2. Preliminaries
A formal context is a triplet
, where
X and
Y are finite sets, and
is a relation from
U to
A. In a formal context
,
U is a finite set of objects,
A is a finite set of attributes or properties, and if
, we simply write
and say that
x has the property
a, or
a is possessed by object
x. The set of all attributes possessed by
and the set of all objects having an attribute
were represented as [
1,
5]:
For a set of objects
,
is the maximal set of properties shared by all objects in
X;
for a set of properties
,
is the maximal set of objects that all have properties in
B; that is,
Let us define two operators between the power sets of U and A as follows:
;
.
Then, obviously, the pair of operators is a contravariant Galois connection between the power sets of U and A.
A pair
,
,
, is called a
formal concept of the context
if
and
[
1,
5].
X is called the
extension of the concept, and
B is called the
extension of the concept.
For a formal context
, the concepts of
are ordered by the following: For two concepts
,
The ordered set of all concepts in is a complete lattice and the ordered set is denoted by and called the formal concept lattice of .
Let U be a universe set and E be a set of parameters with respect to U. In general, parameters are properties or attributes of objects in U.
A pair
is called a
soft set [
14] over
U if
F is a set-valued mapping of
A into
, that is,
is a mapping defined by
for
.
Let
be a non-empty finite set of
objects,
a non-empty finite set of
attributes, and
a soft set. Then, the triple
is called
a soft context [
17].
Let
be a soft context,
denote the power set of
U and
. Then, in [
17],
and
are defined as the following:
- (1)
is a mapping defined as ;
- (2)
is a mapping defined as .
Let and . For and , simply, and .
Then, we showed the following facts:
Theorem 1 ([
17])
. Let be a soft context, and .- (1)
If , then ; if , then ;
- (2)
; ;
- (3)
, ;
- (4)
,
- (5)
, .
In a soft context
, the associated operation
[
17] was induced by
as follows: For each
,
Then, X is called a soft concept in if .
The set of all soft concepts is denoted by . The important properties of soft concepts are obtained as follows:
Theorem 2 ([
17])
. Let be a soft context. Then- (1)
∅, U, are soft concepts.
- (2)
For each , is a soft concept.
- (3)
For each , is a soft concept.
- (4)
X is a soft concept if and only if for some .
- (5)
.
In a soft context , for , consider a set-valued mapping defined by for all . Then, obviously, is a soft set. Since is a soft set, is also a soft context. From now on, we consider only a subset satisfying the soft set which is pure.
Then, naturally, the associated operation , associated mappings and are induced by the soft set as follows:
for each .
for each .
Then, obviously, the following were shown in [
17]:
Lemma 1. ([
17])
Let be a soft context, and . Then- (1)
.
- (2)
.
- (3)
.
Now, for ,
- (1)
X is said to be
dependent [
17] on
if there exist
satisfying
and
,
(
).
- (2)
X is said to be independent of if X is not dependent.
We will denote:
The basic properties of such notions are as follows:
Theorem 3 ([
17])
. Let be a soft context.- (1)
Both ∅ and U are dependent.
- (2)
; .
- (3)
For a non-empty set , there exists such that for .
- (4)
For , there exists satisfying .
3. Consistent Sets of Soft Contexts
We introduce the notion of consistent set on a given soft context, and investigate the characterizations of consistent sets. In general, in a formal context , for , if or , then x could be omitted from , and the structure of concept lattice and reduction are not affected. Dually, for , if or , then a could be omitted from , and the structure of concept lattice and reduction is not affected.
Based on this fact, from now on, we assume that every soft set
is
pure [
18], which is defined as:
,
,
and
for every
.
Definition 1. Let be a soft context and . Then, C is called a consistent set of if .
We recall the notion of base for
in a soft context
defined in [
17]: Let
be a soft context. A subfamily
of
is called a
base for
if it satisfies the following conditions:
- (1)
;
- (2)
For each , there exists such that .
Theorem 4 ([
17])
. For a soft context , the family is a base for . Example 1. Let and . Let us consider a soft context induced by a soft set where the set-valued mapping is defined by For , is a set-valued function defined as follows: Then is a soft set and is a soft context. Furthermore, by Theorem 4, So, , and C is a consistent set of .
Theorem 5. Let be a soft context, and . Then, C is a consistent set of if and only if
Proof. Since and are soft contexts over U, from (5) of Theorem 2, it is obviously obtained. □
Theorem 6. Let be a soft context, and . Then, C is a consistent set of if and only if (1) and (2) .
Proof. Let C be a consistent set of . Then, by Theorem 3, and in general, , , and . So, and .
The converse is obviously obtained. □
Theorem 7. Let be a soft context and . Then, the following are equivalent:
- (1)
C is a consistent set of .
- (2)
is a base of .
Proof. (1) ⇒ (2) Let C be a consistent set of . Then, and by Theorem 4, is a base of . So, is also a base of .
(2) ⇒ (1) For , it is obviously . Now, for the proof of the inclusion relation , let . Then, by hypothesis, there exists such that . Since , . Consequently, and so, C is a consistent set of . □
Corollary 1. Let be a soft context. Then, A is the greatest consistent set of .
Theorem 8 (Fundamental Theorem I of consistent sets). Let be a soft context and . Then, C is a consistent set of if and only if for each , there exists a nonempty subset B of C such that .
Proof. Suppose that C is a consistent set of . For the proof, let . Then, by (3) of Theorem 2, . Since is a base for , there exists such that . Set . Then, and . So, the condition is satisfied.
Conversely, suppose that for each , there exists such that . Now, we show that is a base for . It is sufficient to show that for each , there exists such that . For the proof, let . First, since is a base for , we can take such that . From , it follows that there exists such that . Now, put , and . Then, , and . For each , by hypothesis, there exists such that . From Theorem 1, it follows that . Consequently, put ; then, from , it follows that and . So, the proof is completed. □
Corollary 2. Let be a soft context and . Then, C is a consistent set of if and only if for every nonempty subset B of , there exists a nonempty subset E of C such that .
Proof. Suppose that C is a consistent set. Let . Then, for each , there exists such that . So, and . So, the statement is obtained.
The converse is obvious. □
Theorem 9 ([
17])
. Let be a soft context. For , if a mapping defined by for is surjective, then is a base for . By the above theorem, we obtain the following useful theorem in finding consistent sets:
Theorem 10. Let be a soft context and . If there exists a surjective mapping , then C is a consistent set.
Remark 1. In the above theorem, the converse is not always true as shown in the next example.
Example 2. In Example 1, . Consider a consistent set . Then, sinceit is impossible that there is any surjective mapping defined as follows for . So, in Theorem 10, the converse is not true. Let be a soft context and .
For , d is said to be dependent on A if there exists satisfying : Otherwise, d is said to be independent on A.
We denote:
Example 3. In Example 1, , and . For , and so, e is dependent. From , and g are obviously independent on A. a is also independent on A, since but . For the same reason, b and c are independent on A. Obviously, the following are obtained:
Lemma 2. Let be a soft context. Then
- (1)
and ;
- (2)
if a is dependent, then there exists such that where .
Theorem 11. Let be a soft context. Then, is a consistent set of .
Proof. First, we show that the mapping defined by for is well-defined and surjective. For each , suppose that for . Then, , by (3) of Theorem 3, there exists such that and . Then, and . Thus, and . This implies that , and it is a contradiction. So, the mapping is well defined.
For each , by (4) of Theorem 3, there exists an element such that . If , then by Lemma 2, and there exists such that where . Then, , which contradicts . From this fact, we can say that , and so is surjective.
Finally, by Theorem 10, is a consistent set of . □
Definition 2. For a soft context , put . Then, Example 4. In Example 3, for a soft context where and , we showed that and
For ,
;
and
from , ;
from , ;
from , .
Consequently,
The following is directly obtained:
Lemma 3. For a soft context ,
- (1)
;
- (2)
if and only if and for every .
Theorem 12. Let be a soft context and . Then, for , is a consistent set of .
Proof. We showed that there exists a surjective mapping defined by in the proof of Theorem 11. Now, for , let and consider the inclusion map . Since , there is at least one element such that and . This implies that the composition map is surjective, and by Theorem 10, is a consistent set of . □
Theorem 13. Let be a soft context and . Let C be a consistent set of . Then, for , if and only if is not a consistent set of .
Proof. Let and suppose that is a consistent set of . Then, is a base for . From this fact, for , there exists a nonempty subset such that . This contradicts . So, is not a consistent set of .
Conversely, for , assume that . Then, from , . In case: . By Theorem 12, is a consistent set of , and since , is a consistent set of . By hypothesis, since C is a consistent set of , is also a consistent set of .
In case: . Since is also a consistent set of and , is a consistent set of . □
Theorem 14. Let be a soft context and . If C is a consistent set of and , then (1) : (2) .
Proof. (1) It is sufficient to show that . Suppose that for , . Since , . So, , which contradicts the assumption . Consequently, .
(2) Suppose that . Then, for , since , there is no subset B of C such that where for . So, is not a base for , that is, C is a consistent set of . So, the proof is completed. □
The converse of Theorem 14 is not always true as shown in the next example:
Example 5. Let and . As in Example 1, consider a soft context and the set-valued mapping defined by Then, we showed in Example 3. For , in Example 4, we found that and .
Take . Then, it is easily obtained: Clearly, and . However, for , there is no such that . So, by Theorem 8, C is not a consistent set of .
Remark 2 (Essential Zone of consistent sets). For a soft context , from Theorems 13 and 14, we found that every consistent set has to contain the set . This result is very important information that we should consider when constructing a consistent set of a soft context . Based on this fact, the following fundamental theorem of consistent sets is obtained.
Theorem 15 (Fundamental Theorem II of consistent sets). Let be a soft context and . Then, C is a consistent set of if and only if C satisfies that
- (1)
:
- (2)
For each , .
Proof. Let C be a consistent set of . Then, obviously, (1) is satisfied by Theorem 14. For the proof of (2), suppose that there is some such that . For , let . Then, we know that and . Now, for , there is no such that . This means that there is no such that . So, is not a base for , C is not a consistent set.
For the converse, assume that the two conditions (1) and (2) are satisfied. Put for . Then, for each , since , is a nonempty subset of C, and so . Now, we show that the mapping defined by for is surjective. For the proof of surjection, let . Then, by Theorem 11, there is an element such that . From , in case , it is obviously . In case , by the condition (2), there exists some such that . So, in any case, for , there exists satisfying . Consequently, is surjective and so is a base for , and so, is also a base for . Hence, C is a consistent set of . □
Corollary 3. Let be a soft context and . Then, for each , there exists a nonempty subset B of C such that if and only if C satisfies that
- (1)
;
- (2)
For each , .
Proof. It follows from Theorems 8 and 15. □
Let
be a soft context. In [
17], we defined an order between
as follows: For
,
Then the
infimum ⋏ and
supremum ⋎ in the ordered set
, are defined as follows:
Then, we showed that is a complete lattice. The complete lattice is called the soft concept lattice and simply, denoted by .
Let
and
be two soft concept lattices.
is said to be
finer than
, which is denoted by the following:
If
and
, then these two soft concept lattices are said to be
isomorphic to each other, and denoted as follows
Theorem 16 ([
17])
. Let be a soft context. Then, if and only if . Theorem 17. Let be a soft context. If C is a consistent set of , then .
Proof. For a consistent set , by Theorem 5, if and only if . So, . □
For a formal context
, let us define a soft set
as follows
. Then,
is a soft context. Therefore, every formal context
induces a soft context
. We call
the
associated soft context [
17] induced by a formal context
. Furthermore, for
. Then, denote
. Then,
is a formal context and
is also the
associated soft context induced by a formal context
.
Theorem 18 ([
17])
. Let be a formal context. Then,- (1)
the formal concept lattice is order-isomorphic to ;
- (2)
for the formal concept lattice of ,
Corollary 4. Let be a formal context and . Then,
- (1)
the formal concept lattice is order-isomorphic to ;
- (2)
Proof. (1) For , is a formal context and is the associated soft context of . By Theorem 18, the formal concept lattice is order-isomorphic to .
(2) Since is a formal context and is the formal concept lattice of , it follows from (2) of Theorem 18. □
Let
be a formal context. Then,
D is called
a consistent [
7] of
if there exists a set
such that
.
Finally, we obtain the meaningful theorem for the consistent set of a given formal context:
Theorem 19. Let be a formal context. Then, C is a consistent of if and only if C is a consistent set of the associated soft context .
Proof. Let C be a consistent set of a formal context . Then, by Theorem 18 and Corollary 4, if and only if if and only if . Hence, C is also a consistent set of the associated soft context . □
4. Application
In this section, we apply the Fundamental Theorem II of Consistent Sets (Theorem 15) to obtain a consistent set of a given formal context. We may write instead of a set .
For
and
,
Table 1 shows a formal context
.
From the formal context , naturally, we can define a set-valued mapping as , and from now on, the set-valued mapping is simply denoted by F. Then, is a soft set as follows:
Therefore, we have the associated soft context
(simply,
) induced by a formal context
. Then, by Theorem 4 and
, we can find the family
of all soft concepts as:
Consequently, by Theorem 18 and , the formal concept lattice is easily obtained as the following diagram:
From now on, we will describe the process of obtaining a consistent set of by using Theorem 19.
First, we explain how to obtain a consistent set of the associated soft context through three steps by using Theorem 15.
Step 1: By using the soft set defined above, we obtain the independent set on A in the following way:
- (1)
From ,
; ; ;
; ; :
- (2)
;
;
;
; .
From
and
, it follows that
, and so,
Step 2: We find two classes and of in the following way: For ,
; ;
; ;
.
So, ; ; .
Hence,
Step 3: By using Theorem 15, we construct a consistent set
C of the associated soft context
as follows:
So finally, we obtain a consistent set of the associated soft context .
Furthermore, by Theorem 19, is also a consistent set of a formal context and .
In fact, we can construct the formal concept lattice by using the following facts: For a consistent set of the associated soft context ,
- (1)
is a set-valued mapping defined as follows:
- (2)
.
- (3)
is a mapping defined as follows: For
Finally, by Corollary 4 and ,
where