The category of neutrosophic crisp sets

2017·0.31 MB·English

by  K. Hur, P. K. Lim, J. G. Lee, J. Kim;

Annals of Fuzzy Mathematics and Informatics
FMI
@
Volume 14, No. 1, (July 2017), pp. 43–54
ISSN: 2093–9310 (print version) (cid:13)c Kyung Moon Sa Co.
ISSN: 2287–6235 (electronic version) http://www.kyungmoon.com
http://www.afmi.or.kr
The category of neutrosophic crisp sets
K. Hur, P. K. Lim, J. G. Lee, J. Kim
Received 18 Februry 2017; Revised 1 March 2017; Accepted 19 March 2017
Abstract. We introduce the category NCSet consisting of neutro-
sophic crisp sets and morphisms between them. And we study NCSet in
the sense of a topological universe and prove that it is Cartesian closed
over Set, where Set denotes the category consisting of ordinary sets and
ordinary mappings between them.
2010 AMS Classification: 03E72, 18B05
Keywords: Neutrosophic crisp set, Cartesian closed category, Topological uni-
verse, Neutrosophic crisp set.
Corresponding Author: J. G. Lee ([email protected])
1. Introduction
I
n1965, Zadeh[20]hadintroducedaconceptofafuzzysetasthegeneralization
of a crisp set. In 1986, Atanassove [1] proposed the notion of intuitionistic fuzzy set
as the generalization of fuzzy sets considering the degree of membership and non-
membership. In1998Smarandache[19]introducedtheconceptofaneutrosophicset
considering the degree of membership, the degree of indeterminacy and the degree
of non-membership. Moreover, Salama et al. [15, 16, 18] applied the concept of
neutrosophic crisp sets to topology and relation.
After that time, many researchers [2, 3, 4, 5, 7, 8, 10, 12, 13, 14] have inves-
tigated fuzzy sets in the sense of category theory, for instance, Set(H), Set (H),
f
Set (H), Fuz(H). Among them, the category Set(H) is the most useful one as the
g
”standard” category, because Set(H) is very suitable for describing fuzzy sets and
mappings between them. In particular, Carrega [2], Dubuc [3], Eytan [4], Goguen
[5], Pittes [12], Ponasse [13, 14] had studied Set(H) in topos view-point. However
Hur et al. investigated Set(H) in topological view-point. Moreover, Hur et al. [8]
introduced the category ISet(H) consisting of intuitionistic H-fuzzy sets and mor-
phisms between them, and studied ISet(H) in the sense of topological universe.
Recently, Lim et al [10] introduced the new category VSet(H) and investigated it
in the sense of topological universe.
J.G.Leeetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,43–54
TheconceptofatopologicaluniversewasintroducedbyNel[11], whichimpliesa
Cartesian closed category and a concrete quasitopos. Furthermore the concept has
already been up to effective use for several areas of mathematics.
Inthispaper,first,weobtainsomepropertiesofneutrosophiccrispsetsproposed
bySalamaandSmarandache[17]in2015. Second,weintroducethecategoryNCSet
consisting of neutrosophic crisp sets and morphisms between them. And we prove
that the category NCSet is topological and cotopological over Set (See Theorem
4.6 and Corollary 4.8), where Set denotes the category consisting of ordinary sets
and ordinary mappings between them. Furthermore, we prove that final episinks
in NCSet are preserved by pullbacks(See Theorem 4.10) and NCSet is Cartesian
closed over Set (See Theorem 4.15).
2. Preliminaries
Inthissection,welistsomebasicdefinitionsandwell-knownresultsfrom[6,9,11]
which are needed in the next sections.
Definition2.1([9]). LetAbeaconcretecategoryand((Y ,ξ )) afamilyofobjects
j j J
inAindexedbyaclassJ.ForanysetX,let(f :X →Y ) beasourceofmappings
j j J
indexed by J. Then an A-structure ξ on X is said to be initial with respect to (in
short, w.r.t.) (X,(f ),(Y ,ξ )) , if it satisfies the following conditions:
j j j J
(i) for each j ∈J, f :(X,ξ)→(Y ,ξ ) is an A-morphism,
j j j
(ii) if (Z,ρ) is an A-object and g :Z →X is a mapping such that for each j ∈J,
the mapping f ◦g :(Z,ρ)→(Y ,ξ ) is an A-morphism, then g :(Z,ρ)→(X,ξ) is
j j j
an A-morphism.
In this case, (f :(X,ξ)→(Y ,ξ )) is called an initial source in A.
j j j J
Dual notion: cotopological category.
Result 2.2 ([9], Theorem 1.5). A concrete category A is topological if and only if
it is cotopological.
Result 2.3 ([9], Theorem 1.6). Let A be a topological category over Set, then it is
complete and cocomplete.
Definition 2.4 ([9]). Let A be a concrete category.
(i) The A-fibre of a set X is the class of all A-structures on X.
(ii) A is said to be properly fibred over Set, it satisfies the followings:
(a) (Fibre-smallness) for each set X, the A-fibre of X is a set,
(b) (Terminal separator property) for each singleton set X, the A-fibre of X
has precisely one element,
(c) if ξ and η are A-structures on a set X such that id : (X,ξ) → (X,η) and
id:(X,η)→(X,ξ) are A-morphisms, then ξ =η.
Definition 2.5 ([6]). 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 BA
and a A-morphism e : A×BA → B (called the evaluation) such that for any
A,B
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J.G.Leeetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,43–54
A-objectC andanyA-morphismf :A×C →B,thereexistsauniqueA-morphism
f¯:C →BA such that the diagram commutes:
Definition 2.6 ([6]). 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) A satisfies the fibre-
smallness condition; (c) A has the terminal separator property,
(ii) A is cotopological over Set,
(iii)finalepisinksinAarepreservedbypullbacks,i.e.,foranyepisink(g :X →
j j
Y) and any A-morphism f : W → Y, the family (e : U → W) , obtained by
J j j J
taking the pullback f and g , for each j ∈J, is again a final episink.
j
3. Neutrosophic crisp sets
In [17], Salama and Smarandache introduced the concept of a neutrosophic crisp
set in a set X and defined the inclusion between two neutrosophic crisp sets, the
intersection[union]oftwoneutrosophiccrispsets,thecomplementofaneutrosophic
crisp set, neutrosophic crisp empty [resp., whole] set as more than two types. And
they studied some properties related to neutrosophic crisp set operations. However,
by selecting only one type, we define the inclusion, the intersection [union], and
neutrosophic crisp empty [resp., whole] set again and find some properties.
Definition 3.1. Let X be a non-empty set. Then A is called a neutrosophic crisp
set (in short, NCS) in X if A has the form A=(A ,A ,A ),
1 2 3
where A ,A , and A are subsets of X,
1 2 3
The neutrosophic crisp empty [resp., whole] set, denoted by φ [resp., X ] is an
N N
NCS in X defined by φ = (φ,φ,X) [resp., X = (X,X,φ)]. We will denote the
N N
set of all NCSs in X as NCS(X).
In particular, Salama and Smarandache [17] classified a neutrosophic crisp set as
the followings.
A neutrosophic crisp set A=(A ,A ,A ) in X is called a:
1 2 3
(i) neutrosophic crisp set of Type 1 (in short, NCS-Type 1), if it satisfies
A ∩A =A ∩A =A ∩A =φ,
1 2 2 3 3 1
(ii) neutrosophic crisp set of Type 2 (in short, NCS-Type 2), if it satisfies
A ∩A =A ∩A =A ∩A =φ and A ∪A ∪A =X,
1 2 2 3 3 1 1 2 3
(iii) neutrosophic crisp set of Type 3 (in short, NCS-Type 3), if it satisfies
A ∩A ∩A =φ and A ∪A ∪A =X.
1 2 3 1 2 3
WewilldenotethesetofallNCSs-Type1[resp.,Type2andType3]asNCS (X)
1
[resp., NCS (X) and NCS (X)].
2 3
Definition 3.2. Let A=(A ,A ,A ),B =(B ,B ,B )∈NCS(X). Then
1 2 3 1 2 3
(i) A is said to be contained in B, denoted by A⊂B, if
A ⊂B , A ⊂B and A ⊃B ,
1 1 2 2 3 3
(ii) A is said to equal to B, denoted by A=B, if
A⊂B and B ⊂A,
(iii) the complement of A, denoted by Ac, is an NCS in X defined as:
Ac =(A ,Ac,A ),
3 2 1
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(iv) the intersection of A and B, denoted by A∩B, is an NCS in X defined as:
A∩B =(A ∩B ,A ∩B ,A ∪B ),
1 1 2 2 3 3
(v) the union of A and B, denoted by A∪B, is an NCS in X defined as:
A∪B =(A ∪B ,A ∪B ,A ∩B ).
1 1 2 2 3 3
Let (A ) ⊂NCS(X), where A =(A ,A ,A ). Then
j j∈J j j,1 j,2 j,3
(cid:84) (cid:84)
(vi) the intersection of (A ) , denoted by A (simply, A ), is an NCS
j j∈J j∈J j j
in X defined as:
(cid:92) (cid:92) (cid:92) (cid:91)
A =( A , A , A ),
j j,1 j,2 j,3
(cid:83) (cid:83)
(vii) the the union of (A ) , denoted by A (simply, A ), is an NCS in
j j∈J j∈J j j
X defined as:
(cid:91) (cid:91) (cid:91) (cid:92)
A =( A , A , A ).
j j,1 j,2 j,3
The followings are the immediate results of Definition 3.2.
Proposition 3.3. Let A,B,C ∈NCS(X). Then
(1) φ ⊂A⊂X ,
N N
(2) if A⊂B and B ⊂C, then A⊂C,
(3) A∩B ⊂A and A∩B ⊂B,
(4) A⊂A∪B and B ⊂A∪B,
(5) A⊂B if and only if A∩B =A,
(6) A⊂B if and only if A∪B =B.
Also the followings are the immediate results of Definition 3.2.
Proposition 3.4. Let A,B,C ∈NCS(X). Then
(1) (Idempotent laws): A∪A=A, A∩A=A,
(2) (Commutative laws): A∪B =B∪A, A∩B =B∩A,
(3) (Associative laws): A∪(B∪C)=(A∪B)∪C, A∩(B∩C)=(A∩B)∩C,
(4) (Distributive laws): A∪(B∩C)=(A∪B)∩(A∪C),
A∩(B∪C)=(A∩B)∪(A∩C),
(5) (Absorption laws): A∪(A∩B)=A, A∩(A∪B)=A,
(6) (DeMorgan’s laws): (A∪B)c =Ac∩Bc, (A∩B)c =Ac∪Bc,
(7) (Ac)c =A,
(8) (8a) A∪φ =A, A∩φ =φ ,
N N N
(8b) A∪X =X , A∩X =A,
N N N
(8c) Xc =φ , φc =X ,
N N N N
(8d) in general, A∪Ac (cid:54)=X , A∩Ac (cid:54)=φ .
N N
Proposition 3.5. Let A∈NCS(X) and let (A ) ⊂NCS(X). Then
j j∈J
(1) ((cid:84)A )c =(cid:83)Ac, ((cid:83)A )c =(cid:84)Ac,
j j j j
(cid:83) (cid:83) (cid:84) (cid:84)
(2) A∩( A )= (A∩A ), A∪( A )= (A∪A ).
j j j j
(cid:84) (cid:84) (cid:84) (cid:83)
Proof. (1) A =(A ,A ,A ). Then A =( A , A , A ). Thus
j j,1 j,2 j,3 j j,1 j,2 j,3
((cid:84)A )c =((cid:83)A ,((cid:84)A )c,(cid:84)A )=((cid:83)A ,(cid:83)Ac ,(cid:84)A )=(cid:83)Ac.
j j,3 j,2 j,1 j,3 j,2 j,1 j
Similarly, the second part is proved.
(2) Let A=(A ,A ,A ). Then
1 2 3
(cid:84) (cid:84) (cid:84) (cid:83)
A∪( A )=(A ∪( A ),A ∪( A ),A ∩( A ))
j 1 j,1 2 j,2 3 j,3
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(cid:84) (cid:84) (cid:83)
=( (A ∪A ), (A ∪A ), (A ∩A )
1 j,1 2 j,2 3 j,3
(cid:84)
= (A∪A ).
j
Similarly, the first part is proved. (cid:3)
Definition3.6. Letf :X →Y beamapping,andletA=(A ,A ,A )∈NCS(X)
1 2 3
and B =(B ,B ,B )∈NCS(Y). Then
1 2 3
(i) the image of A under f, denoted by f(A), is an NCS in Y defined as:
f(A)=(f(A ),f(A ),f(A )),
1 2 3
(ii) the preimage of B, denoted by f−1(B), is an NCS in X defined as:
f−1(B)=(f−1(B ),f−1(B ),f−1(B )).
1 2 3
Proposition 3.7. Let f : X → Y be a mapping and let A,B,C ∈ NCS(X),
(A ) ⊂ NCS(X) and D,E,F ∈ NCS(Y), (D ) ⊂ NCS(Y). Then the
j j∈J k k∈K
followings hold:
(1) if B ⊂C, then f(B)⊂f(C) and if E ⊂F, then f−1(E)⊂f−1(F).
(2) A⊂f−1f(A)) and if f is injective, then A=f−1f(A)),
(3) f(f−1(D))⊂D and if f is surjective, then f(f−1(D))=D,
(4) f−1((cid:83)D )=(cid:83)f−1(D ), f−1((cid:84)D )=(cid:84)f−1(D ),
k k k k
(cid:83) (cid:83) (cid:84) (cid:84)
(5) f( A )= f(A ), f( A )⊂ f(A ),
j j j j
(6) f(A) = φ if and only if A = φ and hence f(φ ) = φ , in particular if f
N N N N
is surjective, then f(X )=Y ,
N N
(7) f−1(Y )=Y , f−1(φ )=φ.
N N N
Definition 3.8 ([17]). Let A = (A ,A ,A ) ∈ NCS(X), where X is a set having
1 2 3
at least distinct three points. Then A is called a neutrosophic crisp point (in short,
NCP) in X, if A , A and A are distinct singleton sets in X.
1 2 3
Let A = {p }, A = {p } and A = {p }, where p (cid:54)= p (cid:54)= p ∈ X. Then
1 1 2 2 3 3 1 2 3
A=(A ,A ,A ) is an NCP in X. In this case, we will denote A as p=(p ,p ,p ).
1 2 3 1 2 3
Furthermore, we will denote the set of all NCPs in X as NCP(X).
Definition 3.9. Let A = (A ,A ,A ) ∈ NCS(X) and let p = (p ,p ,p ) ∈
1 2 3 1 2 3
NCP(X). ThenpissaidtobelongtoA,denotedbyp∈A,if{p }⊂A ,{p }⊂A
1 1 2 2
and {p }c ⊃A , i.e., p ∈A , p ∈A and p ∈Ac.
3 3 1 1 2 2 3 3
Proposition 3.10. Let A=(A ,A ,A )∈NCS(X). Then
1 2 3
(cid:91)
A= {p∈NCP(X):p∈A}.
Proof. Let p=(p ,p ,p )∈NCP(X). Then
1 2 3
(cid:83)
{p∈NCP(X):p∈A}
=((cid:83){p ∈X :p ∈A },(cid:83){p ∈X :p ∈A },(cid:84){p ∈X :p ∈Ac}
1 1 1 2 2 2 3 3 3
=A. (cid:3)
Proposition 3.11. Let A = (A ,A ,A ),B = (B ,B ,B ) ∈ NCS(X). Then
1 2 3 1 2 3
A⊂B if and only if p∈B, for each p∈A.
Proof. Suppose A⊂B and let p=(p ,p ,p )∈A. Then
1 2 3
A ⊂B ,A ⊂B ,A ⊃B
1 1 2 2 3 3
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J.G.Leeetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,43–54
and
p ∈A ,p ∈A ,p ∈Ac.
1 1 2 2 3 3
Thus p ∈B ,p ∈B ,p ∈Bc. So p∈B. (cid:3)
1 1 2 2 3 3
Proposition 3.12. Let (A ) ⊂NCS(X) and let p∈NCP(X).
j j∈J
(cid:84)
(1) p∈ A if and only if p∈A for each j ∈J.
j j
(cid:83)
(2) p∈ A if and only if there exists j ∈J such that p∈A .
j j
Proof. Let A =(A ,A ,A ) for each j ∈J and let p=(p ,p ,p ).
j j,1 j,2 j,3 1 2 3
(1) Suppose p∈(cid:84)A . Then p ∈(cid:84)A ,p ∈(cid:84)A ,p ∈(cid:83)Ac . Thus
j 1 j,1 2 j,2 3 j,3
p ∈A ,p ∈A ,p ∈Ac , for each j ∈J. So p∈A for each j ∈J.
1 j,1 2 j,2 3 j,3 j
We can easily see that the sufficient condition holds.
(2) suppose the necessary condition holds. Then there exists j ∈J such that
p ∈A ,p ∈A ,p ∈Ac .
1 j,1 2 j,2 3 j,3
Thus p ∈(cid:83)A ,p ∈(cid:83)A ,p ∈((cid:84)A )c. So p∈(cid:83)A .
1 j,1 2 j,2 3 j,3 j
We can easily prove that the necessary condition holds. (cid:3)
Definition 3.13. Let f : X → Y be an injective mapping, where X,Y are sets
having at least distinct three points. Let p = (p ,p ,p ) ∈ NCP(X). Then the
1 2 3
image of p under f, denoted by f(p), is an NCP in Y defined as:
f(p)=(f(p ),f(p ),f(p )).
1 2 3
Remark 3.14. In Definition 3.13, if either X or Y has two points, or f is not
injective, then f(p) is not an NCP in Y.
Definition 3.15 ([17]). Let A = (A ,A ,A ) ∈ NCS(X) and B = (B ,B ,B ) ∈
1 2 3 1 2 3
NCS(Y). Then the Cartesian product of A and B, denoted by A×B, is an NCS
in X×Y defined as: A×B =(A ×B ,A ×B ,A ×B ).
1 1 2 2 3 3
4. Properties of NCSet
Definition 4.1. Apair(X,A)iscalledaneutrosophiccrispspace(inshort,NCSp),
if A∈NCS(X).
Definition 4.2. Apair(X,A)iscalledaneutrosophiccrispspace-Typej (inshort,
NCSp-Type j), if A∈NCS (X), j =1,2,3.
j
Definition 4.3. Let (X,A ), (Y,A ) be two NCSps or NCSps-Type j, j =1,2,3
X Y
andletf :X →Y beamapping. Thenf :(X,A )→(Y,A )iscalledamorphism,
X Y
if A ⊂f−1(A ), equivalently,
X Y
A ⊂f−1(A ), A ⊂f−1(A ) and A ⊃f−1(A ),
X,1 Y,1 X,2 Y,2 X,3 Y,3
where A =(A ,A ,A ) and A =(A ,A ,A ).
X X,1 X,2 X,3 Y Y,1 Y,2 Y,3
Inparticular,f :(X,A )→(Y,A )iscalledanepimorphism[resp.,amonomor-
X Y
phism and an isomorphism], if it is surjective [resp., injective and bijective].
From Definitions 3.9, 4.3 and Proposition 3.11, it is obvious that
f :(X,A )→(Y,A ) is a morphism
X Y
if and only if
p=(p ,p ,p )∈f−1(A ), for each p=(p ,p ,p )∈A , i.e.,
1 2 3 Y 1 2 3 X
f(p )∈A , f(p )∈A , f(p )∈/ A , i.e.,
1 Y,1 2 Y,2 3 Y,3
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f(p)=(f(p ),f(p ),f(p ))∈A .
1 2 3 Y
The following is an immediate result of Definitions 4.3.
Proposition 4.4. For each NCSp or each NCSps-Type j (X,A ), j = 1,2,3, the
X
identity mapping id:(X,A )→(X,A ) is a morphism.
X X
Proposition 4.5. Let (X,A ), (Y,A ), (Z,A ) be NCSps or NCSps-Type j, j =
X Y Z
1,2,3 and let f : X → Y, g : Y → Z be mappings. If f : (X,A ) → (Y,A )
X Y
and f : (Y,A ) → (Z,A ) are morphisms, then g ◦f : (X,A ) → (Z,A ) is a
Y Z X Z
morphism.
Proof. LetA =(A ,A ,A ),A =(A ,A ,A )andA =(A ,A ,
X X,1 X,2 X,3 Y Y,1 Y,2 Y,3 Z Z,1 Z,2
A ). Then by the hypotheses, A ⊂ f−1(A ) and A ⊂ g−1(A ). Thus by
Z,3 X Y Y Z
Definition 4.3,
A ⊂f−1(A ), A ⊂f−1(A ), A ⊃f−1(A )
X,1 Y,1 X,2 Y,2 X,3 Y,3
and
A ⊂g−1(A ), A ⊂g−1(A ), A ⊃g−1(A ).
Y,1 Z,1 Y,2 Z,2 Y,3 Z,3
So A ⊂f−1(g−1(A )), A ⊂f−1(g−1(A )), A ⊃f−1(g−1(A )).
X,1 Z,1 X,2 Z,2 X,3 Z,3
Hence A ⊂(g◦f)−1(A ), A ⊂(g◦f)−1(A ), A ⊃(g◦f)−1(A ).
X,1 Z,1 X,2 Z,2 X,3 Z,2
Therefore g◦f is a morphism. (cid:3)
From Propositions 4.4 and 4.5, we can form the concrete category NCSet [resp.,
NCSet ] consisting of NCSs [resp., -Type j, j = 1,2,3] and morphisms between
j
them. Every NCSet [resp., NCSet , j =1,2,3]-morphism will be called a NCSet
j
[resp., NCSet , j =1,2,3]-mapping.
j
Theorem 4.6. The category NCSet is topological over Set.
Proof. Let X be any set and let ((X ,A )) be any families of NCSps indexed
j j j∈J
by a class J. Suppose (f : X → (X ,A )) is a source of ordinary mappings. We
j j j J
define the NCS A in X by A =(cid:84)f−1(A ) and A =(A ,A ,A ).
X X j j X X,1 X,2 X,3
Then clearly, A =(cid:84)f−1(A ), A =(cid:84)f−1(A ), A =(cid:83)f−1(A ).
X,1 j j,1 X,2 j j,2 X,3 j j,3
Thus (X,A ) is an NCSp and A ⊂ f−1(A ), A ⊂ f−1(A ) and A ⊃
X X,1 j j,1 X,2 j j,2 X,3
f−1(A ). So each f :(X,A )→(X ,A ) is an NCSet-mapping.
j j,3 j X j j
Now let (Y,A ) be any NCSp and suppose g : Y → X is an ordinary mapping
Y
for which f ◦g :(Y,A )→(X ,A ) is a NCSet-mapping for each j ∈J. Then for
j Y j j
each j ∈J, A ⊂(f ◦g)−1(A )=g−1(f−1(A )). Thus
Y j j j j
(cid:92)
A ⊂(f ◦g)−1(A )=g−1( f−1(A ))=g−1(A ).
Y j j j j X
So g :(Y,A )→(X,A ) is an NCSet-mapping. Hence (f :(X,A )→(X ,A )
Y X j X j j J
is an initial source in NCSet. This completes the proof. (cid:3)
Example 4.7. (1) Let X be a set, let (Y,A ) be an NCSp and let f : X → Y be
Y
an ordinary mapping. Then clearly, there exists a unique NCS A in X for which
X
f :(X,A )→(Y,A ) is an NCSet-mapping. In fact, A =f−1(A ).
X Y X Y
In this case, A is called the inverse image under f of the NCS structure A .
X Y
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(2) Let ((X ,A )) be any family of NCSps and let X = Π X . For each
j j j∈J j∈J j
j ∈J,letpr :X →X betheordinaryprojection. ThenthereexistsauniqueNCS
j j
A in X for which pr :(X,A →(X ,A ) is an NCSet-mapping for each j ∈J.
X j X j j
In this case, A is called the product of (A ) , denoted by
X j j∈J
A =ΠA =(ΠA ,ΠA ,ΠA )
X j j,1 j,2 j,3
and (ΠX ,ΠA ) is called the product NCSp of ((X ,A )) .
j j j j j∈J
In fact, A =(cid:84) pr−1(A ).
X j∈J j j
Inparticular, ifJ ={1,2}, thenA ×A =(A ×A ,A ×A ,A ×A ),
1 2 1,1 2,1 1,2 2,2 1,3 2,3
where A =(A ,A ,A )∈NCS(X ) and A =(A ,A ,A )∈NCS(X ).
1 1,1 1,2 1,3 1 2 2,1 2,2 2,3 2
The following is obvious from Result 2.2. But we show directly it.
Corollary 4.8. The category NCSet is cotopological over Set.
Proof. Let X be any set and let ((X ,A )) be any family of NCSps indexed by a
j j J
class J. Suppose (f :X →X) is a sink of ordinary mappings. We define A as
j j J X
(cid:83)
A = f (A ), where A = (A ,A ,A ) and A = (A ,A ,A ). Then
X j j X X,1 X,2 X,3 j j,1 j,2 j,3
clearly, A ∈NCS(X) and each f :(X ,A )→(X,A ) is an NCSet-mapping.
X j j j X
Now for each NCSp (Y,A ), let g : X → Y be an ordinary mapping for which
Y
eachg◦f :(X ,A )→(Y,A )isanNCSet-mapping. Thenclearlyforeachj ∈J,
j j j Y
A ⊂(g◦f )−1(A ), i.e., A ⊂f−1(g−1(A )).
j j Y j j Y
Thus (cid:83)A ⊂ (cid:83)f−1(g−1(A )). So f ((cid:83)A ) ⊂ f ((cid:83)f−1(g−1(A ))). By Proposi-
j j Y j j j j Y
tion 3.7 and the definition of A ,
X
(cid:91) (cid:91)
f ( A )= f (A )=A
j j j j X
and
(cid:91) (cid:91)
f ( f−1(g−1(A )))= (f ◦f−1)(g−1(A ))=g−1(A ).
j j Y j j Y Y
Hence A ⊂g−1(A ). Therefore g :(X,A )→(Y,A ) is an NCSet-mapping.
X Y X Y
This completes the proof. (cid:3)
The following is proved similarly as the proof of Theorem 4.6.
Corollary 4.9. The category NCSet is topological over Set for j =1,2,3.
j
The following is proved similarly as the proof of Corollary 4.8.
Corollary 4.10. The category NCSet is cotopological over Set for j =1,2,3.
j
Theorem 4.11. Final episinks in NCSet are prserved by pullbacks.
Proof. Let (g : (X ,A ) → (Y,A )) be any final episink in NCSet and let f :
j j j Y J
(W,A )→(Y,A ) be any NCSet-mapping. For each j ∈J, let
W Y
U ={(w,x )∈W ×X :f(w)=g (x )}.
j j j j j
For each j ∈J, we define the NCS A =(A ,A ,A ) in U by:
Uj Uj,1 Uj,2 Uj,3 j
A =A ×A ,A =A ×A ,A =A ×A .
Uj,1 W,1 j,1 Uj,2 W,2 j,2 Uj,3 W,3 j,3
For each j ∈ J, let e : U → W and p : U → X be ordinary projections of U .
j j j j j j
Then clearly,
A ⊂e−1(A ),A ⊂e−1(A ),A ⊃e−1(A )
Uj,1 j W,1 Uj,2 j W,2 Uj,3 j W,3
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and
A ⊂p−1(A ),A ⊂p−1(A ),A ⊃p−1(A ).
Uj,1 j j,1 Uj,2 j j,2 Uj,3 j j,3
Thus A ⊂ e−1(A ) and A ⊂ p−1(A ). So e : (U ,A ) → (W,A ) and
Uj j W Uj j j j j Uj W
p :(U ,A )→(X ,A ) are NCSet-mappings. Moreover, g ◦p =f◦e for each
j j Uj j j h h j
j ∈J, i.e., the diagram is a pullback square in NCSet:
p (cid:45)
(U ,A ) j (X ,A )
j Uj j j
e g
j j
(cid:63) (cid:63)
(cid:45)
(W,A ) (Y,A ).
W Y
f
Nowinordertoprovethat(e ) isanepisinkinNCSet,i.e.,eache issurjective,
j J j
let w ∈ W. Since (g ) is an episink, there exists j ∈ J such that g (x ) = f(w)
j J j j
for some x ∈ X . Thus (w,x ) ∈ U and w = e (w,x ). So (e ) is an episink in
j j j j j j j J
NCSet.
Finally, let us show that (e ) is final in NCSet. Let A∗ be the final structure
j J W
in W w.r.t. (e ) and let w = (w ,w ,w ) ∈ A . Since f : (W,A ) → (Y,A ) is
j J 1 2 3 W W Y
an NCSet-mapping, by Definition 3.9,
w ∈A ∩f−1(A ), w ∈A ∩f−1(A ) and w ∈Ac ∩(f−1(A ))c.
1 W,1 Y,1 2 W,2 Y,2 3 W,3 Y,3
Thus
w ∈A ,f(w )∈A , w ∈A ,f(w )∈A and w ∈Ac ,f(w )∈Ac .
1 W,1 1 Y,1 2 W,2 2 Y,2 3 W,3 3 Y,3
Since (g ) is final,
j J
(cid:91) (cid:91)
w ∈A ,x ∈ A ,
1 W,1 j,1 j,1
J xj,1∈gj−1(f(w))
(cid:91) (cid:91)
w ∈A ,x ∈ A
2 W,2 j,2 j,2
J xj,2∈gj−1(f(w))
and
(cid:92) (cid:92)
w ∈Ac ,x ∈( A )c.
3 W,3 j,3 j,3
J xj,3∈gj−1(f(w))
So (w ,x ) ∈ A , (w ,x ) ∈ A and (w ,x ) ∈ Ac . Since A∗ is the
1 j,1 Uj,1 2 j,2 Uj,2 3 j,3 Uj,1 W
final structure in W w.r.t. (e ) , w ∈ A∗ , i.e., A ⊂ A∗ . On the other hand,
j J W W W
since (e :(U ,A )→(W,A )) is final, 1 :(W,A∗ )→(W,A ) is an NCSet-
j j Uj W J W W W
mapping and thus A∗ ⊂ A . Hence A∗ = A . Therefore (e ) is final. This
W W W W j J
completes the proof. (cid:3)
The following is proved similarly as the proof of Theorem 4.9.
Corollary4.12. FinalepisinksinNCSet areprservedbypullbacks,forJ =1,2,3.
j
For any singleton set {a}, NCS A [resp., NCS-Type j A , for j = 1,2,3]
{a} {a},j
on {a} is not unique, the category NCSet [resp., NCSet , for j = 1,2,3] is not
j
properly fibred over Set. Then by Definition 2.6, Corollary 4.8 and Theorem 4.11
[resp., Corollaries 4.10 and 4.12], we have the following result.
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J.G.Leeetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,43–54
Theorem 4.13. The category NCSet [resp., NCSet , for j = 1,2,3] satisfies
j
all the conditions of a topological universe over Set except the terminal separator
property.
The following is an immediate result of Definitions 3.9 and 3.15.
Proposition 4.14. Let p = (p ,p ,p ),q = (q ,q ,q ) ∈ NCP(X) and let A =
1 2 3 1 2 3
(A ,A ,A ),B = (B ,B ,B ) ∈ NCS(X). Then (p,q) ∈ A × B if and only if
1 2 3 1 2 3
(p ,q ) ∈ A ×B , (p ,q ) ∈ A ×B and (p ,q ) ∈ (A ×B )c, i.e., p ∈ Ac or
1 1 1 1 2 2 2 2 3 3 2 2 3 3
q ∈Bc.
3 3
Theorem 4.15. The category NCSet is Cartesian closed over Set.
Proof. It is clear that NCSet has products by Theorem 4.6. Then it is sufficient to
see that NCSet has exponential objects.
For any NCSps X=(X,A ) and Y =(Y,A ), let YX be the set of all ordinary
X Y
mappings from X to Y. We define the NCS A = (A ,A ,A ) in YX
YX YX,1 YX,2 YX,3
by: for each f = (f ,f ,f ) ∈ YX, f ∈ A if and only if f(x) ∈ A , for each
1 2 3 YX Y
x=(x ,x ,x )∈NCP(X), i.e.,
1 2 3
f ∈A ,f ∈A ,f ∈/ A
1 YX,1 2 YX,2 3 YX,3
if and only if
f (x )∈A ,f (x )∈A ,f (x )∈/ A .
1 1 Y,1 2 2 Y,2 3 3 Y,3
In fact,
A ={f ∈YX :f (x )∈A for each x ∈X},
YX,1 1 1 1 Y,1 1
A ={f ∈YX :f (x )∈A for each x ∈X},
YX,2 2 2 2 Y,2 2
A ={f ∈YX :f (x )∈/ A for some x ∈X}.
YX,3 3 3 3 Y,3 3
Then clearly, (YX,A ) is an NCSp.
YX
Let YX =(YX,A ). Then by the definition of A ,
YX YX
A ⊂f−1(A ), A ⊂f−1(A ) and A ⊃f−1(A ).
YX,1 Y,1 YX,2 Y,2 YX,3 Y,3
We define e :X ×YX →Y by e (x,f)=f(x), for each (x,f)∈X ×YX.
X,Y X,Y
Let(x,f)∈A ×A ,wherex=(x ,x ,x ),f =(f ,f ,f ). ThenbyProposition
X YX 1 2 3 1 2 3
4.14 and the definition of e ,
X,Y
(x ,f )∈A ×A , (x ,f )∈A ×A , (x ,f )∈(A ×A )c
1 1 X,1 YX,1 2 2 X,2 YX,2 3 3 X,3 YX,3
and
e (x ,f )=f (x ), e (x ,f )=f (x ), e (x ,f )=f (x ).
X,Y 1 1 1 1 X,Y 2 2 2 2 X,Y 3 3 3 3
Thus by the definition of A ,
YX
(x ,f )∈f−1(A )×f−1(A ),
1 1 Y,1 Y,1
(x ,f )∈f−1(A )×f−1(A ),
2 2 X,2 X,2
(x ,f )∈(f−1(A )×(f−1(A ))c.
3 3 X,3 X,3
So (x ,f )∈e−1 (A ), (x ,f )∈e−1 (A ) and (x ,f )∈(e−1 (A ))c. Hence
1 1 X,Y Y,1 2 2 X,Y Y,2 3 3 X,Y Y,3
A ×A ⊂e−1 (A ). Therefore e :X×YX →Y is an NCSet-mapping.
X YX X,Y Y X,Y
For any Z=(Z,A )∈NCSet, let h:X×Z→Y be an NCSet-mapping. We
Z
define h¯ : Z → YX by [h¯(z)](x) = h(x,z), for each z ∈ Z and each x ∈ X. Let
(x,z)∈A ×A , where x=(x ,x ,x ) and z =(z ,z ,z ). Since h:X×Z→Y
X Z 1 2 3 1 2 3
is an NCSet-mapping,
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