Notes on Independence of Attributes in Soft Contexts

Won Keun Min

Department of Mathematics, Kangwon National University, Chuncheon, Korea

Received August 24, 2018; Revised September 17, 2019; Accepted September 20, 2019.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Abstract
For the purpose of studying the formal concepts and the reduction in a formal context, we have combined the formal contexts with the soft sets to form soft contexts and proposed the soft concept in a soft context. As a series of studies, it is necessary to investigate specific properties of attributes. For this purpose, we introduce and study the notion of independent and dependent attributes in a given soft context. In particular, we will study the following: (1) Every dependent attribute is generated by some independent attributes: (2) The set of all soft concepts can be completely constructed by independent attributes.

**Keywords**: Formal context, Formal concept, Concept lattice, Soft set, Soft context, Soft concept, Independent attributes

- 1. Introduction
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.

- 2. Preliminaries
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]):$${x}^{*}=\{a\in A|xIa\};\mathrm{\hspace{0.17em}\u200a\u200a}{a}^{*}=\{x\in U|xIa\}.$$ Extending

*x*to a subset*X*⊆*U*and*a*to a subset*B*⊆*A*:$${X}^{*}=\{a\in A|\forall x\in X,xIa\};\mathrm{\hspace{0.17em}\u200a\u200a}{B}^{*}=\{x\in U|\forall b\in B,xIb\}.$$ 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.,$$F:A\to P(U).$$ 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}_{∈}_{A}*F*(*a*) = ∅︀ and*F*(*a*) ≠ ∅︀ for every*a*∈*A*. From now on, we assume that all soft sets are*pure*.Let

*U*= {*x*_{1},*x*_{2}, . . . ,*x*} be a non-empty finite set of_{n}*objects*,*A*= {*a*_{1},*a*_{2}, . . . ,*a*} a non-empty finite set of_{m}*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 asF ^{+}(*B*) = ∩_{b}_{∈}_{B}*F*(*b*);2)

F ^{−}:*P*(*U*) →*P*(*A*) is a mapping defined asF ^{−}(*X*) = {*a*∈*A*:*X*⊆*F*(*a*)}.We will denote

Im (F ^{+}) = {F ^{+}(*B*)|*B*∈*P*(*A*)} andIm (F ^{−}) = {F ^{−}(*X*)|*X*∈*P*(*U*)}.Simply, we denote

F ^{+}({*a*}) =F ^{+}(*a*) for each*a*∈*A*, andF ^{−}({*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*, thenF ^{−}(*Y*) ⊆F ^{−}(*X*); if*C*⊆*D*, thenF ^{+}(*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 Ψand Φ_{F}[15] induced by_{F}F ^{+},F ^{+}as the following ways:For each

*X*∈*P*(*U*),$$\begin{array}{l}{\mathrm{\Psi}}_{F}:P(U)\to P(U)\hspace{0.17em}\text{is\hspace{0.17em}a\hspace{0.17em}mapping\hspace{0.17em}defined\hspace{0.17em}as}\\ {\mathrm{\Psi}}_{F}(X)={\mathbf{F}}^{+}{\mathbf{F}}^{-}(X):\end{array}$$ In the rest of the paper, Ψ is instead of Ψ

when there is no ambiguity._{F}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. Then1) ∅︀,

*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*).

- 3. Independence, Dependence of Attributes
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*). Then for_{a}*d*∈*A*,*d*is said to be*dependent*on*A*if there exists*G*≠ ∅︀ satisfying_{d}*F*(*d*) =F ^{+}(*G*) = ∩_{d}_{a∈Gd}*F*(*a*).Otherwise,

*d*is said to be*independent*on*A*.$$\begin{array}{ll}\text{We\hspace{0.17em}denote:}& {A}_{D}=\{a\in A|a\hspace{0.17em}\text{is\hspace{0.17em}dependent\hspace{0.17em}on\hspace{0.17em}}A\};\\ \hspace{0.17em}& {A}_{I}=\{a\in A|a\hspace{0.17em}\text{is\hspace{0.17em}independent\hspace{0.17em}on\hspace{0.17em}}A\}.\end{array}$$ ### 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$$\begin{array}{l}F(a)=F(d)=\{1,2,4\};\mathrm{\hspace{0.17em}\u200a\u200a}F(b)=\{2,4,5\};\\ F(c)=\{2,4\};\mathrm{\hspace{0.17em}\u200a\u200a}F(e)=\{1,3\};\mathrm{\hspace{0.17em}\u200a\u200a}F(f)=\{1,3,5\}.\end{array}$$ Then

*a*is independent on*A*since*G*= ∅︀, and also_{a}*e*is independent since*G*= {_{e}*f*} ≠ ∅︀ butF ^{+}(*G*) = ∩_{e}_{g∈Ge}*F*(*g*) =*F*(*f*) ≠*F*(*e*). In case of*c*∈*A*, it is dependent since*G*= {_{c}*a*,*b*,*d*} ≠ ∅︀ andF ^{+}(*G*) = ∩_{c}_{g∈Gc}*F*(*g*) =*F*(*c*).Consequently, we have

$${A}_{I}=\{a,b,d,e,f\};\mathrm{\hspace{0.17em}\u200a\u200a}{A}_{D}=\{c\}.$$ Then we easily obtain the following facts:

### Theorem 3.3

Let (

*U*,*A*,*F*) be a soft context. Then1)

*A*∩_{D}*A*= ∅︀;_{I}*A*∪_{D}*A*=_{I}*A*.2)

*a*is independent if and only if either*G*= ∅︀ or if_{a}*G*≠ ∅︀, then_{a}F ^{+}(*G*) = ∩_{a}_{g∈Ga}*F*(*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*b*_{1},*b*_{2}, · · ·,*b*∈_{n}*A*if*F*(*a*) = ∩_{b}_{∈}_{B}*F*(*b*) for some*B*= {*b*_{1},*b*_{2}, · · ·,*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*, there exists_{D}*B*⊆*A*such that_{I}F _{+}(*B*) = ∩_{b}_{∈}_{B}*F*(*b*) =*F*(*d*).**Proof**Suppose that there is a dependent element

*d*of*A*such that it can not be generated only by independent elements of*A*.Put is not generated only by independent elements of

*A*}. Then by hypothesis, is not empty. Furthermore, .For the proof, assume that .

First, pick up one element in , say

*d*_{1}. Then since ,*G*_{d}_{1}= {*a*∈*A*|*F*(*d*_{1}) ⊊*F*(*a*)} is not empty set and . So, we can pick up . Then obviously,*F*(*d*_{1}) ⊊*F*(*d*_{2}).Second, for , let us consider

*G*_{d}_{2}= {*a*∈*A*|*F*(*d*_{2}) ⊊*F*(*a*)}. Then since and*G*_{d}_{2}is the maximal set of generators for*d*_{2}, . Now, pick up . Then obviously,*F*(*d*_{2}) ⊊*F*(*d*_{3}) and so,*F*(*d*_{1}) ⊊*F*(*d*_{2}) ⊊*F*(*d*_{3}).Repeating this process, after a finite number (

*n*− 2), we get an element . Then it satisfies that*F*(*d*_{1}) ⊊, · · · ⊊*F*(*d*_{n}_{−2}) ⊊*F*(*d*_{n}_{−1}) and .Finally, since , we can pick up the last element . Then

*F*(*d*_{1}) ⊊*F*(*d*_{2}) ⊊ · · · ⊊*F*(*d*_{n}_{−1}) ⊊*F*(*d*)._{n}In the last step, since

*d*is the last element in , for_{n}*G*_{dn}= {*a*∈*A*|*F*(*d*) ⊊_{n}*F*(*a*)}, . So, the last element is generated by finitely many independent elements of*A*. From this fact and*F*(*d*_{1}) ⊊, · · · ⊊*F*(*d*_{n}_{−2}) ⊊*F*(*d*_{n}_{−1}) ⊊*F*(*d*), we know that should be generated by finitely many independent elements of_{n}*A*. Consequently, is the empty set. So the proof is completed.### Theorem 3.8

Let (

*U*,*A*,*F*) be a soft context. Then ∩_{a∈AI}*F*(*a*) = ∅︀.**Proof**From Theorem 2.1 and Theorem 3.7, it follows

F ^{+}(*A*)∩_{I}F ^{+}(*A*) =_{D}F ^{+}(*A*) and ∩_{I}_{a∈AI}*F*(*a*) =F ^{+}(*A*) =_{I}F ^{+}(*A*)∩_{I}F ^{+}(*A*) =_{D}F ^{+}(*A*∪_{I}*A*) =_{D}F ^{+}(*A*) = ∩_{a}_{∈}_{A}*F*(*a*). Since the soft set (*F*,*A*) is pure, ∩_{a∈AI}*F*(*a*) = ∅︀.### Theorem 3.9

Let (

*U*,*A*,*F*) be a soft context. Put for*A*⊆_{I}*A*. Then$$s(U,A,F)=\{\cap \mathcal{S}\mid \mathcal{S}\subseteq {\mathcal{A}}_{\mathcal{I}}\}.$$ **Proof**1) Take . Then from the theorem above, . It is obvious that . Take ; . Then clearly, .

2) Let

*X*∈*s*(*U*,*A*,*F*). Then, from (4) of Theorem 2.2, there exists*B*⊆*A*such that*X*=F ^{+}(*B*). Let*C*=*B*∩*A*and_{I}*D*=*B*∩*A*. From_{D}*B*=*C*∪*D*and*C*∩*D*= ∅︀, it follows*X*=F ^{+}(*B*) =F ^{+}(*C*∪*D*) =F ^{+}(*C*) ∩F ^{+}(*D*) =F ^{+}(*C*) ∩ (∩_{d}_{∈}_{D}*F*(*d*)). So, from Theorem 3.7, for each*d*∈*D*⊆*A*, there exists_{D}*E*⊆_{d}*A*such that_{I}F _{+}(*E*) = ∩_{d}_{e∈Ed}*F*(*e*) =*F*(*d*). Put*H*= ∪_{d}_{∈}_{D}*E*. Then_{d}*H*⊆*A*and_{I}F ^{+}(*H*) =F ^{+}(∪_{d}_{∈}_{D}*E*) = ∩_{d}_{d}_{∈}_{D}F ^{+}(*E*) = ∩_{d}_{d}_{∈}_{D}*F*(*d*). It implies*X*=F ^{+}(*B*) =F ^{+}(*C*) ∩ (∩_{d}_{∈}_{D}*F*(*d*)) =F ^{+}(*C*) ∩F ^{+}(*H*) =F ^{+}(*C*∪*H*) for*C*∪*H*⊆*A*. Finally, put ; then and . So the proof is completed._{I}For a formal context (

*U*,*A*,*I*), let us define a soft set*F*:_{I}*A*→*P*(*U*) as follows*F*(_{I}*a*) = {*x*∈*U*: (*x*,*a*) ∈*I*}. Then (*U*,*A*,*F*) is a soft context. So, every formal context (_{I}*U*,*A*,*I*) induces a soft context (*U*,*A*,*F*). In this paper, we call (_{I}*U*,*A*,*F*) the_{I}*associated soft context*induced by a formal context (*U*,*A*,*I*).### 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}**Proof**For the associated soft context (

*U*,*A*,*F*), from Theorem 3.9, it is obvious that_{I}*X*∈ {F ^{+}(*B*)|*B*is any subset of*A*} if and only if . So the fact is obtained._{I}In [1], for a formal context (

*U*,*A*,*I*), The concepts of (*U*,*A*,*I*) are ordered by$$({X}_{1},{B}_{1})\le ({X}_{2},{B}_{2})\iff {X}_{1}\subseteq {X}_{2}(\iff {B}_{1}\supseteq {B}_{2}),$$ where (

*X*_{1},*B*_{1}), (*X*_{2},*B*_{2}) are two concepts.(

*X*_{1},*B*_{1}) is called a*sub-concept*of (*X*_{2},*B*_{2}), and (*X*_{2},*B*_{2}) is called a*super-concept*of (*X*_{1},*B*_{1}). The ordered set of all concepts in (*U*,*A*,*I*) is denoted by*L*(*U*,*A*,*I*) and called*the concept lattice*of (*U*,*A*,*I*), where the infimum and supremum are defined by:$$\begin{array}{l}({X}_{1},{B}_{1})\wedge ({X}_{2},{B}_{2})=({X}_{1}\cap {X}_{2},{({B}_{1}\cup {B}_{2})}^{**}),\\ ({X}_{1},{B}_{1})\vee ({X}_{2},{B}_{2})=({({X}_{1}\cup {X}_{2})}^{**},{B}_{1}\cap {B}_{2}).\end{array}$$ ### Theorem 3.11 ( [15])

Let (

*U*,*A*,*I*) be a formal context. Then$$L(U,A,I)=\{(X,{{\mathbf{F}}_{I}}^{-}(X))|X\hspace{0.17em}is\hspace{0.17em}any\hspace{0.17em}element\hspace{0.17em}of\hspace{0.17em}s(U,A,{F}_{I})\}.$$ 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*in (_{I}*U*,*A*,*F*)}._{I}

- 4. Conclusions
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.

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- Biography
Won Keun Min received the M.S. and the Ph.D. degrees in mathematics from Korea University, Seoul, Korea in 1983 and 1987, respectively. He is currently a professor in the Department of Mathematics, Kangwon National University. His research interests include general topology, fuzzy topology and soft set theory.E-mail: wkmin@kangwon.ac.kr