Interval neutrosophic fuzzy concept lattice representation and interval-similarity measure

Abstract

This paper applies the interval neutrosophic set theory to express fuzzy formal concept. A new interval neutrosophic fuzzy concept lattice is proposed. Furthermore, interval-distance and interval-similarity measure are discussed. We can find some interval neutrosophic fuzzy concepts to meet particular similarity range given by users. The result of this paper may enhance the flexibility of extracting knowledge and expand the capacity of knowledge.

Keywords

Interval neutrosophic set formal concept analysis interval-similarity measure

1 Introduction

Formal concept analysis (FCA) introduced by Wille [1] had been applied in various research fields [2]. Due to express the fuzziness and uncertainty of knowledge, Singh [3, 4] proposed interval-valued fuzzy concept lattice and three-way fuzzy concept lattice.

Neutrosophic set (NS), proposed by Samarandache [5], is characterized by a truth-membership function, an indeterminacy membership function and a falsity membership function independently which are within the real standard or nonstandard unit interval. Thus, NS generalizes the fuzzy set [6], interval-valued fuzzy set [7], intuitionistic fuzzy set [8] and interval-valued intuitionistic fuzzy set [9]. NS can handle incomplete, uncertain and inconsistent information which often appear in real world. For further generalization, Wang [11] introduced the interval neutrosophic set (INS) in which three membership functions are independent and their values are subinterval of [0,1]. There are many applications based on NS and INS, such as database [12], medical diagnosis [13], decision making [14 –24], and image processing [25, 26].

Torra [27] proposed the hesitant fuzzy set. Wei [28] developed some prioritized aggregation operators for aggregating hesitant fuzzy information. These operators can be applied in multiple attribute decision making problems. Furthermore, Wei [29] and Lin [30, 31] proposed some aggregation operators for aggregating hesitant fuzzy linguistic information. Interval-valued intuitionistic fuzzy context was proposed by Xu [32] in which the degree of membership and the degree of non-membership are expressed by interval values instead of real values. Wei [33] also developed interval valued hesitant fuzzy uncertain linguistic aggregation operators. Ye [14] proposed similarity measures between INSs using the weighted Euclidean distance and the weighted Hamming distance.

However, up to now, we find that there are still the following two problems needed to be considered.

How can we construct the corresponding concept lattice when we introduce INS to the formal context?

How can we find multiple concepts similar to the target concepts within a certain similarity range required by users?

To solve these problems, we will do the following works:

Construct a novel concept lattice called interval neutrosophic fuzzy (INF) concept lattice in Section 3. The intent of INF concept is an INS whose truth, falsity, and indeterminacy degree are intervals. Hence, INF concept lattice can represent uncertain, imprecise, incomplete and inconsistent information which exist in the real world.

Interval-similarity measure defined in Section 4 which can help users to find more similar concepts in a certain similarity range.

The rest of this paper is organized as follows. Section 2 reviews basic definitions of FCA [1] and INS [11]. In Section 3, we define a new Galois connection. In addition, we search out the INF concept lattice. Section 4 proposes the interval-distance and interval-similarity measure for finding INF concepts that similar to the target concept within a required similarity range. Section 5 uses two examples to illustrate the results in Sections 3 and 4. Finally, Section 6 concludes this paper and points out our future works.

2 Preliminaries

In this section, we will review some preliminaries for interval number and INS. For more detail, please, interval number is referred to [10] and INS is seen [11].

Definition 2.1. [1] Let φ : P → Q and ψ : Q → P be maps between two ordered sets (P, ≤) and (Q, ≤). Such a pair of maps is called a Galois connection between the ordered sets if

p₁ ≤ p₂ ⇒ φp₁ ≥ φp₂

q₁ ≤ q₂ ⇒ ψq₁ ≥ ψq₂

p ≤ ψφp and q ≤ φψq

Definition 2.2. [10] An interval number D is an interval [a^-, a⁺] with 0 ≤ a^- ≤ a⁺ ≤ 1. The interval [a, a] is identified with the real number a ∈ [0, 1]. D (0, 1) denotes the set of all interval numbers on [0,1]. For the interval numbers $D_{1} = [a_{1}^{-}, b_{1}^{+}]$ and $D_{2} = [a_{2}^{-}, b_{2}^{+}]$ , we can define $\begin{matrix} (1) min (D_{1}, D_{2}) = min ([a_{1}^{-}, b_{1}^{+}], [a_{2}^{-}, b_{2}^{+}]) \\ = [min (a_{1}^{-}, a_{2}^{-}), min (b_{1}^{+}, b_{2}^{+})] \\ (2) \max (D_{1}, D_{2}) = max ([a_{1}^{-}, b_{1}^{+}], [a_{2}^{-}, b_{2}^{+}]) \\ = [max (a_{1}^{-}, a_{2}^{-}), max (b_{1}^{+}, b_{2}^{+})] \\ (3) D_{1} \leq D_{2}, \Leftrightarrow a_{1}^{-} \leq a_{2}^{-} and b_{1}^{+} \leq b_{2}^{+} \\ (4) D_{1} = D_{2} \Leftrightarrow a_{1}^{-} = a_{2}^{-} and b_{1}^{+} = b_{2}^{+} \\ (5) kD = [{ka}_{1}^{-}, {kb}_{1}^{+}] where 0 \leq k \leq 1 . \end{matrix}$

Then, (D (0, 1) , ≤ , ∨ , ∧) is a complete lattice with [0, 0] as the least element and [1, 1] as the great.

Definition 2.3. [11] Let X be a space of points (objects), with a generic in X denoted by x. An INS A is characterized by a truth-membership function T_A, indeterminacy-membership function I_A and falsity-membership F_A. For each point x in X, T_A (x), I_A (x), F_A (x) ⊆ [0, 1]. The set of all INSs on X is denoted by INS(X).

Remark 2.1. From Definition 2.3, an INS A can be expressed as $\begin{matrix} A & = & {〈 x, T_{A} (x), I_{A} (x), F_{A} (x) 〉 | x \in X} \\ = & {〈 x, [\underset{A}{infT} (x), \underset{A}{supT} (x)], \\ [\underset{A}{infI} (x), \underset{A}{supI} (x)], \\ [\underset{A}{infF} (x), \underset{A}{supF} (x)] 〉 | x \in X} . \end{matrix}$

So, the sum of T_A (x) , I_A (x) and F_A (x) satisfies $0 \leq \underset{A}{supT} (x) + \underset{A}{supI} (x) + \underset{A}{supF} (x) \leq 3$ .

For example, $\begin{matrix} A & = & {〈 x_{1}, [0.2, 0.5], [0.4, 0.8], [0.1, 0.4] 〉 \\ 〈 x_{2}, [0.3, 0.7], [0.5, 0.6], [0.7, 0.9] 〉, \\ 〈 x_{3}, [0.6, 0.9], [0.4, 0.7], [0.2, 0.5] 〉} \\ is an INS on X = {x_{1}, x_{2}, x_{3}} . \end{matrix}$

Definition 2.4. [11] Let A and B be two INSs defined on X. A (x) ≤ B (x) if and only if $\begin{matrix} \underset{A}{infT} (x) & \leq & \underset{B}{infT} (x), \underset{A}{supT} (x) \leq \underset{B}{supT} (x), \\ \underset{A}{infI} (x) & \geq & \underset{B}{infI} (x), \underset{A}{supI} (x) \geq \underset{B}{supI} (x), \\ \underset{A}{infF} (x) & \geq & \underset{B}{infF} (x), \underset{A}{supF} (x) \geq \underset{B}{supF} (x) . \end{matrix}$

An INS A is contained in the other INS B, A ⊆ B if and only if A (x) ≤ B (x), for all x in X.

Definition 2.5. [11] (1) The intersection of two INS A and B is an INS C, written as C = A ∩ B, whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by $\begin{matrix} {infT}_{C} (x) & = & \min ({infT}_{A} (x), {infT}_{B} (x)), \\ {supT}_{C} (x) & = & \min ({supT}_{A} (x), {supT}_{B} (x)), \\ {infI}_{C} (x) & = & \max ({infI}_{A} (x), {infI}_{B} (x)), \\ {supI}_{C} (x) & = & \max ({supI}_{A} (x), {supI}_{B} (x)), \\ {infF}_{C} (x) & = & \max ({infF}_{A} (x), {infF}_{B} (x)), \\ {supF}_{C} (x) & = & \max ({supF}_{A} (x), {supF}_{B} (x)), \end{matrix}$ for all x in X.

(2) The union of two INS A and B is an INS C whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by $\begin{matrix} {infT}_{C} (x) & = & \max ({infT}_{A} (x), {infT}_{B} (x)), \\ {supT}_{C} (x) & = & \max ({supT}_{A} (x), {supT}_{B} (x)), \\ {infI}_{C} (x) & = & \min ({infI}_{A} (x), {infI}_{B} (x)), \\ {supI}_{C} (x) & = & \min ({supI}_{A} (x), {supI}_{B} (x)), \\ {infF}_{C} (x) & = & \min ({infF}_{A} (x), {infF}_{B} (x)), \\ {supF}_{C} (x) & = & \min ({supF}_{A} (x), {supF}_{B} (x)), \end{matrix}$ for all x in X.

3 Interval neutrosophic fuzzy concept lattice representation

In this section, we will introduce the INS to the theory of FCA. To construct INF concept lattice, we will give some necessary terms. First, we describe the INF formal context. Second, we provide a pair of operators which will form a Galois connection.

Definition 3.1. (1) A triple $(G, M, \tilde{R})$ is called an INF formal context, if G is an object set and M is an attribute set and $\tilde{R}$ is an INS on G × M where $\tilde{R} (x, m) = 〈 T_{\tilde{R}} (x, m), I_{\tilde{R}} (x, m), F_{\tilde{R}} (x, m) 〉 .$

We denote $V = {\tilde{R} (x, m) | x \in G, m \in M}$ .

(2) Let $K = (G, M, \tilde{R})$ be an INF formal context. For X ⊆ G and B ⊆ M, a pair of operators ^*:ρ (G) → INS (M) and ^*:INS (M) → ρ (G) are defined by $\begin{matrix} X^{*} & = & \tilde{A} \\ = & {〈 m, T_{\tilde{A}} (m), I_{\tilde{A}} (m), F_{\tilde{A}} (m) 〉 | m \in M} \end{matrix}$ where $\begin{matrix} T_{\tilde{A}} (m) & = & \land_{\forall x \in X} T_{\tilde{R}} (x, m), \\ I_{\tilde{A}} (m) & = & \lor_{\forall x \in X} I_{\tilde{R}} (x, m), \\ F_{\tilde{A}} (m) & = & \lor_{\forall x \in X} F_{\tilde{R}} (x, m) . \end{matrix}$

In special case, ∅^* = {〈m, [1, 1] , [0, 0] , [0, 0] 〉|m ∈ M} .

${\tilde{B}}^{*} = {x \in G | \tilde{R} (x, m) \geq \tilde{B} (m), \forall m \in M, \forall \tilde{R} (x, m) \in V}$ where $\tilde{B} (m) = 〈 [0, 0], [1, 1], [1, 1] 〉$ for m ∉ B and we denote $G^{B} = {\tilde{B} | \tilde{B} (m) = \tilde{R} (x, m), x \in G, m \in M}$ .

(3) A pair $(X, \tilde{B})$ is called an INF concept if $X^{*} = \tilde{B}$ and ${\tilde{B}}^{*} = X$ . X and $\tilde{B}$ are called the extent and the intent of $(X, \tilde{B})$ , respectively. The set of all INF concepts is denoted by $B (G, M, \tilde{R})$ .

Next, the above operators will induce a Galois connection between ρ (G) and INS (M) when such sets are ordered with the set inclusion relation. The following proposition shows that the operator (*,*) can form a Galois connection.

Proposition 3.1. Let $K = (G, M, \tilde{R})$ be an INF formal context. The operator (*,*) forms a Galois connection. Specifically, for X₁, X₂, X ⊆ G and B₁, B₂, B ⊆ M, the operator (*,*) has the following properties $\begin{matrix} (1) & X_{1} \subseteq X_{2} \Rightarrow X_{2}^{*} \subseteq X_{1}^{*}, {\tilde{B}}_{1} \subseteq {\tilde{B}}_{2} \Rightarrow {\tilde{B}}_{2}^{*} \subseteq {\tilde{B}}_{1}^{*} \\ (2) & X \subseteq X^{* *}, \tilde{B} \subseteq {\tilde{B}}^{* *} \\ (3) & X^{*} = X^{* * *}, {\tilde{B}}^{*} = {\tilde{B}}^{* * *} \\ (4) & X \subseteq {\tilde{B}}^{*} \Leftrightarrow \tilde{B} \subseteq X^{*} \\ (5) & {(X_{1} \cup X_{2})}^{*} = X_{1}^{*} \cap X_{2}^{*}, {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*} = {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*} \end{matrix}$

Proof. (1) On one hand, suppose $X_{1}^{*} = {\tilde{A}}_{1}$ , $X_{2}^{*} = {\tilde{A}}_{2}$ . According to Definition 3.1 (2), for any m ∈ M, we have $\begin{matrix} {\tilde{A}}_{1} (m) & = & 〈 \land_{\forall x \in X_{1}} T_{\tilde{R}} (x, m), \lor_{\forall x \in X_{1}} I_{\tilde{R}} (x, m), \\ \lor_{\forall x \in X_{1}} F_{\tilde{R}} (x, m) 〉 \\ {\tilde{A}}_{2} (m) & = & 〈 \land_{\forall x \in X_{2}} T_{\tilde{R}} (x, m), \lor_{\forall x \in X_{2}} I_{\tilde{R}} (x, m), \\ \lor_{\forall x \in X_{2}} F_{\tilde{R}} (x, m) 〉 \end{matrix}$

Since X₁ ⊆ X₂, we have the following inequalities: $\begin{matrix} \land_{\forall x \in X_{1}} T_{\tilde{R}} (x, m) \geq \land_{\forall x \in X_{2}} T_{\tilde{R}} (x, m), \\ \lor_{\forall x \in X_{1}} I_{\tilde{R}} (x, m) \leq \lor_{\forall x \in X_{2}} I_{\tilde{R}} (x, m), \\ \lor_{\forall x \in X_{1}} F_{\tilde{R}} (x, m) \leq \lor_{\forall x \in X_{2}} F_{\tilde{R}} (x, m) . \end{matrix}$

Furthermore, by Definition 2.2(3), we can obtain $\begin{matrix} \inf \land_{\forall x \in X_{1}} T_{\tilde{R}} (x, m) \geq \inf \land_{\forall x \in X_{2}} T_{\tilde{R}} (x, m), \\ \sup \land_{\forall x \in X_{1}} T_{\tilde{R}} (x, m) \geq \sup \land_{\forall x \in X_{2}} T_{\tilde{R}} (x, m), \\ \inf \lor_{\forall x \in X_{1}} I_{\tilde{R}} (x, m) \leq \inf \lor_{\forall x \in X_{2}} I_{\tilde{R}} (x, m), \\ \sup \lor_{\forall x \in X_{1}} I_{\tilde{R}} (x, m) \leq \sup \lor_{\forall x \in X_{2}} I_{\tilde{R}} (x, m), \\ \inf \lor_{\forall x \in X_{1}} F_{\tilde{R}} (x, m) \leq \inf \lor_{\forall x \in X_{2}} F_{\tilde{R}} (x, m), \\ \sup \lor_{\forall x \in X_{1}} F_{\tilde{R}} (x, m) \leq \sup \lor_{\forall x \in X_{2}} F_{\tilde{R}} (x, m) . \end{matrix}$

According to Definition 2.4, we can decide ${\tilde{A}}_{2} (m) \leq {\tilde{A}}_{1} (m)$ for any m ∈ M. Thus, ${\tilde{A}}_{2} \subseteq {\tilde{A}}_{1}$ . From the given, we derive $X_{2}^{*} \subseteq X_{1}^{*}$ .

On the other hand, by Definition 3.1(2), we have $\begin{matrix} {\tilde{B}}_{1}^{*} & = & {x \in G | \tilde{R} (x, m) \geq {\tilde{B}}_{1} (m), \\ \forall m \in M, \forall \tilde{R} (x, m) \in V} \\ {\tilde{B}}_{2}^{*} & = & {x \in G | \tilde{R} (x, m) \geq {\tilde{B}}_{2} (m), \\ \forall m \in M, \forall \tilde{R} (x, m) \in V} \end{matrix}$

Since ${\tilde{B}}_{1} \subseteq {\tilde{B}}_{2}$ , we obtain that for any m ∈ M, ${\tilde{B}}_{1} (m) \leq {\tilde{B}}_{2} (m)$ by Definition 2.4. It follows that $\tilde{R} (x, m) \geq {\tilde{B}}_{2} (m)$ implies $\tilde{R} (x, m) \geq {\tilde{B}}_{1} (m)$ for any $\tilde{R} (x, m) \in V$ . Hence, if $x \in {\tilde{B}}_{2}^{*}$ , then $x \in {\tilde{B}}_{1}^{*}$ . That means ${\tilde{B}}_{2}^{*} \subseteq {\tilde{B}}_{1}^{*}$ .

(2) On one hand, assume $X^{*} = \tilde{A}$ . Then from Definition 3.1(2), we easily find $\begin{matrix} X^{* *} & = & {\tilde{A}}^{*} = {x \in G | \tilde{R} (x, m) \\ \geq & \tilde{A} (m), \forall m \in M, \forall \tilde{R} (x, m) \in V} \end{matrix}$ where $\begin{matrix} \tilde{A} (m) & = & 〈 \land_{\forall x \in X} T_{\tilde{R}} (x, m), \lor_{\forall x \in X} I_{\tilde{R}} (x, m), \\ \lor_{\forall x \in X} F_{\tilde{R}} (x, m) 〉 \end{matrix}$

If x ∈ X. Then for any m ∈ M, we have $\tilde{R} (x, m) \geq \tilde{A} (m)$ . Thus, x ∈ X^**. It may be easily seen X ⊆ X^**.

On the other hand, suppose $\tilde{B} \in G^{B}$ . Then from Definition 3.1(2), we easily find $\begin{matrix} {\tilde{B}}^{*} & = & {x \in G | \tilde{R} (x, m) \geq \tilde{B} (m), \forall m \in M, \\ \forall \tilde{R} (x, m) \in V} \end{matrix}$

It follows $\tilde{R} (x, m) \geq \tilde{B} (m)$ for any $x \in {\tilde{B}}^{*}$ . For every m ∈ M, we can get $\begin{matrix} {\tilde{B}}^{* *} (m) & = & 〈 \land_{\forall x \in {\tilde{B}}^{*}} T_{\tilde{R}} (x, m), \lor_{\forall x \in {\tilde{B}}^{*}} I_{\tilde{R}} (x, m), \\ \lor_{\forall x \in {\tilde{B}}^{*}} F_{\tilde{R}} (x, m) 〉 \\ \geq & 〈 T_{\tilde{R}} (x, m), I_{\tilde{R}} (x, m), F_{\tilde{R}} (x, m) 〉 \\ = & \tilde{B} (m) \end{matrix}$

From Definition 2.4, we can obtain $\tilde{B} \subseteq {\tilde{B}}^{* *}$ .

(3) We easily obtained the needed results from the items (1) and (2).

(4) If $X \subseteq {\tilde{B}}^{*}$ . Then we easily determine ${\tilde{B}}^{* *} \subseteq X^{*}$ from the item (1). In addition, we find $\tilde{B} \subseteq {\tilde{B}}^{* *}$ using the item (2). Thus, there is $\tilde{B} \subseteq X^{*}$ .

(5) It is easily seen ${({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*} \subseteq {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*}$ . Moreover, we have $x \in {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*} \Leftrightarrow (x \in {\tilde{B}}_{1}^{*}) \land (x \in {\tilde{B}}_{2}^{*}) .$

Thus, by Definition 3.1(2), we find $\tilde{R} (x, m_{1}) \geq {\tilde{B}}_{1} (m_{1})$ for any m₁ ∈ B₁ and $\tilde{R} (x, m_{2}) \geq {\tilde{B}}_{2} (m_{2})$ for any m₂ ∈ B₂. We have $\tilde{R} (x, m) \geq ({\tilde{B}}_{1} \cup {\tilde{B}}_{2}) (m)$ for any m ∈ B₁ ∪ B₂ from Definition 2.5(2). Hence, $x \in {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*}$ . So we have ${({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*} \supseteq {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*}$ . Combining with ${({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*} \subseteq {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*}$ , we derive ${({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*} = {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*}$ . Analogously, we can get ${(X_{1} \cup X_{2})}^{*} = X_{1}^{*} \cap X_{2}^{*}$ . □

We can obtain that the operator (*,*) forms a Galois connection from Definition 2.1. In addition, we will show that there is an order relation on the set of all INF concepts. What is more, all INF concepts ordered by this relation will form a complete lattice by the following conclusion.

Theorem 3.1. Let $(X_{1}, {\tilde{B}}_{1})$ and $(X_{2}, {\tilde{B}}_{2})$ be INF concept where X₁, X₂ ⊆ G, B₁, B₂ ⊆ M. We have the following properties

$(X_{1}^{* *}, X_{1}^{*})$ and $({\tilde{B}}_{1}^{*}, {\tilde{B}}_{1}^{* *})$ are INF concepts.

$(X_{1} \cap X_{2}, ({\tilde{B}}_{1} \cup {\tilde{B}}_{2})^{* *})$ and $({(X_{1} \cup X_{2})}^{* *}, {\tilde{B}}_{1} \cap {\tilde{B}}_{2})$ are INF concepts.

If $(X_{1}, {\tilde{B}}_{1})$ and $(X_{2}, {\tilde{B}}_{2})$ are ordered by $(X_{1}, {\tilde{B}}_{1}) ≼ (X_{2}, {\tilde{B}}_{2}) \Leftrightarrow X_{1} \subseteq X_{2} \Leftrightarrow {\tilde{B}}_{2} \subseteq {\tilde{B}}_{1} .$ Then, ≼ is an order relation. In this case, $(X_{1}, {\tilde{B}}_{1})$ is a subconcept of $(X_{2}, {\tilde{B}}_{2})$ .

The set of all INF concepts of $(G, M, \tilde{R})$ ordered by this relation is denoted by $\underline{B} (G, M, \tilde{R})$ and called INF conceptlattice.

$\underline{B} (G, M, \tilde{R})$ is a complete lattice if the infimum and supremum are given by $\begin{matrix} (X_{1}, {\tilde{B}}_{1}) \land (X_{2}, {\tilde{B}}_{2}) \\ = (X_{1} \cap X_{2}, {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{* *}), \\ (X_{1}, {\tilde{B}}_{1}) \lor (X_{2}, {\tilde{B}}_{2}) \\ = ({(X_{1} \cup X_{2})}^{* *}, {\tilde{B}}_{1} \cap {\tilde{B}}_{2}) . \end{matrix}$

Proof. (1) By Definition 3.1(3) and Proposition 3.1(3), we can easily find that $(X_{1}^{* *}, X_{1}^{*})$ and $({\tilde{B}}_{1}^{*}, {\tilde{B}}_{1}^{* *})$ are INF concepts.

(2) On one hand, since $(X_{1}, {\tilde{B}}_{1})$ and $(X_{2}, {\tilde{B}}_{2})$ are INF concepts, we have $X_{1}^{*} = {\tilde{B}}_{1}$ , ${\tilde{B}}_{1}^{*} = X_{1}$ , $X_{2}^{*} = {\tilde{B}}_{2}$ and ${\tilde{B}}_{2}^{*} = X_{2}$ from Definition 3.1(3). Thus we obtain $(X_{1} \cap X_{2})^{*} = {({\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*})}^{*} = {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{* *}$ .

On the other hand, we have ${({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{* * *} = {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{*} = {\tilde{B}}_{1}^{*} \cap {\tilde{B}}_{2}^{*} = X_{1} \cap X_{2}$ by the items (3) and (5) in Proposition 3.1. Thus, we can derive $(X_{1} \cap X_{2}, {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{* *}) \in B (G, M, \tilde{R})$ from Definition 3.1(3).

Similarly, we can also get $((X_{1} \cup X_{2})^{* *}, {\tilde{B}}_{1} \cap {\tilde{B}}_{2})$ is an INF concept.

(3) In order to prove $(B (G, M, \tilde{R}), ≼)$ is an ordered set, we need to prove that the relation ≼ satisfies reflexivity, antisymmetry and transitivity.

First, $(X_{1}, {\tilde{B}}_{1}) ≼ (X_{1}, {\tilde{B}}_{1})$ obviously holds.

Second, if $(X_{1}, {\tilde{B}}_{1}) ≼ (X_{2}, {\tilde{B}}_{2})$ and $(X_{2}, {\tilde{B}}_{2}) ≼ (X_{1}, {\tilde{B}}_{1})$ , then we get X₁ = X₂ and ${\tilde{B}}_{1} = {\tilde{B}}_{2}$ . So $(X_{1}, {\tilde{B}}_{1}) = (X_{2}, {\tilde{B}}_{2})$ .

Next, if $(X_{1}, {\tilde{B}}_{1}) ≼ (X_{2}, {\tilde{B}}_{2})$ and $(X_{2}, {\tilde{B}}_{2}) ≼ (X_{3}, {\tilde{B}}_{3})$ , then we have X₁ ⊆ X₂ ⊆ X₃ and ${\tilde{B}}_{3} \subseteq {\tilde{B}}_{2} \subseteq {\tilde{B}}_{1}$ . So, $(X_{1}, {\tilde{B}}_{1}) ≼ (X_{3}, {\tilde{B}}_{3})$ holds. We can see that $\underline{B} (G, M, \tilde{R})$ is exactly an ordered set.

(4) We may easily decide $(X_{1} \cap X_{2}, {({\tilde{B}}_{1} \cup {\tilde{B}}_{2})}^{* *}) \in B (G, M, \tilde{R})$ according to the item (2) and X₁ ∧ X₂ = X₁ ∩ X₂. So, we can find that $(X_{1} \cap X_{2}, ({\tilde{B}}_{1} \cup {\tilde{B}}_{2})^{* *})$ is exactly the largest subconcpet of the INF concepts $(X_{1}, {\tilde{B}}_{1})$ and $(X_{2}, {\tilde{B}}_{2})$ .

The formula for the supremum is obtained correspondingly. Thus, $\underline{B} (G, M, \tilde{R})$ is a complete lattice. □

Remark 3.1. In fact, Proposition 3.1 and Theorem 3.1 give an answer to (P1).

To make much clear for obtaining INF concepts, we will give an algorithm for generating all INF concepts as follows. Let $(G, M, \tilde{R})$ be an INF formal context defined in Definition 3.1 where |G| = m and |M| = n.

Algorithm 1.

Input: INF formal context $(G, M, \tilde{R})$

Output: INF concept lattice $\underline{B} (G, M, \tilde{R})$

Compute all subset X_i (i = 1, 2, …, 2^m) of object set G.

For each X_i, calculate attribute set ${\tilde{B}}_{i}$ using Galois connection i.e. ${\tilde{B}}_{i} = X_{i}^{*}$ .

Compute ${\tilde{B}}_{i}^{*}$ . If ${\tilde{B}}_{i}^{*} = X_{i}$ , then $(X_{i}, {\tilde{B}}_{i})$ is an INF concept. If ${\tilde{B}}_{i}^{*} \supseteq X_{i}$ , then $({\tilde{B}}_{i}^{*}, {\tilde{B}}_{i})$ is an INF concept.

We can obtain INF concept lattice $\underline{B} (G, M, \tilde{R})$ by the order relation “≼” defined in Theorem 3.1.

In the field of three-way concept analysis, the result in Singh [4] is a better one. However, we need to notice the following facts: based on single-valued neutrosophic set, Singh [4] proposed Three-way fuzzy concept lattice. The extent (intent) of the three-way fuzzy concept is a single-valued neutrosophic set on object (attribute) set. However, INF concept lattice proposed in this paper is based on INS. The intent of INF concept is an INS on attribute set, and the extent of INF concept is the power set of object set. In other words, the notion of INF concept in this paper is different with three-way fuzzy concept in Singh [4]. Hence, the result provided by this paper is also different from Singh [4]. That is to say, all of results in this paper are new and valuable to consider and read.

4 Interval-similarity measure between interval neutrosophic sets

Distance measures such as Euclidean distance and Hamming distance [14] are often expressed as a real number on [0,1]. This distance does not reflect how the distance changes on the element. We may use interval-distance which consists of the minimum and the maximum distance to express this distance variation.

Definition 4.1. Let INS (X) be all INSs on the discourse domain X ={ x₁, x₂, …, x_n }. For A, B ∈ INS (X), the interval-distance between A and B is denoted by $\begin{matrix} D (A, B) \\ = [d_{\min} (A, B), d_{\max} (A, B)] \\ = (d_{\inf} (A, B) \land d_{\sup} (A, B), \\ d_{\inf} (A, B) \lor d_{\sup} (A, B)], \end{matrix}$ where d_inf (A, B) and d_sup (A, B) are defined by $\begin{matrix} d_{\inf} (A, B) \\ = {\begin{matrix} \frac{1}{3 n} \sum_{i = 1}^{n} (\underset{A}{infT} (x_{i}) - \underset{B}{infT} (x_{i}))^{2} \\ + (\underset{A}{infI} (x_{i}) - \underset{B}{infI} (x_{i}))^{2} \\ + (\underset{A}{infF} (x_{i}) - \underset{B}{infF} (x_{i}))^{2} \end{matrix}}^{\frac{1}{2}}, \\ d_{\sup} (A, B) \\ = {\begin{matrix} \frac{1}{3 n} \sum_{i = 1}^{n} (\underset{A}{supT} (x_{i}) - \underset{B}{supT} (x_{i}))^{2} \\ + (\underset{A}{supI} (x_{i}) - \underset{B}{supI} (x_{i}))^{2} \\ + (\underset{A}{supF} (x_{i}) - \underset{B}{supF} (x_{i}))^{2} \end{matrix}}^{\frac{1}{2}} . \end{matrix}$

D (A, B) reaches the maximum value $\tilde{1} = [1, 1]$ such that A ={ 〈 x_i, [1, 1] , [0, 0] , [0, 0] 〉 |x_i ∈ X } and B ={ 〈 x_i, [0, 0] , [1, 1] , [1, 1] 〉 |x_i ∈ X }. If A = B, the D (A, B) reaches the minimum value $\tilde{0} = [0, 0]$ .

Therefore, interval-distance can be regarded as an interval number. We can see that interval-distance measures the distance between INSs in a more accurate way.

Remark 4.1. We will use an example to explain why exploit the ∧ and ∨ operator between d_inf (A, B) and d_sup (A, B) in the definition of D (A, B).

Sometimes, d_inf (A, B) > d_sup (A, B). For example, $\begin{matrix} A & = & {〈 x_{1}, [0.1, 0.3], [0.1, 0.3], [0.1, 0.3] 〉}, \\ B & = & {〈 x_{1}, [0.2, 0.3], [0.2, 0.3], [0.2, 0.3] 〉}, \end{matrix}$ $\begin{matrix} d_{\inf} (A, B) & = & \sqrt{\frac{1}{3} [{(0.1 - 0.2)}^{2} + {(0.1 - 0.2)}^{2} + {(0.1 - 0.2)}^{2}]} = 0.1, \\ d_{\sup} (A, B) & = & \sqrt{\frac{1}{3} [{(0.3 - 0.3)}^{2} + {(0.3 - 0.3)}^{2} + {(0.3 - 0.3)}^{2}]} = 0 . \end{matrix}$

So we have d_inf (A, B) > d_sup (A, B).

The following conclusion will show that interval-distance defined in this paper is reasonable.

Theorem 4.1. The interval-distance D (A, B) between INSs A and B satisfies the following properties

$\tilde{0} \leq D (A, B) \leq \tilde{1}$

D (A, B) = D (B, A)

if A = B, then $D (A, B) = \tilde{0}$

if A ⊆ B ⊆ C, then D (A, B) ≤ D (A, C) and D (B, C) ≤ D (A, C).

Proof. According to Definition 4.1, we can easily find that D (A, B) has the properties (1) - (3). So, we only need to prove the property (4).

If A ⊆ B ⊆ C. Then for any x_i ∈ X, we can get the following inequalities from Definition 2.4. $\begin{matrix} {infT}_{A} (x_{i}) & \leq & {infT}_{B} (x_{i}) \leq {infT}_{C} (x_{i}), \\ {supT}_{A} (x_{i}) & \leq & {supT}_{B} (x_{i}) \leq {supT}_{C} (x_{i}), \\ {infI}_{A} (x_{i}) & \geq & {infI}_{B} (x_{i}) \geq {infI}_{C} (x_{i}), \\ {supI}_{A} (x_{i}) & \geq & {supI}_{B} (x_{i}) \geq {supI}_{C} (x_{i}), \\ {infF}_{A} (x_{i}) & \geq & {infF}_{B} (x_{i}) \geq {infF}_{C} (x_{i}), \\ {supF}_{A} (x_{i}) & \geq & {supF}_{B} (x_{i}) \geq {supF}_{C} (x_{i}) . \end{matrix}$

Moreover, we also obtain $\begin{matrix} {({infT}_{A} (x_{i}) - {infT}_{B} (x_{i}))}^{2} \\ \leq {({infT}_{A} (x_{i}) - {infT}_{C} (x_{i}))}^{2}, \\ {({supT}_{A} (x_{i}) - {supT}_{B} (x_{i}))}^{2} \\ \leq {({supT}_{A} (x_{i}) - {supT}_{C} (x_{i}))}^{2} \\ {({infI}_{A} (x_{i}) - {infI}_{B} (x_{i}))}^{2} \\ \leq {({infI}_{A} (x_{i}) - {infI}_{C} (x_{i}))}^{2}, \\ {({supI}_{A} (x_{i}) - {supI}_{B} (x_{i}))}^{2} \\ \leq {({supI}_{A} (x_{i}) - {supI}_{C} (x_{i}))}^{2}, \\ {({infF}_{A} (x_{i}) - {infF}_{B} (x_{i}))}^{2} \\ \leq {({infF}_{A} (x_{i}) - {infF}_{C} (x_{i}))}^{2}, \\ {({supF}_{A} (x_{i}) - {supF}_{B} (x_{i}))}^{2} \\ \leq {({supF}_{A} (x_{i}) - {supF}_{C} (x_{i}))}^{2} . \end{matrix}$

This follows $\begin{matrix} d_{\inf} (A, B) & \leq & d_{\inf} (A, C), \\ d_{\sup} (A, B) & \leq & d_{\sup} (A, C) . \end{matrix}$

Thus, we have $\begin{matrix} d_{\inf} (A, B) \land d_{\sup} (A, B) \\ \leq d_{\inf} (A, C) \land d_{\sup} (A, C), \\ d_{\inf} (A, B) \lor d_{\sup} (A, B) \\ \leq d_{\inf} (A, C) \lor d_{\sup} (A, C) . \end{matrix}$

That follows that d_min (A, B) ≤ d_min (A, C) and d_max (A, B) ≤ d_max (A, C). By Definition 2.2(3) the conclusion D (A, B) ≤ D (A, C) holds.

D (B, C) ≤ D (A, C) can also be obtained by the means of the above proof. Thus, the property (4) is obtained. □

It is well known that the distance measure often induces similarity measure [34 –36]. Based on the relationship of distance measure and similarity measure, we may propose interval-similarity measure correspondent with the interval-distance. Furthermore, we give its properties similar to that for interval-distance.

Definition 4.2. Let INS (X) be all INSs on the discourse domain X ={ x₁, x₂, …, x_n }. For A, B ∈ INS (X), the interval-similarity measure between A and B is defined by $SIM (A, B) = [1 - d_{\max} (A, B), 1 - d_{\min} (A, B)] .$

Theorem 4.2. The interval-similarity measure SIM (A, B) between INS A and B possesses the following properties

$\tilde{0} \leq SIM (A, B) \leq \tilde{1}$

SIM (A, B) = SIM (B, A)

if A = B, then $SIM (A, B) = \tilde{0}$

if A ⊆ B ⊆ C, then SIM (A, C) ≤ D (A, B) and D (A, C) ≤ D (B, C).

Proof. According to Definition 4.2 and Theorem 4.1, we can easily obtain (1) - (4) by analogous way in the proof of Theorem 4.1.

Because interval-similarity measure is also regarded as an interval number and users often hope to find some INF concepts analogous to the target concept within a required similarity range, we will propose inclusion measure between interval numbers to represent the degree of satisfaction on the required similarity range. Therefore, we present the set-theoretic on interval numbers with the assistance of Definition 2.2 as follows.

Definition 4.3. Let D₁ = [a^-, a⁺] and D₂ = [b^-, b⁺] be interval numbers such that a^- ∨ b^- ≤ a⁺ ∧ b⁺. We have the set-theoretic on D (0, 1) as follows

D₃ = D₁ ∩ D₂ = [a^- ∨ b^-, a⁺ ∧ b⁺]

D₄ = D₁ ∪ D₂ = [a^- ∧ b^-, a⁺ ∨ b⁺]

D₁ ⊆ D₂ ⇔ a^- ≥ b^- and a⁺ ≤ b⁺

$D_{1}^{C} = [1 - a^{+}, 1 - a^{-}]$

Definition 4.4. Let D₁ = [a^-, a⁺] and D₂ = [b^-, b⁺] be the interval numbers such that a^- ∨ b^- < a⁺ ∧ b⁺. The inclusion measure between D₁ and D₂ is denoted by $C (D_{1}, D_{2}) = \frac{| D_{1} \cap D_{2} |}{| D_{1} |}$ where |D₁| = a⁺ - a^- is the cardinality of D₁. If a^- ∨ b^- ≥ a⁺ ∧ b⁺, then C (D₁, D₂) = 0.

We can easily see that the higher (lower) the inclusion measure, the larger (smaller) the intersection of the two interval numbers. When D₁ ⊆ D₂, the maximum of inclusion measure is 1. The following properties show that the inclusion measure can reflect inclusion relationship between interval numbers to a certain extent.

Theorem 4.3. Let D₁, D₂, D₃ be interval number. The inclusion measure has the following properties $\begin{matrix} (1) 0 \leq C (D_{1}, D_{2}) \leq 1 \\ (2) D_{1} \subseteq D_{2} \Rightarrow C (D_{1}, D_{2}) = 1 \\ (3) D_{1} \subseteq D_{2} \subseteq D_{3} \Rightarrow C (D_{3}, D_{1}) \leq C (D_{2}, D_{1}) \end{matrix}$

Proof. For convenience, we denote D₁ = [a^-, a⁺], D₂ = [b^-, b⁺] and D₃ = [c^-, c⁺].

(1) If a^- ∨ b^- ≥ a⁺ ∧ b⁺, we can get C (D₁, D₂) = 0 according to Definition 4.4.

If D₁ ⊆ D₂, then D₁ ∩ D₂ = [a^- ∨ b^-, a⁺ ∧ b⁺] = [a^-, a⁺] from the items (1) and (3) in Definition 4.3. So C (D₁, D₂) = 1.

When a^- ∨ b^- < a⁺ ∧ b⁺, then 0 < C (D₁, D₂) <1. Thus 0 ≤ C (D₁, D₂) ≤ 1 holds.

(2) If D₁ ⊆ D₂, then D₁ ∩ D₂ = [a^-, a⁺] from the items (1) and (3) in Definition 4.3.We can obtain C (D₁, D₂) = 1.

(3) If D₁ ⊆ D₂ ⊆ D₃, according to Definition 4.4, we have $C (D_{3}, D_{1}) = \frac{| D_{3} \cap D_{1} |}{| D_{3} |} = \frac{| D_{1} |}{| D_{3} |} \leq \frac{| D_{1} |}{| D_{2} |} = \frac{| D_{2} \cap D_{1} |}{| D_{2} |} = C (D_{2}, D_{1})$ .

Remark 4.2. The Theorems 4.2 and 4.3 are the answer to (P2).

5 Examples

This section will illustrate the results in Section 3 (Section 4) with Example 1 (Example 2).

Example 1. The INF formal context $K = (G, M, \tilde{R})$ with G ={ x₁, x₂, x₃, x₄ } and M ={ a, b, c, d }. $\tilde{R}$ is shown as Table 1. We will obtain all INF concepts step by step as the process in Algorithm 1.

Table 1
Interval neutrosophic fuzzy formal context $K = (G, M, \tilde{R})$

a b c d

x ₁ 〈 [0 . 4, 0 . 7] , [0 . 2, 0 . 5] , [0 . 2, 0 . 3]〉〈 [0 . 3, 0 . 5] , [0 . 3, 0 . 7] , [0 . 5, 0 . 8]〉〈 [0 . 3, 0 . 5] , [0 . 4, 0 . 6] , [0 . 1, 0 . 3]〉〈 [0 . 4, 0 . 6] , [0 . 2, 0 . 5] , [0 . 3, 0 . 9]〉

x ₂ 〈 [0 . 4, 0 . 7] , [0 . 3, 0 . 5] , [0 . 2, 0 . 6]〉〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 9] , [0 . 6, 0 . 9]〉〈 [0 . 3, 0 . 5] , [0 . 4, 0 . 6] , [0 . 2, 0 . 5]〉〈 [0 . 2, 0 . 6] , [0 . 4, 0 . 5] , [0 . 2, 0 . 3]〉

x ₃ 〈 [0 . 3, 0 . 6] , [0 . 3, 0 . 8] , [0 . 2, 0 . 3]〉〈 [0 . 2, 0 . 4] , [0 . 8, 0 . 9] , [0 . 5, 0 . 8]〉〈 [0 . 2, 0 . 4] , [0 . 4, 0 . 9] , [0 . 3, 0 . 5]〉〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 6] , [0 . 4, 0 . 9]〉

x ₄ 〈 [0 . 4, 0 . 7] , [0 . 2, 0 . 5] , [0 . 5, 0 . 6]〉〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 6] , [0 . 3, 0 . 8]〉〈 [0 . 4, 0 . 8] , [0 . 4, 0 . 7] , [0 . 3, 0 . 9]〉〈 [0 . 4, 0 . 6] , [0 . 3, 0 . 4] , [0 . 4, 0 . 5]〉

	a	b	c	d
x ₁	〈 [0 . 4, 0 . 7] , [0 . 2, 0 . 5] , [0 . 2, 0 . 3]〉	〈 [0 . 3, 0 . 5] , [0 . 3, 0 . 7] , [0 . 5, 0 . 8]〉	〈 [0 . 3, 0 . 5] , [0 . 4, 0 . 6] , [0 . 1, 0 . 3]〉	〈 [0 . 4, 0 . 6] , [0 . 2, 0 . 5] , [0 . 3, 0 . 9]〉
x ₂	〈 [0 . 4, 0 . 7] , [0 . 3, 0 . 5] , [0 . 2, 0 . 6]〉	〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 9] , [0 . 6, 0 . 9]〉	〈 [0 . 3, 0 . 5] , [0 . 4, 0 . 6] , [0 . 2, 0 . 5]〉	〈 [0 . 2, 0 . 6] , [0 . 4, 0 . 5] , [0 . 2, 0 . 3]〉
x ₃	〈 [0 . 3, 0 . 6] , [0 . 3, 0 . 8] , [0 . 2, 0 . 3]〉	〈 [0 . 2, 0 . 4] , [0 . 8, 0 . 9] , [0 . 5, 0 . 8]〉	〈 [0 . 2, 0 . 4] , [0 . 4, 0 . 9] , [0 . 3, 0 . 5]〉	〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 6] , [0 . 4, 0 . 9]〉
x ₄	〈 [0 . 4, 0 . 7] , [0 . 2, 0 . 5] , [0 . 5, 0 . 6]〉	〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 6] , [0 . 3, 0 . 8]〉	〈 [0 . 4, 0 . 8] , [0 . 4, 0 . 7] , [0 . 3, 0 . 9]〉	〈 [0 . 4, 0 . 6] , [0 . 3, 0 . 4] , [0 . 4, 0 . 5]〉

Step 1. All subsets of G are as follows: $\begin{matrix} X_{0} & = & \emptyset, X_{1} = {x_{1}}, X_{2} = {x_{2}}, \\ X_{3} & = & {x_{3}}, X_{4} = {x_{4}}, X_{5} = {x_{1}, x_{2}}, \\ X_{6} & = & {x_{1}, x_{3}}, X_{7} = {x_{1}, x_{4}}, X_{8} = {x_{2}, x_{3}}, \\ X_{9} & = & {x_{2}, x_{4}}, X_{10} = {x_{3}, x_{4}}, \\ X_{11} & = & {x_{1}, x_{2}, x_{3}}, X_{12} = {x_{1}, x_{2}, x_{4}}, \\ X_{13} & = & {x_{1}, x_{3}, x_{4}}, X_{14} = {x_{2}, x_{3}, x_{4}}, \\ X_{15} & = & {x_{1}, x_{2}, x_{3}, x_{4}} . \end{matrix}$

Step 2. For convenience, we take X₃ ={ x₃ } to illustrate the procedure of this method. Calculate attribute set ${\tilde{B}}_{3}$ i.e. $\begin{matrix} {\tilde{B}}_{3} = X_{3}^{*} & = & {〈 a, [0.3, 0.6], [0.3, 0.8], [0.2, 0.3] 〉, \\ 〈 b, [0.2, 0.4], [0.8, 0.9], [0.5, 0.8] 〉, \\ 〈 c, [0.2, 0.4], [0.4, 0.9], [0.3, 0.5] 〉, \\ 〈 d, [0.4, 0.5], [0.4, 0.6], [0.4, 0.9] 〉} . \end{matrix}$

Step 3. Compute ${\tilde{B}}_{3}^{*}$ i.e. ${\tilde{B}}_{3}^{*} = {x_{1}, x_{3}} \neq X_{3}$ .

Then, $({\tilde{B}}_{3}^{*}, {\tilde{B}}_{3}) = (x_{1} x_{3}, {\tilde{B}}_{3})$ is an INF concept. Consequently, all INF concepts are as follows: $\begin{matrix} C_{0} & = & (\emptyset, {\tilde{B}}_{0}), C_{1} = (x_{1}, {\tilde{B}}_{1}), \\ C_{2} & = & (x_{2}, {\tilde{B}}_{2}), C_{3} = (x_{1} x_{3}, {\tilde{B}}_{3}), \\ C_{4} & = & (x_{4}, {\tilde{B}}_{4}), C_{5} = (x_{1} x_{2}, {\tilde{B}}_{5}), \\ C_{6} & = & (x_{1} x_{4}, {\tilde{B}}_{6}), C_{7} = (x_{2} x_{4}, {\tilde{B}}_{7}), \\ C_{8} & = & (x_{1} x_{2} x_{3}, {\tilde{B}}_{8}), C_{9} = (x_{1} x_{2} x_{4}, {\tilde{B}}_{9}), \\ C_{10} & = & (x_{1} x_{3} x_{4}, {\tilde{B}}_{10}), C_{11} = (x_{1} x_{2} x_{3} x_{4}, {\tilde{B}}_{11}) . \end{matrix}$

The intensions of them are as follows. $\begin{matrix} {\tilde{B}}_{0} & = & {〈 a, [1, 1], [0, 0], [0, 0] 〉, \\ 〈 b, [1, 1], [0, 0], [0, 0] 〉, \\ 〈 c, [1, 1], [0, 0], [0, 0] 〉, \\ 〈 d, [1, 1], [0, 0], [0, 0] 〉}, \\ {\tilde{B}}_{1} & = & {〈 a, [0.4, 0.7], [0.2, 0.5], [0.2, 0.3] 〉, \\ 〈 b, [0.3, 0.5], [0.3, 0.7], [0.5, 0.8] 〉, \\ 〈 c, [0.3, 0.5], [0.4, 0.6], [0.1, 0.3] 〉, \\ 〈 d, [0.4, 0.6], [0.2, 0.5], [0.3, 0.9] 〉}, \\ {\tilde{B}}_{2} & = & {〈 a, [0.4, 0.7], [0.3, 0.5], [0.2, 0.6] 〉, \\ 〈 b, [0.4, 0.5], [0.4, 0.9], [0.6, 0.9] 〉, \\ 〈 c, [0.3, 0.5], [0.4, 0.6], [0.2, 0.5] 〉, \\ 〈 d, [0.2, 0.6], [0.4, 0.5], [0.2, 0.3] 〉}, \\ {\tilde{B}}_{3} & = & {〈 a, [0.3, 0.6], [0.3, 0.8], [0.2, 0.3] 〉, \\ 〈 b, [0.2, 0.4], [0.8, 0.9], [0.5, 0.8] 〉, \\ 〈 c, [0.2, 0.4], [0.4, 0.9], [0.3, 0.5] 〉, \\ 〈 d, [0.4, 0.5], [0.4, 0.6], [0.4, 0.9] 〉}, \\ {\tilde{B}}_{4} & = & {〈 a, [0.4, 0.7], [0.2, 0.5], [0.5, 0.6] 〉, \\ 〈 b, [0.4, 0.5], [0.4, 0.6], [0.3, 0.8] 〉, \\ 〈 c, [0.4, 0.8], [0.4, 0.7], [0.3, 0.9] 〉, \\ 〈 d, [0.4, 0.6], [0.3, 0.4], [0.4, 0.5] 〉}, \\ {\tilde{B}}_{5} & = & {〈 a, [0.4, 0.7], [0.3, 0.5], [0.2, 0.6] 〉, \\ 〈 b, [0.3, 0.5], [0.4, 0.9], [0.6, 0.9] 〉, \\ 〈 c, [0.3, 0.5], [0.4, 0.6], [0.2, 0.5] 〉, \\ 〈 d, [0.2, 0.6], [0.4, 0.5], [0.3, 0.9] 〉}, \\ {\tilde{B}}_{6} & = & {〈 a, [0.4, 0.7], [0.2, 0.5], [0.5, 0.6] 〉, \\ 〈 b, [0.3, 0.5], [0.4, 0.7], [0.5, 0.8] 〉, \\ 〈 c, [0.3, 0.5], [0.4, 0.7], [0.3, 0.9] 〉, \\ 〈 d, [0.4, 0.6], [0.3, 0.5], [0.4, 0.9] 〉}, \\ {\tilde{B}}_{7} & = & {〈 a, [0.4, 0.7], [0.3, 0.5], [0.5, 0.6] 〉, \\ 〈 b, [0.4, 0.5], [0.4, 0.9], [0.6, 0.9] 〉, \\ 〈 c, [0.3, 0.5], [0.4, 0.7], [0.3, 0.9] 〉, \\ 〈 d, [0.2, 0.6], [0.4, 0.5], [0.4, 0.5] 〉}, \\ {\tilde{B}}_{8} & = & {〈 a, [0.3, 0.6], [0.3, 0.8], [0.2, 0.6] 〉, \\ 〈 b, [0.2, 0.4], [0.8, 0.9], [0.6, 0.9] 〉, \\ 〈 c, [0.2, 0.4], [0.4, 0.9], [0.3, 0.5] 〉, \\ 〈 d, [0.2, 0.5], [0.4, 0.6], [0.4, 0.9] 〉}, \\ {\tilde{B}}_{9} & = & {〈 a, [0.4, 0.7], [0.3, 0.5], [0.5, 0.6] 〉, \\ 〈 b, [0.3, 0.5], [0.4, 0.9], [0.6, 0.9] 〉, \\ 〈 c, [0.3, 0.5], [0.4, 0.7], [0.3, 0.9] 〉, \\ 〈 d, [0.2, 0.6], [0.4, 0.5], [0.4, 0.9] 〉}, \\ {\tilde{B}}_{10} & = & {〈 a, [0.3, 0.6], [0.3, 0.8], [0.5, 0.6] 〉, \\ 〈 b, [0.2, 0.4], [0.8, 0.9], [0.5, 0.8] 〉, \\ 〈 c, [0.2, 0.4], [0.4, 0.9], [0.3, 0.9] 〉, \\ 〈 d, [0.4, 0.5], [0.4, 0.6], [0.4, 0.9] 〉}, \\ {\tilde{B}}_{11} & = & {〈 a, [0.3, 0.6], [0.3, 0.8], [0.5, 0.6] 〉, \\ 〈 b, [0.2, 0.4], [0.8, 0.9], [0.6, 0.9] 〉, \\ 〈 c, [0.2, 0.4], [0.4, 0.9], [0.3, 0.9] 〉, \\ 〈 d, [0.2, 0.5], [0.4, 0.6], [0.4, 0.9] 〉} . \end{matrix}$

Step 4. All INF concepts are ordered by relation “≼”. The obtained INF concept lattice is shown in Fig. 1.

Fig. 1

INF concept lattice.

Example 2. We consider the interval-similarity measure between INSs when find similar concepts.

Hence, we do not need to consider the extents of the INF concepts.

Let INF formal context be defined as Example 1. Suppose required similarity range is [0.837, 0.879] and the INF target concept is $C^{*} = (X^{*}, {\tilde{B}}^{*})$ given by users where $\begin{matrix} {\tilde{B}}^{*} & = & {〈 a, [0.3, 0.5], [0.4, 0.6], [0.3, 0.7] 〉, \\ 〈 b, [0.3, 0.6], [0.5, 0.7], [0.5, 0.7] 〉, \\ 〈 c, [0.4, 0.6], [0.2, 0.3], [0.4, 0.8] 〉, \\ 〈 d, [0.2, 0.5], [0.4, 0.6], [0.2, 0.7] 〉} . \end{matrix}$

To facilitate observation, the interval-distance and interval-similarity measure between ${\tilde{B}}^{*}$ and ${\tilde{B}}_{i}$ (i = 0, 1, 2, … , 11) and the inclusion measure between the obtained interval-similarity measure and the required similarity range are shown as Table 2. We can easily to see that ${\tilde{B}}_{6}$ , followed by ${\tilde{B}}_{9}, {\tilde{B}}_{7}, {\tilde{B}}_{5}$ , is closest to ${\tilde{B}}^{*}$ within the similarity range [0.837, 0.879].

Table 2

Interval-distance, interval-similarity measure and inclusion measure

Interval-distance	Interval-similarity measure	Inclusion measure
$D ({\tilde{B}}_{0}, {\tilde{B}}^{*}) = [0.510, 0.593]$	$SIM ({\tilde{B}}_{0}, {\tilde{B}}^{*}) = [0.407, 0.490]$	$C (SIM ({\tilde{B}}_{0}, {\tilde{B}}^{*}), α) = 0.00$
$D ({\tilde{B}}_{1}, {\tilde{B}}^{*}) = [0.165, 0.230]$	$SIM ({\tilde{B}}_{1}, {\tilde{B}}^{*}) = [0.770, 0.835]$	$C (SIM ({\tilde{B}}_{1}, {\tilde{B}}^{*}), α) = 0.00$
$D ({\tilde{B}}_{2}, {\tilde{B}}^{*}) = [0.111, 0.208]$	$SIM ({\tilde{B}}_{2}, {\tilde{B}}^{*}) = [0.792, 0.889]$	$C (SIM ({\tilde{B}}_{2}, {\tilde{B}}^{*}), α) = 0.43$
$D ({\tilde{B}}_{3}, {\tilde{B}}^{*}) = [0.155, 0.180]$	$SIM ({\tilde{B}}_{3}, {\tilde{B}}^{*}) = [0.820, 0.845]$	$C (SIM ({\tilde{B}}_{3}, {\tilde{B}}^{*}), α) = 0.32$
$D ({\tilde{B}}_{4}, {\tilde{B}}^{*}) = [0.155, 0.262]$	$SIM ({\tilde{B}}_{4}, {\tilde{B}}^{*}) = [0.738, 0.845]$	$C (SIM ({\tilde{B}}_{4}, {\tilde{B}}^{*}), α) = 0.07$
$D ({\tilde{B}}_{5}, {\tilde{B}}^{*}) = [0.111, 0.182]$	$SIM ({\tilde{B}}_{6}, {\tilde{B}}^{*}) = [0.818, 0.889]$	$C (SIM ({\tilde{B}}_{6}, {\tilde{B}}^{*}), α) = 0.59$
$D ({\tilde{B}}_{6}, {\tilde{B}}^{*}) = [0.144, 0.163]$	$SIM ({\tilde{B}}_{5}, {\tilde{B}}^{*}) = [0.837, 0.856]$	$C (SIM ({\tilde{B}}_{5}, {\tilde{B}}^{*}), α) = 1.00$
$D ({\tilde{B}}_{7}, {\tilde{B}}^{*}) = [0.125, 0.180]$	$SIM ({\tilde{B}}_{7}, {\tilde{B}}^{*}) = [0.820, 0.875]$	$C (SIM ({\tilde{B}}_{7}, {\tilde{B}}^{*}), α) = 0.69$
$D ({\tilde{B}}_{8}, {\tilde{B}}^{*}) = [0.158, 0.243]$	$SIM ({\tilde{B}}_{8}, {\tilde{B}}^{*}) = [0.757, 0.842]$	$C (SIM ({\tilde{B}}_{8}, {\tilde{B}}^{*}), α) = 0.05$
$D ({\tilde{B}}_{9}, {\tilde{B}}^{*}) = [0.122, 0.180]$	$SIM ({\tilde{B}}_{10}, {\tilde{B}}^{*}) = [0.820, 0.878]$	$C (SIM ({\tilde{B}}_{10}, {\tilde{B}}^{*}), α) = 0.70$
$D ({\tilde{B}}_{10}, {\tilde{B}}^{*}) = [0.163, 0.223]$	$SIM ({\tilde{B}}_{9}, {\tilde{B}}^{*}) = [0.777, 0.837]$	$C (SIM ({\tilde{B}}_{9}, {\tilde{B}}^{*}), α) = 0.00$
$D ({\tilde{B}}_{11}, {\tilde{B}}^{*}) = [0.155, 0.229]$	$SIM ({\tilde{B}}_{11}, {\tilde{B}}^{*}) = [0.771, 0.845]$	$C (SIM ({\tilde{B}}_{11}, {\tilde{B}}^{*}), α) = 0.10$

Analysis. Ye [14] proposed similarity measures between two different INSs using the weighted Euclidean distance. In order to illustrate the significance of interval-similarity measure, we will compare the interval-similarity measure obtained by interval-distance with the similarity measure obtained by the normalized Euclidean distance. In a given similarity range [0.837, 0.879], we can attain that any of ${\tilde{B}}_{5}, {\tilde{B}}_{6}, {\tilde{B}}_{7}$ and ${\tilde{B}}_{9}$ is similar to ${\tilde{B}}^{*}$ if we utilize the method in Ye [14]. We can summarize all of comparisons between our method and that in Ye [14] as Table 3.

Table 3

Comparison of different methods

Inclusion measure	Similarity by Ye [14]
$C (SIM ({\tilde{B}}_{6}, {\tilde{B}}^{*}), α) = 1.00$	$S ({\tilde{B}}_{6}, {\tilde{B}}^{*}) = 0.846$
$C (SIM ({\tilde{B}}_{9}, {\tilde{B}}^{*}), α) = 0.70$	$S ({\tilde{B}}_{9}, {\tilde{B}}^{*}) = 0.846$
$C (SIM ({\tilde{B}}_{7}, {\tilde{B}}^{*}), α) = 0.69$	$S ({\tilde{B}}_{7}, {\tilde{B}}^{*}) = 0.845$
$C (SIM ({\tilde{B}}_{5}, {\tilde{B}}^{*}), α) = 0.59$	$S ({\tilde{B}}_{5}, {\tilde{B}}^{*}) = 0.849$

Particularly, using the method in Ye [14], we can find ${\tilde{B}}_{5}$ is closest to ${\tilde{B}}^{*}$ ; ${\tilde{B}}_{6}$ and ${\tilde{B}}_{9}$ have the same similarity degree as ${\tilde{B}}^{*}$ . By the method described in this paper, the similarity degree between ${\tilde{B}}_{6}$ to ${\tilde{B}}^{*}$ is larger than that between ${\tilde{B}}_{9}$ to ${\tilde{B}}^{*}$ . Based on the inclusion measure between interval numbers, we can find similar concepts more precisely with interval-similarity measure described in thispaper.

6 Conclusion

This paper proposes the INF formal context and constructs the corresponding INF concept lattice which will extend the capacity of concept. This result gives an answer to (P1). Based on the inclusion measure between interval numbers, Section 4 gives interval-similarity measure to find some INF concepts similar to the target concept within required similarity range. The needed answer for (P2) is shown as Theorems 4.2 and 4.3. The results of this paper extend the theory of concept lattice, although there are some problems in the applications. In further research, we will explore much more algorithms for attribute reduction under the isomorphic sense and methods of multiple attribute decision making with neutrosophic information.

Footnotes

Acknowledgments

This paper is granted by National Nature Science Foundation of China (61572011).

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