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Article

A Study on Cubic H-Relations in a Topological Universe Viewpoint

1
Division of Applied Mathematics, Nanoscale Science and Technology Institute, Wonkwang University, Iksan 54538, Korea
2
Department of Applied Mathematics, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Korea
3
School of Mathematics, Shandong University of Technology, Zibo 255049, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 482; https://doi.org/10.3390/math8040482
Submission received: 18 February 2020 / Revised: 22 March 2020 / Accepted: 27 March 2020 / Published: 1 April 2020
(This article belongs to the Special Issue Fuzziness and Mathematical Logic )

Abstract

:
We introduce the concrete category CRel P ( H ) [resp. CRel R ( H ) ] of cubic H-relational spaces and P-preserving [resp. R-preserving] mappings between them and study it in a topological universe viewpoint. In addition, we prove that it is Cartesian closed over Set . Next, we introduce the subcategory CRel P , R ( H ) [resp. CRel R , R ( H ) ] of CRel P ( H ) [resp. CRel R ( H ) ] and investigate it in the sense of a topological universe. In particular, we obtain exponential objects in CRel P , R ( H ) [resp. CRel R , R ( H ) ] quite different from those in CRel P ( H ) [resp. CRel R ( H ) ].

1. Introduction

In 1984, Nel [1] introduced the concept of a topological universe which implies quasitopos [2]. Its notion has already been put to effective use several areas of mathematics in [3,4,5]. After then, Kim et al. [6] and Lee et al. [7] constructed the category NSet ( H ) of neutrosophic H-sets and morphisms between them and the category NCSet ( H ) of neutrosophic crisp sets and morphisms between them, and they studied each category in the sense of a topological universe. On the other hand, Cerruti [8] constructed the category of L-fuzzy relations and obtained some of its properties. Hur [9,10] [resp. Hur et al. [11] and Lim et al [12] formed the category Rel ( H ) of H-fuzzy relational spaces [resp. IRel ( H ) of H-intuitionistic fuzzy relational spaces and VRel ( H ) of vague relational spaces] and each category was investigated in topological universe viewpoint.
In 2012, Jun et al. [13] introduced the notion of a cubic set and investigated some of its properties. After that time, Ahn and Ko [14] studied cubic subalgebras and filters of C I -algebras. Akram et al. [15] applied the concept of cubic sets to K U -algebras. Jun et al. [16] dealt with cubic structures of ideals of B C I -algebras. Jun and Khan [17] found some properties of cubic ideals in semigroups. Jun et al. [18] studied cubic subgroups. Zeb et al. [19] defined the notion of a cubic topology and investigated some of its properties. Recently, Mahmood et al. [20] dealt with multicriteria decision making based on cubic sets. Rashed et al. [21] applied the concept of cubic sets to graph theory. Yaqoob et al. [22] introduced the notion of a cubic finite switchboard state machine and studied its various properties. Ma et al. [23] define a cubic relation on H v -LA-semigroup and investigated some of its properties. Kim et al. [24] defined cubic relations and obtained some their properties.
In this paper, we study the category of cubic relations and morphisms between them in the sense of a topological universe proposed by Nel. First, we define the concept of a cubic H-relational space for a Heyting algebra H and introduce the concrete category CRel P ( H ) [resp. CRel R ( H ) ] of cubic H-relational spaces and P-preserving [resp. R-preserving] mappings between them, and obtained some categorical structures and give examples. In particular, we prove that the category CRel P ( H ) [resp. CRel R ( H ) ] is Cartesian closed over Set , where Set denotes the category consisting of ordinary sets and ordinary mappings between them. Next, we introduce the subcategory CRel P , R ( H ) [resp. CRel R , R ( H ) ] of CRel P ( H ) [resp. CRel R ( H ) ] and investigate it in the sense of a topological universe. In particular, we obtain exponential objects in CRel P , R ( H ) [resp. CRel R , R ( H ) ] quite different from those in CRel P ( H ) [resp. CRel R ( H ) ].

2. Preliminaries

In this section, we list some basic definitions for category theory which are needed in the next sections. Let us recall that a concrete category is a category of sets which are endowed with an unspecified structure. Refer to [25] for the notions of a topological category and a cotopological category.
Definition 1
([25]). Let A be a concrete category.
(i) 
The A -fiber of a set X is the class of all A -structures on X.
(ii) 
A is said to be properly fibered over Set , if it satisfies the following:
(a) 
(Fiber-smallness) for each set X, the A -fiber of X is a set,
(b) 
(Terminal separator property) for each singleton set X, the A -fiber of X has precisely one element,
(c) 
if ξ and η are A -structures on a set X such that i d : ( X , ξ ) ( X , η ) and i d : ( X , η ) ( X , ξ ) are A -morphisms, then ξ = η .
Definition 2
([26]). A category A is said to be Cartesian closed, if it satisfies the following conditions:
(i) 
for each A -object A and B, there exists a product A × B in A ,
(ii) 
exponential objects exist in A, i.e., for each A -object A, the functor A × : A A has a right adjoint, i.e., for any A -object B, there exist an A -object B A and a A -morphism e A , B : A × B A B (called the evaluation) such that for any A -object C and any A -morphism f : A × C B , there exists a unique A -morphism f ¯ : C B A such that e A , B ( 1 A × f ¯ ) = f , i.e., the diagram commutes:
    Mathematics 08 00482 i001
Definition 3
([1]). A category A is called a topological universe over Set if it satisfies the following conditions:
(i) 
A is well-structured, i.e., (a) A is concrete category; (b) fiber-smallness condition; (c) A has the terminal separator property,
(ii) 
A is cotopological over Set ,
(iii) 
final episinks in A are preserved by pullbacks, i.e., for any episink ( g j : X j Y ) J and any A -morphism f : W Y , the family ( e j : U j W ) J , obtained by taking the pullback f and g j , for each j J , is again a final episink.
Now refer to [13,27,28,29,30,31,32,33,34] for the concepts of fuzzy sets, fuzzy relations, interval-valued fuzzy sets and interval-valued fuzzy relations, neutrosophic crisp sets, neutrosophic sets and operation between them, respectively.

3. Properties of the Categories HRel P ( H ) and HRel R ( H )

In this section, first, we write the concept of a cubic set introduced by Jun et al. [13] (Also, see [13] for the equality A = B and orders A B , A B for any cubic sets A , B , the complement A c of a cubic set A , and the unions A B , A B and intersections A B , A B of two cubic sets A , B ). Next, we introduce the category CRel P ( H ) [resp. CRel R ( H ) ] 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 CSet P ( H ) [resp. CSet R ( H ) ] (See [35]).
Throughout this section and next section, H denotes a complete Heyting algebra (Refer to [36,37] for its definition) and [ H ] denotes the set of all closed subintervals of H.
Definition 4
([13]). Let X be a nonempty set. Then a complex mapping A = < A , λ > : X [ I ] × I is called a cubic set in X, where I = [ 0 , 1 ] and [ I ] be the set of all closed subintervals of I.
A cubic set A = < A , λ > in which A ( x ) = 0 and λ ( x ) = 1 (resp. A ( x ) = 1 and λ ( x ) = 0 ) for each x X is denoted by 0 ¨ (resp. 1 ¨ ).
A cubic set B = < B , μ > in which B ( x ) = 0 and μ ( x ) = 0 (resp. B ( x ) = 1 and μ ( x ) = 1 ) for each x X is denoted by 0 ^ (resp. 1 ^ ). In this case, 0 ^ (resp. 1 ^ ) will be called a cubic empty (resp. whole) set in X.
We denote the set of all cubic sets in X by ( [ I ] × I ) X .
Definition 5.
Let X be a nonempty set. Then a complex mapping R = < R , λ > : X × X [ H ] × H is called a cubic H-relation in X. The pair ( X , R ) 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. ( [ H ] × H ) X × X . In fact, each member R = < R , λ > ( [ H ] × H ) X × Y is a cubic H-set in X × X (See [35]).
Definition 6.
Let ( X , R X ) = ( X , < R X , λ X > ) and ( Y , R Y ) = ( Y , < R Y , λ Y > ) be two cubic H-relational spaces. Then a mapping f : ( X , R X ) ( Y , R Y ) is called:
(i) 
a P-order preserving mapping, if it satisfies the following condition:
R X R Y f 2 = < R Y f 2 , λ Y f 2 > , i . e . , for each ( x , y ) X × X ,
< [ R X ( x , y ) , R X + ( x , y ) ] , λ ( x , y ) >
P < [ R Y ( f ( x ) , f ( y ) ) , R Y + ( f ( x ) , f ( y ) ) ] , λ Y ( f ( x ) , f ( y ) ) > , i . e . ,
R X ( x , y ) ( R Y f 2 ) ( x , y ) , R X + ( x , y ) ( R Y + f 2 ) ( x , y ) , λ X ( x , y ) ( λ Y f 2 ) ( x , y ) ,
(ii) 
a R-order preserving mapping, if it satisfies the following condition:
R X R Y f 2 = < R Y f 2 , λ Y f 2 > , i . e . , for each ( x , y ) X × X ,
< [ R X ( x , y ) , R X + ( x , y ) ] , λ ( x , y ) >
R < [ R Y ( f ( x ) , f ( y ) ) , R Y + ( f ( x ) , f ( y ) ) ] , λ Y ( f ( x ) , f ( y ) ) > , i . e . ,
R X ( x , y ) ( R Y f 2 ) ( x , y ) , R X + ( x , y ) ( R Y + f 2 ) ( x , y ) , λ X ( x , y ) ( λ Y f 2 ) ( x , y ) ,
where f 2 = f × f .
Proposition 1.
Let ( X , R X ) = ( X , < R X , λ X > ) , ( Y , R Y ) = ( Y , < R Y , λ Y > ) and ( Z , R Z ) = ( Z , < R Z , λ Z > ) be three cubic H- relational spaces.
(1) 
The identity mapping 1 X : ( X , R X ) ( X , R X ) is a P-order [resp. R-oder] preserving mapping.
(2) 
If f : ( X , R X ) ( Y , R Y ) and g : ( Y , R Y ) ( Z , R Z ) are P-preserving [resp. R-preserving] mappings, then g f : ( X X , R X ) ( Z , R Z ) 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 f : ( X , R X ) ( Y , R Y ) and g : ( Y , R Y ) ( Z , R Z ) are P-preserving mappings and let ( x , y ) X × X . Then
R X ( x , y ) = < [ R X ( x , y ) , R X + ( x , y ) ] , λ X ( x , y ) >
P < [ ( R Y f 2 ) ( x , y ) , R Y + f 2 ) ( x , y ) ] , λ Y f 2 ) ( x , y ) >
[ Since f is a P-preserving mapping]
= < [ R Y ( f ( x ) , f ( y ) ) , R Y + ( f ( x ) , f ( y ) ) ] , λ Y ( f ( x ) , f ( y ) ) >
P [ R Z ( g ( f ( x ) ) , g ( f ( y ) ) ) , R Z + ( g ( f ( x ) ) , g ( f ( y ) ) ) ] , λ Z ( g ( f ( x ) ) , g ( f ( y ) ) ) >
[ Since g is a P-preserving mapping]
= [ R Z ( g f ) 2 ( x , y ) , R Z + ( g f ) 2 ( x , y ) ] , λ Z ( g f ) 2 ( x , y ) > .
Thus, R X R Z ( g f ) 2 . So g f 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 CRel P ( H ) [resp. CRel R ( H ) ]. Then from Proposition 1, we can easily see that CRel P ( H ) [resp. CRel R ( H ) ] forms a concrete category. In the sequel, a P-preserving [resp. R-preserving] mapping between any two cubic H-spaces will be called a CRel P ( H ) -mapping [resp. CRel R ( H ) -mapping].
Lemma 1.
The category CRel P ( H ) [resp. CRel R ( H ) ] is topological over Set .
Proof. 
Let X be a set and let ( X j , R j ) j J = ( X j , < R j , λ j > ) be any family of cubic H-relational spaces indexed by a class J. Suppose ( f j : X X j ) J be a source of mappings. We define a mapping R X , P = < R X , P , λ X , P > : X × X [ H ] × H as follows: for each ( x , y ) X × X ,
R X , P ( x ) = [ j J ( R j f j 2 ) ] ( x , y ) , i . e . ,
R X , P ( x , y ) = < [ j J R j ( f j ( x ) , f j ( y ) ) , j J R j + ( f j ( x ) , f j ( y ) ) , j J λ j ( f j ( x ) , f j ( y ) ) > .
Then clearly, for each j J and ( x , y ) X × X ,
< [ R X , P ( x , y ) , R X , P + ( x , y ) ] , λ X , P ( x , y ) > P < [ R j ( f j ( x ) , f j ( y ) ) , R j + ( f j ( x ) , f j ( y ) ) , λ j ( f j ( x ) , f j ( y ) ) > .
Thus, R X , P R j f j 2 , for each j J . So f j : ( X , R X , P ) ( X j , R j ) is a CRel P ( H ) -mapping, for each j J .
For any object ( Y , R Y ) = ( Y , < R Y , λ Y ) , let g : Y X be any mapping for which f j g : ( Y , R Y ) ( X j , R j ) is a CRel P ( H ) -mapping, for each j J and let ( y , y ) Y × Y . Then for each j J ,
R Y ( y , y ) P [ R j ( f j g ) 2 ] ( y , y ) = [ ( R j f j 2 ) g 2 ] ( y , y ) , i . e . ,
< [ R Y ( y , y ) , R Y + ( y , y ) ] , λ Y ( y , y ) >
P < [ ( R j f j 2 ) ( g ( y ) , g ( y ) ) , ( R j + f j 2 ) ( g ( y ) , g ( y ) ] , ( λ j f j 2 ) ( g ( y ) , g ( y ) > .
Thus,
< [ R Y ( y , y ) , R Y + ( y , y ) ] , λ Y ( y , y ) >
P < [ j J ( R j f j 2 ) ( g ( y ) , g ( y ) , j J ( R j f j 2 ) ( g ( y ) , g ( y ) ] ,
j J ( λ j f j 2 ) ( g ( y ) , g ( y ) >
= [ j J ( R j f j ) ] ( g ( y ) , g ( y )
= ( R X , P g 2 ) ( y , y ) . [By the definition of R X , P ]
So R Y R X , P g 2 . Hence g : ( Y , R Y ) ( X , R X , P ) is a CRel P ( H ) -mapping. Therefore ( f j : ( X , R X , P ) ( X j , R j ) ) J is an initial source in CRel P ( H ) .
Now define a mapping R X , R = < R X , R , λ X , R > : X × X [ H ] × H as below: for each ( x , y ) X × X ,
R X , R ( x ) = [ j J ( R j f j 2 ) ] ( x , y ) , i . e . , R X , R ( x , y ) = < [ j J R j ( f j ( x ) , f j ( y ) ) , j J R j + ( f j ( x ) , f j ( y ) ) , j J λ j ( f j ( x ) , f j ( y ) ) > .
Then clearly, for each j J and ( x , y ) X × X ,
< [ R X , R ( x , y ) , R X , R + ( x , y ) ] , λ X , R ( x , y ) > R < [ R j ( f j ( x ) , f j ( y ) ) , R j + ( f j ( x ) , f j ( y ) ) ] , λ j ( f j ( x ) , f j ( y ) ) > .
Thus, R X , R R j f j 2 , for each j J . So f j : ( X , R X , R ) ( X j , R j ) is a CRel R ( H ) -mapping, for each j J .
For any object ( Y , R Y ) = ( Y , < R Y , λ Y ) , let g : Y X be any mapping for which f j g : ( Y , R Y ) ( X j , R j ) is a CRel R ( H ) -mapping, for each j J and let ( y , y ) Y × Y . Then for each j J ,
R Y ( y , y ) R [ R j ( f j g ) 2 ] ( y , y ) = [ ( R j f j 2 ) g 2 ] ( y , y ) , i . e . ,
< [ R Y ( y , y ) , R Y + ( y , y ) ] , λ Y ( y , y ) > R < [ ( R j f j 2 ) ( g ( y ) , g ( y ) ) , ( R j + f j 2 ) ( g ( y ) , g ( y ) ] , ( λ j f j 2 ) ( g ( y ) , g ( y ) > .
Thus,
< [ R Y ( y , y ) , R Y + ( y , y ) ] , λ Y ( y , y ) >
R < [ j J ( R j f j 2 ) ( g ( y ) , g ( y ) , j J ( R j f j 2 ) ( g ( y ) , g ( y ) ] ,
j J ( λ j f j 2 ) ( g ( y ) , g ( y ) >
= [ j J ( R j f j ) ] ( g ( y ) , g ( y )
= ( R X , R g 2 ) ( y , y ) . [By the definition of R X , R ]
So R Y R X , R g 2 . Hence g : ( Y , R Y ) ( X , R X , R ) is a CRel R ( H ) -mapping. Therefore ( f j : ( X , R X , R ) ( X j , R j ) ) J is an initial source in CRel R ( H ) . This completes the proof. □
Example 1.
(1) (Inverse image of a cubicH-relation) Let X be a set, let ( Y , R Y ) = ( Y , < R Y , λ Y > ) be a cubic H-relational space and let f : X Y be a mapping. Then there exists a unique initial cubic H-relation of P-order type R X , P [resp. R-order type R X , R ] in X for which f : ( X , R X , P ) ( Y , R Y ) is a CRel P ( H ) -mapping [resp. f : ( X , R X , R ) ( Y , R Y ) is a CRel R ( H ) -mapping]. In fact,
R X , P = R Y f 2 = < R Y f 2 , λ Y f 2 > = R X , R .
In this case, R X , P [resp. R X , R ] is called the inverse image under f of the cubic H-relation R Y in Y.
In particular, if X Y and f : X Y is the inclusion mapping, then the inverse image R X , P [resp. R X , R ] of R Y under f is called a cubic H-subrelation of ( Y , R Y ) . In fact,
R X , P ( x , y ) = R Y ( x , y ) = R X , R ( x , y ) , for each ( x , y ) X × X .
(2) (CubicH-product relation) Let ( ( X j , R j ) ) j J = ( ( X j , < R j , λ j > ) ) j J be any family of cubic H-relational spaces and let X = Π j J X j . For each j J , let p r j : X X j be the ordinary projection. Then there exists a unique cubic H-relation of P-order type, R X , P in X for which p r j : ( X , R X , P ) ( X j , R j ) is a CRel P ( H ) -mapping, for each j J . In this case, R X , P is called the cubic H-product relation of ( R j ) j J and ( X , R X , P ) is called the cubic H-product relational space of ( ( X j , R j ) ) j J , and denoted as the following, respectively:
R X , P = Π j J R j
and
( X , R X , P ) = ( Π j J X j , Π j J R j ) = ( Π j J X j , < Π j J R j , Π j J λ j > ) .
In fact, R X , P ( x ) = [ j J ( R j p r j ) ] ( x , y ) , for each ( x , y ) X × X .
Similarly, there exists a unique cubic H-relation of R-order type, R X , R in X for which p r j : ( X , R X , R ) ( X j , R j ) is a CRel R ( H ) -mapping, for each j J . In this case, R X , R is called the cubic H-product * relation of ( R j ) j J and ( X , R X , R ) is called the cubic H-product * relational space of ( ( X j , R j ) ) j J , and denoted as the following, respectively:
R X , R = Π j J * R j
and
( X , R X , R ) = ( Π j J X j , Π j J * R j ) = ( Π j J X j , < Π j J R j , Π j J * λ j > ) .
In fact, R X , R ( x , y ) = [ j J ( R j p r j ) ] ( x , y ) , for each ( x , y ) X × X .
In particular, if J = { 1 , 2 } , then for each ( x 1 , y 1 ) , ( x 2 , y 2 ) X 1 × X 2 ,
( R 1 × R 2 ) ( ( x 1 , y 1 ) , ( x 2 , y 2 ) )
= < [ R 1 ( x 1 , x 2 ) R 2 ( y 1 , y 2 ) , R 1 + ( x 1 , x 2 ) R 2 + ( y 1 , y 2 ) ] , λ 1 ( x 1 , x 2 ) λ 2 ( y 1 , y 2 ) >
and
( R 1 × * R 2 ) ( ( x 1 , y 1 ) , ( x 2 , y 2 ) )
= < [ R 1 ( x 1 , x 2 ) R 2 ( y 1 , y 2 ) , R 1 + ( x 1 , x 2 ) R 2 + ( y 1 , y 2 ) ] , λ 1 ( x 1 , x 2 ) λ 2 ( y 1 , y 2 ) > .
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 CRel P ( H ) [resp. CRel R ( H ) ] is complete and cocomplete over Set .
Furthermore, we can easily see that CRel P ( H ) [resp. CRel R ( H ) ] 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 CRel P ( H ) [resp. CRel R ( H ) ] is cotopological.
Lemma 2.
The category CRel P ( H ) [resp. CRel R ( H ) ] is cotopological over Set .
Proof. 
Let X be any set and let ( X j , R j ) j J = ( X j , < R j , λ j > ) be any family of cubic H-relational spaces indexed by a class J. Suppose ( f j : X j X ) j J is a sink of mappings. We define a mapping R X , P = < R X , P , λ X , P > : X × X [ H ] × H as follows: for each ( x , y ) X × X ,
R X , P ( x , y ) = ( j J ( x j , y j ) f 2 ( x , y ) R j ) ( x j , y j ) = j J ( x j , y j ) f 2 ( x , y ) R j ( x j , y j ) .
Then we can easily see that
f j : ( X j , R j ) ( X , R X , P ) is a CRel P ( H ) mapping , for each j J .
For any cubic H-relational space ( Y , R Y ) = ( Y , < R Y , λ Y > ) , let g : X Y be any mapping such that g f j : ( X j , R j ) ( Y , R Y ) is a CRel P ( H ) -mapping, for each j J and let ( x , y ) X × X . Then for each j J and each ( x j , y j ) f j 2 ( x , y ) ,
R j ( x j , y j )
= < [ R j ( x j , y j ) , [ R j + ( x j , y j ) ] , λ j ( x j , y j ) >
P < [ ( R Y ( g f j ) 2 ) ( x j , y j ) , ( R Y + ( g f j ) 2 ) ( x j , y j ) ] , ( λ Y ( g f j ) 2 ) ( x j , y j ) >
= < [ ( R Y g 2 ) ( f j ( x j ) , f j ( y j ) ) , ( R Y + g 2 ) ( f j ( x j ) , f j ( y j ) ) ] , ( λ Y g 2 ) ( f j ( x j ) , f j ( y j ) ) >
= < [ ( R Y g 2 ) ( x , y ) , ( R Y + g 2 ) ( x , y ) , ( λ Y g 2 ) ( x , y ) >
= ( R Y g 2 ) ( x , y ) .
Thus, by the definition of R X , P , R X , P ( x , y ) P ( R Y g 2 ) ( x , y ) . So R X , P R Y g 2 . Hence g : ( X , R X , P ) ( Y , R Y ) is a CRel P ( H ) -mapping. Therefore CRel P ( H ) is cotopological over Set .
Now we define a mapping R X , R = < R X , R , λ X , R > : X × X [ H ] × H as follows: for each ( x , y ) X × X ,
R X , R ( x , y )
= ( j J ( x j , y j ) f 2 ( x , y ) R j ) ( x j , y j )
= < [ j J ( x j , y j ) f 2 ( x ) R j ( x j , y j ) , j J ( x j , y j ) f 2 ( x , y ) R j + ( x j , y j ) ] ,
j J x j f 2 ( x , y ) λ j ( x j , y j ) > .
Then we can easily see that
f j : ( X j , R j ) ( X , R X , R ) is a CRel R ( H ) mapping , for each j J .
For any cubic H-relational space ( Y , R Y ) = ( Y , < R Y , λ Y > ) , let g : X Y be any mapping such that g f j : ( X j , R j ) ( Y , R Y ) is a CRel R ( H ) -mapping, for each j J and let ( x , y ) X × X . Then for each j J and each ( x j , y j ) f j 2 ( x , y ) ,
R j ( x j , y j )
= < [ R j ( x j , y j ) , [ R j + ( x j , y j ) ] , λ j ( x j , y j ) >
R < [ ( R Y ( g f j ) 2 ) ( x j , y j ) , ( R Y + ( g f j ) 2 ) ( x j , y j ) ] , ( λ Y ( g f j ) 2 ) ( x j , y j ) >
= < [ ( R Y g 2 ) ( f j ( x j ) , f j ( y j ) ) , ( R Y + g 2 ) ( f j ( x j ) , f j ( y j ) ) ] , ( λ Y g 2 ) ( f j ( x j ) , f j ( y j ) ) >
= < [ ( R Y g 2 ) ( x , y ) , ( R Y + g 2 ) ( x , y ) ] , ( λ Y g 2 ) ( x , y ) >
= ( R Y g 2 ) ( x , y ) .
Thus, by the definition of R X , R , R X , R ( x , y ) R ( R Y g 2 ) ( x , y ) . So R X , R R Y g 2 . Hence g : ( X , R X , R ) ( Y , R Y ) is a CRel R ( H ) -mapping. Therefore CRel R ( H ) is cotopological over Set . This completes the proof. □
Example 2.
(CubicH-quotient relation) Let ( X , R ) = ( X , < R , λ > ) be a cubic H-relational space, let ∼ be an equivalence relation on X and let π : X X / be the canonical mapping. We define a mapping R X / , P : X / × X / [ H ] × H as below: for each ( [ x ] , [ y ] ) X / × X / ,
R X / , P ( [ x ] , [ y ] )
= [ ( x , y ) π 2 ( [ x ] , [ y ] ) R ] ( x , y )
= < [ ( x , y ) π 21 ( [ x ] , [ y ] ) R ( x , y ) , ( x , y ) π 2 ( [ x ] , [ y ] ) R + ( x , y ) ] ,
( x , y ) π 2 ( [ x ] , [ y ] ) λ ( x , y ) > .
Then we can easily see that R X / , P is a cubic H-relation in X / . Furthermore, π : ( X , R ) ( X / , R X / , P ) is a CRel P ( H ) -mapping. Thus, R X / , P is the final cubic H-relation in X / .
Now we define a mapping R X / , R : X / × X / [ H ] × H as follows: for each ( [ x ] , [ y ] ) X / × X / ,
R X / , R ( [ x ] ) = [ ( x , y ) π 2 ( [ x ] , [ y ] ) R ] ( x , y )
= < [ ( x , y ) π 2 ( [ x ] , [ y ] ) R ( x , y ) , ( x , y ) π 2 ( [ x ] , [ y ] ) R + ( x , y ) ] ,
( x , y ) π 2 ( [ x ] , [ y ] ) λ ( x , y ) > .
Then we can easily see that R X / , R is a cubic H-relation in X / . Furthermore, π : ( X , R ) ( X / , A X / , R ) is a Crel R ( H ) -mapping. Thus, R X / , R is the final cubic H-relation in X / .
In this case, R X / , P [resp. A X / , R ] is called the cubic H-quotient [resp. H-quotient * ] relation in X induced by ∼.
Definition 7
([38]). Let A be a concrete category and let f , g : A B be two A -morphisms. Then a pair ( E , e ) is called an equalizer in A of f and g, if the following conditions hold:
(i) 
e : E A is an A -morphism,
(ii) 
f e = g e ,
(iii) 
for any A -morphism e : E A such that f e = g e , there exists a unique A -morphism e ¯ : E E such that e = e e ¯ .
In this case, we say that A has equalizers.
Dual notion: Coequalizer.
Proposition 2.
The category CRel P ( H ) [resp. CRel R ( H ) ] has equalizers.
Proof. 
Let f , g : ( X , R X ) ( Y , R Y ) be two CRel P ( H ) -mappings, where R X = < R X , λ X > and R Y = < R Y , λ Y > . Let E = { a X : f ( a ) = g ( a ) } and define a mapping R E , P : E × E [ H ] × H as follows: for each ( a , b ) E × E ,
R E , P ( a , b ) = R X ( a , b ) ) = < [ R X ( a , b ) , R X + ( a , b ) ] , λ X ( a , b ) > .
Then clearly, R E , P is a cubic H-relation in E and R E , P R X . Consider the inclusion mapping i : E X . Then clearly, i : ( E , A P , E ) ( X , A ) is a CSet P ( H ) -mapping and f i = g i .
Let k : ( E , R E ) ( X , A X ) be a CRel P ( H ) -mapping such that f k = g k . We define a mapping k ¯ : E E as follows: for each e E ,
k ¯ ( e ) = i 1 k ( e ) .
Then clearly, k = i k ¯ .
Let ( e , f ) E × E . Since k : ( E , R E ) ( X , R E , P ) is a CRel P ( H ) -mapping,
R E , P ( k ¯ ) 2 ( e , f ) = R E , P ( k ¯ ) 2 ( e , f )
= R E , P ( i 2 k 2 ( e , f ) )
= R E , P k 2 ( e , f )
P R E ( e , f ) .
Thus, R E R E , P ( k ¯ ) 2 . So k ¯ : ( E , R E ) ( E , R E , P ) is a CRel P ( H ) -mapping.
Now in order to prove the uniqueness of k ¯ , let r ¯ : E E such that i r ¯ = k . Then r ¯ = i 1 k = k ¯ . Thus, k ¯ is unique. Hence CRel P ( H ) has equalizers.
Similarly, we can prove that CRel R ( H ) has the equalizer R E , P . □
For two cubic H-relations R X = < R X , λ X > in X and R Y = < R Y , λ Y > in Y, the product of P-order type [resp. R-order type], denoted by R X × P Y Y [resp. R X × R R Y ], is a cubic H-relation in X × Y defined by: for any ( x , y ) , ( x , y ) X × Y ,
( R X × P R Y ) ( ( x , y ) , ( x , y ) ) = < R X ( x , x ) R Y ( y , y ) , λ X ( x , x ) λ X ( y , y ) >
[resp. ( R X × R R ) Y ( ( x , y ) , ( x , y ) ) = < R X ( x , x ) R Y ( y , y ) , λ X ( x , x ) λ X ( y , y ) > ].
Lemma 3.
Final episinks in CRel P ( H ) [resp. CRel R ( H ) ] are preserved by pullbacks.
Proof. 
Let ( g j : ( X j , R j ) ( Y , R Y ) ) j J be any final episink in CRel P ( H ) and let f : ( W , R W ) ( Y , R Y ) be any CRel P ( H ) -mapping, where R j = < R j , λ j > , R Y = < R Y , λ Y > and R W = < R W , λ W > . For each j J , let
U j = { ( w , x j ) W × X j : f ( w ) = g j ( x j ) }
and let us define a mapping R U j , P = < R U j , P , λ U j , P > : U j × U j [ H ] × H as follows: for each ( ( w , x j ) , ( w , x j ) ) U j × U j ,
R U j , P ( ( w , x j ) , ( w , x j ) )
= ( R W × P R j ) U j × U j ( ( w , x j ) , ( w , x j ) )
= ( R W × P R j ) ( ( w , x j ) , ( w , x j ) )
= < R W ( w , w ) R j ( x j , x j ) , λ W ( w , w ) λ j ( x j , x j ) >
= < ( R W × R j ) ( ( w , x j ) , ( w , x j ) ) , ( λ W × λ j ) ( ( w , x j ) , ( w , x j ) ) > , i.e.,
R U j , P = < R W × R j U j × U j , λ W × λ j U j × U j > .
For each j J , let e j : U j W and p j : U j X j be the usual projections. Then clearly, e j : ( U j , R U j , P ) ( W , R W ) and p j : ( U j , R U j , P ) ( X j , R j ) are CRel P ( H ) -mappings and g j p j = f e j , for each j J . Thus, we have the following pullback square in CRel P ( H ) :
    Mathematics 08 00482 i002
We will prove that ( e j : ( U j , R U j , P ) ( W , R W ) ) j J is a final episink in CRel P ( H ) . Let w W . Since ( g j ) j J is an episink in CSet P ( H ) , there is j J such that g j ( x j ) = f ( w ) , for some x j X j . Thus, ( w , x j ) U j and e j ( w , x j ) = w . So ( e j ) j J is an episink in CRel P ( H ) .
Finally, let us show that ( e j ) J is final in CRel P ( H ) . Let R W * be the final structure in W regarding ( e j ) j J and let ( w , w ) W × W . Then
R W ( w , w ) = < R W ( w , w ) , λ W ( w , w ) >
= < R W ( w , w ) R W ( w , w ) , λ W ( w , w ) λ W ( w , w ) >
P < R W ( w , w ) R Y f 2 ( w , w ) , λ W ( w , w ) λ Y f 2 ( w , w ) >
[ Since f : ( W , R W ) ( Y , R Y ) is a CRel P ( H ) -mapping]
= < R W ( w , w ) [ j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) R j ( x j , x j ) ] ,
λ W ( w , w ) [ j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) λ j ( x j , x j ) ] >
[ Since ( g j : ( R j , R j ) ( Y , R Y ) ) j J is a final episink in CRel P ( H ) ]
= < j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) [ R W ( w , w ) R j ( x j , x j ) ] ] ,
j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) [ λ W ( w , w ) λ j ( x j , x j ) ] >
= < j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ R W ( w , w ) R j ( x j , x j ) ] ] ,
j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ λ W ( w , w ) λ j ( x j , x j ) ] >
= < j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ R U j , P ( ( w , x j , ( w , x j ) ] ,
j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ λ U j , P ( ( w , x j , ( w , x j ) ] >
= R W * ( w , w ) .
Thus, R W R W * . Since ( e j : ( U j , R U j ) ( W , R W ) ) j J is final, 1 W : ( W , R W * ) ( W , R W ) is a CRel P ( H ) -mapping. So R W * R W . Hence R W * = R W . Therefore ( e j ) j J is final.
Now we define a mapping R U j , R = < R U j , R , λ U j , R > : U j [ H ] × H as follows: for each ( ( w , x j ) , ( w , x j ) ) U j × U j ,
R U j , R ( ( w , x j ) , ( w , x j ) )
= ( R W × R R j ) U j × U j ( ( w , x j ) , ( w , x j ) )
= ( R W × R R j ) ( ( w , x j ) , ( w , x j ) )
= < R W ( w , w ) R j ( x j , x j ) , λ W ( w , w ) λ j ( x j , x j ) > .
For each j J , let e j : U j W and p j : U j X j be the usual projections. Then we can similarly prove that final episinks in Rel R ( H ) are preserved by pullbacks. This completes the proof. □
For any singleton set { a } , since the cubic set R { a } in { a } is not unique, the category CRel ( H ) is not properly fibered over Set . Then from Definitions 1 and 3, Lemmas 2 and 3, we have the following result.
Theorem 1.
The category CRel P ( H ) [resp. CRel R ( H ) ] satisfies all the conditions of a topological universe over Set except the terminal separator property.
Theorem 2.
The category CRel P ( H ) [resp. CRel R ( H ) ] is Cartesian closed over Set .
Proof. 
From Lemma 1, it is clear that CRel P ( H ) [resp. CRel R ( H ) ] has products. Then it is sufficient to prove that CRel P ( H ) [resp. CRel R ( H ) ] has exponential objects.
For any cubic H-relational spaces X = ( X , R X ) = ( X , < R X , λ X > ) and Y = ( Y , R Y ) = ( Y , < R Y , λ Y > ) , let Y X be the set of all ordinary mappings from X to Y. We define two mappings R Y X : Y X × Y X [ H ] and λ Y X : Y X × Y X H as follows: for each ( f , g ) Y X × Y X ,
R Y X ( f , g ) = { h H : R X ( x , y ) h R Y ( f ( x ) , f ( y ) ) , for each ( x , y ) X × X } and
λ Y X ( f , g ) = { h H : λ X ( x , y ) h λ Y ( f ( x ) , f ( y ) ) , for each ( x , y ) X × X } . Then clearly, A Y X = < A Y X , λ Y X > is a cubic H-relation in Y X . Moreover, by the definitions of R Y X and λ Y X ,
R X ( x , y ) R Y X ( f , g ) R Y ( f ( x ) , f ( y ) ) , R X + ( x , y ) R Y X + ( f , g ) R Y ( f ( x ) , f ( y ) )
and
λ X ( x , y ) λ Y X ( f , g ) λ Y ( f ( x ) , f ( y ) ) ,
for each ( x , y ) X × X .
Let Y X = ( Y X , R Y X ) and let us define a mapping e X , Y : X × Y X Y as follows: for each ( x , f ) X × Y X ,
e X , Y ( x , f ) = f ( x ) .
Let ( x , f ) , ( y , g ) X × Y X . Then
( R X × P R Y X ) ( ( x , f ) , ( y , g ) ) = R X ( x , y ) R Y X ( f , g )
R Y ( f ( x ) , f ( y ) )
= R Y e X , Y 2 ( ( x , f ) , ( y , g ) ) ,
[ By the definition of e X , Y ]
( R X + × P R Y X + ) ( ( x , f ) , ( y , g ) ) = R X + ( x , y ) R Y X + ( f , g )
A Y + ( f ( x ) , f ( y ) )
= A Y + e X , Y 2 ( ( x , f ) , ( y , g ) ) and
( λ X × P λ Y X ) ( ( x , f ) , ( y , g ) ) = λ X ( x , y ) λ Y X ( f , g )
λ Y ( f ( x ) , f ( y ) )
= λ Y e X , Y 2 ( ( x , f ) , ( y , g ) ) .
Thus, e X , Y : X × P Y X Y is a CRel P ( H ) -mapping, where X × P Y X = ( X × Y X , < A X × P A Y X , λ X × P λ Y X > ) .
For any cubic H-relational space Z = ( Z , < A Z , λ Z > ) , let k : X × P Z Y be a CRel P ( H ) -mapping. We define a mapping k ¯ : Z Y X as follows: for each z Z and each x X ,
[ k ¯ ( z ) ] ( x ) = k ( x , z ) .
Then we can prove that k ¯ is a unique CRel P ( H ) -mapping such that e X , Y ( 1 X × k ¯ ) = k .
Now we define two mappings R Y X , R : Y X × Y X [ H ] and λ Y X , R : Y X × Y X H as follows: for each ( f , g ) Y X × Y X and each x X ,
R Y X , R ( f , g ) = R Y X , P ( f , g )
and
λ Y X , R ( f , g ) = { h H : λ X ( x , y ) h λ Y ( f ( x ) , f ( y ) ) , for each ( x , y ) X × X } .
Then clearly, R Y X , R = < R Y X , R , λ Y X , R > is a cubic H-relation in Y X . Moreover, by the definitions of R Y X , R and λ Y X , R ,
R X ( x , y ) R Y X , R ( f , g ) R Y ( f ( x ) , f ( y ) )
and
λ X ( x , y ) λ Y X , R ( f , g ) λ Y ( f ( x ) , f ( y ) ) ,
for each x X . Let Y X = ( Y X , R Y X , R ) and let us define a mapping e X , Y : X × Y X Y as follows: for each ( x , f ) X × Y X ,
e X , Y ( x , f ) = f ( x ) .
Let ( x , f ) , ( y , g ) X × Y X . Then by the definitions of R Y X , R and λ Y X , R , we have the followings:
( R X × R A Y X , R ) ( ( x , f ) , ( y , g ) ) R Y e X , Y 2 ( ( x , f ) , ( y , g ) )
and
( λ X × R λ P , Y X ) ( ( x , f ) , ( y , g ) ) λ Y e X , Y 2 ( ( x , f ) , ( y , g ) ) .
Thus, R X × R R Y X , R R Y e X , Y 2 . So e X , Y : X × R Y X Y is a CRel R ( H ) -mapping, where X × R Y X = ( X × Y X , < R X × R R Y X , R , λ X × R λ Y X , R > ) .
For any cubic H-relational space Z = ( Z , < R Z , λ Z > ) , let k : X × R Z Y be a CRel R ( H ) -mapping. We define a mapping k ¯ : Z Y X as follows: for each z Z and each x X ,
[ k ¯ ( z ) ] ( x ) = k ( x , z ) .
Then we can prove that k ¯ is a unique CRel R ( H ) -mapping such that
e X , Y ( 1 X × k ¯ ) = k .
This completes the proof. □
Remark 1.
The category CRel P ( H ) [resp. CRel R ( H ) ] is not a topos (See [39] for its definition), since it has no subobject classifier.
Example 3.
Let I = { 0 , 1 } be two points chain, respectively and let X = { a } . Let R 1 and R 2 be the cubic H-relations in X defined by:
R 1 ( a ) = < 0 , 0 > a n d R 2 ( a ) = < 1 , 1 > .
Let 1 X : ( X , R 1 ) ( X , R 2 ) be the identity mapping. Then clearly, 1 X is both monomorphism and epimorphism in CRel P ( H ) [resp. CRel R ( H ) ]. However, 1 X is not an isomorphism in CRel P ( H ) [resp. CRel R ( H ) ]. Thus, CRel ( H ) has no subobject classifier.

4. The Categories CRel P , R ( H ) and CRel R , R ( H )

In this section, we obtain two subcategories CRel P , R ( H ) and CRel R , R ( H ) of CRel P ( H ) and CRel R ( H ) , respectively which are topological universes over Set .
It is interesting that final structures and exponential objects in CRel P , R ( H ) [resp. CRel R , R ( H ) ] are shown to be quite different from those in CRel P ( H ) [resp. CRel R ( H ) ].
First of all, we list two well-known results.
Result 1 (Theorem 2.5 [25]). Let A be a well-powered and co(well-powered) topological category. Then the followings are equivalent:
(1)
B is bireflective in A ,
(2)
B is closed under the formation of initial sources, i.e., for any initial source ( f j : A A j ) j J in A with A j B for each j J , then A B .
Result 2 (Theorem 2.6 [25]). If A is a topological category and B is a bireflective subcategory of A , then B is also a topological category. Moreover, every source in B which is initial in A is initial in B .
Definition 8.
Let X be a nonempty set and let R = < R , λ > be a cubic H-relation in X. Then R is said to be reflexive, if R and λ are reflexive, i.e., R ( x , x ) = 1 and λ ( x , x ) = 1 , for each x X .
The class of all cubic H-reflexive relational spaces and CRel P ( H ) -mappings [resp. CRel R ( H ) -mappings between them forms a subcategory of CRel P ( H ) [resp. CRel R ( H ) ] denoted by CRel P , R ( H ) [resp. CRel R , R ( H ) ].
The following is the immediate result of Definitions 1 and 8.
Lemma 4.
The category CRel P , R ( H ) [resp. CRel R , R ( H ) ] is properly fibered over Set .
Lemma 5.
The category CRel P , R ( H ) [resp. CRel R , R ( H ) ] is closed under the formation of initial sources in The category CRel P ( H ) [resp. CRel R ( H ) ]
Proof. 
Let f j : ( X , R X , P ) ( X j , R j ) ) j J be an initial source in CRel P ( H ) such that each ( X j , R j ) belongs to CRel P , R ( H ) , where ( X , R X , P ) = ( X , < R X , P , λ X , P > ) and ( X j , R j ) = ( X j , < R j , λ j > ) . Let x X and let j J . Since R j and λ j are reflexive, R j f j 2 ( x , x ) = 1 and λ j f j 2 ( x , x ) = 1 . Then
R X , P ( x , x ) = j J R j f j 2 ( x , x ) = 1 and λ X , P ( x , x ) = j J λ j f j 2 ( x , x ) = 1 .
Thus, R X , P ( x , x ) = < 1 , 1 > . So R X , P is reflexive.
Now let f j : ( X , R X , R ) ( X j , R j ) ) j J be an initial source in CRel R ( H ) such that each ( X j , R j ) belongs to CRel R , R ( H ) . Then clearly, for each x X ,
R X , R ( x , x ) = R X , P ( x , x ) = 1 and λ X , R ( x , x ) = j J λ j f j 2 ( x , x ) = 1 .
Thus, R X , R ( x , x ) = < 1 , 1 > . So R X , R is reflexive. This completes the proof. □
From Results 1, 2 and Lemma 5, we have the followings.
Proposition 3.
(1) The category CRel P , R ( H ) [resp. CRel R , R ( H ) ] is a bireflective subcategory of CRel P ( H ) [resp. CRel R ( H ) ].
(2) The category CRel P , R ( H ) [resp. CRel R , R ( H ) ] is topological over Set .
It is well-known that a category A is topological if and only if it is cotopological. Then by (2) of the above Proposition, the category CRel P , R ( H ) [resp. CRel R , R ( H ) ] is cotopological over Set . However, we will prove that CRel P , R ( H ) [resp. CRel R , R ( H ) ] is cotopological over Set , directly.
Lemma 6.
the category CRel P , R ( H ) [resp. CRel R , R ( H ) ] has final structure over Set .
Proof. 
Let X be a nonempty set and let ( ( X j , R j ) ) = ( ( X j , < R j , λ j > ) j J be any family of cubic H-relational spaces indexed by a class J. We define two mappings R X , P : X [ H ] and λ X , P : X H , respectively as below: for each ( x , y ) X × X ,
R X , P ( x , y ) = { j J ( x j , y j ) f j 2 ( x , y ) R j ( x j , y j )    if ( x , y ) ( X × X X ) 1 if ( x , y ) X
and
λ X , P ( x , y ) = j J ( x j , y j ) f j 2 ( x , y ) λ j ( x j , y j )    if ( x , y ) ( X × X X ) 1 if ( x , y ) X ,
where X = { ( x , x ) : x X } . Then clearly, R X , P is the cubic H-reflexive relation in X given by: for each ( x , y ) X × X ,
R X , P ( x , y ) = j J ( x j , y j ) f j 2 ( x , y ) R j ( x j , y j )    i f ( x , y ) ( X × X X ) < 1 , 1 > if ( x , y ) X .
Moreover, we can easily check that ( X , R X , P ) = ( X , < R X , P , λ X , P > ) is a final structure in CRel P , R ( H ) . Thus, ( f j : ( X j , R j ) ( X , R X , P ) ) j J is a final sink in CRel P , R ( H ) .
Now we define two mappings R X , R : X [ H ] and λ X , R : X H , respectively as follows: for each ( x , y ) X × X ,
R X , R ( x , y ) = R X , P ( x , y )
and
λ X , R ( x , y ) = j J ( x j , y j ) f j 2 ( x , y ) λ j ( x j , y j )    if ( x , y ) ( X × X X ) 1 if ( x , y ) X .
Then clearly, R X , R is the cubic H-reflexive relation in X given by: for each ( x , y ) X × X ,
R X , R ( x , y ) = j J ( x j , y j ) f j 2 ( x , y ) R j ( x j , y j )    if ( x , y ) ( X × X X ) < 1 , 1 > if ( x , y ) X .
Moreover, we can easily show that ( f j : ( X j , R j ) ( X , R X , R ) ) j J is a final sink in CRel R , R ( H ) . □
Lemma 7.
Final episinks in CRel P , R ( H ) [resp. CRel R , R ( H ) ] are preserved by pullbacks.
Proof. 
Let ( g j : ( X j , R j ) ( Y , R Y , P ) ) j J be any final episink in CRel P , R ( H ) and let f : ( W , R W ) ( Y , R Y , P ) be any CRel P ( H ) -mapping, where ( W , R W ) is a cubic H-reflexive relational space. For each j J , let us take U j , R U j , P , e j and p j as in the first proof of Lemma 3. Then we can easily check that CRel P , R ( H ) is closed under the formation of pullbacks in CRel P ( H ) . Thus, it is enough to prove that ( e j ) j J is final.
Suppose R W * is the final cubic H-relation in W regarding ( e j ) j J and let ( w , w ) ( W × W X ) . Then
R W ( w , w ) = < R W ( w , w ) , λ W ( w , w ) >
= < R W ( w , w ) R W ( w , w ) , λ W ( w , w ) λ W ( w , w ) >
P < R W ( w , w ) R Y f 2 ( w , w ) , λ W ( w , w ) λ Y f 2 ( w , w ) >
[ Since f : ( W , R W ) ( Y , R Y ) is a CRel P ( H ) -mapping]
= < R W ( w , w ) [ j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) R j ( x j , x j ) ] ,
λ W ( w , w ) [ j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) λ j ( x j , x j ) ] >
[ Since ( g j : ( R j , R j ) ( Y , R Y ) ) j J is a final episink in CRel P ( H ) ]
= < j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) [ R W ( w , w ) R j ( x j , x j ) ] ] ,
j J ( x j , x j ) g j 2 ( f ( w ) , f ( w ) ) [ λ W ( w , w ) λ j ( x j , x j ) ] >
= < j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ R W ( w , w ) R j ( x j , x j ) ] ] ,
j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ λ W ( w , w ) λ j ( x j , x j ) ] >
= < j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ R U j , P ( ( w , x j , ( w , x j ) ] ,
j J ( ( w , x j ) , ( w , x j ) ) e j 2 ( w , w ) [ λ U j , P ( ( w , x j , ( w , x j ) ] >
= R W * ( w , w ) .
Thus, R W R W * . On the other hand, by a similar argument in the first proof of Lemma 3, R W * R W on W × W W . So R W * = R W on W × W W . Now let ( w , w ) W . Then clearly, R W * ( w , w ) = < 1 , 1 > = R W ( w , w ) . Thus, R W * = R W on W . Hence R W * = R W on W.
Now for each j J , let us R U j , R = < R U j , R , λ U j , R > : U j [ H ] × H be the mapping as in the second proof of Lemma 3. Then we can similarly prove that final episinks in Rel R , R ( H ) 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 CRel P , R ( H ) [resp. CRel R , R ( H ) ] is a topological universe over Set . In particular, CRel P , R ( H ) [resp. CRel R , R ( H ) ] is Cartesian closed over Set (See [1]) and a concrete quasitopos(See [40]).
In [41], Noh obtained exponential objects in Rel ( I ) , where Rel ( I ) denotes the category of fuzzy relations. By applying his construction of an exponential object in Rel ( I ) to the category CRel P , R ( H ) [resp. CRel R , R ( H ) ], we have the following.
Proposition 4.
The category CRel P , R ( H ) [resp. CRel R , R ( H ) ] has an exponential object.
Proof. 
For any X = ( X , R X ) = ( X , < R X , λ X > , Y = ( Y , R Y ) = ( X , < R Y , λ Y > ) Ob ( CRel P , R ( H ) ) and let Y X = hom ( X , Y ) . For any ( f , g ) Y X × Y X , let
D ( f , g ) = { ( x , y ) X × X : R X ( x , y ) > R Y ( f ( x ) , g ( y ) ) , λ X ( x , y ) > λ Y ( f ( x ) , g ( y ) ) } .
We define a mapping R Y X , P = < R Y X , P , λ Y X , P > : Y X × Y X [ H ] × H as follows: for each ( f , g ) Y X × Y X ,
R Y X , P ( f , g ) = < ( x , y ) D ( f , g ) R Y ( f ( x ) , f ( y ) ) , ( x , y ) D ( f , g ) λ Y ( f ( x ) , f ( y ) ) > if D ( f , g ) ϕ < 1 , 1 > if D ( f , g ) = ϕ .
Then by the definition of D ( f , g ) , D ( f , f ) = ϕ , for each f Y X . Thus, R Y X , P ( f , f ) = < 1 , 1 > , for each f Y X . So R Y X , P is a cubic H-reflexive relation in Y X .
Let Y X = ( Y X , R Y X , P ) and we define the mapping e X , Y : X × P Y X Y as follows: for each ( a , f ) X × Y X ,
e X , Y ( a , f ) = f ( a ) .
Let ( a , f ) , ( b , g ) X × Y X .
Case 1: Suppose D ( f , g ) = ϕ . Then
( R X × P R Y X , P ) ( ( a , f ) , ( b , g ) )
= < R X ( a , b ) R Y X , P ( f , g ) , λ X ( a , b ) λ Y X , P ( f , g ) >
= < R X ( a , b ) , λ X ( a , b ) >
[ By the definition of R Y X , P , R Y X , P ( f , g ) = 1 , λ Y X , P ( f , g ) = 1 ]
P < R Y ( f ( x ) , g ( y ) ) , λ Y ( f ( x ) , g ( y ) ) > [Since D ( f , g ) = ϕ ]
= < R Y e X , Y 2 ( ( a , f ) , ( b , g ) ) .
Case 2: Suppose D ( f , g ) ϕ . Then
( R X × P R Y X , P ) ( ( a , f ) , ( b , g ) )
= < R X ( a , b ) [ ( x , y ) D ( f , g ) R Y ( f ( x ) , f ( y ) ) ] ,
λ X ( a , b ) [ ( x , y ) D ( f , g ) λ Y ( f ( x ) , f ( y ) ) ] >
P < R Y ( f ( x ) , g ( y ) ) , λ Y ( f ( x ) , g ( y ) ) >
= < R Y e X , Y 2 ( ( a , f ) , ( b , g ) ) .
Thus, in either case, R X × R Y X , P R Y e X , Y 2 . So e X , Y is a CRel P ( H ) -mapping.
Let Z = ( Z , R Z ) = ( Z , < R Z , λ Z > ) be any cubic H-reflexive relational space and let h : X × Z Y be any CRel P ( H ) -mapping. We define the mapping h ¯ : Z Y X as follows: for each c Z and each a X ,
[ h ¯ ( c ) ] ( a ) = h ( a , c ) .
Let c Z and let a , b X . Then
R Y [ h ¯ ( c ) ] 2 ( a , b )
= R Y ( [ h ¯ ( c ) ] ( a ) , [ h ¯ ( c ) ] ( b ) )
= < R Y ( [ h ¯ ( c ) ] ( a ) , [ h ¯ ( c ) ] ( b ) ) , λ Y ( [ h ¯ ( c ) ] ( a ) , [ h ¯ ( c ) ] ( b ) ) >
= < R Y ( h ( a , c ) , h ( b , c ) ) , λ Y ( h ( a , c ) , h ( b , c ) ) >
= < R Y h 2 ( h ( a , c ) , h ( b , c ) ) , λ Y h 2 ( h ( a , c ) , h ( b , c ) ) >
= R Y h 2 ( ( a , c ) , ( b , c ) )
P ( R X × P R Z ) ( ( a , c ) , ( b , c ) )
= < ( R X × P R Z ) ( ( a , c ) , ( b , c ) ) , ( λ X t i m e s P λ Z ) ( ( a , c ) , ( b , c ) ) >
= < R X ( a , b ) R Z ( c , c ) , λ X ( a , b ) λ Z ( c , c ) >
= < R X ( a , b ) , λ X ( a , b ) > [Since R Z is reflexive]
= R X ( a , b ) .
Thus, R X R Y [ h ¯ ( c ) ] 2 . So h ¯ ( c ) : X Y is a CRel P ( H ) -mapping. Hence h ¯ is well-defined. Let c , c Z .
Case 1: Suppose D ( h ¯ ( c ) , h ¯ ( c ) ) = ϕ . Then
R Y X , P h ¯ 2 ( c , c ) = R Y X , P ( h ¯ ( c ) , h ¯ ( c ) )
= < 1 , 1 > [By the definition of R Y X , P ]
P R Z ( c , c ) .
Case 2: Suppose D ( h ¯ ( c ) , h ¯ ( c ) ) ϕ . Then
R Y X , P ( h ¯ ( c ) , h ¯ ( c ) ) = < R Y X , P ( h ¯ ( c ) , h ¯ ( c ) ) , λ Y X , P ( h ¯ ( c ) , h ¯ ( c ) ) >
= < ( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) R Y ( [ h ¯ ( c ) ] ( a ) , [ h ¯ ( c ) ] ( b ) ) ,
( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) λ Y ( [ h ¯ ( c ) ] ( a ) , [ h ¯ ( c ) ] ( b ) ) >
= < ( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) R Y ( h ( a , c ) , h ( b , c ) ) ,
( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) λ Y ( h ( a , c ) , h ( b , c ) ) >
P < ( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) [ R X ( a , b ) R Z ( c , c ) ] ,
( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) [ λ X ( a , b ) λ Z ( c , c ) ] > .
On one hand, for any ( a , b ) D ( h ¯ ( c ) , h ¯ ( c ) ) ,
R X ( a , b ) > R Y ( [ h ¯ ( c ) ] ( a ) , [ h ¯ ( c ) ] ( b ) )
= R Y ( h ( a , c ) , h ( b , c ) )
R X ( a , b ) R Z ( c , c ) .
Thus, R X ( a , b ) > R Z ( c , c ) . Similarly, we have λ X ( a , b ) > λ Z ( c , c ) . So
R Y X , P ( h ¯ ( c ) , h ¯ ( c ) ) P R Z ( c , c ) .
Hence in either cases, R Z R Y X , P h ¯ 2 . Therefore h ¯ is a CRel P ( H ) -mapping. Furthermore, h ¯ is unique and e X , Y ( 1 X × h ¯ ) = h .
Now for any X = ( X , R X ) = ( X , < R X , λ X > , Y = ( Y , R Y ) = ( X , < R Y , λ Y > ) Ob ( CRel R , R ( H ) ) and let Y X = hom ( X , Y ) . For any ( f , g ) Y X × Y X , let
D ( f , g ) = { ( x , y ) X × X : R X ( x , y ) > R Y ( f ( x ) , g ( y ) ) , λ X ( x , y ) < λ Y ( f ( x ) , g ( y ) ) } .
We define a mapping R Y X , R = < R Y X , R , λ Y X , R > : Y X × Y X [ H ] × H as follows: for each ( f , g ) Y X × Y X ,
R Y X , R ( f , g )
= < ( x , y ) D ( f , g ) R Y ( f ( x ) , f ( y ) ) , ( x , y ) D ( f , g ) λ Y ( f ( x ) , f ( y ) ) > if D ( f , g ) ϕ < 1 , 1 > if D ( f , g ) = ϕ .
Then we can easily check that R Y X , R is a cubic H-reflexive relation in Y X . Moreover, by the similar argument of the above proof, we can show that R Y X , R is an exponential object in Y X . This completes the proof. □
Remark 2.
(1) We can see that exponential objects in CRel P , R ( H ) [resp. CRel R , R ( H ) ] is quite different from those in CRel P ( H ) [resp. CRel R ( H ) ] constructed in Theorem 1.
(2) The category CRel P , R ( H ) [resp. CRel R , R ( H ) ] has no subject classifier.
Example 4.
Let H = { 0 , 1 } be the two points chain and let X = { a , b } . Let R 1 , P = < R 1 , P , λ 1 , P > and R 2 , P = < R 2 , P , λ 2 , P > be cubic H-reflexive relations in X given by:
R 1 , P ( a , a ) = R 1 , P ( b , b ) = < 1 , 1 > , R 1 , P ( a , b ) = R 1 , P ( b , a ) = < 0 , 0 >
and
R 2 , P ( a , a ) = R 2 , P ( b , b ) = < 1 , 1 > , R 2 , P ( a , b ) = R 2 , P ( b , a ) = < 1 , 1 > .
Let 1 X : ( X , R 1 , P ) ( X , R 2 , P ) be the identity mapping. Then clearly, 1 X is both monomorphism and epimorphism in CRel P ( H ) . However, 1 X is not an isomorphism in CRel P ( H ) .

5. Conclusions

We constructed the concrete category CRel P ( H ) [resp. CRel R ( H ) ] of cubic H-relational spaces and P-preserving [resp. R-preserving] mappings between them and studied it in the sense of a topological universe. In particular, we proved that it is Cartesian closed over Set . Next, We introduced the category CRel P , R ( H ) [resp. CRel R , R ( H ) ] of cubic H-reflexive relational spaces and P-preserving [resp. R-preserving] mappings between them and investigated it in a viewpoint of a topological universe. In particular, we obtained exponential objects in CRel P , R ( H ) [resp. CRel R , R ( H ) ] quite different from those in CRel P , R ( H ) [resp. CRel R , R ( H ) ]. Also we proved that CRel P ( H ) [resp. CRel R ( H ) ] is a topological universe but CRel P ( H ) [resp. CRel R ( H ) ] not a topological universe. In the future, we will expect one to study some full subcategories of the category CRel P ( H ) [resp. CRel R ( H ) ].

Author Contributions

Creation and mathematical ideas, J.-G.L. and K.H.; writing–original draft preparation, J.-G.L. and K.H.; writing–review and editing, X.C. and K.H.; funding acquisition, J.-G.L. All authors have read and agreed to the published version of the manuscript.

Funding

This paper was supported by Wonkwang University in 2020.

Conflicts of Interest

The authors declare no conflict of interest.

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Lee, J.-G.; Hur, K.; Chen, X. A Study on Cubic H-Relations in a Topological Universe Viewpoint. Mathematics 2020, 8, 482. https://doi.org/10.3390/math8040482

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Lee J-G, Hur K, Chen X. A Study on Cubic H-Relations in a Topological Universe Viewpoint. Mathematics. 2020; 8(4):482. https://doi.org/10.3390/math8040482

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Lee, Jeong-Gon, Kul Hur, and Xueyou Chen. 2020. "A Study on Cubic H-Relations in a Topological Universe Viewpoint" Mathematics 8, no. 4: 482. https://doi.org/10.3390/math8040482

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