International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(1): 50-60
Published online March 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.1.50
© The Korean Institute of Intelligent Systems
Won Keun Min
Department of Mathematics, Kangwon National University, Chuncheon, Korea
Correspondence to :
Won Keun Min (wkmin@kangwon.ac.kr)
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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Attribute reduction of a formal context is one of the main aims in formal concept analysis. Several methods have been proposed to deal with attribute reduction in an object-oriented concept lattice; however, these methods are inevitably very complicated. To overcome this problem, we study the notion of consistent sets and attribute reduction in object-oriented soft concept lattices in a soft context. Using independent attributes and object-oriented soft concepts, we study the characterization of consistent sets and attribute reductions of object-oriented soft concept lattices. In particular, we introduce two classes of independent attributes for object-oriented soft concepts, 1_{m} and 2_{m}, to construct effective attribute reductions for object-oriented soft concept lattices. Subsequently, we investigate the relationship between the two classes and attribute reduction in object-oriented soft concept lattices. Finally, we investigate meaningful information on how to construct attribute reductions for object-oriented soft concept lattices. Additionally, we apply this information to find attribute reductions in the object-oriented concept lattices of formal contexts.
Keywords: Formal context, Object-oriented concept, Soft context, Soft concept, Object-oriented soft concept, Consistent set, Attribute reduction
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
Similar to Yao’s idea [9], we propose a new notion,
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
A formal context is a triplet (
For
For
Hereafter, for every formal context (
In the formal context (
Yao [9] introduced a new concept called the
Let (
Then a pair (
For any object-oriented concept (
Then, (
Let
In other words, for
We assume that every soft set (
Let
In [15], for each
(1) is defined as ;
(2) is defined as .
Simply, we denote: For
Let (
(1) If
(2) ;
(3) ;
(4) .
Let us consider an operator defined as follows. For each
Subsequently,
Let (
Now, we recall the notion of order on
For an ordered set (
Subsequently, (
The complete lattice (
Let
If
In this section, we introduce the notion of a consistent set (simply,
Let (
Let (
From
Let
Then, (
Then
For
Then, (
Thus,
Recall the notion of base [28] for
(1) .
(2) For each
For a soft context (
Let (
Let
Suppose that is the base of
Let (
Suppose that
Conversely, suppose that for each
As ℰ ⊆ ℱ
Let (
In [31], we studied the notions of
Let (
We denote:
Then, we show that ℳ= {
Thus, the following theorem is obtained:
Let (
From Theorem 3.7, we find that
For a soft context (
In Example 3.3, for a soft context (
For
Therefore,
Consequently, 1
For a soft context (
(1)
(2)
Let (
For each
Let (
For
Conversely, for
In the case:
We introduced the notions of dependence and independence for the object-oriented soft concept in [32] as follows: Let (
(1)
(2)
Note that
Let (
Let
However, the opposite result was obtained.
Let (
From the above results, we obtain the following theorem for
Let (
In the theorem above, the converse is not always true, as shown in the following example:
In Example 3.3,
it is impossible that there is any surjective mapping
Let (
(1) 1
(2) For each
Let
For (2), suppose there is some
Conversely, suppose that conditions (1) and (2) are satisfied. For each
Using Theorem 3.15, we demonstrate that the mapping
Condition (2) in Theorem 3.17 is essential, as shown in the following example:
For Example 3.3, we found that:
Take
In addition,
However, for
For
Let (
For a soft context (
It is evident from Theorem 3.2.
Every
Let (
For any soft context (
There is at least one
Let (
(1)
(2)
(3) The mapping
(1) ⇔ (2) For each
(2) ⇒ (3) Let
(3) ⇒ (2) Suppose that the mapping
Finally, we obtain the following useful theorem on constructing an
Let (
(1) 1
(2) For each
Suppose that
For the proof of condition (2), we consider the following two cases. (i) Suppose that there is some
Conversely, suppose that for
For Example 3.3, we found that
Consider
In Example 4.7, we consider an
Then 1
Next, we take
Consequently, by Theorem 4.6,
Let (
Let (
(1) ;
(2) ;
(3) For an object-oriented formal concept (
Let (
It is obtained from Theorem 5.1.
Hereafter, we may write
Let us consider the formal context (
Then the object-oriented concept lattice
Now we can define a soft set (
and
Note that
For 2
To apply Theorem 4.5, we select two elements,
Finally, we construct an
Similarly, we construct three
(2)
(3)
(4)
From Theorem 5.2, we know that
We now describe the process of finding
First, define the soft set (
Then by Theorem 5.1, we obtain
where
Consequently, from Theorem 5.2, we have
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 (
is an object-oriented discernibility attribute set of (
is an object-oriented discernibility matrix of the formal context (
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
No potential conflict of interest relevant to this article was reported.
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(1): 50-60
Published online March 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.1.50
Copyright © The Korean Institute of Intelligent Systems.
Won Keun Min
Department of Mathematics, Kangwon National University, Chuncheon, Korea
Correspondence to:Won Keun Min (wkmin@kangwon.ac.kr)
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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Attribute reduction of a formal context is one of the main aims in formal concept analysis. Several methods have been proposed to deal with attribute reduction in an object-oriented concept lattice; however, these methods are inevitably very complicated. To overcome this problem, we study the notion of consistent sets and attribute reduction in object-oriented soft concept lattices in a soft context. Using independent attributes and object-oriented soft concepts, we study the characterization of consistent sets and attribute reductions of object-oriented soft concept lattices. In particular, we introduce two classes of independent attributes for object-oriented soft concepts, 1_{m} and 2_{m}, to construct effective attribute reductions for object-oriented soft concept lattices. Subsequently, we investigate the relationship between the two classes and attribute reduction in object-oriented soft concept lattices. Finally, we investigate meaningful information on how to construct attribute reductions for object-oriented soft concept lattices. Additionally, we apply this information to find attribute reductions in the object-oriented concept lattices of formal contexts.
Keywords: Formal context, Object-oriented concept, Soft context, Soft concept, Object-oriented soft concept, Consistent set, Attribute reduction
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
Similar to Yao’s idea [9], we propose a new notion,
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
A formal context is a triplet (
For
For
Hereafter, for every formal context (
In the formal context (
Yao [9] introduced a new concept called the
Let (
Then a pair (
For any object-oriented concept (
Then, (
Let
In other words, for
We assume that every soft set (
Let
In [15], for each
(1) is defined as ;
(2) is defined as .
Simply, we denote: For
Let (
(1) If
(2) ;
(3) ;
(4) .
Let us consider an operator defined as follows. For each
Subsequently,
Let (
Now, we recall the notion of order on
For an ordered set (
Subsequently, (
The complete lattice (
Let
If
In this section, we introduce the notion of a consistent set (simply,
Let (
Let (
From
Let
Then, (
Then
For
Then, (
Thus,
Recall the notion of base [28] for
(1) .
(2) For each
For a soft context (
Let (
Let
Suppose that is the base of
Let (
Suppose that
Conversely, suppose that for each
As ℰ ⊆ ℱ
Let (
In [31], we studied the notions of
Let (
We denote:
Then, we show that ℳ= {
Thus, the following theorem is obtained:
Let (
From Theorem 3.7, we find that
For a soft context (
In Example 3.3, for a soft context (
For
Therefore,
Consequently, 1
For a soft context (
(1)
(2)
Let (
For each
Let (
For
Conversely, for
In the case:
We introduced the notions of dependence and independence for the object-oriented soft concept in [32] as follows: Let (
(1)
(2)
Note that
Let (
Let
However, the opposite result was obtained.
Let (
From the above results, we obtain the following theorem for
Let (
In the theorem above, the converse is not always true, as shown in the following example:
In Example 3.3,
it is impossible that there is any surjective mapping
Let (
(1) 1
(2) For each
Let
For (2), suppose there is some
Conversely, suppose that conditions (1) and (2) are satisfied. For each
Using Theorem 3.15, we demonstrate that the mapping
Condition (2) in Theorem 3.17 is essential, as shown in the following example:
For Example 3.3, we found that:
Take
In addition,
However, for
For
Let (
For a soft context (
It is evident from Theorem 3.2.
Every
Let (
For any soft context (
There is at least one
Let (
(1)
(2)
(3) The mapping
(1) ⇔ (2) For each
(2) ⇒ (3) Let
(3) ⇒ (2) Suppose that the mapping
Finally, we obtain the following useful theorem on constructing an
Let (
(1) 1
(2) For each
Suppose that
For the proof of condition (2), we consider the following two cases. (i) Suppose that there is some
Conversely, suppose that for
For Example 3.3, we found that
Consider
In Example 4.7, we consider an
Then 1
Next, we take
Consequently, by Theorem 4.6,
Let (
Let (
(1) ;
(2) ;
(3) For an object-oriented formal concept (
Let (
It is obtained from Theorem 5.1.
Hereafter, we may write
Let us consider the formal context (
Then the object-oriented concept lattice
Now we can define a soft set (
and
Note that
For 2
To apply Theorem 4.5, we select two elements,
Finally, we construct an
Similarly, we construct three
(2)
(3)
(4)
From Theorem 5.2, we know that
We now describe the process of finding
First, define the soft set (
Then by Theorem 5.1, we obtain
where
Consequently, from Theorem 5.2, we have
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 (
is an object-oriented discernibility attribute set of (
is an object-oriented discernibility matrix of the formal context (
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
Table 1 . Soft context.
- | |||||||
---|---|---|---|---|---|---|---|
1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
4 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |
Table 2 . Formal context.
- | |||||||
---|---|---|---|---|---|---|---|
1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |
2 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
3 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |
4 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
5 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Won Keun Min
International Journal of Fuzzy Logic and Intelligent Systems 2019; 19(3): 158-162 https://doi.org/10.5391/IJFIS.2019.19.3.158