For the purpose of the study of hierarchical structures based on a binary relation between objects and attributes, Wille introduced FCA (formal concept analysis) [1] in 1982 and investigated the notions of context, formal concept, and concept lattice. A formal context, a type of information system, is represented in a tabular form of an object-attribute value relationship [2–5]. A formal concept is presented in pairs consisting of objects and attributes. The order relationship between two formal concepts is well defined, and it is well known that through this order relationship the collection of all formal concepts is a complete lattice. It is simply called the concept lattice [5]. Formal concept analysis has been widely applied to many information systems research fields, and many studies are actively conducted to apply problems in real-world situations. [5–11].

The concept of soft sets was introduced by Molodtsov in 1999 [12] for the purpose of dealing with complex problems and uncertainties: Let U be an initial universe set (simply, universe set) and A be a collection of characteristics or properties of objects in U. A pair (F,A) is called a soft set over U if F is a mapping of A into the set of all subsets of the set U. For the soft set theory, Ali et al. [13] proposed new operations that complements the concepts defined by Maji in [14].

In [15], we constructed the soft context combining the notions of formal contexts and soft sets as set-valued mappings. Additionally, we introduced and investigated the notions of soft concepts and soft concepts lattice that are closely related to formal concepts and concept lattices in FCA.

As a series of studies in [15], we would like to investigate the specific properties of attributes. In this paper, we introduce and study the notions of independent and dependent attributes in a given soft context. In particular, we will show the following two facts: 1) Every dependent attribute is generated by some independent attributes: 2) The set of all soft concepts can be completely constructed by independent attributes.

We recall some basic definitions of formal concept analysis used in this paper. A formal context is a triplet (U, A, I), where U is a non-empty finite set of objects, A is a nonempty finite set of attributes, and I is a relation between U and A. In the formal context (U, A, I), for a pair of elements x ∈ U and a ∈ A, if (x, a) ∈ I, we write xIa. A set of attributes with an object x ∈ U and a set of objects with an attribute a ∈ A can be represented as ( [1, 5]):

Extending x to a subset X ⊆ U and a to a subset B ⊆ A:

A pair (X,B) of two sets X ⊆ U and B ⊆ A is called a formal concept of the context (U, A, I) if X = B^* and B = X^* (see [1, 5]). X and B are called the extent and the intent of the concept, respectively.

Let U be an initial universe set (simply, universe set) and E be a collection of all possible parameters with respect to U, where parameters are the characteristics or properties of objects in U. We will call E the set of parameters with respect to U. Let P(U) denote the power set and A ⊆ E.

A pair (F,A) is called a soft set [12] over U if F is a set-valued mapping of A into the set of all subsets of the set U, i.e.,

In other words, the soft set is a parameterized family of subsets of the set U. Every set F(e), for e ∈ A ⊆ E, from this family may be considered as the set of e-elements of the soft set (F,A), or as the set of e-approximate elements of the soft set.

We call a soft set (F,A) is pure if ∩_a∈AF(a) = ∅︀ and F(a) ≠ ∅︀ for every a ∈ A. From now on, we assume that all soft sets are pure.

Let U = {x₁, x₂, . . . , x_n} be a non-empty finite set of objects, A = {a₁, a₂, . . . , a_m} a non-empty finite set of attributes, and F : A → P(U) a soft set. Then the triple (U, A, F) is called a soft context [15].

Let (U, A, F) be a soft context. Then for a soft set (F,A),

1) F⁺ : P(A) → P(U) is a mapping defined as F⁺(B) = ∩_b∈BF(b);

2) F⁻ : P(U) → P(A) is a mapping defined as F⁻(X) = {a ∈ A : X ⊆ F(a)}.

We will denote Im(F⁺) = {F⁺(B)|B ∈ P(A)} and Im(F⁻) = {F⁻(X)|X ∈ P(U)}.

Simply, we denote F⁺({a}) = F⁺(a) for each a ∈ A, and F⁻({x}) = F⁻(x) for each x ∈ U.

Theorem 2.1 ( [15])

Let (U, A, F) be a soft context, X, Y ⊆ U and C,D ⊆ A. Then we have the following things:

1) If X ⊆ Y, then F⁻(Y ) ⊆ F⁻(X); if C ⊆ D, then F⁺(D) ⊆ F⁺(C);

2) X ⊆ F⁺F⁻(X); C ⊆ F⁻F⁺(C);

3) F⁻(X∪Y ) = F⁻(X)∩F⁻(Y ), F⁺(C∪D) = F⁺(C)∩ F⁺(D);

4) F⁻(X) = F⁻F⁺F⁻(X), F⁺(C) = F⁺F⁻F⁺(C);

5) F⁻(X) ∪ F⁻(Y ) ⊆ F⁻(X ∩ Y ), F⁺(C) ∪ F⁺(D) ⊆ F⁺(C ∩ D).

In a soft context (U, A, F), let us define two associated operations Ψ_F and Φ_F [15] induced by F⁺,F⁺ as the following ways:

For each X ∈ P(U),

In the rest of the paper, Ψ is instead of Ψ_F when there is no ambiguity.

Let (U, A, F) be a soft context and X ∈ P(U). Then X is called a soft concept [15] in (U, A, F) if Ψ(X) = F⁺F⁻(X) = X. The set of all soft concepts will be denoted by s(U, A, F).

Theorem 2.2 ( [15])

Let (U, A, F) be a soft context. Then

1) ∅︀, U, Ψ(X) are soft concepts.

2) For each B ⊆ A, F⁺(B) is a soft concept.

3) For each a ∈ A, F(a) is a soft concept.

4) X is a soft concept if and only if X = F⁺(B) for some B ∈ P(A).

5) Im(F⁺) = s(U, A, F).

For a soft context (U, A, F), we introduce and study the notions of dependent and independent in A. We will show that every formal concept of formal context is represented by independent elements of the associated soft context.

From now on, we will denote |X| the cardinal number of any set X.

Definition 3.1

Let (U, A, F) be a soft context. Put G_a(A) = {g ∈ A | F(a) ⊊ F(g)} ⊆ A (simply, G_a). Then for d ∈ A, d is said to be dependent on A if there exists G_d ≠ ∅︀ satisfying F(d) = F⁺(G_d) = ∩_a∈GdF(a).

Otherwise, d is said to be independent on A.

Example 3.2

Let U = {1, 2, 3, 4, 5} and A = {a, b, c, d, e, f}. Let us consider a soft set F : A → P(U) defined by

Then a is independent on A since G_a = ∅︀, and also e is independent since G_e = {f} ≠ ∅︀ but F⁺(G_e) = ∩_g∈GeF(g) = F(f) ≠ F(e). In case of c ∈ A, it is dependent since G_c = {a, b, d} ≠ ∅︀ and F⁺(G_c) = ∩_g∈GcF(g) = F(c).

Consequently, we have

Then we easily obtain the following facts:

Theorem 3.3

Let (U, A, F) be a soft context. Then

1) A_D ∩ A_I = ∅︀; A_D ∪ A_I = A.

2) a is independent if and only if either G_a = ∅︀ or if G_a ≠ ∅︀, then F⁺(G_a) = ∩_g∈GaF(g) ≠ F(a).

Definition 3.4

Let (U, A, F) be a soft context. For a ∈ A, we say that the element a is generated by finitely many elements b₁, b₂, · · ·, b_n ∈ A if F(a) = ∩_b∈BF(b) for some B = {b₁, b₂, · · ·, b_n} ⊆ A, and b ∈ B is called generator for a.

Example 3.5

In Example 3.2, for c ∈ A, a, b and d are generators for c.

Lemma 3.6

Let (U, A, F) be a soft context. For d ∈ A_D, G_d = {g ∈ A | F(d) ⊊ F(g)} is the maximal set of generators.

Theorem 3.7

Let (U, A, F) be a soft context. Then every dependent element of A is generated by finitely many independent elements of A, that is, for each d ∈ A_D, there exists B ⊆ A_I such that F₊(B) = ∩_b∈BF(b) = F(d).

Theorem 3.8

Let (U, A, F) be a soft context. Then ∩_a∈AIF(a) = ∅︀.

Theorem 3.9

Let (U, A, F) be a soft context. Put for A_I ⊆ A. Then

Theorem 3.10

Let (U, A, I) be a formal context. Then for the associated soft context (U, A, F_I ), s(U, A, F_I) = {F⁺(B)|B is any subset of A_I}.

Theorem 3.11 ( [15])

Let (U, A, I) be a formal context. Then

By Theorem 3.10 and Theorem 3.11, the following theorem is obtained:

Theorem 3.12

Let (U, A, I) be a formal context. Then

L(U, A, I) = {(F⁺(B), F⁻F⁺(B))|B is any subset of A_I in (U, A, F_I )}.

We introduced the notion of independent and dependent attributes in a given soft context. Then we showed that every dependent attribute is generated by some independent attributes in a given soft context. In the next research, we will study special properties of the independent attributes, and characterizations for soft concepts and soft concept lattice by using a nonempty finite set of attributes. Furthermore, the results of this paper will be applied to the reduction of formal concepts and the research of the formal concept analysis.