{↑_{I}}} \cup {h_{i}^{↑_{F}} \cap h_{j}^{↑_{I}}} \cup {h_{i}^{↑_{I}} \cap h_{j}^{↑_{T}}} \cup {h_{i}^{↑_{I}} \cap h_{j}^{↑_{F}}}$ .

Proof. (1) Let o ∈ M, if $o \in h_{i}^{↑_{T}} \cap h_{j}^{↑_{T}}$ , then T_o (h_i) ≥ δ, F_o (h_i) < θ and T_o (h_j) ≥ δ, F_o (h_j) < θ, so h_i ∈ o^↓_T and h_j ∈ o^↓_T. By the definition, we have φ_o (h_i, h_j) = 1. Similarly, if $o \in h_{i}^{↑_{I}} \cap h_{j}^{↑_{I}}$ , then T_o (h_i) < θ, F_o (h_i) < θ, I_o (h_i) ≥ σ and T_o (h_j) < θ, F_o (h_j) < θ, I_o (h_j) ≥ σ, so h_i ∈ o^↓_I and h_j ∈ o^↓_I, then φ_o (h_i, h_j) = 1. If $o \in h_{i}^{↑_{F}} \cap h_{j}^{↑_{F}}$ , then T_o (h_i) < θ, F_o (h_i) ≥ δ and T_o (h_j) < θ, F_o (h_j) ≥ δ, so h_i ∈ o^↓_F and h_j ∈ o^↓_F, then φ_o (h_i, h_j) = 1. So h_i and h_j are united about issue o. Therefore, $α (h_{i}, h_{j}) = {h_{i}^{↑_{T}} \cap h_{j}^{↑_{T}}} \cup {h_{i}^{↑_{I}} \cap h_{j}^{↑_{I}}} \cup {h_{i}^{↑_{F}} \cap h_{j}^{↑_{F}}}$ .

(2) Let o ∈ M, if $o \in h_{i}^{↑_{T}} \cap h_{j}^{↑_{F}}$ , then T_o (h_i) ≥ δ, F_o (h_i) < θ and T_o (h_j) < θ, F_o (h_j) ≥ δ, so h_i ∈ o^↓_T and h_j ∈ o^↓_F. By the definition, we have φ_o (h_i, h_j) = -1. Similarly, if $o \in h_{i}^{↑_{F}} \cap h_{j}^{↑_{T}}$ , then T_o (h_i) < θ, F_o (h_i) ≥ δ and T_o (h_j) ≥ δ, F_o (x_j) < θ, so h_i ∈ o^↓_F and h_j ∈ o^↓_T. By the definition, we have φ_o (h_i, h_j) = -1. So h_i and h_j are conflicted about issue o. Therefore, $β (h_{i}, h_{j}) = {h_{i}^{↑_{T}} \cap h_{j}^{↑_{F}}} \cup {h_{i}^{↑_{F}} \cap h_{j}^{↑_{T}}}$ .

(3) It can be directly obtained from the first two conclusions. □

In conflict analysis, we are mainly interested in two things. One is to find out the relationships between the agents involved in the conflict, and the other is how to resolve conflicts and find the optimal strategy that satisfies agents as many as possible. We will discuss the optimal strategy, Firstly, we assign a value to each agent’s attitude to each issue, and call all the values as the values of attribute. Then we calculate the optimal strategy by computing the values attribute.

Definition 29. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system. ∀h ∈ G, o_i ∈ M, the object-induced neutrosophic three-way concept constructed by h is (h^μν, (h^↑_T, h^↑_I, h^↑_F)), the value of attribute is defined as follows:

(1) If o_i ∈ h^↑_T, then o_i (h) = 1.

(2) If o_i ∈ h^↑_F, then o_i (h) = -1.

(3) If o_i ∈ h^↑_I, when T_o (h) > F_o (h), then o_i (h) = 1, when T_o (h) < F_o (h), then o_i (h) = -1, when T_o (h) = F_h (h), then o_i (h) = 0.

(4) If o_i ∉ h^↑_T, h^↑_I, h^↑_F, then o_i (h) = 0.

This definition assigns the value of attribute of the strongly supported attitude to 1 and the value of attribute of the strongly objected attitude to -1. The neutral ones can be divided into three categories: when o_i ∈ h^↑_I, determined by T_o (h) and F_o (h), if T_o (h) > F_o (h), then the value of attribute is 1, and when T_o (h) < F_o (h), the value of attribute is -1, when T_o (h) = F_o (h) or o_i ∉ h^↑_T, h^↑_I, h^↑_F, the value of attribute is 0.

Definition 30. Let $\dot{F} = (G, M, \dot{I})$ be a neutrosophic conflict information system, where G ={ h₁, h₂, …, h_m } is the set of all agents, M={o₁, o₂, … , o_n } is the set of all issues. ∀o_p ∈ M, for any positive region POS (h_i), h_q ∈ POS (h_i), then the optimal strategy Y ={ o₁ (Y) , o₂ (Y) , …, o_n (Y) } based on the positive region POS (h_i) is defined as follows: $o_{p} (Y) = {\begin{matrix} + 1, \\ - 1, \\ 0, \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} if \sum_{h_{q} \in {POS (h}_{i})} o_{p} (h_{q}) > 0, \\ if \sum_{h_{q} \in {POS (h}_{i})} o_{p} (h_{q}) < 0, \\ if \sum_{h_{q} \in {POS (h}_{i})} o_{p} (h_{q}) = 0 . \end{matrix}$

In this definition, o_p (Y) = +1 means that in the optimal strategy, the final attitude of all agents in the positive region POS (h_i) towards the issue o_p is supportive, o_p (Y) = -1 means that the final attitude of all agents in the positive region POS (h_i) on the issue o_p is opposed. If o_p (Y) = 0, it means that all agents in the positive region POS (h_i) on the issue o_p is neutral.

Example 10. For the neutrosophic conflict information system in Example 7, let $t_{1} = \frac{1}{2}, t_{2} = \frac{2}{3}$ . the distances between h₁, h₂, h₃ and h₄ are:

$ζ (h_{1}, h_{1}) = \frac{\frac{0 + 0 + 0}{2}}{3} = 0$ , $ζ (h_{1}, h_{2}) = \frac{\frac{0 + 1 + 1}{2}}{3} = \frac{1}{3}$ , $ζ (h_{1}, h_{3}) = \frac{\frac{2 + 1 + 1}{2}}{3} = \frac{2}{3}$ , $ζ (h_{1}, h_{4}) = \frac{\frac{0 + 2 + 1}{2}}{3} = \frac{1}{2}$ , $ζ (h_{1}, h_{5}) = \frac{\frac{0 + 1 + 1}{2}}{3} = \frac{1}{3}$ , $ζ (h_{1}, h_{6}) = \frac{\frac{2 + 0 + 1}{2}}{3} = \frac{1}{2}$ , $ζ (h_{2}, h_{2}) = \frac{\frac{0 + 0 + 0}{2}}{3} = 0$ , $ζ (h_{2}, h_{3}) = \frac{\frac{2 + 0 + 0}{2}}{3} = \frac{1}{3}$ , $ζ (h_{2}, h_{4}) = \frac{\frac{0 + 1 + 0}{2}}{3} = \frac{1}{6}$ , $ζ (h_{2}, h_{5}) = \frac{\frac{0 + 0 + 0}{2}}{3} = 0$ , $ζ (h_{2}, h_{6}) = \frac{\frac{2 + 1 + 0}{2}}{3} = \frac{1}{2}$ , $ζ (h_{3}, h_{3}) = \frac{\frac{0 + 0 + 0}{2}}{3} =$ 0, $ζ (h_{3}, h_{4}) = \frac{\frac{2 + 1 + 0}{2}}{3} = \frac{1}{2}$ , $ζ (h_{3}, h_{5}) = \frac{\frac{2 + 0 + 0}{2}}{3} = \frac{1}{3}$ , $ζ (h_{3}, h_{6}) = \frac{\frac{0 + 1 + 0}{2}}{3} = \frac{1}{6}$ , $ζ (h_{4}, h_{4}) = \frac{\frac{0 + 0 + 0}{2}}{3} = 0$ , $ζ (h_{4}, h_{5}) = \frac{\frac{0 + 1 + 0}{2}}{3} = \frac{1}{6}$ , $ζ (h_{4}, h_{6}) = \frac{\frac{2 + 2 + 0}{2}}{3} = \frac{2}{3}$ , $ζ (h_{5}, h_{5}) = \frac{\frac{0 + 0 + 0}{2}}{3} = 0$ , $ζ (h_{5}, h_{6}) = \frac{\frac{2 + 1 + 0}{2}}{3} = \frac{1}{2}$ , $ζ (h_{6}, h_{6}) = \frac{\frac{0 + 0 + 0}{2}}{3} = 0$ .

The union classes are: [h₁] _POS = {h₁, h₂, h₅}, [h₂] _POS = {h₁, h₂, h₃, h₄, h₅}, [h₃] _POS = {h₂, h₃, h₅, h₆}, [h₄] _POS = {h₂, h₄, h₅}, [h₅] _POS = {h₁, h₂, h₃, h₄, h₅}, [h₆] _POS = {h₃, h₆}.

The neutral classes are: [h₁] _NEU ={ h₄, h₆ }, [h₂] _NEU ={ h₆ }, [h₃] _NEU ={ h₄ }, [h₄] _NEU ={ h₁, h₃ }, [h₅] _NEU ={ h₆ }, [h₆] _NEU ={ h₁, h₂, h₅ }.

The conflict classes are: [h₁] _NEG ={ h₃ }, [h₃] _NEG ={ h₁ }, [h₄] _NEG ={ h₆ }, [h₆] _NEG ={ h₄ }.

The positive regions are: POS (h₁) = POS₁ (h₂) = POS₁ (h₅) ={ h₁, h₂, h₅ }, POS₂ (h₂) = POS₁ (h₃) = POS₂ (h₅) ={ h₂, h₃, h₅ }, POS₃ (h₂) = POS (x₄) = POS₃ (h₅) ={ h₂, h₄, h₅ }, POS₂ (h₃) = POS (x₆) ={ h₃, h₆ }.

The neutral regions are: NEU₁ (h₁) = NEU₁ (h₄) ={ h₁, h₄ }, NEU₂ (h₁) = NEU₁ (h₆) ={ h₁, h₆ }, NEU (h₂) = NEU₂ (h₆) ={ h₂, h₆ }, NEU (h₃) = NEU₂ (h₄) ={ h₃, h₄ }, NEU (h₅) = NEU₃ (h₆) ={ h₅, h₆ }.

The negative regions are: NEG (h₁) = NEG (h₃) ={ h₁, h₃ }, NEG (h₄) = NEG (h₆) ={ h₄, h₆ }.

Then for the positive region POS (h₁) = POS₁ (h₂) = POS₁ (h₅) ={ h₁, h₂, h₅ }, the optimal strategy is Y₁ ={ + 1, + 1, - 1 }. That means h₁, h₂ and h₅ want to go to Beojong and Shanghai, but do not want to go to Chengdu.

For the positive region POS₂ (h₂) = POS₁ (h₃) = POS₂ (h₅) ={ h₂, h₃, h₅ }, the optimal strategy is Y₂ ={ + 1, - 1, - 1 }. That means h₂, h₃ and h₅ want to go to Beojong, but do not want to go to Shanghai or Chengdu.

For the positive region POS₃ (h₂) = POS (h₄) = POS₃ (h₅) ={ h₂, h₄, h₅ }, the optimal strategy is Y₃ ={ + 1, - 1, - 1 }. That means h₂, h₄ and h₅ want to go to Beojong, but do not want to go to Shanghai or Chengdu.

For the positive region POS₂ (h₃) = POS (h₆) ={ h₃, h₆ }, the optimal strategy is Y₄ ={ - 1, 0, - 1 }. That means h₃ and h₆ do not want to go to Beijng or Chengdu, but they are neutral about whether to go to Shanghai or not.

6 Conclusion

This paper mainly studies the neutrosophic three-way concept lattice to solve the problems in conflict analysis with the agents who have “selective phobia”. When expressing the situation of “the agent’s attitude towards the problem is that everything is fine” with the neutrosophic number (x₁, x₂, x₃), x₁, x₂ and x₃ should all be greater than a threshold. We think the situation of “everyting is fine” can be ignored, but the neutral attitude is to have one’s own choice. Thus, this is reflected in the three sets of approximation operators which we gave. The neutrosophic three-way concept lattice extends the original binary relationship between objects and attributes to neutrosophic relations. We propose the object-induced neutrosophic three-way concept lattices and the attribute-induced neutrosophic three-way concept lattices. In addition, based on the candidate neutrosophic three-way concepts and the redundant neutrosophic three-way concepts, a method for constructing the neutrosophic three-way concept lattice is given. Finally, we apply the neutrosophic three-way concept lattices to conflict analysis, and propose the related optimal strategy.

Footnotes

Acknowledgment

This work has been partially supported by the National Natural Science Foundation of China (Grant No. 61976130).

References

Wille

, Restrueturing lattice theory: An approach based on hierarchies of concepts, In: Rival I, ed. Ordered Sets, Dordrecht-Boston: Reidel, 1982, 445–470.

Gediga

and Duntsch

, Modal-style operators in qualitative data analysis[C], Proceedings of the 2002 IEEE International Conference on Data Mining, Maebashi City, Dec 9–12, 2002. Piscataway: IEEE, 2002:155–162.

Yao

, Concept lattices in rough set theory[C], Proceedings of the 2004 IEEE Annual Meeting of the Fuzzy Information, Banff, Jun 27–30, 2004. Piscataway: IEEE, 2004:796–801.

Yao

, A comparative study of formal concept analysis and rough set theory in data analysis[C], Proceedings of the 4th International Conference on Rough Sets and Current Trends in Computing, Uppsala, Jun 1–5, 2004. Berlin: Springer, 2004:59–68.

, Wei

and Yao

, Three-way formal concept analysis, in: Rough Sets and Knowledge Technology-Proceedings of the 9th International Conference, RSKT 2014, Shanghai, China, October 24–26, 2014, 2014, pp. 732–741.

and Wang

, Approximate concept construction with three-waydecisions and attribute reduction in incomplete contexts, Knowledge-Based Systems 91 (2016), 165–178.

Yao

, Interval sets and three-way concept analysis in incompletecontexts, International Journal of Machine Learning andCybernetics 8(1) (2017), 3–20.

, Qian

and Wei

, The connections between three-way and classical concept lattices, Knowledge-Based Systems, 2016, 143–151.

Yang

, Lu

, Jia

and Li

, Constructing three-way conceptlattice based on the composite of classical lattices, International Journal of Approximate Reasoning 2020(121) (2020), 174–186.

10.

Qian

, Zhao

and Wang

, Research on construction methods andalgorithms of three-way concept lattices based on isomorphismtheory, Journal of Zhejiang University (Science Edition) 47(3) (2020), 322–328.

11.

Burusco

and Fuentes

G.R.

, The study of the L-fuzzy conceptlattice, Mathware & Soft Computing 1(3) (1994), 209–218.

12.

, Yang

and Zhang

, Fuzzy three-way formal concept analysisand concept-cognitive learning, Journal of Northwest University(Natural Science Edition) 50(04) (2020), 516–528.

13.

, Wei

, Ren

and Zhao

, Pythagorean fuzzy three-wayconcept lattice [J], Journal of Shandong University (ScienceEdition) 55(11) (2020), 58–65.

14.

Yao

, Interval sets and three-way concept analysis in incompletecontexts, International Journal of Machine Learning andCybernetics 8(1) (2017), 3–20.

15.

Zhao

, Miao

and Hu

, On relationship between three-wayconcept lattices, Information Sciences 538 (2020), 396–414.

16.

Smarandache

, Neutrosophic: Neutrosophic Probability, Set and Logic; American Researcher Press: Rehoboth, DE, USA, 1999.

17.

Wang

, Samarandache

, Zhang

Y.Q.

and Sunderraman

, Interval Neutrosophic Sets and Logic: Theory and Applications in Conputing; Hexis: Phoenix, AZ, USA, 2005.

18.

Wang

, Samarandache

, Zhang

Y.Q.

and Sunderraman

, Singlevalued neutrosophic sets, Multispace Multistruct 4(2010), 410–413.

19.

, Similarity measures between interval neutrosophic sets andtheir applications in multicriteria decision-making, J IntellFuzzy Syst 26 (2014), 165–172.

20.

, A multicriteria decision-making method using aggregationoperators for simplified neutrosophic sets, J Intell FuzzySyst 26 (2014), 2459–2466.

21.

Garg

, Novel neutrality aggregation operator-based multiattributegroup decision-making method for single-valued neutrosophicnumbers[J], Soft Computing: A Fusion of Foundations,Methodologies and Applications 24(14) (2020), 10327–10349.

22.

Garg Nancy

, Multiple attribute decision making based onimmediate probabilities aggregation operators forsingle-valued and interval neutrosophic sets[J], Journal ofApplied Mathematics & Computing 63(1-2) (2020), 619–653.

23.

Garg Nancy

, Algorithms for single-valued neutrosophic decisionmaking based on TOPSIS and clustering methods with new distancemeasure[J], AIMS Mathematics 5(3) (2020), 2671–2693.

24.

Singh

P.K.

, Three-way fuzzy concept lattice representation usingneutrosophic set[J], International Journal of Machine Learning& Cybernetics 8(1) (2017), 69–79.

25.

Singh

P.K.

, Complex neutrosophic concept lattice and itsapplications to air quality analysis, Chaos, Solitons andFractals: the Interdisciplinary Journal of Nonlinear Science, andNonequilibrium and Complex Phenomena 109 (2018), 206–213.

26.

Singh

P.K.

, Three-way n-valued neutrosophic concept lattice atdifferent granulation, International Journal of MachineLearning and Cybernetics 9(11) (2018), 1839–1855.

27.

Mao

and Ling

, Interval neutrosophic fuzzy concept latticerepresentation and interval-similarity measure, Journal ofIntelligent & Fuzzy Systems 33(2) (2017), 957–967.

28.

Pawlak

, An inquiry into anatomy of conflicts, Inf Sci 109 (1998), 65–78.

29.

Lang

, Miao

, Zhang

and Yao

, Conflict Analysis for Pythagorean Fuzzy Information Systems, International Joint Conference on Rough Sets, 2017, 359–367.

30.

Lang

, Miao

and Cai

, Conflict Analysis for Pythagorean Fuzzy Information Systems with Group Decision Making, Information Sciences, 2017, 185–207.

31.

Lang

, Miao

and Fujita

, Three-way Group Conflict Analysis Based on Pythagorean Fuzzy Set Theory, IEEE Transactions on Fuzzy Systems, 2019.

32.

Sun

, Ma

and Zhao

, Rough set-based conflict analysis modeland method over two universes, Inf Sci 372 (2016), 111–125.

33.

Fan

, Qi

and Wei

, A conflict analysis model based on three-way decisions, Springer Verlag, 2018, 522–532.

34.

Yao

, Three-way conflict analysis: Reformulations and extensionsof the Pawlak model, Knowledge-Based Systems 180(2019), 26–37.

35.

Sun

, Chen

, Zhang

and Ma

, Three-way decision makingapproach to conflict analysis and resolution using probabilisticrough set over two universes, Information Sciences 507(2020), 809–822.

36.

Lang

, A general conflict analysis model based on three-waydecision, International Journal of Machine Learning andCybernetics 11(5) (2020), 1083–1094.

37.

Zhi

, Qi

, Qian

and Ren

, Conflict analysis under one-voteveto based on approximate three-way concept lattice, Information Sciences 516 (2020), 316–330.

38.

Smarandache

, Neutrosophic set-a generialization of theintuitionistics fuzzy sets, Int J Pure Appl Math 24(3) (2005), 287–297.

39.

Borzooei

R.A.

, Farahani

and Moniri

, Neutrosophic deductivefilters on BL-algebras, J Intell Fuzzy Syst 26 (2014), 2993–3004.

40.

Belohlavek

, Fuzzy Galois Connections, Math Logic Quart 45(4) (1999), 497–504.

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

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Neutrosophic set

2.2 Pawlak’s conflict analysis model

Table 1 An information system of Example 1 a 1 a 2 a 3 a 4 a 5 x 1 +1 +1 –1 –1 +1 x 2 –1 +1 +1 –1 +1 x 3 +1 –1 –1 –1 0 x 4 –1 0 –1 +1 +1

3 Neutrosophic three-way concept lattice

3.1 The object-induced neutrosophic three-way concept lattices

4.2 Type-I constructing the attribute-induced neutrosophic three-way concept lattices

4.3 Type-II 2 constructing the object-induced neutrosophic three-way concept lattices

4.4 Type-II 2 constructing the attribute-induced neutrosophic three-way concept lattices

5 Conflict analysis model based on neutrosophic three-way concept lattices

Table 3 Positive formal context F ̇ T = ( G , M , I ̇ T ) o 1 o 2 o 3 h 1 1 1 0 h 2 1 0 0 h 3 0 0 0 h 4 1 0 0 h 5 1 0 0 h 6 0 1 0

Footnotes

Acknowledgment

References

Table 1
An information system of Example 1

a ₁ a ₂ a ₃ a ₄ a ₅

x ₁ +1 +1 –1 –1 +1

x ₂ –1 +1 +1 –1 +1

x ₃ +1 –1 –1 –1 0

x ₄ –1 0 –1 +1 +1

Table 3
Positive formal context ${\dot{F}}_{T} = (G, M, {\dot{I}}_{T})$

o ₁ o ₂ o ₃

h ₁ 1 1 0

h ₂ 1 0 0

h ₃ 0 0 0

h ₄ 1 0 0

h ₅ 1 0 0

h ₆ 0 1 0