Matrix-based reduction approach for one-sided fuzzy three-way concept lattices

Abstract

Three-way concept analysis is a mathematical model of the combination of formal concept analysis and three-way decision, and knowledge discovery plays a significant impact on formal fuzzy contexts since such datasets are frequently encountered in real life. In this paper, a novel type of one-sided fuzzy three-way concept lattices is presented in a given formal fuzzy context with its complement, in which a ternary classification is available. In such case, we comprehensively explore the connections between the proposed models and classical fuzzy concept lattices among elements, sets, and orders. Furthermore, approaches to granular matrix-based reductions are investigated, by which granular consistent sets, and granular reducts via discernibility Boolean matrices are tectonically put forward. At last, the demonstrated results are performed by several experiments which enrich the research of three-way concept analysis.

Keywords

Formal fuzzy contexts granular reduction one-sided fuzzy concept lattices three-way decision

1 Introduction

Formal concept analysis (FCA briefly), proposed by Wille [1] in 1982, is a theoretical thinking for knowledge discovery and data mining. Formal context and formal concept are two fundamental notions of FCA. A formal context is a triple (U, A, I), where U is a set of objects, A is a set of attributes, and I is a binary relation over U and A. A formal concept, is a pair (X, B) consisting of a set of objects, which is the maximal set of objects occupying all properties in B, and a set of attributes, which is the maximal set of attributes shared by all objects in X. At this point, FCA has been widely applied to pattern recognition, expert system and other fields in the last decade [2–6 , 37–39].

As a regular rule, given a formal context (U, A, I), for x ∈ U and a ∈ A, a binary relation I with a specific form of the cross is 1 if (x, a) ∈ I and 0 otherwise. However, in the real world, the binary relation is universally fuzzy rather than crisp so that the corresponding context is called formal fuzzy context. Consequently, the fuzzy concept lattices manifested a pleasing personality on the academic stages, and different kinds of fuzzy concepts have been explored under different semantics. For instance, the generalization of FCA to fuzzy sets has been proposed by [7] to overcome uncertainty and vagueness in serial data. Afterwards, the “one-sided fuzzy concept” is originated from the work of Yahia [8] and Kraji [9]. More specifically, a one-sided fuzzy concept (X, B) explicitly has two following forms: “X is crisp and B is fuzzy” and “X is fuzzy and B is crisp”. Subsequently, Shao et al. [10] investigated granular reduct approach to preserve the object granules from crisp-fuzzy concept. Based on fuzzy-crisp concept lattice, Li et al. [11] studied the attribute characteristics and reduction method so that the representation of formal fuzzy context is more succinct. After that, from the viewpoint of graph theory, Mao et al. [12] presented the judgment theorems of meet-irreducible elements.

FCA, a preference model, only focuses on commonly-shared information. Nevertheless, we sometimes need not only positive information, but also negative information to make a decision in some elections. To overcome this problem, three-way formal concept analysis (3WCA) is the outcome of mixing together FCA and three-way decision (3WD) [16–18 , 40]. In [19, 20], the significant breakthrough is that the attribute-induced extension (or the object-induced intension) of each three-way concept has separately a positive and a negative part. Therefore, three-way concepts can supply much more information than classical concepts. Besides, Yao [21] characterized 3WCA by representing partially-known concepts in incomplete formal contexts based on interval sets. Li et al. [22] obtained the three-way cognitive learning process via multi-granularity so as to remember three-way cognitive concepts from a given clue. In [24], Qian et al. studied dividing and conquering strategy from the communities of apposition and subposition of formal contexts. Incidentally, as the significant examples of crisp-intuitionistic formal concepts [13], three-way dual concepts [16], and L-fuzzy concepts [27] can also be considered as a category of ternary classification to some extent.

It is worth noting that knowledge reduction in 3WCA has gained growing interests in recent years [28 –32]. Ren et al. [28] formulated four kinds of attribute reductions and discussed their relationships introduced in [19, 20]. From the perspective of irreducible elements, Li et al. [29] examined two models to construct approximate concepts and studied attribute reduction in incomplete contexts based on 3WD. In [32], Zhang et al. provided a new insight in three-way reducts by a dependency degree. At present, although knowledge reduction of 3WCA has attracted more and more attentions, the research that applying the idea of 3WD to formal fuzzy contexts is still in the early stage [25 –27], especially for one-sided formal fuzzy contexts. Within the neutrosophic context, Singh [25] formulated the three-way fuzzy concept lattice. He et al. [27] put forward L-fuzzy concept analysis for 3WD with the assistant of the fuzzy logic theory and 3WCA.

It should be pointed out that few results about one-sided fuzzy three-way concept lattices have been done. Motivated by this problem, in terms of a formal fuzzy context, we will introduce a general approach to construct fuzzy three-way concept lattice. Specifically speaking, a fuzzy set $\tilde{A}$ consists of three-wise disjoint subsets, namely the fuzzy positive region ${POS}_{X}^{\tilde{A}} = X^{▽}$ , the fuzzy negative region ${NEG}_{X}^{\tilde{A}} = X^{\bar{▽}}$ , and the fuzzy boundary region ${BND}_{X}^{\tilde{A}} = 1 - X^{▽} - X^{\bar{▽}}$ , while the crisp-intuitionistic fuzzy concept lattice represented by [31] occupies different three-wise disjoint subsets. A prominent difference about the fuzzy negative region between the Pang’s work [31] and one-sided fuzzy three-way lattice lies in that the former is comprised of the degree of non-membership rather than the degree of membership. That is, the intuitionistic fuzzy negative region is $1 - X^{▽}$ when intuitionistic fuzzy sets weaken to fuzzy sets. And meanwhile, stimulated by [4], proposals in this paper depend on the one-sided fuzzy three-way concept lattices which is different from the attribute reduction with respect to 3WCA. The demonstrated results can reflect more information than fuzzy concept lattices and provide a novel cornerstone by granular matrix for 3WCA.

This paper is organized as follows. In the next Section, some basic terminologies of formal fuzzy contexts and matrix theory throughout the paper are first reviewed. Section 3 gives the notions of one-sided fuzzy three-way concept lattices. In this case, the connections between fuzzy three-way concept lattices and fuzzy concept lattices are thoroughly discussed in Section 4. Then the attribute discernibility Boolean matrix for attribute reducts is presented in Section 5. Analogously, Section 6 and 7 develop an approach to obtain object reducts and numerical experiments are conducted to verify the effectiveness of the proposed method. Finally, we conclude this paper with a summary and further work in Section 8.

2 Preliminaries

In this section, we briefly recall some basic terminologies which are frequently used in the main discussion.

2.1 Formal fuzzy contexts

Definition 1. [33] Let P and Q be two posets, a pair (φ, ψ) of mappings φ: P → Q and ψ : Q → P is a Galois connection between P and Q if

p₁ ≤ _Pp₂ ⇒ φ (p₂) ≤ _Qφ (p₁) for any p₁, p₂ ∈ P;

q₁ ≤ _Qq₂ ⇒ ψ (q₂) ≤ _Pψ (q₁) for all q₁, q₂ ∈ Q;

p ≤ _Pψ (φ (p)) and q ≤ _Qφ (ψ (q)) for any (p, q) ∈ P × Q.

Lemma 1. [34] Let P and Q be two posets. Then (M, ≤ _M) is seen as a poset if the Cartesian product M = P × Q occupies the following conclusion: (p₁, q₁) ≤ _M (p₂, q₂) ⇔ p₁ ≤ _Pp₂ and q₁ ≤ _Qq₂.

Let U be a nonempty set. The set of all subsets of U and the set of all fuzzy sets on U are defined by $P (U)$ and $F (U)$ , respectively. Furthermore, let $\tilde{A}$ , $\tilde{B}$ be fuzzy sets on U. $\tilde{A} \subseteq \tilde{B}$ if and only if $\tilde{A} (x) \leq \tilde{B} (x)$ , ∀x ∈ U, and the operations ∪, ∩ and c on $F (U)$ are, respectively, denoted as follows: $\begin{matrix} (\tilde{A} \cup \tilde{B}) (x) = \tilde{A} (x) \lor \tilde{B} (x); \\ (\tilde{A} \cap \tilde{B}) (x) = \tilde{A} (x) \land \tilde{B} (x); \\ {\tilde{A}}^{c} (x) = 1 - \tilde{A} (x) . \end{matrix}$

A triplet $(U, A, \tilde{I})$ is called a formal fuzzy context, where U and A are, respectively, non-empty finite sets of objects and attributes, and $\tilde{I} \in F (U \times A)$ is a fuzzy binary relation between U and A.

Thereinafter, the one-sided formal fuzzy concepts were described respectively by [8, 9]. Specifically, a pattern (X, B) has two categories: one is that X is crisp and B is fuzzy, X is fuzzy and B is crisp as another.

Definition 2. [8] Let $(U, A, \tilde{I})$ be a formal fuzzy context. For X ⊆ U and $\tilde{B} \in F (A)$ , two operators $▽ : P (U) \to F (A)$ and $△ : F (A) \to P (U)$ are given by: $\begin{matrix} X^{▽} (a) = ⋀_{x \in X} \tilde{I} (x, a), a \in A, \\ {\tilde{B}}^{△} = {x \in U : \tilde{B} (a) \leq \tilde{I} (x, a), \forall a \in A} . \end{matrix}$

Property 1. [8] For X, X₁, X₂ ⊆ U, and $\tilde{B}, {\tilde{B}}_{1}, {\tilde{B}}_{2} \in F (A)$ , the following properties hold:

$X_{1} \subseteq X_{2} \Rightarrow X_{2}^{▽} \subseteq X_{1}^{▽}$ , ${\tilde{B}}_{1} \subseteq {\tilde{B}}_{2} \Rightarrow {\tilde{B}}_{2}^{△} \subseteq {\tilde{B}}_{1}^{△}$ ;

X ⊆ X^▽△, $\tilde{B} \subseteq {\tilde{B}}^{△ ▽}$ ;

X^▽ = X^▽△▽, ${\tilde{B}}^{△} = {\tilde{B}}^{△ ▽ △}$ ;

$(X_{1} \cup X_{2})^{▽} = X_{1}^{▽} \cap X_{2}^{▽}$ , $({\tilde{B}}_{1} \cup {\tilde{B}}_{2})^{△} = {\tilde{B}}_{1}^{△} \cap {\tilde{B}}_{2}^{△}$ .

Therefore, a crisp-fuzzy concept is a pair $(X, \tilde{B}) \in P (U) \times F (A)$ satisfying $X^{▽} = \tilde{B}$ and ${\tilde{B}}^{△} = X$ . We denote all crisp-fuzzy concepts in $(U, A, \tilde{I})$ as $OB (U, \tilde{A}, \tilde{I})$ .

Example 1. Table 1 outlines a formal fuzzy context $(U, A, \tilde{I})$ , in which U = {1, 2, 3, 4} and A = {a, b, c, d, e}. Then $OB (U, \tilde{A}, \tilde{I})$ is shown in Fig. 1.

Fig. 1

$OB (U, \tilde{A}, \tilde{I})$ .

Table 1

Formal fuzzy context $(U, A, \tilde{I})$

$\tilde{I}$	a	b	c	d	e
1	0.9	0.7	0.2	0.9	0.8
2	0.8	0.8	0.8	0.3	0.2
3	0.1	0.2	0.1	0.8	0.2
4	0.1	0.3	0.8	0.1	0.2

Definition 3. Given a formal fuzzy context $(U, A, \tilde{I})$ , for $\tilde{X} \in F (U)$ , and B ⊆ A, two operators $▽ : P (A) \to F (U)$ and $△ : F (U) \to P (A)$ are, respectively, given by: $\begin{matrix} B^{▽} (x) = ⋀_{a \in B} \tilde{I} (x, a), x \in U, \\ {\tilde{X}}^{△} = {a \in A : \tilde{X} (x) \leq \tilde{I} (x, a), \forall x \in U} . \end{matrix}$

Property 2. For $\tilde{X}, {\tilde{X}}_{1}, {\tilde{X}}_{2} \in F (U)$ , and B, B₁, B₂ ⊆ A, the following properties hold:

${\tilde{X}}_{1} \subseteq {\tilde{X}}_{2} \Rightarrow {\tilde{X}}_{2}^{△} \subseteq {\tilde{X}}_{1}^{△}$ , $B_{1} \subseteq B_{2} \Rightarrow B_{2}^{▽} \subseteq B_{1}^{▽}$ ;

$\tilde{X} \subseteq {\tilde{X}}^{△ ▽}$ , B ⊆ B^▽△;

${\tilde{X}}^{△} = {\tilde{X}}^{△ ▽ △}$ , B^▽ = B^▽△▽;

$({\tilde{X}}_{1} \cup {\tilde{X}}_{2})^{△} = {\tilde{X}}_{1}^{△} \cap {\tilde{X}}_{2}^{△}$ , $(B_{1} \cup B_{2})^{▽} = B_{1}^{▽} \cap B_{2}^{▽}$ .

Evidently, $(\tilde{X}, B)$ is a fuzzy-crisp concept if ${\tilde{X}}^{△} = B$ and $B^{▽} = \tilde{X}$ . All fuzzy-crisp concepts form a complete lattice, and is denoted as $AB (\tilde{U}, A, \tilde{I})$ .

Example 2. A formal fuzzy context $(U, A, \tilde{I})$ is shown in Table 2, where U = {1, 2, 3, 4, 5} and A = {a, b, c}. Then the corresponding $AB (\tilde{U}, A, \tilde{I})$ is listed in Fig. 2.

Fig. 2

$AB (\tilde{U}, A, \tilde{I})$ .

Table 2

Formal fuzzy context $(U, A, \tilde{I})$

$\tilde{I}$	a	b	c
1	0.9	0.8	0.1
2	0.7	0.2	0.3
3	0.2	0.8	0.1
4	0.9	0.3	0.7
5	0.8	0.2	0.9

2.2 Fuzzy relation and basic notions of matrix

Let U = {x₁, x₂, ⋯ , x_n} and X ⊆ U, the characteristic function is denoted as λ_X = [λ_X (x₁) , λ_X (x₂) , ⋯ , λ_X (x_n)], where $λ_{X} (x_{i}) = {\begin{matrix} 1, & x_{i} \in X, \\ 0, & x_{i} \notin X . \end{matrix}$

Specifically, λ_X (x_i) assigns 1 to an element x_i belonging to X, or else 0.

Let U = {u₁, u₂, ⋯ , u_n} and V = {v₁, v₂, ⋯ , v_m} be two universes. If $\tilde{R} \in F (U \times V)$ , $\tilde{R}$ is called a fuzzy relation matrix. For simplicity, $\tilde{R} = [r_{ij}]_{n \times m}$ is defined as follows:

$\tilde{R} = [\begin{matrix} r_{11} & r_{12} & \dots & r_{1 m} \\ r_{21} & r_{22} & \dots & r_{2 m} \\ \dots & \dots & \dots & \dots \\ r_{n 1} & r_{n 2} & \dots & r_{nm} \end{matrix}] .$

Let A = [a_ij] _n×m, B = [b_ij] _n×m and C = [c_ij] _m×t be three fuzzy relation matrices. Then the following operations hold:

A ≤ B iff a_ij ≤ b_ij, i = 1, 2, ⋯ , n, j = 1, 2, ⋯ , m;

A < B iff A ≤ B and A ≠ B;

A · C = [d_ij] _n×t, where d_ij = ⋁ _1≤k≤m [a_ik ∧ c_kj];

A^c = [1 - a_ij] _n×m.

Definition 4. [4] Let A = [a_ij] _n×m, B = [b_ij] _m×p and C = [c_ij] _n×m be fuzzy relation matrices. The formulas are defined as follows: $a_{ik} \otimes b_{kj} = {\begin{matrix} 0, & a_{ik} > b_{kj}, \\ 1, & a_{ik} \leq b_{kj} . \end{matrix}$ and A ⊗ B = (d_ij) _n×p, in which d_ij = ⋀ _1≤k≤m (a_ik ⊗ b_kj). $a_{ij} ⊙ c_{ij} = {\begin{matrix} 1, & a_{ij} > c_{ij}, \\ 0, & a_{ij} \leq c_{ij} . \end{matrix}$ and A ⊙ C = (e_ij) _n×m, where e_ij = a_ij ⊙ c_ij.

3 Formal fuzzy contexts based on three-way decision

In this section, we mainly introduce the notions of one-sided fuzzy three-way concept lattices. In what follows, the corresponding properties will be explicitly discussed.

3.1 Crisp-fuzzy three-way concept lattices

Inspired by [19, 28], we first introduce the notion of the negative operator.

Let $(U, A, \tilde{I})$ be a formal fuzzy context. For X ⊆ U and $\tilde{B} \in F (A)$ , two negative derivation operators $\bar{▽} : P (U) \to F (A)$ and $\bar{△} : F (A) \to P (U)$ are deemed respectively by: $\begin{matrix} X^{\bar{▽}} (a) = ⋀_{x \in X} {\tilde{I}}^{c} (x, a), a \in A, \\ {\tilde{B}}^{\bar{△}} = {x \in U : \tilde{B} (a) \leq {\tilde{I}}^{c} (x, a), \forall a \in A} . \end{matrix}$

Property 3. For X, X₁, X₂ ⊆ U, and $\tilde{B}, {\tilde{B}}_{1}, {\tilde{B}}_{2} \in F (A)$ . Then the following properties hold:

$X_{1} \subseteq X_{2} \Rightarrow X_{2}^{\bar{▽}} \subseteq X_{1}^{\bar{▽}}$ , ${\tilde{B}}_{1} \subseteq {\tilde{B}}_{2} \Rightarrow {\tilde{B}}_{2}^{\bar{△}} \subseteq {\tilde{B}}_{1}^{\bar{△}}$ ;

$X \subseteq X^{\bar{▽} \bar{△}}$ , $\tilde{B} \subseteq {\tilde{B}}^{\bar{△} \bar{▽}}$ ;

$X^{\bar{▽}} = X^{\bar{▽} \bar{△} \bar{▽}}$ , ${\tilde{B}}^{\bar{△}} = {\tilde{B}}^{\bar{△} \bar{▽} \bar{△}}$ ;

$(X_{1} \cup X_{2})^{\bar{▽}} = X_{1}^{\bar{▽}} \cap X_{2}^{\bar{▽}}$ , $({\tilde{B}}_{1} \cup {\tilde{B}}_{2})^{\bar{△}} = {\tilde{B}}_{1}^{\bar{△}} \cap {\tilde{B}}_{2}^{\bar{△}}$ .

Proof. It is straightforward based on Property 1.□

In what follows, we will investigate fuzzy three-way operators via integrating the positive operator with the negative operator.

Definition 5. Given a formal fuzzy context $(U, A, \tilde{I})$ , for X ⊆ U, and $\tilde{B}, \tilde{C} \in F (A)$ , two associated three-way operators are respectively defined as: $X^{↑} = (X^{▽} (a), X^{\bar{▽}} (a)) = (⋀_{x \in X} \tilde{I} (x, a), ⋀_{x \in X} {\tilde{I}}^{c} (x, a)), a \in A,$ (1) the crisp-fuzzy three-way operator and $(\tilde{B}, \tilde{C})^{↓} = {\tilde{B}}^{△} \cap {\tilde{C}}^{\bar{△}} = {x \in U : \forall a \in A, \tilde{B} (a) \leq \tilde{I} (x, a), and, \tilde{C} (a) \leq {\tilde{I}}^{c} (x, a)} .$ (2) the crisp-fuzzy inverse operator.

Proposition 1. For X₁, X₂, X ⊆ U, and $\tilde{B}, {\tilde{B}}_{1}, {\tilde{B}}_{2}, \tilde{C}, \tilde{C_{1}}, \tilde{C_{2}} \in F (A)$ , the following properties hold:

$X_{1} \subseteq X_{2} \Rightarrow X_{2}^{↑} \subseteq X_{1}^{↑}$ , $({\tilde{B}}_{1}, {\tilde{C}}_{1}) \subseteq ({\tilde{B}}_{2}, {\tilde{C}}_{2}) \Rightarrow ({\tilde{B}}_{2}, {\tilde{C}}_{2})^{↓} \subseteq ({\tilde{B}}_{1}, {\tilde{C}}_{1})^{↓}$ ;

X ⊆ X^↑↓, $(\tilde{B}, \tilde{C}) \subseteq (\tilde{B}, \tilde{C})^{↓ ↑}$ ;

X^↑ = X^↑↓↑, $(\tilde{B}, \tilde{C})^{↓} = (\tilde{B}, \tilde{C})^{↓ ↑ ↓}$ ;

$X \subseteq (\tilde{B}, \tilde{C})^{↓} \Leftrightarrow (\tilde{B}, \tilde{C}) \subseteq X^{↑}$ ;

$(X_{1} \cup X_{2})^{↑} = X_{1}^{↑} \cap X_{2}^{↑}$ , $(({\tilde{B}}_{1}, {\tilde{C}}_{1}) \cup ({\tilde{B}}_{2}, {\tilde{C}}_{2}))^{↓} = ({\tilde{B}}_{1}, {\tilde{C}}_{1})^{↓} \cap ({\tilde{B}}_{2}, {\tilde{C}}_{2})^{↓}$ ;

The pair (↑ , ↓) constructs a Galois connection between $(P (U), \subseteq)$ and $(F (A) \times F (A), \leq)$ .

Proof.

If X₁ ⊆ X₂, we have $⋀_{x \in X_{2}} \tilde{I} (x, a) \leq ⋀_{x \in X_{1}} \tilde{I} (x, a)$ , and $⋀_{x \in X_{2}} {\tilde{I}}^{c} (x, a) \leq ⋀_{x \in X_{1}} {\tilde{I}}^{c} (x, a)$ for any a ∈ A, which yields $X_{2}^{↑} \subseteq X_{1}^{↑}$ . In addition, suppose $({\tilde{B}}_{1}, {\tilde{C}}_{1}) \subseteq ({\tilde{B}}_{2}, {\tilde{C}}_{2})$ , we obtain ${\tilde{B}}_{1} \subseteq {\tilde{B}}_{2}$ and ${\tilde{C}}_{1} \subseteq {\tilde{C}}_{2}$ , i.e., ${\tilde{B}}_{1} (a) \leq {\tilde{B}}_{2} (a)$ and ${\tilde{C}}_{1} (a) \leq {\tilde{C}}_{2} (a)$ for all a ∈ A. It follows that ${\tilde{B}}_{2} (a) \leq \tilde{I} (x, a)$ and ${\tilde{C}}_{2} (a) \leq {\tilde{I}}^{c} (x, a)$ implies ${\tilde{B}}_{1} (a) \leq \tilde{I} (x, a)$ and ${\tilde{C}}_{1} (a) \leq {\tilde{I}}^{c} (x, a)$ for any a ∈ A, respectively. Thus, $({\tilde{B}}_{2}, {\tilde{C}}_{2})^{↓} \subseteq ({\tilde{B}}_{1}, {\tilde{C}}_{1})^{↓}$ .

For any x ∈ X, it follows from Eq.(1) that $X^{▽} (a) = ⋀_{x \in X} \tilde{I} (x, a) \leq \tilde{I} (x, a)$ and $X^{\bar{▽}} (a) = ⋀_{x \in X} {\tilde{I}}^{c} (x, a) \leq {\tilde{I}}^{c} (x, a)$ , respectively. By Eq.(2), we have $x \in (X^{▽}, X^{\bar{▽}})^{↓} = X^{↑ ↓}$ . That is X ⊆ X^↑↓. Similarly, we can prove that $(\tilde{B}, \tilde{C}) \subseteq (\tilde{B}, \tilde{C})^{↓ ↑}$ .

By items 1 and 2, we know X ⊆ X^↑↓ ⇒ X^↑↓↑ ⊆ X^↑ and X^↑ ⊆ X^↑↓↑. Thus, X^↑ = X^↑↓↑. Likewise, $(\tilde{B}, \tilde{C})^{↓} = (\tilde{B}, \tilde{C})^{↓ ↑ ↓}$ can be proved.

Based on item 1, $X \subseteq (\tilde{B}, \tilde{C})^{↓}$ yields $(\tilde{B}, \tilde{C})^{↓ ↑} \subseteq X^{↑}$ . It follows from item 2 that $(\tilde{B}, \tilde{C}) \subseteq (\tilde{B}, \tilde{C})^{↓ ↑}$ . Thus, $(\tilde{B}, \tilde{C}) \subseteq X^{↑}$ . Then, we conclude that $X \subseteq (\tilde{B}, \tilde{C})^{↓} \Rightarrow (\tilde{B}, \tilde{C}) \subseteq X^{↑}$ . Analogously, we can prove the inverse inclusion.

$(X_{1} \cup X_{2})^{↑} = (⋀_{x \in X_{1} \cup X_{2}} \tilde{I} (x, a), ⋀_{x \in X_{1} \cup X_{2}} {\tilde{I}}^{c} (x, a)) = (⋀_{i \in {1, 2}} (⋀_{x \in X_{i}} \tilde{I} (x, a)), ⋀_{i \in {1, 2}} (⋀_{x \in X_{i}} {\tilde{I}}^{c} (x, a))) = X_{1}^{↑} \cap X_{2}^{↑}$ . On the other hand, $\forall x \in (({\tilde{B}}_{1}, {\tilde{C}}_{1}) \cup ({\tilde{B}}_{2}, {\tilde{C}}_{2}))^{↓} \Leftrightarrow \forall x \in ({\tilde{B}}_{1} \cup {\tilde{B}}_{2}, {\tilde{C}}_{1} \cup {\tilde{C}}_{2})^{↓} \Leftrightarrow \forall a \in A, ({\tilde{B}}_{1} \cup {\tilde{B}}_{2}) (a) \leq \tilde{I} (x, a)$ , and $\forall a \in A, ({\tilde{C}}_{1} \cup {\tilde{C}}_{2}) (a) \leq {\tilde{I}}^{c} (x, a) \Leftrightarrow \forall i \in {1, 2}, {\tilde{B}}_{i} (a) \leq \tilde{I} (x, a)$ and ${\tilde{C}}_{i} (a) \leq {\tilde{I}}^{c} (x, a) \Leftrightarrow ({\tilde{B}}_{1}, {\tilde{C}}_{1})^{↓} \cap ({\tilde{B}}_{2}, {\tilde{C}}_{2})^{↓}$ .

By Definition 1, it follows immediately.□

Definition 6. For X ⊆ U, and

\tilde{B}, \tilde{C} \in F (A)

, a pair

(X, (\tilde{B}, \tilde{C}))

is called a crisp-fuzzy three-way concept (OFT-concept) if

X = (\tilde{B}, \tilde{C})^{↓}

and

(\tilde{B}, \tilde{C}) = X^{↑}

, where X and

(\tilde{B}, \tilde{C})

are named respectively the extension and the intension of

(X, (\tilde{B}, \tilde{C}))

For any OFT-concepts $(X_{1}, ({\tilde{B}}_{1}, {\tilde{C}}_{1}))$ and $(X_{2}, ({\tilde{B}}_{2}, {\tilde{C}}_{2}))$ , $(X_{1}, ({\tilde{B}}_{1}, {\tilde{C}}_{1})) \leq (X_{2}, ({\tilde{B}}_{2}, {\tilde{C}}_{2}))$ if and only if X₁ ⊆ X₂ (or equivalently, $({\tilde{B}}_{2}, {\tilde{C}}_{2}) \subseteq ({\tilde{B}}_{1}, {\tilde{C}}_{1})$ ). Correspondingly, the collection of all the OFT-concepts forms a complete lattice, which is the so-called crisp-fuzzy three-way concept lattice (OFT-concept lattice) by interpreting as $OFTL (U, \tilde{A}, \tilde{I})$ , in which the equipped supremum and infimum are represented as follows: $\begin{matrix} (X_{1}, ({\tilde{B}}_{1}, {\tilde{C}}_{1})) \lor (X_{2}, ({\tilde{B}}_{2}, {\tilde{C}}_{2})) = \\ ((X_{1} \cup X_{2})^{↑ ↓}, ({\tilde{B}}_{1}, {\tilde{C}}_{1}) \cap ({\tilde{B}}_{2}, {\tilde{C}}_{2})); \\ (X_{1}, ({\tilde{B}}_{1}, {\tilde{C}}_{1})) \land (X_{2}, ({\tilde{B}}_{2}, {\tilde{C}}_{2})) = \\ (X_{1} \cap X_{2}, (({\tilde{B}}_{1}, {\tilde{C}}_{1}) \cup ({\tilde{B}}_{2}, {\tilde{C}}_{2}))^{↓ ↑}) . \end{matrix}$

Example 3. Consider a formal fuzzy context in Table 1. Then the corresponding lattice $OFTL (U, \tilde{A}, \tilde{I})$ is listed in Fig. 3.

Fig. 3

$OFTL (U, \tilde{A}, \tilde{I})$ .

In Fig. 3, the intensions $O {\tilde{A}}_{i} (i \in Λ)$ of all OFT-concepts are depicted as follows:

$O {\tilde{A}}_{1} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.1}, e^{0.2}), (a^{0.1}, b^{0.2}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{2} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.1}, e^{0.2}), (a^{0.2}, b^{0.2}, c^{0.2}, d^{0.2}, e^{0.8})), O {\tilde{A}}_{3} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.1}, e^{0.2}), (a^{0.1}, b^{0.3}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{4} : ((a^{0.1}, b^{0.3}, c^{0.2}, d^{0.1}, e^{0.2}), (a^{0.1}, b^{0.2}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{5} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.3}, e^{0.2}), (a^{0.1}, b^{0.2}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{6} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.1}, e^{0.2}), (a^{0.9}, b^{0.7}, c^{0.2}, d^{0.2}, e^{0.8})),$

$O {\tilde{A}}_{7} : ((a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.2}), (a^{0.2}, b^{0.2}, c^{0.2}, d^{0.7}, e^{0.8})), O {\tilde{A}}_{8} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.3}, e^{0.2}), (a^{0.2}, b^{0.2}, c^{0.2}, d^{0.2}, e^{0.8})), O {\tilde{A}}_{9} : ((a^{0.1}, b^{0.3}, c^{0.2}, d^{0.1}, e^{0.2}), (a^{0.1}, b^{0.3}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{10} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.8}, e^{0.2}), (a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{11} : ((a^{0.8}, b^{0.7}, c^{0.2}, d^{0.3}, e^{0.2}), (a^{0.1}, b^{0.2}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{12} : ((a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.2}), (a^{0.9}, b^{0.7}, c^{0.2}, d^{0.9}, e^{0.8})), O {\tilde{A}}_{13} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.8}, e^{0.2}), (a^{0.9}, b^{0.8}, c^{0.9}, d^{0.2}, e^{0.8})), O {\tilde{A}}_{14} : ((a^{0.8}, b^{0.8}, c^{0.8}, d^{0.3}, e^{0.2}), (a^{0.2}, b^{0.2}, c^{0.2}, d^{0.7}, e^{0.8})), O {\tilde{A}}_{15} : ((a^{0.9}, b^{0.7}, c^{0.2}, d^{0.9}, e^{0.8}), (a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{16} : ((a^{1}, b^{1}, c^{1}, d^{1}, e^{1}), (a^{1}, b^{1}, c^{1}, d^{1}, e^{1})),$

3.2 Fuzzy-crisp three-way concept lattices

One can also explore the derivation operators by exchanging attributes and objects in the above formulated notion to search fuzzy-crisp three-way concepts.

Let $(U, A, \tilde{I})$ be a formal fuzzy context. For $\tilde{X} \in F (U)$ and B ⊆ A, two negative derivation operators $\bar{▽} : P (A) \to F (U)$ and $\bar{△} : F (U) \to P (A)$ are respectively defined as follows: $\begin{matrix} B^{\bar{▽}} (x) = ⋀_{a \in B} {\tilde{I}}^{c} (x, a), x \in U, \\ {\tilde{X}}^{\bar{△}} = {a \in A : \tilde{X} (x) \leq {\tilde{I}}^{c} (x, a), \forall x \in U} . \end{matrix}$

Property 4. For $\tilde{X}, {\tilde{X}}_{1}, {\tilde{X}}_{2} \in F (U)$ , and B, B₁, B₂ ⊆ A, then the following properties hold:

${\tilde{X}}_{1} \subseteq {\tilde{X}}_{2} \Rightarrow {\tilde{X}}_{2}^{\bar{△}} \subseteq {\tilde{X}}_{1}^{\bar{△}}$ , $B_{1} \subseteq B_{2} \Rightarrow B_{2}^{\bar{▽}} \subseteq B_{1}^{\bar{▽}}$ ;

$\tilde{X} \subseteq {\tilde{X}}^{\bar{△} \bar{▽}}$ , $B \subseteq B^{\bar{▽} \bar{△}}$ ;

${\tilde{X}}^{\bar{△}} = {\tilde{X}}^{\bar{△} \bar{▽} \bar{△}}$ , $B^{\bar{▽}} = B^{\bar{▽} \bar{△} \bar{▽}}$ ;

$({\tilde{X}}_{1} \cup {\tilde{X}}_{2})^{\bar{△}} = {\tilde{X}}_{1}^{\bar{△}} \cap {\tilde{X}}_{2}^{\bar{△}}$ , $(B_{1} \cup B_{2})^{\bar{▽}} = B_{1}^{\bar{▽}} \cap B_{2}^{\bar{▽}}$ .

Proof. It follows immediately from Property 2.□

Definition 7. For B ⊆ A, and $\tilde{X}, \tilde{Y} \in F (U)$ , two associated three-way operators are given by: $B^{↑} = (B^{▽} (x), B^{\bar{▽}} (x)) = (⋀_{a \in B} \tilde{I} (x, a), ⋀_{a \in B} {\tilde{I}}^{c} (x, a)), x \in U,$ (3) the fuzzy-crisp three-way operator and $(\tilde{X}, \tilde{Y})^{↓} = {\tilde{X}}^{△} \cap {\tilde{Y}}^{\bar{△}} = {a \in A : \forall x \in U, \tilde{X} (x) \leq \tilde{I} (x, a), and, \tilde{Y} (x) \leq {\tilde{I}}^{c} (x, a)} .$ (4) the fuzzy-crisp inverse operator.

Proposition 2. For B, B₁, B₂ ⊆ A, and $\tilde{X}, {\tilde{X}}_{1}, {\tilde{X}}_{2}, \tilde{Y}, {\tilde{Y}}_{1}, {\tilde{Y}}_{2} \in F (U)$ , the following properties hold:

$B_{1} \subseteq B_{2} \Rightarrow B_{2}^{↑} \subseteq B_{1}^{↑}$ , $({\tilde{X}}_{1}, {\tilde{Y}}_{1}) \subseteq ({\tilde{X}}_{2}, {\tilde{Y}}_{2}) \Rightarrow ({\tilde{X}}_{2}, {\tilde{Y}}_{2})^{↓} \subseteq ({\tilde{X}}_{1}, {\tilde{Y}}_{1})^{↓}$ ;

B ⊆ B^↑↓, $(\tilde{X}, \tilde{Y}) \subseteq (\tilde{X}, \tilde{Y})^{↓ ↑}$ ;

B^↑ = B^↑↓↑, $(\tilde{X}, \tilde{Y})^{↓} = (\tilde{X}, \tilde{Y})^{↓ ↑ ↓}$ ;

$B \subseteq (\tilde{X}, \tilde{Y})^{↓} \Leftrightarrow (\tilde{X}, \tilde{Y}) \subseteq B^{↑}$ ;

$(B_{1} \cup B_{2})^{↑} = B_{1}^{↑} \cap B_{2}^{↑}$ , $(({\tilde{X}}_{1}, {\tilde{Y}}_{1}) \cup ({\tilde{X}}_{2}, {\tilde{Y}}_{2}))^{↓} = ({\tilde{X}}_{1}, {\tilde{Y}}_{1})^{↓} \cap ({\tilde{X}}_{2}, {\tilde{Y}}_{2})^{↓}$ ;

The pair (↑ , ↓) forms a Galois connection between $(P (A), \subseteq)$ and $(F (U) \times F (U), \leq)$ .

Proof. We can proceed analogously to the proof from Proposition 1.□

Definition 8. For B ⊆ A, and $\tilde{X}, \tilde{Y} \in F (U)$ , if $B^{↑} = (\tilde{X}, \tilde{Y})$ and $(\tilde{X}, \tilde{Y})^{↓} = B$ , then $((\tilde{X}, \tilde{Y}), B)$ is called a fuzzy-crisp three-way concept (AFT-concept for short), where B and $(\tilde{X}, \tilde{Y})$ are respectively called the intension and the extension of $((\tilde{X}, \tilde{Y}), B)$ .

Meanwhile, the set of all AFT-concepts is naturally equipped with a partial order ≤ denoted as follows: $\begin{matrix} (({\tilde{X}}_{1}, {\tilde{Y}}_{1}), B_{1}) \leq (({\tilde{X}}_{2}, {\tilde{Y}}_{2}), B_{2}) \Leftrightarrow \\ ({\tilde{X}}_{1}, {\tilde{Y}}_{1}) \subseteq ({\tilde{X}}_{2}, {\tilde{Y}}_{2}) \Leftrightarrow B_{2} \subseteq B_{1} . \end{matrix}$

It is straightforward that the collection of all AFT-concepts is exactly a complete lattice, which is named as fuzzy-crisp three-way concept lattice (AFT-concept lattice), and is denoted by $AFTL (\tilde{U}, A, \tilde{I})$ , where the supremum and infimum are defined as follows: $\begin{matrix} (({\tilde{X}}_{1}, {\tilde{Y}}_{1}), B_{1}) \lor (({\tilde{X}}_{2}, {\tilde{Y}}_{2}), B_{2}) = \\ ((({\tilde{X}}_{1}, {\tilde{Y}}_{1}) \cup ({\tilde{X}}_{2}, {\tilde{Y}}_{2}))^{↓ ↑}, B_{1} \cap B_{2}); \\ (({\tilde{X}}_{1}, {\tilde{Y}}_{1}), B_{1}) \land (({\tilde{X}}_{2}, {\tilde{Y}}_{2}), B_{2}) = \\ ((({\tilde{X}}_{1}, {\tilde{Y}}_{1}) \cap ({\tilde{X}}_{2}, {\tilde{Y}}_{2})), (B_{1} \cup B_{2})^{↑ ↓}) . \end{matrix}$

Example 4. Continuation from Example 2. The corresponding $AFTL (\tilde{U}, A, \tilde{I})$ is shown in Fig. 4.

Fig. 4

$AFTL (\tilde{U}, A, \tilde{I})$ .

In Fig. 4, the extensions $A {\tilde{O}}_{i} (i \in Λ)$ of $AFTL (\tilde{U}, A, \tilde{I})$ are computed as follows:

$A {\tilde{O}}_{1} : ((1^{1}, 2^{1}, 3^{1}, 4^{1}, 5^{1}), (1^{1}, 2^{1}, 3^{1}, 4^{1}, 5^{1})), A {\tilde{O}}_{2} : ((1^{0.9}, 2^{0.7}, 3^{0.2}, 4^{0.9}, 5^{0.8}), (1^{0.1}, 2^{0.3}, 3^{0.8}, 4^{0.1}, 5^{0.2})), A {\tilde{O}}_{3} : ((1^{0.8}, 2^{0.2}, 3^{0.8}, 4^{0.3}, 5^{0.2}), (1^{0.2}, 2^{0.8}, 3^{0.2}, 4^{0.7}, 5^{0.8})), A {\tilde{O}}_{4} : ((1^{0.1}, 2^{0.3}, 3^{0.1}, 4^{0.7}, 5^{0.9}), (1^{0.9}, 2^{0.7}, 3^{0.9}, 4^{0.3}, 5^{0.1})), A {\tilde{O}}_{5} : ((1^{0.8}, 2^{0.2}, 3^{0.2}, 4^{0.3}, 5^{0.2}), (1^{0.1}, 2^{0.3}, 3^{0.2}, 4^{0.1}, 5^{0.2})), A {\tilde{O}}_{6} : ((1^{0.1}, 2^{0.3}, 3^{0.1}, 4^{0.7}, 5^{0.8}), (1^{0.1}, 2^{0.3}, 3^{0.8}, 4^{0.1}, 5^{0.1})), A {\tilde{O}}_{7} : ((1^{0.1}, 2^{0.2}, 3^{0.1}, 4^{0.3}, 5^{0.2}), (1^{0.2}, 2^{0.7}, 3^{0.2}, 4^{0.3}, 5^{0.1})), A {\tilde{O}}_{8} : ((1^{0.1}, 2^{0.2}, 3^{0.1}, 4^{0.3}, 5^{0.2}), (1^{0.1}, 2^{0.3}, 3^{0.2}, 4^{0.1}, 5^{0.1})) .$

4 The connections between the one-sided and one-sided fuzzy three-way concept lattices

In this section, we will investigate the relationships between one-sided fuzzy three-way concept lattices and classical fuzzy concept lattices.

4.1 The relationships between OFT-concept lattice and fuzzy concept lattice

Theorem 1. Let $(U, A, \tilde{I})$ be a formal fuzzy context. If $(X, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})$ and $(Y, \tilde{C}) \in OB (U, \tilde{A}, {\tilde{I}}^{c})$ , then $(X, (\tilde{B}, X^{\bar{▽}}))$ and $(Y, (Y^{▽}, \tilde{C}))$ are respectively OFT-concepts.

Proof. Since $(X, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})$ , we have $X^{▽} = \tilde{B}$ and ${\tilde{B}}^{△} = X$ . Applying Property 3, we conclude from $X \subseteq X^{\bar{▽} \bar{△}}$ that $X \cap X^{\bar{▽} \bar{△}} = X$ . In fact, $(\tilde{B}, X^{\bar{▽}})^{↓} = {\tilde{B}}^{△} \cap X^{\bar{▽} \bar{△}} = X$ . Thus, $(X, (\tilde{B}, X^{\bar{▽}})) \in OFTL (U, \tilde{A}, \tilde{I})$ . Analogously, it is obvious that $(Y, (Y^{▽}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})$ since $(Y, \tilde{C}) \in OB (U, \tilde{A}, {\tilde{I}}^{c})$ .□

Theorem 2. Let $(U, A, \tilde{I})$ be a formal fuzzy context. If $(X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})$ , then $({\tilde{B}}^{△}, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})$ and $({\tilde{C}}^{\bar{△}}, \tilde{C}) \in OB (U, \tilde{A}, {\tilde{I}}^{c})$ .

Proof. Since $(X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})$ , we obtain $X^{↑} = (X^{▽}, X^{\bar{▽}})$ and $(\tilde{B}, \tilde{C})^{↓} = {\tilde{B}}^{△} \cap {\tilde{C}}^{\bar{△}} = X$ . Then $X^{▽} = \tilde{B}$ and $X^{\bar{▽}} = \tilde{C}$ . Henceforth, $({\tilde{B}}^{△}, \tilde{B}) = (X^{▽ △}, X^{▽}) \in OB (U, \tilde{A}, \tilde{I})$ and $({\tilde{C}}^{\bar{△}}, \tilde{C}) = (X^{\bar{▽} \bar{△}}, X^{\bar{▽}}) \in OB (U, \tilde{A}, {\tilde{I}}^{c})$ , respectively.□

Hereafter, we shall analyze the connections among $OB (U, \tilde{A}, \tilde{I})$ , $OB (U, \tilde{A}, {\tilde{I}}^{c})$ and $OFTL (U, \tilde{A}, \tilde{I})$ . Succinctly, the following expressions are defined as: ${OB}_{E} (U, \tilde{A}, \tilde{I}) = {X : (X, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})}, {OB}_{I} (U, \tilde{A}, \tilde{I}) = {\tilde{B} : (X, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})}, {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I}) = {\tilde{B} : (X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})}, {OFTL}_{I}^{-} (U, \tilde{A}, \tilde{I}) = {\tilde{C} : (X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})} .$ Proposition 3. For a given formal fuzzy context $(U, A, \tilde{I})$ , and its complementary context $(U, A, {\tilde{I}}^{c})$ , then

${OB}_{E} (U, \tilde{A}, \tilde{I}) \subseteq {OFTL}_{E} (U, \tilde{A}, \tilde{I})$ ,

${OB}_{E} (U, \tilde{A}, {\tilde{I}}^{c}) \subseteq {OFTL}_{E} (U, \tilde{A}, \tilde{I})$ ,

${OB}_{I} (U, \tilde{A}, \tilde{I}) = {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I})$ ,

${OB}_{I} (U, \tilde{A}, {\tilde{I}}^{c}) = {OFTL}_{I}^{-} (U, \tilde{A}, \tilde{I})$ .

Proof.

For any $X \in {OB}_{E} (U, \tilde{A}, \tilde{I})$ , we know $(X, X^{▽}) \in OB (U, \tilde{A}, \tilde{I})$ . It follows from Theorem 1 that $(X, (X^{▽}, X^{\bar{▽}})) \in OFTL (U, \tilde{A}, \tilde{I})$ . Thus, $X \in {OFTL}_{E} (U, \tilde{A}, \tilde{I})$ .

The proof is straightforward from Theorem 1.

For any $\tilde{B} \in {OB}_{I} (U, \tilde{A}, \tilde{I})$ , we obtain $({\tilde{B}}^{△}, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})$ . By Theorem 1, we have $({\tilde{B}}^{△}, (\tilde{B}, {\tilde{B}}^{△ \bar{▽}})) \in OFTL (U, \tilde{A}, \tilde{I})$ , which implies that $\tilde{B} \in {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I})$ . Therefore, ${OB}_{I} (U, \tilde{A}, \tilde{I}) \subseteq {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I})$ .

Next we prove the inverse inclusion. For any $\tilde{B} \in {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I})$ , there exists X ⊆ U and $\tilde{C} \in F (A)$ such that $(X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})$ . From Theorem 2, it suffices to show that $({\tilde{B}}^{△}, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})$ . Thus, $\tilde{B} \in {OB}_{I} (U, \tilde{A}, \tilde{I})$ and ${OB}_{I} (U, \tilde{A}, \tilde{I}) \supseteq {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I})$ . Accordingly, ${OB}_{I} (U, \tilde{A}, \tilde{I}) = {OFTL}_{I}^{+} (U, \tilde{A}, \tilde{I})$ .

Analogously to the proof of item 3, and it follows from Theorem 1 and 2 that ${OB}_{I} (U, \tilde{A}, {\tilde{I}}^{c}) = {OFTL}_{I}^{-} (U, \tilde{A}, \tilde{I})$ .□

Hence, it is apparent that ${OB}_{E} (U, \tilde{A}, \tilde{I}) \cup {OB}_{E} (U, \tilde{A}, {\tilde{I}}^{c}) \subseteq {OFTL}_{E} (U, \tilde{A}, \tilde{I})$ . However, ${OFTL}_{E} (U, \tilde{A}, \tilde{I})$ fails to ${OB}_{E} (U, \tilde{A}, \tilde{I}) \cup {OB}_{E} (U, \tilde{A}, {\tilde{I}}^{c})$ . Such result is illustrated by the following counterexample.

Example 5. Continuation from Example 1. The corresponding formal concept lattice $OB (U, \tilde{A}, {\tilde{I}}^{c})$ is depicted in Fig. 5. We see from Fig. 3 that ${2, 3} \in {OFTL}_{E} (U, \tilde{A}, \tilde{I})$ , while ${2, 3} \notin {OB}_{E} (U, \tilde{A}, \tilde{I})$ and ${2, 3} \notin {OB}_{E} (U, \tilde{A}, {\tilde{I}}^{c})$ from Figs. 1 and 5, respectively. Thus, ${2, 3} \notin {OB}_{E} (U, \tilde{A}, \tilde{I}) \cup {OB}_{E} (U, \tilde{A}, {\tilde{I}}^{c})$ .

Fig. 5

$OB (U, \tilde{A}, {\tilde{I}}^{c})$ .

Theorem 3. Let $(U, A, \tilde{I})$ be a formal fuzzy context, and ${\tilde{I}}^{c} \in F (U \times A) ∖ {\tilde{I}}$ . Then we have

There exists a ∧-preserving order embedding of $OB (U, \tilde{A}, \tilde{I})$ in $OFTL (U, \tilde{A}, \tilde{I})$ ,

There exists a ∧-preserving order embedding of $OB (U, \tilde{A}, {\tilde{I}}^{c})$ in $OFTL (U, \tilde{A}, \tilde{I})$ .

Proof.

For $(X, \tilde{B}) \in OB (U, \tilde{A}, \tilde{I})$ , we define a subjective mapping $φ : OB (U, \tilde{A}, \tilde{I}) \to OFTL (U, \tilde{A}, \tilde{I})$ such that $φ ((X, \tilde{B})) = (X, (\tilde{B}, X^{\bar{▽}}))$ . For any $(X, \tilde{B}), (Y, \tilde{C}) \in OB (U, \tilde{A}, \tilde{I})$ , we have $φ ((X, \tilde{B})) \leq φ ((Y, \tilde{C}))$ if $(X, \tilde{B}) \leq (Y, \tilde{C})$ . Furthermore, it follows that $φ ((X, \tilde{B}) \land (Y, \tilde{C})) = φ ((X \cap Y, (\tilde{B} \cup \tilde{C})^{△ ▽}))$ , and $φ ((X, \tilde{B})) \land φ ((Y, \tilde{C})) = (X, (\tilde{B}, X^{\bar{▽}})) \land (Y, (\tilde{C}, Y^{\bar{▽}})) = (X \cap Y, ((\tilde{B}, X^{\bar{▽}}) \cup (\tilde{C}, Y^{\bar{▽}}))^{↓ ↑})$ . Thus, $φ ((X, \tilde{B}) \land (Y, \tilde{C})) = φ ((X, \tilde{B})) \land φ ((Y, \tilde{C}))$ . This implies that φ is a ∧-preserving order embedding of $OB (U, \tilde{A}, \tilde{I})$ in $OFTL (U, \tilde{A}, \tilde{I})$ .

Analogously to the proof of item 1, we define $ψ ((Y, \tilde{C})) = (Y, (Y^{▽}, \tilde{C}))$ for any $(Y, \tilde{C}) \in OB (U, \tilde{A}, {\tilde{I}}^{c})$ . It is easily to see that $ψ ((X, \tilde{B}) \land (Y, \tilde{C})) = ψ ((X, \tilde{B})) \land ψ ((Y, \tilde{C}))$ .□

As shown blew, the mapping φ is not a ∨-preserving order embedding of $OB (U, \tilde{A}, \tilde{I})$ in $OFTL (U, \tilde{A}, \tilde{I})$ . An illustrative example is used to show this idea.

Example 6. Table 3 depicts a formal fuzzy context $(U, A, \tilde{I})$ , in which U = {1, 2, 3} and A = {a, b, c, d, e}. The related concept lattices are shown in Figs. 6 and 7. We know that $(2, (a^{0.8}, b^{0.2}, c^{0.8}, d^{0.3}, e^{0.2})), (3, (a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.9})) \in OB (U, \tilde{A}, \tilde{I})$ from Fig. 6, and $φ ((2, (a^{0.8}, b^{0.2}, c^{0.8}, d^{0.3}, e^{0.2})) \lor (3, (a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.9}))) = φ ((U, (a^{0.1}, b^{0.2}, c^{0.1}, d^{0.3}, e^{0.2}))) = (U, O {\tilde{A}}_{1})$ . However, $φ ((2, (a^{0.8}, b^{0.2}, c^{0.8}, d^{0.3}, e^{0.2}))) \lor φ ((3, (a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.9}))) = (2, O {\tilde{A}}_{6}) \lor (3, O {\tilde{A}}_{5}) = (23, O {\tilde{A}}_{2})$ . To sum up, φ is not necessarily ∨-preserving mapping.

Table 3

Formal fuzzy context $(U, A, \tilde{I})$

$\tilde{I}$	a	b	c	d	e
1	0.9	0.7	0.2	0.9	0.8
2	0.8	0.2	0.8	0.3	0.2
3	0.1	0.3	0.1	0.7	0.9

Fig. 6

$OB (U, \tilde{A}, \tilde{I})$ .

In Fig. 7, the intensions $O {\tilde{A}}_{i} (i \in Λ)$ of $OFTL (U, \tilde{A}, \tilde{I})$ are computed as follows:

$O {\tilde{A}}_{1} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.3}, e^{0.2}), (a^{0.1}, b^{0.3}, c^{0.2}, d^{0.1}, e^{0.1})), O {\tilde{A}}_{2} : ((a^{0.1}, b^{0.2}, c^{0.1}, d^{0.3}, e^{0.2}), (a^{0.2}, b^{0.7}, c^{0.2}, d^{0.3}, e^{0.1})), O {\tilde{A}}_{3} : ((a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.8}), (a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.1})), O {\tilde{A}}_{4} : ((a^{0.8}, b^{0.2}, c^{0.2}, d^{0.3}, e^{0.2}), (a^{0.1}, b^{0.3}, c^{0.2}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{5} : ((a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.9}), (a^{0.9}, b^{0.7}, c^{0.9}, d^{0.3}, e^{0.1})), O {\tilde{A}}_{6} : ((a^{0.8}, b^{0.2}, c^{0.8}, d^{0.3}, e^{0.2}), (a^{0.2}, b^{0.8}, c^{0.2}, d^{0.7}, e^{0.8})), O {\tilde{A}}_{7} : ((a^{0.9}, b^{0.7}, c^{0.2}, d^{0.9}, e^{0.8}), (a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.2})), O {\tilde{A}}_{8} : ((a^{1}, b^{1}, c^{1}, d^{1}, e^{1}), (a^{1}, b^{1}, c^{1}, d^{1}, e^{1})) .$

Fig. 7

$OFTL (U, \tilde{A}, \tilde{I})$ .

Theorem 4. There exists a ∨-preserving order embedding of $OFTL (U, \tilde{A}, \tilde{I})$ in $K = OB (U, \tilde{A}, \tilde{I}) \times OB (U, \tilde{A}, {\tilde{I}}^{c})$ .

Proof. Suppose that $(X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})$ , we denote a subjective mapping $ω : OFTL (U, \tilde{A}, \tilde{I}) \to K$ such that $ω ((X, (\tilde{B}, \tilde{C}))) = (({\tilde{B}}^{△}, \tilde{B}), ({\tilde{C}}^{\bar{△}}, \tilde{C}))$ . Moreover, $ω ((X, (\tilde{B}, \tilde{C})) \lor (Y, (\tilde{D}, \tilde{E}))) = ω (((X \cup Y)^{↑ ↓}, (\tilde{B}, \tilde{C}) \cap (\tilde{D}, \tilde{E}))) = ω (((X \cup Y)^{↑ ↓}, (\tilde{B} \cap \tilde{D}, \tilde{C} \cap \tilde{E}))) = (((\tilde{B} \cap \tilde{D})^{△}, \tilde{B} \cap \tilde{D}), ((\tilde{C} \cap \tilde{E})^{\bar{△}}, \tilde{C} \cap \tilde{E})) = ((({\tilde{B}}^{△} \cup {\tilde{D}}^{△})^{▽ △}, \tilde{B} \cap \tilde{D}), (({\tilde{C}}^{\bar{△}} \cup {\tilde{E}}^{\bar{△}})^{\bar{▽} \bar{△}}, \tilde{C} \cap \tilde{E})) = (({\tilde{B}}^{△}, \tilde{B}) \lor ({\tilde{D}}^{△}, \tilde{D}), ({\tilde{C}}^{\bar{△}}, \tilde{C}) \lor ({\tilde{E}}^{\bar{△}}, \tilde{E})) = (({\tilde{B}}^{△}, \tilde{B}), ({\tilde{C}}^{\bar{△}}, \tilde{C})) \lor (({\tilde{D}}^{△}, \tilde{D}), ({\tilde{E}}^{\bar{△}}, \tilde{E})) = ω ((X, (\tilde{B}, \tilde{C}))) \lor ω ((Y, (\tilde{D}, \tilde{E})))$ . Hence, ω is a ∨-preserving mapping of $OFTL (U, \tilde{A}, \tilde{I})$ in $K$ . Furthermore, if $(X, (\tilde{B}, \tilde{C})) \leq (Y, (\tilde{D}, \tilde{E})) \Leftrightarrow (\tilde{D}, \tilde{E}) \subseteq (\tilde{B}, \tilde{C}) \Leftrightarrow \tilde{D} \subseteq \tilde{B}$ and $\tilde{E} \subseteq \tilde{C} \Leftrightarrow ({\tilde{B}}^{△}, \tilde{B}) \leq ({\tilde{D}}^{△}, \tilde{D})$ and $({\tilde{C}}^{\bar{△}}, \tilde{C}) \leq ({\tilde{E}}^{\bar{△}}, \tilde{E}) \Leftrightarrow (({\tilde{B}}^{△}, \tilde{B}), ({\tilde{C}}^{\bar{△}}, \tilde{C})) \leq (({\tilde{D}}^{△}, \tilde{D}), ({\tilde{E}}^{\bar{△}}, \tilde{E})) \Leftrightarrow ω ((X, (\tilde{B}, \tilde{C}))) \leq ω ((Y, (\tilde{D}, \tilde{E})))$ . Thus, ω is an order embedding. It conclude that ω is a ∨-preserving order embedding of $OFTL (U, \tilde{A}, \tilde{I})$ in $K$ .□

Nevertheless, the mapping ω defined in Theorem 4 is not necessarily ∧-preserving.

Example 7. Continuation from Example 6. From Figs. 6–8, it follows that $(23, O {\tilde{A}}_{2}), (13, O {\tilde{A}}_{3}) \in OFTL (U, \tilde{A}, \tilde{I})$ . We can obtain $ω ((23, O {\tilde{A}}_{2})) = ((U, (a^{0.1}, b^{0.2}, c^{0.1}, d^{0.3}, e^{0.2}))$ , (23, (a^0.2, b^0.7, c^0.2, d^0.3, e^0.1))), and $ω ((13, O {\tilde{A}}_{3})) = ((13, (a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.8})), (13, (a^{0.1}, b^{0.3}, c^{0.8}, d^{0.1}, e^{0.1})))$ . Then $ω ((23, O {\tilde{A}}_{2})) \land ω ((13, O {\tilde{A}}_{3})) = ((13, (a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.8}))$ , (3, (a^0.9, b^0.7, c^0.9, d^0.3, e^0.1))). However, $ω ((23, O {\tilde{A}}_{2}) \land (13, O {\tilde{A}}_{3})) = ω ((3, O {\tilde{A}}_{5})) = ((3, (a^{0.1}, b^{0.3}, c^{0.1}, d^{0.7}, e^{0.9})), (3, (a^{0.9}, b^{0.7}, c^{0.9}, d^{0.3}, e^{0.1})))$ . Therefore, $ω ((23, O {\tilde{A}}_{2}) \land (13, O {\tilde{A}}_{3})) \neq ω ((23, O {\tilde{A}}_{2})) \land ω ((13, O {\tilde{A}}_{3}))$ .

Fig. 8

$OB (U, \tilde{A}, {\tilde{I}}^{c})$ .

4.2 The relationships between AFT-concept lattice and fuzzy concept lattice

In this subsection, the proofs of process will be omitted because most of the proofs are analogous to that in subsection 4.1.

Theorem 5. Let $(U, A, \tilde{I})$ be a formal fuzzy context. If $(\tilde{X}, B) \in AB (\tilde{U}, A, \tilde{I})$ and $(\tilde{Y}, C) \in AB (\tilde{U}, A, {\tilde{I}}^{c})$ , then $((\tilde{X}, B^{\bar{▽}}), B)$ and $((C^{▽}, \tilde{Y}), C)$ are respectively AFT-concepts.

Theorem 6. Let $(U, A, \tilde{I})$ be a formal fuzzy context. If $((\tilde{X}, \tilde{Y}), B) \in AFTL (\tilde{U}, A, \tilde{I})$ , then $(\tilde{X}, {\tilde{X}}^{△}) \in AB (\tilde{U}, A, \tilde{I})$ and $(\tilde{Y}, {\tilde{Y}}^{\bar{△}}) \in AB (\tilde{U}, A, {\tilde{I}}^{c})$ , respectively.

For simplicity, the symbols are defined as follows: ${AB}_{E} (\tilde{U}, A, \tilde{I}) = {\tilde{X} : (\tilde{X}, B) \in AB (\tilde{U}, A, \tilde{I})}, {AB}_{I} (\tilde{U}, A, \tilde{I}) = {B : (\tilde{X}, B) \in AB (\tilde{U}, A, \tilde{I})}, {AFTL}_{E}^{+} (\tilde{U}, A, \tilde{I}) = {\tilde{X} : ((\tilde{X}, \tilde{Y}), B) \in AFTL (\tilde{U}, A, \tilde{I})}, {AFTL}_{E}^{-} (\tilde{U}, A, \tilde{I}) = {\tilde{Y} : ((\tilde{X}, \tilde{Y}), B) \in AFTL (\tilde{U}, A, \tilde{I})} .$ Proposition 4. For a given formal fuzzy context $(U, A, \tilde{I})$ , and its complementary context $(U, A, {\tilde{I}}^{c})$ , then

${AB}_{I} (\tilde{U}, A, \tilde{I}) \subseteq {AFTL}_{I} (\tilde{U}, A, \tilde{I})$ ,

${AB}_{I} (\tilde{U}, A, {\tilde{I}}^{c}) \subseteq {AFTL}_{I} (\tilde{U}, A, \tilde{I})$ ,

${AB}_{E} (\tilde{U}, A, \tilde{I}) = {AFTL}_{E}^{+} (\tilde{U}, A, \tilde{I})$ ,

${AB}_{E} (\tilde{U}, A, {\tilde{I}}^{c}) = {AFTL}_{E}^{-} (\tilde{U}, A, \tilde{I})$ .

Hereinafter, the relationships between $AB (\tilde{U}, A, \tilde{I})$ , $AB (\tilde{U}, A, {\tilde{I}}^{c})$ , and $AFTL (\tilde{U}, A, \tilde{I})$ are investigated from the communities of their algebraic structures.

Theorem 7. Let $(U, A, \tilde{I})$ be a formal fuzzy context, and ${\tilde{I}}^{c} \in F (U \times A) \ {\tilde{I}}$ . Then we have

There exists a ∨-preserving order embedding of $AB (\tilde{U}, A, \tilde{I})$ in $AFTL (\tilde{U}, A, \tilde{I})$ ,

There exists a ∨-preserving order embedding of $AB (\tilde{U}, A, {\tilde{I}}^{c})$ in $AFTL (\tilde{U}, A, \tilde{I})$ .

In Theorem 7, we denote $φ : AB (\tilde{U}, A, \tilde{I}) \to AFTL (\tilde{U}, A, \tilde{I})$ such that $φ ((\tilde{X}, B)) = ((\tilde{X}, B^{\bar{▽}}), B)$ . Note that φ is a ∨-preserving order embedding. Homoplastically, $ψ ((\tilde{Y}, C)) = ((C^{▽}, \tilde{Y}), C)$ is also a ∨-preserving order embedding.

Theorem 8. There exists a ∧-preserving order embedding of $AFTL (\tilde{U}, A, \tilde{I})$ in $K = AB (\tilde{U}, A, \tilde{I}) \times AB (\tilde{U}, A, {\tilde{I}}^{c})$ .

In the above Theorem, we define $ω : AFTL (\tilde{U}, A, \tilde{I}) \to K$ such that $ω (((\tilde{X}, \tilde{Y}), B)) = ((\tilde{X}, {\tilde{X}}^{△}), (\tilde{Y}, {\tilde{Y}}^{\bar{△}}))$ . It is easily verified that ω is a ∧-preserving order embedding of $AFTL (\tilde{U}, A, \tilde{I})$ in $K$ .

In fact, the mapping φ and ψ proposed in Theorem 7 are not necessarily ∧-preservings. Likewise, the mapping ω presented by Theorem 8 is not ∨-preserving.

Example 8. Continuation from Example 2. Fig. 9 shows its complementary concept lattice $AB (\tilde{U}, A, {\tilde{I}}^{c})$ .

Fig. 9

$AB (\tilde{U}, A, {\tilde{I}}^{c})$ .

Notice that $((1^{0.8}, 2^{0.2}, 3^{0.8}, 4^{0.3}, 5^{0.2}), b) \in AB (\tilde{U}, A, \tilde{I})$ and $((1^{0.1}, 2^{0.3}, 3^{0.1}, 4^{0.7}, 5^{0.9}), c) \in AB (\tilde{U}, A, \tilde{I})$ according to Fig. 2. Moreover, from Fig. 4, it follows that φ (((1^0.8, 2^0.2, 3^0.8, 4^0.3, 5^0.2) , b) ∧ ((1^0.1, 2^0.3, 3^0.1, 4^0.7, 5^0.9) , c)) = $φ (((1^{0.1}, 2^{0.2}, 3^{0.1}, 4^{0.3}, 5^{0.2}), A)) = (A {\tilde{O}}_{8}, A)$ . However, $φ (((1^{0.8}, 2^{0.2}, 3^{0.8}, 4^{0.3}, 5^{0.2}), b)) \land φ (((1^{0.1}, 2^{0.3}, 3^{0.1}, 4^{0.7}, 5^{0.9}), c)) = (A {\tilde{O}}_{3}, b) \land (A {\tilde{O}}_{4}, c) = (A {\tilde{O}}_{7}, bc)$ . Thus, φ (((1^0.8, 2^0.2, 3^0.8, 4^0.3, 5^0.2) , b) ∧ ((1^0.1, 2^0.3, 3^0.1, 4^0.7, 5^0.9) , c)) ≠ φ (((1^0.8, 2^0.2, 3^0.8, 4^0.3, 5^0.2) , b)) ∧ φ (((1^0.1, 2^0.3, 3^0.1, 4^0.7, 5^0.9) , c)). We conclude that φ is not necessarily ∧-preserving.

Furthermore, it shows $ω ((A {\tilde{O}}_{6}, ac)) \lor ω ((A {\tilde{O}}_{7}, bc)) = (((1^{0.1}, 2^{0.3}, 3^{0.1}, 4^{0.7}, 5^{0.8}), ac), ((1^{0.9}, 2^{0.7}, 3^{0.9}, 4^{0.3}, 5^{0.1}), c))$ . Nevertheless, $ω ((A {\tilde{O}}_{6}, ac) \lor (A {\tilde{O}}_{7}, bc)) = ω ((A {\tilde{O}}_{4}, c)) = ((1^{0.1}, 2^{0.3}, 3^{0.1}, 4^{0.7}, 5^{0.9}), c), (1^{0.9}, 2^{0.7}, 3^{0.9}, 4^{0.3}, 5^{0.1}), c)$ . Obviously, $ω ((A {\tilde{O}}_{6}, ac)) \lor ω ((A {\tilde{O}}_{7}, bc)) \neq ω ((A {\tilde{O}}_{6}, ac) \lor (A {\tilde{O}}_{7}, bc))$ . Thereby, the mapping ω is not ∨-preserving.

5 Granular matrix-based reduction in crisp-fuzzy three-way concept lattices

In this section, a formal fuzzy context $(U, A, \tilde{I})$ is seen as a dimensional numerical table, in which the row and column correspond to object and attribute, respectively. Therefore, a formal fuzzy context is called a fuzzy relation matrix $M_{\tilde{I}} = (r_{ij})_{n \times m}$ , where |U| = n and |A| = m, respectively. Furthermore, we shall propose an explicitly systematic approach to characterize the granular matrix-based reduction of formal fuzzy context based on 3WD.

5.1 Matrix presentations of OFT-concept lattices

Given a formal fuzzy context, each row of numerical table is actually $x^{▽} (a) = \tilde{I} (x, a)$ for x ∈ U. Similarly, in its complementary one, each row of numerical table is clearly $x^{\bar{▽}} (a) = {\tilde{I}}^{c} (x, a)$ for any x ∈ U. For a matrix M, we denote the i-th row as M (i, :), the j-th column as M (: , j), and their cross as M (i, j).

Proposition 5. Given a formal fuzzy context $(U, A, \tilde{I})$ with the fuzzy relation matrix $M_{\tilde{I}}$ , then $OM = (M_{\tilde{I}} \otimes M_{\tilde{I}}^{T}) \land (\sim M_{\tilde{I}} \otimes (\sim M_{\tilde{I}}^{T}))$ (5) where i-th row of OM is $λ_{x_{i}^{↑ ↓}}$ .

Proof. $\begin{matrix} x_{i}^{↑ ↓} = {x_{j} \in U : \forall a \in A, x_{i}^{▽} (a) \leq \tilde{I} (x_{j}, a), \\ and, \forall a \in A, x_{i}^{\bar{▽}} (a) \leq {\tilde{I}}^{c} (x_{j}, a)} \\ = {x_{j} \in U : \forall a \in A, x_{i}^{▽} (a) \otimes \tilde{I} (x_{j}, a) = 1, \\ and, \forall a \in A, x_{i}^{\bar{▽}} (a) \otimes {\tilde{I}}^{c} (x_{j}, a) = 1} \\ = {x_{j} \in U : ⋀_{a \in A} x_{i}^{▽} (a) \otimes \tilde{I} (x_{j}, a) = 1, \\ and, ⋀_{a \in A} x_{i}^{\bar{▽}} (a) \otimes {\tilde{I}}^{c} (x_{j}, a) = 1} \\ = {x_{j} \in U : (M_{\tilde{I}} (i, :) \otimes M_{\tilde{I}}^{T} (j, :)) \land (\sim M_{\tilde{I}} (i, :) \\ \otimes (\sim M_{\tilde{I}}^{T} (j, :))) = 1} . \end{matrix}$ Therefore, $λ_{x_{i}^{↑ ↓}} = (M_{\tilde{I}} (i, :) \otimes M_{\tilde{I}}^{T}) \land (\sim M_{\tilde{I}}^{T} (i, :) \otimes (\sim M_{\tilde{I}}^{T}))$ .□

Definition 9. Let $(U, A, \tilde{I})$ be a formal fuzzy context with a Boolean matrix OM. Denote $x_{i}^{↑ ↓}$ for each x_i ∈ U as $λ_{x_{i}^{↑ ↓}}$ . Namely, $λ_{x_{i}^{↑ ↓}} = OM (i, :)$ . Then OM is called an object granular matrix.

The procedure for obtaining object granular matrix OM is exactly given in Algorithm 1. With respect to matrices M₁ and M₂, Steps 2-7 actually characterize a strategy to calculate OM, which can be done in O (|U|²|A|). Henceforth, its time complexity is O (|U|²|A|).

Algorithm 1 Computing object granular matrix based on fuzzy relation matrix (COGM)
Input: A formal fuzzy context $(U, A, \tilde{I})$ with fuzzy relation matrix $M_{\tilde{I}} = (r_{ij})_{n \times m}$ .
Output: Object granular matrix OM.
1: Let OM = φ;
2: Fori = 1 to ndo
3: Create matrices B and D consisting of a n-by-1 tilling of copies of $M_{\tilde{I}} (i, :)$ and $M_{\tilde{I}}^{c} (i, :)$ ;
4: Let $C = B - M_{\tilde{I}}$ and $E = D - M_{\tilde{I}}^{c}$ and replace the location greater than 0 in matrix C and E by 1, and otherwise by 0, respectively;
5: Calculate row vectors H and J such that element in them are the minimum of related row in 1 - C and 1 - E, respectively;
6: M₁ ← [M₁ ; H] and M₂ ← [M₂ ; J];
7: end For
8: OM ← [OM ; min (M₁, M₂)];
9: Return OM.

Example 9. Table 4 exhibits a formal fuzzy context $(U, A, \tilde{I})$ , where U = {x₁, x₂, x₃, x₄, x₅} and A = {a₁, a₂, a₃, a₄, a₅}. By computation, we have $OM = (M_{\tilde{I}} \otimes M_{\tilde{I}}^{T}) \land (\sim M_{\tilde{I}} \otimes (\sim M_{\tilde{I}}^{T})) =$ $[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] .$

Table 4

Formal fuzzy context $(U, A, \tilde{I})$

$\tilde{I}$	a ₁	a ₂	a ₃	a ₄	a ₅
x ₁	0.9	0.7	0.2	0.9	0.8
x ₂	0.8	0.8	0.8	0.3	0.2
x ₃	0.1	0.2	0.1	0.8	0.2
x ₄	0.1	0.3	0.8	0.1	0.2
x ₅	0.8	0.8	0.8	0.1	0.2

5.2 Attribute matrix-based reduction in OFT-concept lattices

Note that $(X, (\tilde{B}, \tilde{C})) = ⋁_{x \in X} (x^{↑ ↓}, x^{↑})$ for any $(X, (\tilde{B}, \tilde{C})) \in OFTL (U, \tilde{A}, \tilde{I})$ . More specifically, {x^↑↓ : x ∈ U} is regarded as the information granular of $OFTL (U, \tilde{A}, \tilde{I})$ .

Definition 10. Let $(U, A, \tilde{I})$ be a formal fuzzy context with an object granular matrix OM. For B ⊆ A, B is called a granular attribute consistent set if OM = OM_B. Furthermore, if B is a granular attribute consistent set and any C ⊂ B is not a granular attribute consistent set, then B is a granular attribute reduct. The core set w.r.t. the intersection of all granular attribute reducts Red is denoted as Core = ⋂ _B∈RedB.

In short, a granular attribute consistent set is minimal subset preserving the information granular ability. For brevity, we denote a copy attribute matrix as G_B generated from a n-by-1 tilling of copies of λ_B.

Example 10. Continuation from Example 9, assume that B = {a₁, a₂, a₃, a₅}, the copy attribute matrix of B is given by: $G_{B} = [\begin{matrix} 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 \end{matrix}] .$

For B ⊆ A, and x, y ∈ U, we define $R_{B} = {(x, y) \in U \times U : \tilde{I} (x, a) \leq \tilde{I} (y, a), \forall a \in B}, R_{B}^{★} = {(x, y) \in U \times U : {\tilde{I}}^{c} (x, a) \leq {\tilde{I}}^{c} (y, a), \forall a \in B} .$ R_B and $R_{B}^{★}$ are two partial orders on the object set, in which (x, y) ∈ R_B and $(x, y) \in R_{B}^{★}$ mean that y is not less than x with respect to all attributes in B, respectively.

For each x ∈ U, their granule of knowledge induced by R_B and $R_{B}^{★}$ are $R_{B} = {y \in U : (x, y) \in R_{B}}, [x]_{R_{B}^{★}} = {y \in U : (x, y) \in R_{B}^{★}} .$

Lemma 2. [10] Let $(U, A, \tilde{I})$ be a formal fuzzy context. For B ⊆ A and x ∈ U, then [x] _{R
_B} = x^▽_B△_B.

Proposition 6. For any X ⊆ U and B ⊆ A, then X^↑↓ ⊆ X^↑_B↓_B.

Proof.

$x \in X^{↑ ↓} \Leftrightarrow X^{▽ △} \cap X^{\bar{▽} \bar{△}} \Leftrightarrow \forall a \in A, X^{▽} (a) \leq \tilde{I} (x, a), and, X^{\bar{▽}} (a) \leq {\tilde{I}}^{c} (x, a) \Rightarrow \forall a \in B, X^{▽_{B}} (a) \leq \tilde{I} (x, a), and, X^{\bar{▽_{B}}} (a) \leq {\tilde{I}}^{c} (x, a) \Leftrightarrow x \in X^{▽_{B} △_{B}} \cap X^{{\bar{▽}}_{B} {\bar{△}}_{B}} \Leftrightarrow x \in X^{↑_{B} ↓_{B}} .$ □

Corollary 1. Let $(U, A, \tilde{I})$ be a formal fuzzy context and B ⊆ A. OM and OM_B are respectively object granular matrices. Then OM ≤ OM_B.

Proof. It is straightforward based on Proposition 6.□

Proposition 7. Let $(U, A, \tilde{I})$ be a formal fuzzy context and B ⊆ A. B is a granular attribute consistent set iff OM_B ≤ OM.

Proof. It is immediately from Definition 10 and Corollary 1.□

Proposition 8. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the fuzzy relation matrix $M_{\tilde{I}}$ . Then a ∈ A is a core iff OM_A-{a}≰OM.

Proof. Follows directly by applying Definition 10 and Proposition 7.□

Example 11. Continuation from Example 9. Assume that B = {a₁, a₂, a₃, a₅}, this implies that ${OM}_{B} = ((M_{\tilde{I}} \land G_{B}) \otimes (M_{\tilde{I}}^{T} \land G_{B}^{T})) \land$

$(\sim (M_{\tilde{I}} \land G_{B}) \otimes (\sim (M_{\tilde{I}}^{T} \land G_{B}^{T}))) =$

$[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \end{matrix}] ≰ OM .$ Therefore, a₄ is a core attribute.

Proposition 8 manifests an approach to find all core attributes. Based on this discussion, we blew construct the attribute discernibility Boolean matrix in OFT-concept lattice.

Definition 11. Given a formal fuzzy context $(U, A, \tilde{I})$ with the fuzzy relation matrix $M_{\tilde{I}}$ and the object granular matrix OM, define a discernibility attribute Boolean vector as follows: if OM (i, j) =0, $λ_{D} (x_{i}, x_{j}) = M_{\tilde{I}} (i, :) ⊙ M_{\tilde{I}} (j, :) \cup (\sim M_{\tilde{I}} (i, :) ⊙ (\sim M_{\tilde{I}} (j, :))) .$ Then $Q = (λ_{D} (x_{i}, x_{j}) : OM (i, j) = 0 \land | λ_{D} (x_{i}, x_{j}) | \neq | A |) .$ where Q is called the attribute discernibility Boolean matrix.

Henceforth, x_i and x_j are discernible under each $λ_{D} (x_{i}, x_{j})$ under the premise of OM (i, j) =0.

We next employ the attribute discernibility Boolean matrix to characterize the attribute matrix-based reduction.

Theorem 9. Given a formal fuzzy context $(U, A, \tilde{I})$ with the fuzzy relation matrix $M_{\tilde{I}}$ and B ⊆ A, then B is a granular attribute consistent set if and only if $Q \cdot λ_{B}^{T} = (1, 1, \dots, 1)_{1 \times q}^{T}$ , where q is the number of rows with respect to Q.

Proof. Assume that $Q (x_{i}, x_{j}) = {a \in A : \tilde{I} (x_{i}, a) > \tilde{I} (x_{j}, a), or, {\tilde{I}}^{c} (x_{i}, a) > {\tilde{I}}^{c} (x_{j}, a)}$ is the discernibility attribute set.

Necessity. We only have to prove that B ∩ Q (x_i, x_j) ≠ φ for any Q (x_i, x_j) ≠ φ. If B is a granular attribute consistent set, then OM = OM_B. For any OM (i, j) =0, there exists a ∈ A such that $\tilde{I} (x_{i}, a) > \tilde{I} (x_{j}, a)$ or ${\tilde{I}}^{c} (x_{i}, a) > {\tilde{I}}^{c} (x_{j}, a)$ , which implies that x_j ∉ [x_i] _{R
_A} or $x_{j} \notin [x_{i}]_{R_{A}^{★}}$ . It follows from Lemma 2 that $x_{j} \notin x_{i}^{▽ △} \cap x_{i}^{\bar{▽} \bar{△}}$ . With the fact of OM = OM_B from Definition 10, we have $x_{j} \notin x_{i}^{▽_{B} △_{B}} \cap x_{i}^{\bar{▽_{B}} \bar{△_{B}}}$ , that is, $x_{j} \notin x_{i}^{▽_{B} △_{B}}$ or $x_{j} \notin x_{i}^{{\bar{▽}}_{B} {\bar{△}}_{B}}$ . Thus, there exists c ∈ B such that $\tilde{I} (x_{i}, c) > \tilde{I} (x_{j}, c)$ or ${\tilde{I}}^{c} (x_{i}, c) > {\tilde{I}}^{c} (x_{j}, c)$ , which implies that OM_B (i, j) =0. This means c ∈ Q (x_i, x_j). Hence, B ∩ Q (x_i, x_j) ≠ φ. Accordingly, $λ_{D} (x_{i}, x_{j})$ is a row of the attribute discernibility Boolean matrix Q, which implies that $Q \cdot λ_{B}^{T} = (1, 1, \dots, 1)_{1 \times q}^{T}$ .

Sufficiency. With the fact of $Q \cdot λ_{B}^{T} = (1, 1, \dots, 1)_{1 \times q}^{T}$ , it holds that $λ_{D} (x_{i}, x_{j}) \cdot λ_{B}^{T} = 1$ for any OM (i, j) =0, that is Q (x_i, x_j) ∩ B ≠ φ for any Q (x_i, x_j) ≠ φ. There is c ∈ B such that $\tilde{I} (x_{i}, c) > \tilde{I} (x_{j}, c)$ or ${\tilde{I}}^{c} (x_{i}, c) > {\tilde{I}}^{c} (x_{j}, c)$ , which means that OM_B (i, j) =0. Since OM ≤ OM_B, we conclude that OM = OM_B. Therefore, B is a granular attribute consistent set. We complete the proof.□

Algorithm 2 A heuristic algorithm for finding a reduct of $(U, A, \tilde{I})$ based on matrix theory (AHRM)
Input: A formal fuzzy context $(U, A, \tilde{I})$ with fuzzy relation matrix $M_{\tilde{I}} = (r_{ij})_{n \times m}$ .
Output: A granular reduct Red.
1: Initialize Red ← φ, and compute the attribute discernibility Boolean matrix Q;
2: whileQ≠ ∅ do
3: if \|Q (i, :) \|=1 and Q (i, j) =1 do
4: Red ← Red ∪ {a_j} and compute K (a_j). Then Q = Q - ∪ _{a_j∈Red}K (a_j);
5: else find the i-th column of Q such that \|Q (: , t) \| = max {\|Q (: , i) \| : i ≤ \|A\| - \|Red\|};
6: Red ← Red ∪ {a_t}, where a_t is the attribute corresponding to Q (: , t);
7: Q = Q - K (a_t) and remove all rows of Q such that its t-th column is 1;
8: end if
9: end while
10: Return Red

Theorem 10. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the attribute discernibility Boolean matrix Q. Then a_j ∈ A is a core iff there exists i ≤ q such that $Q (i, :) \cdot λ_{A - {a_{j}}}^{T} = 0$ .

Proof. It is obvious according to Theorem 9.□

Denote K (a_j) = {Q (k, :) : Q (k, j) =1}. Then we shall formulate a heuristic model for a reduct via granular matrix. Algorithm 2 puts forward an approach to compute a granular attribute reduct. In Step 1, the attribute discernibility Boolean matrix is generated and its time complexity is O (|U|²|A|). Steps 2-9 are to compute a granular attribute reduct, which can be done in O (|Q||A|), where |Q| means that the row numbers of Q. Therefore, the time complexity of Algorithm 2 is O (max {|U|²|A|, |Q||A|}).

Example 12. Continuation from Example 9. From Definition 11, we have $λ_{D} (x_{i}, x_{j}) = (1, 1, 1, 1, 0)$ when OM (2, 3) =0. Moreover, if $| λ_{D} (x_{i}, x_{j}) | = 0$ or $| λ_{D} (x_{i}, x_{j}) | = 5$ , this corresponding row $λ_{D} (x_{i}, x_{j})$ will be deleted. Hence, the attribute discernibility Boolean matrix is listed as: $Q = [\begin{matrix} 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{matrix}] .$ From Q, notice that Q (3, 4) =1 and |Q (3, :) |=1 for a₄ ∈ A. Thus, a₄ is a core attribute, and K (a₄) is depicted as: $K (a_{4}) = [\begin{matrix} 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \end{matrix}] .$ Furthermore, the attribute discernibility Boolean matrix is updated as follows: $Q = [\begin{matrix} 1 & 1 & 0 & 0 & 0 \end{matrix}] .$ Due to |Q (: , 1) | = |Q (: , 2) |=1, we select attribute a₁ stochastically, and add to Red. Then Q is updated to empty. In sum up, Red = {a₁, a₄} is a granular reduct.

6 Granular matrix-based reduction in fuzzy-crisp three-way concept lattices

In duality, one can study object reduction of formal fuzzy contexts based on 3WD, and the corresponding definitions and theorems can be discussed likewise.

6.1 Matrix presentations of AFT-concept lattices

Proposition 9. Given a formal fuzzy context $(U, A, \tilde{I})$ with the fuzzy relation matrix $M_{\tilde{I}}$ , then $AM = (M_{\tilde{I}}^{T} \otimes M_{\tilde{I}}) \land (\sim M_{\tilde{I}}^{T} \otimes (\sim M_{\tilde{I}}))$ (6) in which the i-th row of AM is $λ_{a_{i}}^{↑ ↓}$ .

Proof. $\begin{matrix} a_{i}^{↑ ↓} = {a_{j} \in A : \forall x \in U, a_{i}^{▽} (x) \leq \tilde{I} (x, a_{j}), \\ and, \forall x \in U, a_{i}^{\bar{▽}} (x) \leq {\tilde{I}}^{c} (x, a_{j})} \\ = {a_{j} \in A : \forall x \in U, a_{i}^{▽} (x) \otimes \tilde{I} (x, a_{j}) = 1, \\ and, \forall x \in U, a_{i}^{\bar{▽}} (x) \otimes {\tilde{I}}^{c} (x, a_{j}) = 1} \\ = {a_{j} \in A : ⋀_{x \in U} a_{i}^{▽} (x) \otimes \tilde{I} (x, a_{j}) = 1, \\ and, ⋀_{x \in U} a_{i}^{\bar{▽}} (x) \otimes {\tilde{I}}^{c} (x, a_{j}) = 1} \\ = {a_{j} \in A : (M_{\tilde{I}}^{T} (:, i) \otimes M_{\tilde{I}} (:, j)) \land (\sim M_{\tilde{I}}^{T} (:, i) \\ \otimes (\sim M_{I} (:, j))) = 1} . \end{matrix}$ Therefore, $λ_{a_{i}^{↑ ↓}} = (M_{\tilde{I}}^{T} (:, i) \otimes M_{\tilde{I}}) \land (\sim M_{\tilde{I}}^{T} (:, i) \otimes (\sim M_{\tilde{I}}))$ .□

Definition 12. Let $(U, A, \tilde{I})$ be a formal fuzzy context with a Boolean matrix AM. Define $a_{i}^{↑ ↓}$ for each a_i ∈ A as $λ_{a_{i}^{↑ ↓}}$ , that is, $λ_{a_{i}^{↑ ↓}} = AM (i, :)$ . Furthermore, AM is called an attribute granular matrix.

Example 13. A formal fuzzy context $(U, A, \tilde{I})$ is exhibited in Table 5, where U = {x₁, x₂, x₃, x₄, x₅} and A = {a₁, a₂, a₃, a₄, a₅}. Applying Proposition 9, one haves $AM = (M_{\tilde{I}}^{T} \otimes M_{\tilde{I}}) \land (\sim M_{\tilde{I}}^{T} \otimes (\sim M_{\tilde{I}})) =$

$[\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] .$

Table 5

Formal fuzzy context $(U, A, \tilde{I})$

$\tilde{I}$	a ₁	a ₂	a ₃	a ₄	a ₅
x ₁	0.9	0.8	0.1	0.1	0.8
x ₂	0.7	0.8	0.2	0.3	0.8
x ₃	0.2	0.8	0.1	0.8	0.8
x ₄	0.9	0.3	0.8	0.1	0.1
x ₅	0.8	0.2	0.2	0.2	0.2

6.2 Object matrix-based reduction in AFT-concept lattices

Definition 13. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the attribute granular matrix AM and X ⊆ U. X is called a granular object consistent set if AM = AM_X. Furthermore, if X is a granular object consistent set, and Y is not a granular object consistent set for any Y ⊂ X, then we say that X is a granular object reduct. The core set w.r.t. the intersection of all granular object reducts Red is given by Core = ⋂ _X∈RedX.

Notice that $((\tilde{X}, \tilde{Y}), B) = ⋀_{a \in B} (a^{↑}, a^{↑ ↓})$ for any $((\tilde{X}, \tilde{Y}), B) \in AFTL (\tilde{U}, A, \tilde{I})$ . Concretely, {a^↑↓ : a ∈ A} is interpreted as the information granular.

For brevity, we denote a copy object matrix as J_X generated from a m-by-1 tilling of copies of λ_X. Likewise, we can analogously employed the object discernibility Boolean matrix.

Proposition 10. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the attribute granular matrix AM. For X ⊆ U, X is a granular object consistent set if and only if AM_X ≤ AM.

Proposition 11. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the attribute granular matrix AM. Then the object x ∈ U is a core iff AM_U-{x}≰AM.

Definition 14. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the fuzzy relation matrix $M_{\tilde{I}}$ , and the attribute granular matrix AM. Define a discernibility object Boolean vector as follows: if AM (i, j) =0 $λ_{Z} (a_{i}, a_{j}) = M_{\tilde{I}} (:, i) ⊙ M_{\tilde{I}} (:, j) \cup (\sim M_{\tilde{I}} (:, i) ⊙ (\sim M_{\tilde{I}} (:, j))) .$ Then $Z = (λ_{Z} (a_{i}, a_{j}) : AM (i, j) = 0 \land | λ_{Z} (a_{i}, a_{j}) | \neq | U |) .$ where Z is called the object discernibility Boolean matrix.

Theorem 11. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the fuzzy relation matrix $M_{\tilde{I}}$ and X ⊆ U. Then X is a granular object consistent set iff $Z \cdot λ_{X}^{T} = (1, 1, \dots, 1)_{1 \times t}^{T}$ , where t means that the number of rows with respect to Z.

Theorem 12. Let $(U, A, \tilde{I})$ be a formal fuzzy context with the object discernibility Boolean matrix Z. Then x_j ∈ U is a core if and only if there is i ≤ t satisfying $Z (i, :) \cdot λ_{U - {x_{j}}}^{T} = 0$ .

7 Numerical experiments

In this section, we conduct some numerical experiments to verify the effectiveness and efficiency of our model. We mainly focus on examining the ability for the algorithms by discussing the granular reduct and the whole running time.

7.1 Data sets

The experiments are performed on a personal computer with Windows 10 and an Intel(R) Core(TM) i5-6200 CPU @ 2.30GHz with 8GB of memory. The algorithms are implemented by Matlab 9.3.

We choose 8 data sets from UCI machine learning repository [41], which are depicted in Table 6.

Table 6
Description of the 8 data sets

NO. Data sets Objects Features

1 Ecoli 336 7

2 Iono 351 34

3 Soy 47 21

4 Zoo 101 16

5 Rice 104 5

6 Flag 194 19

7 Heart 270 13

8 German 1000 21

NO.	Data sets	Objects	Features
1	Ecoli	336	7
2	Iono	351	34
3	Soy	47	21
4	Zoo	101	16
5	Rice	104	5
6	Flag	194	19
7	Heart	270	13
8	German	1000	21

7.2 Comparison on finding a granular reduct in OFT-concept lattices

Lin et al. [4] and Shao et al. [10] proposed a reduction algorithm to compute a granular reduction in formal fuzzy context. Therefore, we compare the proposed algorithm AHRM with the reduction algorithm (called HGRM and CAGR) in [4] and [10].

The whole running time and cardinality of reducts with respect to data sets are displayed in Table 7 and 8. We can observe that the algorithm AHRM is very efficient especially for the data sets with large objects. The results show that AHRM runs faster than CAGR. For example, the time consumptions on Ecoli, Iono, Heart, German can be achieved in a certain number of seconds, whereas CAGR consume a large time on most data sets. Then from the above experimental analysis, we can determine that AHRM is computationally efficient on all data sets. In addition, as seen in Table 8, the cardinalities of AHRM reducts are less than corresponding those of HGRM and CAGR reducts. To sum up, we conclude that the algorithm AHRM is more efficient.

Table 7
Comparison of running times of AHRM and CAGR

No. Datasets Running times

AHRM CAGR

1 Ecoli 0.86 191.86

2 Iono 4.51 1121.43

3 Soy 0.01 0.08

4 Zoo 0.05 1.89

5 Rice 0.02 0.45

6 Flag 0.41 61.54

7 Heart 0.81 157.13

8 German 54.51 6897.45

No.	Datasets	Running times
1	Ecoli	0.86	191.86
2	Iono	4.51	1121.43
3	Soy	0.01	0.08
4	Zoo	0.05	1.89
5	Rice	0.02	0.45
6	Flag	0.41	61.54
7	Heart	0.81	157.13
8	German	54.51	6897.45

Table 8

Comparison of |Red| of AHRM, HGRM and CAGR

Datasets	\|Red\|
	AHRM	HGRM	CAGR
Ecoli	4	6	6
Iono	7	31	31
Soy	10	15	15
Zoo	16	14	14
Rice	2	5	5
Flag	8	19	19
Heart	3	13	13
German	7	21	21

8 Discussion and conclusion

Fuzzy three-way concept analysis is an important issue which can be implied to deal with uncertainty in knowledge representation. In this paper, we have introduced some notions and properties of OFT-concept lattice and AFT-concept lattice. By the proposed discussion, the connections between one-sided fuzzy three-way concept lattices and classical fuzzy concept lattices have been investigated from the communities of elements, sets, and orders, respectively. Furthermore, the object and attribute granular matrix are comprehensively presented based on granular matrices. Then we have studied the problems of reducts in formal fuzzy context based on 3WD. More specifically, we have calculated a reduct by using the discernibility Boolean matrix. Corresponding problems might be considered. The novel knowledge reduction methods and reduction algorithms should be discussed in the future.

Footnotes

Acknowledgments

This work is supported by grants from the National Natural Science Foundation of China (No. 11871259, No. 11701258, and No. 61379021), and the Natural Science Foundation of Fujian Province in China (No. 2019J01748, No. 2017J01507, and No. 2020J02043).

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Matrix-based reduction approach for one-sided fuzzy three-way concept lattices

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Formal fuzzy contexts

3 Formal fuzzy contexts based on three-way decision

3.1 Crisp-fuzzy three-way concept lattices

4.1 The relationships between OFT-concept lattice and fuzzy concept lattice

5.1 Matrix presentations of OFT-concept lattices

6 Granular matrix-based reduction in fuzzy-crisp three-way concept lattices

6.1 Matrix presentations of AFT-concept lattices

7 Numerical experiments

7.1 Data sets

Table 6 Description of the 8 data sets NO. Data sets Objects Features 1 Ecoli 336 7 2 Iono 351 34 3 Soy 47 21 4 Zoo 101 16 5 Rice 104 5 6 Flag 194 19 7 Heart 270 13 8 German 1000 21

Table 7 Comparison of running times of AHRM and CAGR No. Datasets Running times AHRM CAGR 1 Ecoli 0.86 191.86 2 Iono 4.51 1121.43 3 Soy 0.01 0.08 4 Zoo 0.05 1.89 5 Rice 0.02 0.45 6 Flag 0.41 61.54 7 Heart 0.81 157.13 8 German 54.51 6897.45

Footnotes

Acknowledgments

References

Table 6
Description of the 8 data sets

NO. Data sets Objects Features

1 Ecoli 336 7

2 Iono 351 34

3 Soy 47 21

4 Zoo 101 16

5 Rice 104 5

6 Flag 194 19

7 Heart 270 13

8 German 1000 21

Table 7
Comparison of running times of AHRM and CAGR

No. Datasets Running times

AHRM CAGR

1 Ecoli 0.86 191.86

2 Iono 4.51 1121.43

3 Soy 0.01 0.08

4 Zoo 0.05 1.89

5 Rice 0.02 0.45

6 Flag 0.41 61.54

7 Heart 0.81 157.13

8 German 54.51 6897.45