3. Properties of the Categories and
In this section, first, we write the concept of a cubic set introduced by Jun et al. [
13] (Also, see [
13] for the equality
and orders
for any cubic sets
the complement
of a cubic set
, and the unions
and intersections
of two cubic sets
). Next, we introduce the category
[resp.
] consisting of all cubic
H-relational spaces and all P-preserving [resp. R-preserving] mappings between any two cubic
H-relational spaces and it has the similar structures as those of
[resp.
] (See [
35]).
Throughout this section and next section,
H denotes a complete Heyting algebra (Refer to [
36,
37] for its definition) and
denotes the set of all closed subintervals of
H.
Definition 4 ([
13])
. Let X be a nonempty set. Then a complex mapping is called a cubic set in X, where and be the set of all closed subintervals of I.A cubic set in which and (resp. and ) for each is denoted by (resp. ).
A cubic set in which and (resp. and ) for each is denoted by (resp. ). In this case, (resp. ) will be called a cubic empty (resp. whole) set in X.
We denote the set of all cubic sets in X by .
Definition 5. Let X be a nonempty set. Then a complex mapping is called a cubic H-relation in X. The pair is called a cubic H-relational space. In particular, a cubic H-relation from X to X is called a H-relation in or on X. We will denote the set of all cubic H-relations in X as resp. . In fact, each member is a cubic H-set in (See [35]). Definition 6. Let and be two cubic H-relational spaces. Then a mapping is called:
- (i)
a P-order preserving mapping, if it satisfies the following condition: - (ii)
a R-order preserving mapping, if it satisfies the following condition:where .
Proposition 1. Let , and be three cubic H- relational spaces.
- (1)
The identity mapping is a P-order [resp. R-oder] preserving mapping.
- (2)
If and are P-preserving [resp. R-preserving] mappings, then is a P-preserving [resp. R-preserving] mapping.
Proof. (1) The proof follows from the definitions of P-orders and R-orders, and identity mappings.
(2) Suppose
and
are P-preserving mappings and let
. Then
f is a P-preserving mapping]
g is a P-preserving mapping]
.
Thus, . So is a P-preserving mapping. □
We will denote the collection consisting of all cubic H-relational spaces and all P-preserving [resp. R-preserving] mappings between any two cubic H-relational spaces as [resp. ]. Then from Proposition 1, we can easily see that [resp. ] forms a concrete category. In the sequel, a P-preserving [resp. R-preserving] mapping between any two cubic H-spaces will be called a -mapping [resp. -mapping].
Lemma 1. The category [resp. ] is topological over .
Proof. Let
X be a set and let
be any family of cubic
H-relational spaces indexed by a class
J. Suppose
be a source of mappings. We define a mapping
as follows: for each
,
Then clearly, for each
and
,
Thus, , for each . So is a -mapping, for each .
For any object
, let
be any mapping for which
is a
-mapping, for each
and let
. Then for each
,
Thus,
[By the definition of ]
So Hence is a -mapping. Therefore is an initial source in .
Now define a mapping
as below: for each
,
Then clearly, for each
and
,
Thus, , for each . So is a -mapping, for each .
For any object
, let
be any mapping for which
is a
-mapping, for each
and let
. Then for each
,
Thus,
[By the definition of ]
So Hence is a -mapping. Therefore is an initial source in . This completes the proof. □
Example 1. (1) (Inverse image of a cubicH-relation) Let X be a set, let be a cubic H-relational space and let be a mapping. Then there exists a unique initial cubic H-relation of P-order type [resp. R-order type ] in X for which is a -mapping [resp. is a -mapping]. In fact, In this case, [resp. ] is called the inverse image under f of the cubic H-relation in Y.
In particular, if and is the inclusion mapping, then the inverse image [resp. ] of under f is called a cubic H-subrelation of . In fact, (2) (CubicH-product relation) Let be any family of cubic H-relational spaces and let . For each , let be the ordinary projection. Then there exists a unique cubic H-relation of P-order type, in X for which is a -mapping, for each . In this case, is called the cubic H-product relation of and is called the cubic H-product relational space of , and denoted as the following, respectively:and In fact, , for each .
Similarly, there exists a unique cubic H-relation of R-order type, in X for which is a -mapping, for each . In this case, is called the cubic H-product relation of and is called the cubic H-product relational space of , and denoted as the following, respectively:and In fact, , for each .
In particular, if , then for each ,
and
.
The following is obvious from Lemma 3.9 and Theorem 1.6 in [
25] or Proposition in
Section 1 in [
38].
Corollary 1. The category [resp. ] is complete and cocomplete over .
Furthermore, we can easily see that
[resp.
] is well-powered and cowell-powered. It is well-known that a concrete category is topological if and only if it is cotopological (See Theorem 1.5 in [
25]). However, we prove directly that
[resp.
] is cotopological.
Lemma 2. The category [resp. ] is cotopological over .
Proof. Let
X be any set and let
be any family of cubic
H-relational spaces indexed by a class
J. Suppose
is a sink of mappings. We define a mapping
as follows: for each
,
Then we can easily see that
For any cubic H-relational space , let be any mapping such that is a -mapping, for each and let . Then for each and each ,
Thus, by the definition of , So . Hence is a -mapping. Therefore is cotopological over .
Now we define a mapping as follows: for each ,
Then we can easily see that
For any cubic H-relational space , let be any mapping such that is a -mapping, for each and let . Then for each and each ,
Thus, by the definition of , So . Hence is a -mapping. Therefore is cotopological over . This completes the proof. □
Example 2. (CubicH-quotient relation) Let be a cubic H-relational space, let ∼ be an equivalence relation on X and let be the canonical mapping. We define a mapping as below: for each ,
Then we can easily see that is a cubic H-relation in . Furthermore, is a -mapping. Thus, is the final cubic H-relation in .
Now we define a mapping as follows: for each ,
Then we can easily see that is a cubic H-relation in . Furthermore, is a -mapping. Thus, is the final cubic H-relation in .
In this case, [resp. ] is called the cubic H-quotient [resp. H-quotient] relation in X induced by ∼.
Definition 7 ([
38])
. Let be a concrete category and let be two -morphisms. Then a pair is called an equalizer in of f and g, if the following conditions hold:- (i)
is an -morphism,
- (ii)
,
- (iii)
for any -morphism such that , there exists a unique -morphism such that .
In this case, we say that has equalizers.
Dual notion: Coequalizer.
Proposition 2. The category [resp. ] has equalizers.
Proof. Let
be two
-mappings, where
and
. Let
and define a mapping
as follows: for each
,
Then clearly, is a cubic H-relation in E and . Consider the inclusion mapping . Then clearly, is a -mapping and .
Let
be a
-mapping such that
. We define a mapping
as follows: for each
,
Then clearly, .
Let . Since is a -mapping,
Thus, . So is a -mapping.
Now in order to prove the uniqueness of , let such that . Then Thus, is unique. Hence has equalizers.
Similarly, we can prove that has the equalizer . □
For two cubic
H-relations
in
X and
in
Y, the product of P-order type [resp. R-order type], denoted by
[resp.
], is a cubic
H-relation in
defined by: for any
,
[resp. ].
Lemma 3. Final episinks in [resp. ] are preserved by pullbacks.
Proof. Let
be any final episink in
and let
be any
-mapping, where
,
and
. For each
, let
and let us define a mapping
as follows: for each
,
, i.e.,
For each , let and be the usual projections. Then clearly, and are -mappings and , for each . Thus, we have the following pullback square in :
We will prove that is a final episink in . Let . Since is an episink in , there is such that , for some . Thus, and . So is an episink in .
Finally, let us show that is final in . Let be the final structure in W regarding and let . Then
is a -mapping]
is a final episink in ]
.
Thus, . Since is final, is a -mapping. So . Hence . Therefore is final.
Now we define a mapping as follows: for each ,
For each , let and be the usual projections. Then we can similarly prove that final episinks in are preserved by pullbacks. This completes the proof. □
For any singleton set , since the cubic set in is not unique, the category is not properly fibered over . Then from Definitions 1 and 3, Lemmas 2 and 3, we have the following result.
Theorem 1. The category [resp. ] satisfies all the conditions of a topological universe over except the terminal separator property.
Theorem 2. The category [resp. ] is Cartesian closed over .
Proof. From Lemma 1, it is clear that [resp. ] has products. Then it is sufficient to prove that [resp. ] has exponential objects.
For any cubic H-relational spaces and , let be the set of all ordinary mappings from X to Y. We define two mappings and as follows: for each ,
and
Then clearly,
is a cubic
H-relation in
. Moreover, by the definitions of
and
,
and
for each
.
Let
and let us define a mapping
as follows: for each
,
Let Then
the definition of ]
and
.
Thus, is a -mapping, where .
For any cubic
H-relational space
, let
be a
-mapping. We define a mapping
as follows: for each
and each
,
Then we can prove that is a unique -mapping such that .
Now we define two mappings
and
as follows: for each
and each
,
and
Then clearly,
is a cubic
H-relation in
. Moreover, by the definitions of
and
,
and
for each
Let
and let us define a mapping
as follows: for each
,
Let
Then by the definitions of
and
, we have the followings:
and
Thus, So is a -mapping, where .
For any cubic
H-relational space
, let
be a
-mapping. We define a mapping
as follows: for each
and each
,
Then we can prove that
is a unique
-mapping such that
This completes the proof. □
Remark 1. The category [resp. ] is not a topos (See [39] for its definition), since it has no subobject classifier. Example 3. Let be two points chain, respectively and let . Let and be the cubic H-relations in X defined by: Let be the identity mapping. Then clearly, is both monomorphism and epimorphism in [resp. ]. However, is not an isomorphism in [resp. ]. Thus, has no subobject classifier.
4. The Categories and
In this section, we obtain two subcategories and of and , respectively which are topological universes over .
It is interesting that final structures and exponential objects in [resp. ] are shown to be quite different from those in [resp. ].
First of all, we list two well-known results.
Result 1 (Theorem 2.5 [
25]). Let
be a well-powered and co(well-powered) topological category. Then the followings are equivalent:
- (1)
is bireflective in ,
- (2)
is closed under the formation of initial sources, i.e., for any initial source in with for each , then .
Result 2 (Theorem 2.6 [
25]). If
is a topological category and
is a bireflective subcategory of
, then
is also a topological category. Moreover, every source in
which is initial in
is initial in
.
Definition 8. Let X be a nonempty set and let be a cubic H-relation in X. Then is said to be reflexive, if R and λ are reflexive, i.e., and , for each .
The class of all cubic H-reflexive relational spaces and -mappings [resp. -mappings between them forms a subcategory of [resp. ] denoted by [resp. ].
The following is the immediate result of Definitions 1 and 8.
Lemma 4. The category [resp. ] is properly fibered over .
Lemma 5. The category [resp. ] is closed under the formation of initial sources in The category [resp. ]
Proof. Let
be an initial source in
such that each
belongs to
, where
and
. Let
and let
. Since
and
are reflexive,
and
. Then
Thus, . So is reflexive.
Now let
be an initial source in
such that each
belongs to
. Then clearly, for each
,
Thus, . So is reflexive. This completes the proof. □
From Results 1, 2 and Lemma 5, we have the followings.
Proposition 3. (1) The category [resp. ] is a bireflective subcategory of [resp. ].
(2) The category [resp. ] is topological over .
It is well-known that a category is topological if and only if it is cotopological. Then by (2) of the above Proposition, the category [resp. ] is cotopological over . However, we will prove that [resp. ] is cotopological over , directly.
Lemma 6. the category [resp. ] has final structure over .
Proof. Let
X be a nonempty set and let
be any family of cubic
H-relational spaces indexed by a class
J. We define two mappings
and
, respectively as below: for each
,
and
where
. Then clearly,
is the cubic
H-reflexive relation in
X given by: for each
,
Moreover, we can easily check that is a final structure in . Thus, is a final sink in .
Now we define two mappings
and
, respectively as follows: for each
,
and
Then clearly,
is the cubic
H-reflexive relation in
X given by: for each
,
Moreover, we can easily show that is a final sink in . □
Lemma 7. Final episinks in [resp. ] are preserved by pullbacks.
Proof. Let be any final episink in and let be any -mapping, where is a cubic H-reflexive relational space. For each , let us take , , and as in the first proof of Lemma 3. Then we can easily check that is closed under the formation of pullbacks in . Thus, it is enough to prove that is final.
Suppose is the final cubic H-relation in W regarding and let . Then
is a -mapping]
is a final episink in ]
.
Thus, . On the other hand, by a similar argument in the first proof of Lemma 3, on . So on . Now let . Then clearly, . Thus, on . Hence on W.
Now for each , let us be the mapping as in the second proof of Lemma 3. Then we can similarly prove that final episinks in are preserved by pullbacks. This completes the proof. □
The following is the immediate result of Lemma 4, Proposition 3 (2) and Lemma 7.
Theorem 3. The category [resp. ] is a topological universe over . In particular, [resp. ] is Cartesian closed over (See [
1])
and a concrete quasitopos(See [
40]).
In [
41], Noh obtained exponential objects in
, where
denotes the category of fuzzy relations. By applying his construction of an exponential object in
to the category
[resp.
], we have the following.
Proposition 4. The category [resp. ] has an exponential object.
Proof. For any
and let
. For any
, let
We define a mapping as follows: for each ,
Then by the definition of , , for each . Thus, , for each . So is a cubic H-reflexive relation in .
Let
and we define the mapping
as follows: for each
,
Let .
Case 1: Suppose . Then
the definition of , , ]
[Since ]
.
Case 2: Suppose . Then
.
Thus, in either case, . So is a -mapping.
Let
be any cubic
H-reflexive relational space and let
be any
-mapping. We define the mapping
as follows: for each
and each
,
Let and let . Then
[Since is reflexive]
.
Thus, . So is a -mapping. Hence is well-defined. Let .
Case 1: Suppose . Then
[By the definition of ]
.
Case 2: Suppose . Then
.
On one hand, for any ,
.
Thus,
. Similarly, we have
. So
Hence in either cases, . Therefore is a -mapping. Furthermore, is unique and .
Now for any
and let
. For any
, let
We define a mapping as follows: for each ,
Then we can easily check that is a cubic H-reflexive relation in . Moreover, by the similar argument of the above proof, we can show that is an exponential object in . This completes the proof. □
Remark 2. (1) We can see that exponential objects in [resp. ] is quite different from those in [resp. ] constructed in Theorem 1.
(2) The category [resp. ] has no subject classifier.
Example 4. Let be the two points chain and let . Let and be cubic H-reflexive relations in X given by:and Let be the identity mapping. Then clearly, is both monomorphism and epimorphism in . However, is not an isomorphism in .