Formal concept analysis, introduced by Wille [1], is a useful tool for researching the information structures and information contained in data. Formal concept analysis mainly deals with formal concepts and concept lattices induced by a binary relationship between a set of attributes and object attributes. As a fundamental tool for information analysis, knowledge representation and knowledge extraction in a given dataset, the notion of concept lattice has been applied in many fields related to knowledge and information systems, data mining and decision-making [2–11]. Until recently, studies combining formal concept analysis with other analysis tools have been extensively conducted [7, 8, 12–24].

Molodtsov [25] introduced the concept of soft sets as a mathematical tool to handle the complexity and uncertainty of different types of information. Maji et al. [26] introduced operations for soft set theory. Ali et al. [27] proposed new concepts of operations that modified the basic operations introduced by Maji et al. [26]. We propose a soft context [29] by combining the concepts of formal context and a soft set defined by set-valued mapping. Furthermore, we introduced and studied new concepts called soft concepts and soft concept lattices.

Yao [9] introduced a new concept called the object-oriented formal concept in a formal context using the notion of approximation operations and rough sets. An object-oriented concept lattice is a combination of rough sets and formal concept analysis. The author studied the relationships between approximation operations and formal concept analysis and proposed several methods for attribute reduction that can be applied to formal concepts constructed using approximation operations.

Similar to Yao’s idea [9], we propose a new notion, object-oriented soft concept (simply called m-concepts), that combines object-oriented formal concepts and soft sets in [15]. This is closely related to the object-oriented concept of the formal context. Moreover, we introduced the notion of order between two m-concepts and showed that the set of all m-concepts with order is a complete lattice. The set of all m-concepts of order is called the object-oriented soft concept lattice [29] in a soft context, which is also closely related to the object-oriented concept lattice of formal contexts.

Much useful information regarding formal contexts can be obtained from an object-oriented concept lattice. Ma et al. [7] introduced an object-oriented discernibility matrix for an object-oriented concept lattice, and studied its properties. Furthermore, they proposed an approach for object-oriented reduction of an object-oriented concept lattice based on an object-oriented discernibility matrix. However, the process is complex. Therefore, in this study, we intend to utilize object-oriented soft concepts to effectively remove unnecessary and redundant attributes.

For this purpose, we introduce the notion of a consistent set and a reduction for an object-oriented soft concept lattice in a soft context and simply call them OOS-consistent sets and OOS-reductions, respectively. We first study some basic properties of OOS-consistent sets in a soft context and then characterize OOS-consistent sets using m-independent attributes. In particular, for this characterization study, we introduce and study two classes, 1_m and 2_m in set M_I of all m-independent attributes in a given soft context. In addition, we show that every OOS-consistent set can be reduced to the smallest OOS-consistent set, which is called OOS-reduction. We then investigated the relationship between the two classes and OOS-reductions. Finally, we provide meaningful information on how to construct OOS-reductions using two classes, 1_m and 2_m. Furthermore, by applying this fact, we propose a method for effectively constructing attribute reductions of object-oriented concept lattices of formal contexts and explain this process with an example.

A formal context is a triplet (U, A, I), where U is a non-empty finite set of objects, A is a non-empty finite set of attributes, and I is the relationship between U and A. Let (U, A, I) be the formal context. For a pair of elements x ∈ U and a ∈ A, if (x, a) ∈ I, then object x has attribute a.

For X ⊆ U and B ⊆ A, we denote operators X^*, B^* [1, 18] as follows:

For x ∈ U and a ∈ A, we simply denote {x}^* and {a}^* as x^* and a^*, respectively. Then

Hereafter, for every formal context (U, A, I) it is assumed that for x ∈ U, x^* ≠ ∅ or x^* ≠ A, and for a ∈ A, a^* ≠ ∅ or a^* ≠ U.

In the formal context (U, A, I), a pair (X, B) of two sets X ⊆ U and B ⊆ A is called a formal concept of (U, A, I) if X = B^* and B = X^*, where X and B are the extent and intent of the formal concept, respectively.

Yao [9] introduced a new concept called the object-oriented formal concept in a formal context using the notion of approximation operations.

Let (U, A, I) be a formal context in formal concept analysis, where U is a finite non-empty set of objects, A is a finite non-empty set of attributes, and I is a binary relation between U and A.

Then a pair (X, Y ), X ⊆ U, Y ⊆ A, is called an object-oriented formal concept if X = Y^⋄ and Y = X^□. The set of all object-oriented concepts is denoted by

For any object-oriented concept (X₁, Y₁), (X₂, Y₂) ∈ C_o(U, A, I), the binary relationship ⪯ is defined as follows: (X₁, Y₁) ⪯ (X₂, Y₂) if and only if X₁ ⊆ X₂.

Then, (C_o(U, A, I), ⪯) is a partial order set called the object-oriented concept lattice. We denote the partial-order set (C_o(U, A, I), ⪯) by L_o(U, A, I).

Let U be a universal set and A be a collection of properties of the objects in U. A pair (F, A) is called a soft set [25] over U if F is a set-valued mapping of A to 2^U; that is,

In other words, for a ∈ A, every set F(a) may be considered as the set of a-elements of the soft set (F, A).

We assume that every soft set (F, A) is pure [30], which is defined as ∩_a∈AF(a) = ∅, ∪_a∈AF(a) = U, F(a) ≠ U and F(a) ≠ ∅ for each a ∈ A.

Let U = {x₁, x₂, . . . , x_m} be a non-empty finite set of objects, A = {a₁, a₂, . . . , a_n} be a non-empty finite set of attributes, and F : A → 2^U be a soft set. Then, the triple (U, A, F) is called a soft context [28].

In [15], for each B ∈ 2^A and X ∈ 2^U, we defined two mappings as the following:

• (1) is defined as ;

• (2) is defined as .

Simply, we denote: For a ∈ A and and . Clearly, for a ∈ A.

Theorem 2.1 ([15])

Let (U, A, F) be a soft context, X, Y ⊆ U and B, C ⊆ A. Then we have

• (1) If X ⊆ Y, then ; if B ⊆ C, then ;

• (2) ;

• (3) ;

• (4) .

Let us consider an operator defined as follows. For each X ∈ 2^U in a soft context (U, A, F), is an operator defined by .

Subsequently, X is called an object-oriented soft concept (simply, m-concept) [15] in (U, A, F) if . The set of all m-concepts is denoted as m(U, A, F).

Theorem 2.2 ([15])

Let (U, A, F) be a soft context. Then, for X ⊆ U, X is an m-concept if and only if there is some B ⊆ A such that . Further, .

Now, we recall the notion of order on m(U, A, F) defined in [29] as follows: For X, Y ∈ m(U, A, F),

X is called a sub-m-concept of Y, and Y is called the super-m-concept of X.

For an ordered set (m(U, A, F), ⪯), the infimum ∧ and supremum ∨ are defined as

Subsequently, (m(U, A, F), ⪯,∧,∨) denotes the complete lattice.

The complete lattice (m(U, A, F), ⪯,∧,∨) is called the m-concept lattice (or the object-oriented soft concept lattice) and is simply denoted by mL(U, A, F).

Let mL(U, B, F) and mL(U, C, G) be two m-concept lattices. mL(U, B, F) is said to be finer than mL(U, C, G), and is denoted by

If mL(U, B, F) ≤ mL(U, C, G) and mL(U, C, G) ≤ mL(U, B, F), then the two m-concept lattices are said to be isomorphic with each other and are denoted by

In this section, we introduce the notion of a consistent set (simply, OOS-consistent set) for an object-oriented soft concept lattice in a soft context, which is also closely related to the object-oriented consistent set in the formal context. In particular, we introduce the notion of two classes 1_m and 2_m in the family M_I of all m-independent attributes and investigate the characteristics of OOS-consistent sets using two classes 1_m and 2_m.

Definition 3.1

Let (U, A, F) be a soft context and C ⊆ A. Subsequently, C is called a consistent set for object-oriented soft lattices (simply, an OOS-consistent set) of (U, A, F) if mL(U, A, F) ≅ mL(U, C, F|_C).

Theorem 3.2

Let (U, A, F) be a soft context and C ⊆ A. Then C is an OOS-consistent set of (U, A, F) if and only if m(U, A, F) = m(U, C, F|_C).

Example 3.3

Let U = {1, 2, 3, 4} and A = {a, b, c, d, e, f, g}. Let us consider a soft context (U, A, F) as shown in Table 1.

Then, (F, A) is a soft set, as follows:

Then

For C = {a, b, d, e} ⊆ A, F|_C : C → P(U) is a set-valued function defined as F(a) = {2, 3}; F(b) = {1, 2, 4}; F(d) = {1, 2}; F(e) = {2, 4}.

Then, (F|_C, C) is a soft set, and (U, C, F|_C) is a soft context. Furthermore,

Thus, m(U, C, F|_C) = m(U, A, F), and according to Theorem 3.2, C is an OOS-consistent set of (U, A, F).

Recall the notion of base [28] for mC(U, A, F): Let (U, A, F) be the soft context. A subfamily of m(U, A, F) is called a base for m(U, A, F) if it satisfies the following conditions.

• (1) .

• (2) For each X ∈ m(U, A, F), there exists such that .

For a soft context (U, A, F), family is the base of m(U, A, F). Thus, A is an OOS-consistent set, which is the largest OOS-consistent set of soft contexts (U, A, F). We now examine their general characteristics.

Theorem 3.4

Let (U, A, F) be a soft context and C ⊆ A. Then C is an OOS-consistent set if and only if is the base of m(U, A, F).

Theorem 3.5

Let (U, A, F) be a soft context and C ⊆ A. Then, C is an OOS-consistent set of (U, A, F) if and only if for each e ∈ A – C, there exists a non-empty subset B of C such that ∪_b∈BF(b) = F(e).

Corollary 3.6

Let (U, A, F) be a soft context and C ⊆ A. Then, C is an OOS-consistent set of (U, A, F) if and only if, for every non-empty subset B of A – C, there exists a non-empty subset E of C such that ∪_b∈EF(e) = ∪_b∈BF(b).

In [31], we studied the notions of m-dependent and m-independent attributes in a given soft context. In addition, we demonstrated that the family of all m-independent attributes induces a base for the set of all m-concepts in a soft context.

Let (U, A, F) be the soft context. Let M_a = {g ∈ A | F(a) ⫌ F(g)}. Then for d ∈ A, d is said to be m-dependent on A if M_d ≠ ∅ satisfies F(d) = ∪_a∈MdF(a). Otherwise, d is considered to be m-independent of A.

We denote:

Then, we show that ℳ= {F(a) | a ∈ M_I} is the base for m(U, A, F) in [31].

Thus, the following theorem is obtained:

Theorem 3.7

Let (U, A, F) be a soft context. Subsequently, M_I is an OOS-consistent set of (U, A, F).

From Theorem 3.7, we find that M_I ⊆ A is a trivial OOS-consistent set but that the set is not always the smallest OOS-consistent set. Therefore, using OOS-independent attributes, we want to investigate a method for finding an OOS-consistent set that is smaller than M_I. To determine how to construct an OOS-consistent set, we first introduce the next two classes, 1_m and 2_m of M_I. We then investigate the relationship between the two classes and OOS-consistent sets.

Definition 3.8

For a soft context (U, A, F), let n(a) = |{b ∈ M_I | F(a) = F(b)}|. Then

Example 3.9

In Example 3.3, for a soft context (U, A, F) where U = {1, 2, 3, 4} and A = {a, b, c, d, e, f, g}, we showed that M_I = {a, d, e, f, g} and M_D = {b, c}.

For M_I = {a, d, e, f, g},

Therefore, n(e) = 1; n(d) = n(f) = n(a) = n(g) = 2.

Consequently, 1_m(A) = {e}; 2_m(A) = {a, d, f, g}.

Lemma 3.10

For a soft context (U, A, F),

• (1) M_I = 1_m(A) ∪ 2_m(A);

• (2) a ∈ 1_m(A) if and only if F(a) ≠ F(e) for every e ∈ A – {a}.

Theorem 3.11

Let (U, A, F) be a soft context and 2_m(A) ≠ ∅. Subsequently, for x ∈ 2_m(A), M_I – {x} is an OOS-consistent set of (U, A, F).

Theorem 3.12

Let (U, A, F) be a soft context and C ⊆ A. If C is an OOS-consistent set of (U, A, F), then for e ∈ C and e ∈ 1_m(C) if and only if C – {e} is not an OOS-consistent set of (U, A, F).

Theorem 3.13

Let (U, A, F) be a soft context, C ⊆ A and C ≠ ∅. Then, C is an m-consistent set of (U, A, F) if and only if (1) mD = mD_C and (2) mI = mI_C.

Theorem 3.14 ([32])

Let (U, A, F) be a soft context. Then, for D ⊆ A, if the mapping ϕ : D → mI defined by ϕ(d) = F(d) for d ∈ D is surjective, then ℱ_D = {F(d) | d ∈ D} is the base for m(U, A, F).

From the above results, we obtain the following theorem for OOS-consistent sets.

Theorem 3.15

Let (U, A, F) be a soft context and C ⊆ A. If there exists a surjective mapping ϕ : C → mI defined by ϕ(d) = F(d) for d ∈ C, then C is an OOS-consistent set.

In the theorem above, the converse is not always true, as shown in the following example:

Example 3.16

In Example 3.3, mI = {{1, 2}, {2, 3}, {2, 4}}. Consider a consistent set C₁ = {a, b, d, e}. Then because

it is impossible that there is any surjective mapping ϕ : C₁ → mI defined as follows ϕ(d) = F(d) for d ∈ C₁. Therefore, in Theorem 3.15, the converse is not always true.

Theorem 3.17

Let (U, A, F) be a soft context and C ⊆ A. Then, C ⊆ M_I is an OOS-consistent of (U, A, F) if and only if C satisfies

• (1) 1_m(A) ⊆ 1_m(C),

• (2) For each a ∈ 2_m(A), {b ∈ M_I | F(b) = F(a)}∩C ≠ ∅.

Example 3.18

For Example 3.3, we found that:

M_I = {a, d, e, f, g}, 1_m(A) = {e} and 2_m(A) = {a, d, f, g} (See Example 3.9).

Take C = {a, g, c, e} ⊆ A. Thus, the following can be easily obtained.

In addition,

However, for d ∈ 2_m(A), {b ∈ M_I | F(b) = F(d)} ∩ C = ∅.

For d ∈ A–C, there is no any B ⊆ C such that ∪_b∈BF(b) = F(d). Thus, C is not an OOS-consistent set (U, A, F).

Definition 4.1

Let (U, A, F) be a soft context and C ⊆ A. If C is an OOS-consistent set of (U, A, F) and for each c ∈ C, mL(U, A, F) ≇ mL(U, C – {c}, F|_C–{c}), then the OOS-consistent set C is called an OOS-reduction of (U, A, F).

Theorem 4.2

For a soft context (U, A, F), let C be an OOS-consistent set of (U, A, F). Then, C is called an OOS-reduction of (U, A, F) if and only if m(U, A, F) ≠ m(U, C – {c}, F|_C–{c}) for each c ∈ C.

Theorem 4.3

Every OOS-consistent set of a soft context can be reduced to an OOS-reduction.

Theorem 4.4

There is at least one OOS-reduction for any soft context.

Theorem 4.5

Let (U, A, F) be a soft context and C an OOS-consistent set of (U, A, F). Then, the following are equivalent:

• (1) C is the OOS-reduction of (U, A, F).

• (2) C = 1_m(C).

• (3) The mapping ϕ : C → mI defined by ϕ(c) = F(c) for c ∈ C is bijective.

Theorem 4.6

Let (U, A, F) be a soft context and C ⊆ A. Then, C is an OOS-reduction of (U, A, F) if and only if C satisfies

• (1) 1_m(A) ⊆ 1_m(C) ⊆ C ⊆ M_I ;

• (2) For each a ∈ 2_m(A), |{b ∈ M_I | F(b) = F(a)} ∩ C| = 1.

Example 4.7

For Example 3.3, we found that M_I = {a, d, e, f, g}, 1_m(A) = {e} and 2_m(A) = {a, d, f, g} (See Example 3.9).

Consider C = 1_m(A) ∪ {a, d} = {a, d, e}. Then, from Theorem 4.6, C is an OOS-reduction of (U, A, F).

Remark 4.8

In Example 4.7, we consider anOOS-consistent set C₁ = {a, d, e, f}. Then, (F|_C1, C₁) is a soft set, as follows:

Then 1_m(C₁) = {a, e} ≠ C₁. Thus, C₁ is not an OOS-reduction.

Next, we take C = C₁ – {f} = {a, d, e}. Then (F|_C, C) is a soft set as follows:

C is an OOS-consistent set of (U, F, A) and 1_m(C) = {a, d, e} = C.

Consequently, by Theorem 4.6, C is an OOS-reduction which is reduced by an OOS-consistent set C₁ of (U, F, A).

Let (U, A, I) be the formal context. We define the soft set F_I : A → P(U) as F_I (a) = {x ∈ U : (x, a) ∈ I}. Then, (U, A, F_I ) is the associated soft context induced by formal context (U, A, I) [28].

Theorem 5.1 ([32])

Let (U, A, I) be a formal context. Then, for the associated soft context (U, A, F_I ) induced by the formal context (U, A, I),

• (1) ;

• (2) ;

• (3) For an object-oriented formal concept (X, Y ), X is an m-concept in the associated soft context (U, A, F_I ) induced by (U, A, I).

Theorem 5.2

Let (U, A, I) be a formal context. Subsequently, the object-oriented formal concept lattice L_o(U, A, I) is order isomorphic to the object-oriented soft-concept lattice mL(U, A, F_I ) of the associated soft context (U, A, F_I ).

Remark 5.3

Let us consider the formal context (U, A, I) presented in Table 2, where U = {1, 2, 3, 4, 5}, A = {a, b, c, d, e, f, g}.

Then the object-oriented concept lattice L_o(U, A, I) can be represented as shown in Figure 1.

Now we can define a soft set (F, A) induced by I as follows:

and

Note that

For 2_m(A), they can be separated as

To apply Theorem 4.5, we select two elements, a, e ∈ 2_m(A) where a ∈ {a, g} and e ∈ {e, f}, respectively.

Finally, we construct an OOS-reduction as follows:

Similarly, we construct three OOS-reductions, as follows:

• (2) C₂ = 1_m(A) ∪ {a} ∪ {f} = {a, b, c, f};

• (3) C₃ = 1_m(A) ∪ {g} ∪ {e} = {b, c, e, g};

• (4) C₄ = 1_m(A) ∪ {g} ∪ {f} = {b, c, g, f}.

From Theorem 5.2, we know that C_i (i = 1, 2, 3, 4) is also a reduction in the object-oriented formal concept lattice L_o(U, A, I).

We now describe the process of finding L_o(U, C, I|_C) using an OOS-reduction C = C₁ = {a, b, c, e} of the associated soft context (U, A, F_I ).

First, define the soft set (F|_C, C) as follows:

Then by Theorem 5.1, we obtain

where FI←(X)={a∣F(a)⊆X for a∈C}.

Consequently, from Theorem 5.2, we have L_o(U, A, I) ≅ L_o(U, C, I|_C) as shown in Figure 2.

Remark 5.4

To obtain the object-oriented consistent set object-oriented reductions of an object-oriented concept lattice of the formal context in a traditional way, Ma et al. [7] defined the object-oriented discernibility matrix of an object-oriented concept lattice as follows:

Let (U, A, I) be a formal context and (X_i, B_i), (X_j, B_j) ∈ L_o(U, A, I).

is an object-oriented discernibility attribute set of (X_i, B_i) and (X_j, B_j). Furthermore,

is an object-oriented discernibility matrix of the formal context (U, A, I).

They then studied its properties and proposed an approach for the object-oriented reduction of an object-oriented concept lattice based on an object-oriented discernibility matrix. We know that the method proposed in Remark 5.3 above was to reduce attributes using a soft set. Thus, its fundamental difference from Ma’s method [7] is that it does not use the concept of the discernibility matrix.

In this study, in order to effectively obtain the object-oriented consistent set object-oriented reductions of an object-oriented concept lattice of the formal context, the notion of consistent sets and attribute reductions of object-oriented soft concept lattices in a soft context was introduced. In particular, we investigated the construction of consistent sets and attribute reductions of object-oriented soft concept lattices using two classes, 1_m and 2_m of independent attributes for object-oriented soft concepts. Finally, we propose a method for constructing attribute reductions for object-oriented concept lattices in formal contexts without using the notion of a discernibility matrix. In future research, we will study a method for obtaining OOS-consistent sets and attribute reductions for object-oriented soft-concept lattices in a soft-decision context. We will also apply this method to find consistent object-oriented sets in object-oriented concept lattices in a formal decision context.

Table 1. Soft context.

Table 2. Formal context.