On prime spaces of neutrosophic extended triplet groups

Definition 1

[15] Let F and G be two functors between categories C and D . A triple ( F , η , G ) is called a natural transformation, if the function η : O b ( C ) → ℳ o r ( D ) satisfies the following conditions:

1. for any C ∈ O b ( C ) , η ( C ) (usually denoted by η C ) : F ( C ) → G ( C ) is a morphism of D ;

2. for any f ∈ hom C ( C , C ′ ) , the following diagram commutes:

Definition 2

[15] Let C and D be two categories, and let F : C → D and G : D → C be two functors. We say ( F , G ) forms an adjunction if there is a natural transformation η : id → GF such that for any object C of C and D of D and any morphism f : C → G ( D ) , there exists a unique morphism g : F ( C ) → D such that the following diagram commutes:

Definition 3

[15] A functor F : C → D is called an equivalence of categories if there are

1. a functor G : D → C ;

2. a family of isomorphisms φ C : C → G ( F ( C ) ) indexed by the objects of C with the property that for any morphism f : C → C ′ of C , G ( F ( f ) ) = φ C ′ ∘ f ∘ φ C − 1 ;

3. a family of isomorphisms ϕ D : D → F ( G ( D ) ) indexed by the objects of D with the property that for any morphism g : D → D ′ of D , F ( G ( g ) ) = ϕ D ′ ∘ g ∘ ϕ D − 1 .

Definition 4

[7] Let N be a non-empty set together with a binary operation ∗ . Then N is called a neutrosophic extended triplet group or NETG for short, if ( N , ∗ ) is a semigroup and for any a ∈ N , there exist a neutral of “ a ” (denoted by neut ( a ) ) and an opposite of “ a ” (denoted by anti ( a ) ) such that neut ( a ) ∈ N , anti ( a ) ∈ N and:

Proposition 1

[9] Let ( N , ∗ ) be a NETG. Then for every a ∈ N , the following statements hold:

1. neut ( a ) is unique;

2. neut ( a ) ∗ neut ( a ) = neut ( a ) ;

3. neut ( neut ( a ) ) = neut ( a ) .

Definition 5

[16] A semigroup ( S , ∗ ) will be called completely regular if there exists a unary operation a ↦ a − 1 on S with properties: ( a − 1 ) − 1 = a , a ∗ a − 1 ∗ a = a , a ∗ a − 1 = a − 1 ∗ a .

Proposition 2

[12] Let ( N , ∗ ) be a NETG. Then ∀ a ∈ N , ∀ p , q ∈ { anti ( a ) } , p ∗ neut ( a ) ∈ { anti ( a ) } , and p ∗ neut ( a ) = q ∗ neut ( a ) = neut ( a ) ∗ q .

Theorem 1

[12] Let ( N , ∗ ) be a groupoid. Then ( N , ∗ ) is a NETG if and only if ( N , ∗ ) is a completely regular semigroup.

Remark 1

In the proof of Theorem 1, [12] defines a unary operation a ↦ a − 1 by a − 1 = anti ( a ) ∗ neut ( a ) in a NETG ( N , ∗ ) . Then Proposition 2 indicates that this unary operation is well defined, which means a − 1 is determined uniquely for every element a of ( N , ∗ ) and a − 1 ∈ { anti ( a ) } . Moreover, the following properties:

have also been verified and a − 1 is called the inverse element of a in [12]. Therefore, in the following, we will regard a − 1 to be anti ( a ) ∗ neut ( a ) for every element a of a NETG ( N , ∗ ) , and it holds obviously that a − 1 ∗ a = a ∗ a − 1 = neut ( a ) .

Definition 6

[9] Let ( N , ∗ ) be a NETG. A non-empty subset S ⊆ N is called a NT-subgroup of N if it satisfies two conditions:

1. a ∗ b ∈ S for all a , b ∈ S ;

2. { anti ( a ) } ∩ S ≠ ∅ for all a ∈ S .

Proposition 3

[9] Let ( N , ∗ ) be a weak commutative NETG. Then for all a , b ∈ N , neut ( a ) ∗ neut ( b ) = neut ( b ∗ a ) and anti ( a ) ∗ anti ( b ) ∈ { anti ( b ∗ a ) } .

Proposition 4

[13] Let ( N , ∗ ) be a NETG, then ∀ a ∈ N , [ neut ( a ) ] − 1 = neut ( a ) = neut ( a − 1 ) . Moreover, if ( N , ∗ ) is a weak commutative NETG, then ∀ a , b ∈ N , ( a ∗ b ) − 1 = b − 1 ∗ a − 1 .

Definition 7

[14] Let ( N , ∗ ) be a NETG. A non-empty subset S ⊆ N is called an ideal of N if for all s ∈ S and a ∈ N ,

1. s ∗ a ∈ S and a ∗ s ∈ S ;

2. { anti ( s ) } ∩ S ≠ ∅ .

Proposition 5

[14] Let ( N , ∗ ) be a weak commutative NETG, then for every non-empty subset X ⊆ N , ⟨ X ⟩ = { neut ( x ) ∗ y − 1 : x ∈ X , y ∈ N } .

Definition 8

[17] Let X be a topological space. A subset Y ⊆ X is called super-compact if for any family of open subsets { U λ } λ ∈ Λ with Y ⊆ ⋃ λ ∈ Λ U λ , there exists λ 0 ∈ Λ such that Y ⊆ U λ 0 . The set of all super-compact open sets of X is denoted by SO ( X ) .

It is clear that ∅ is a super-compact open set.

Definition 9

[3] A topological space X is called a semigroup-spectral space (or SS-space for short), if it satisfies the following conditions:

1. X is T 0 ;

2. SO ( X ) is closed under finite intersection and forms a base of X ;

3. for any closed set A , if ℬ is a subset of SO ( X ) satisfying ℬ is closed under finite intersection and for any U ∈ ℬ , A ∩ U ≠ ∅ , then A ∩ ( ⋂ U ∈ ℬ U ) ≠ ∅ .

Proposition 6

[3] Let S be a commutative semigroup with a zero element 0, and I , J , I α be ideals of S, α ∈ Λ . Define h ( A ) ≔ { P ∈ PS ( S ) ∣ A ⊆ P } for any A ⊆ S . Then

1. h ( S ) = ∅ , h ( 0 ) = PS ( S ) ;

2. h ( I ) ∪ h ( J ) = h ( I ∩ J ) ;

3. ⋂ { h ( I α ) : α ∈ Λ } = h ( ⋃ { I α : α ∈ Λ } ) .

Theorem 2

[3] Let X be a topological space. Then X is an SS-space if and only if X is homeomorphic to the space PS ( SO ( X ) ) , in which ( SO ( X ) , ∩ ) is a commutative semigroup with the zero element ∅ .

Theorem 3

[3] The category CIS is dually equivalent to the category SST .

Example 1

Consider Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } under multiplication ⋅ modulo 6, then ( Z 6 , ⋅ ) is a NETG, in which neut ( 0 ) = 0 , { anti ( 0 ) } = { 0 , 1 , 2 , 3 , 4 , 5 } ; neut ( 1 ) = 1 , { anti ( 1 ) } = { 1 } ; neut ( 2 ) = 4 , { anti ( 2 ) } = { 2 , 5 } ; neut ( 3 ) = 3 , { anti ( 3 ) } = { 1 , 3 , 5 } ; neut ( 4 ) = 4 , { anti ( 4 ) } = { 1 , 4 } ; neut ( 5 ) = 1 , { anti ( 5 ) } = { 5 } . There are many NT-subgroups of Z 6 , and we can enumerate some: S 1 = { 0 } , S 2 = { 1 } , S 3 = { 3 } , S 4 = { 4 } , S 5 = { 0 , 1 } , S 6 = { 0 , 3 } , S 7 = { 0 , 4 } , S 8 = { 2 , 4 } , S 9 = { 1 , 3 } , S 10 = { 1 , 5 } , S 11 = { 0 , 2 , 4 } , S 12 = { 0 , 2 , 3 , 4 } . We also can list out all ideals of Z 6 : I 1 = { 0 } , I 2 = { 0 , 2 , 4 } , I 3 = { 0 , 3 } , I 4 = { 0 , 2 , 3 , 4 } , I 5 = Z 6 , and I 1 is the smallest ideal. Moreover, I 2 = { 0 , 2 , 4 } , I 3 = { 0 , 3 } , and I 4 = { 0 , 2 , 3 , 4 } are prime ideals.

Example 2

Let L be a frame and j : L → L a nucleus on L , which is a closure operator satisfying j ( a ∧ b ) = j ( a ) ∧ j ( b ) for all a , b ∈ L . We define an operation ∗ on L by: a ∗ b = a ∧ j ( b ) . Then for any a , b , c ∈ L , ( a ∗ b ) ∗ c = a ∧ j ( b ) ∧ j ( c ) and a ∗ ( b ∗ c ) = a ∧ j ( b ∧ j ( c ) ) = a ∧ j ( b ) ∧ j ( c ) , which implies ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) . Since a ∗ a = a ∧ j ( a ) = a , by viewing neut ( a ) = a and a ∈ { anti ( a ) } , we conclude ( L , ∗ ) is a NETG, but not a commutative semigroup.

Example 3

Consider ( Z 6 , ∗ ) in which ∗ is defined as a ∗ b = 3 a + b (mod 6) for all a , b ∈ Z 6 . Then ( Z 6 , ∗ ) is a NETG which is given in Table 1.

Example 4

[9] Let N = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } and define an operation ∗ on N as in Table 2. Then ( N , ∗ ) is a weak commutative NETG, in which neut ( 1 ) = 1 , { anti ( 1 ) } = { 1 } ; neut ( 2 ) = 1 , { anti ( 2 ) } = { 2 } ; neut ( 3 ) = 1 , { anti ( 3 ) } = { 3 } ; neut ( 4 ) = 1 , { anti ( 4 ) } = { 4 } ; neut ( 5 ) = 1 , { anti ( 5 ) } = { 6 } ; neut ( 6 ) = 1 , { anti ( 6 ) } = { 5 } ; neut ( 7 ) = 7 , { anti ( 7 ) } = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } . Since 3 ∗ 2 ≠ 2 ∗ 3 , we can see ( N , ∗ ) is not a commutative semigroup. It is clear that I 1 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } and I 2 = { 7 } are two ideals of N , and I 2 is not only the smallest ideal but also a prime ideal of N .

Example 5

Consider the set of integers Z and the number multiplication ⋅ , then ( Z , ⋅ ) is a commutative semigroup. For any m ∈ Z , m ⋅ 1 = 1 ⋅ m = m , but there is no integer multiplied by m equal to 1. Therefore, ( Z , ⋅ ) is not a NETG, not to say a weak commutative NETG.

Lemma 1

Let ( N , ∗ ) be a weak commutative NETG, then for any a ∈ N , S a ≔ { a m : m is a positive integer } ∪ { ( a − 1 ) n : n is a positive integer } ∪ { neut ( a ) } is a NT-subgroup.

Proof

From Propositions 1, 3, and 4, we can conclude for any positive integer n , ( a n ) − 1 = ( a − 1 ) n and neut ( a n ) = [ neut ( a ) ] n = neut ( a ) . Hence, by Proposition 4, S a satisfies Definition 6(1). Moreover, by Remark 1 and Proposition 4, [ ( a − 1 ) n ] − 1 = [ ( a n ) − 1 ] − 1 = a n ∈ S a . Then it is easy to see that S a is a NT-subgroup.□

Theorem 4

Let ( N , ∗ ) be a weak commutative NETG. Then the following statements hold:

1. if I is an ideal and A is a NT-subgroup of N which satisfies A ∩ I = ∅ , then I is contained in an ideal P, which is maximal with respect to the property of not meeting A, and P is prime;

2. for any proper ideal I and a ∉ I , there exists a prime ideal containing I but not a;

3. every ideal of N is the intersection of prime ideals containing it.

Proof

(1) Assume that I is an ideal and A ∩ I = ∅ . Let Π = { J ∈ Id ( N ) ∣ I ⊆ J and A ∩ J = ∅ } , then I ∈ Π . It is clear that Π satisfies Zorn’s lemma. Let P be a maximal element of Π , then A ∩ P = ∅ . Since P ≠ N , we can choose a , b ∈ N such that a ∉ P and b ∉ P , then P ⊆ P ∪ ⟨ a ⟩ ∈ Id ( N ) and P ⊆ P ∪ ⟨ b ⟩ ∈ Id ( N ) . By the maximality of P , we have A ∩ ( P ∪ ⟨ a ⟩ ) ≠ ∅ and A ∩ ( P ∪ ⟨ b ⟩ ) ≠ ∅ , then there are elements x , y such that x ∈ A ∩ ( P ∪ ⟨ a ⟩ ) and y ∈ A ∩ ( P ∪ ⟨ b ⟩ ) . Since A ∩ P = ∅ , only one case is possible: x ∈ A ∩ ⟨ a ⟩ and y ∈ A ∩ ⟨ b ⟩ . Hence, there exist c , d ∈ N such that x = neut ( a ) ∗ c − 1 and y = neut ( b ) ∗ d − 1 , then x ∗ y = [ neut ( a ) ∗ c − 1 ] ∗ [ neut ( b ) ∗ d − 1 ] = [ neut ( a ) ∗ neut ( b ) ] ∗ ( c − 1 ∗ d − 1 ) = neut ( a ∗ b ) ∗ ( d ∗ c ) − 1 . Since x ∗ y ∈ A , we have neut ( a ∗ b ) ∗ ( d ∗ c ) − 1 ∉ P . Thus, to avoid contradiction, a ∗ b must not exist in P , which implies P is a prime ideal.

(2) Let I be a proper ideal, then there exists a ∈ N such that a ∉ I . Thus, by Lemma 1, S a is a NT-subgroup and S a ∩ I = ∅ . Then by (1), we have there exists a prime ideal P such that I ⊆ P and S a ∩ P = ∅ . So a ∉ P .

(3) Let I be an ideal. If I ≠ N , then by (2), { J ∈ Prim ( N ) ∣ I ⊆ J } ≠ ∅ . Obviously, I ⊆ ⋂ { J ∈ Prim ( N ) ∣ I ⊆ J } . Suppose that a ∉ I , then by Lemma 1, S a is a NT-subgroup with S a ∩ I = ∅ . So by (1), there exists a prime ideal P with I ⊆ P and S a ∩ P = ∅ , which implies a ∉ P . Thus, a ∉ ⋂ { J ∈ Prim ( N ) ∣ I ⊆ J } . Consequently, ⋂ { J ∈ Prim ( N ) ∣ I ⊆ J } = I . If I = N , then { J ∈ Prim ( N ) ∣ I ⊆ J } = ∅ , so I = N = ⋂ ∅ = ⋂ { J ∈ Prim ( N ) ∣ I ⊆ J } .□

Lemma 2

Let ( N , ∗ ) be a weak commutative NETG and I 0 be the smallest ideal of N. Then

1. for any x , y ∈ N , c ( x ) ⊆ c ( x ∗ y ) and c ( x ) ⊆ c ( y ∗ x ) ;

2. for every a ∈ N , c ( a ) = c ( a − 1 ) , which means o ( a ) = o ( a − 1 ) ;

3. if { A i } i ∈ Λ is a family of subsets of N, we have c ( ⋃ i ∈ Λ A i ) = ⋂ i ∈ Λ c ( A i ) ;

4. for any x , y ∈ N , o ( x ∗ y ) = o ( x ) ∩ o ( y ) ;

5. ( Prim ( N ) , τ ) is a T 0 -space;

6. for every x ∈ I 0 , c ( x ) = Prim ( N ) , and o ( x ) = ∅ ;

7. for any A ⊆ N , c ( A ) = c ( ⟨ A ⟩ ) , which means o ( A ) = o ( ⟨ A ⟩ ) .

Proof

We only prove (4) and (5) here, and the other statements are true obviously.

(4) For any x , y ∈ N and P ∈ Prim ( N ) , since x ∈ P or y ∈ P ⇔ x ∗ y ∈ P , we obtain x ∉ P and y ∉ P ⇔ x ∗ y ∉ P , which means P ∈ o ( x ) ∩ o ( y ) ⇔ P ∈ o ( x ∗ y ) . So o ( x ∗ y ) = o ( x ) ∩ o ( y ) .

(5) Let I , J ∈ Prim ( N ) and I ≠ J . Then I ⊈ J or J ⊈ I . If I ⊈ J , then J ∈ o ( I ) and I ∉ o ( I ) . If J ⊈ I , then I ∈ o ( J ) and J ∉ o ( J ) . Hence, ( Prim ( N ) , τ ) is a T 0 -space.□

Theorem 5

Let ( N , ∗ ) be a weak commutative NETG. Then the following statements hold:

1. for any x ∈ N , o ( x ) is a super-compact open set in ( Prim ( N ) , τ ) ;

2. a subset X ⊆ Prim ( N ) is a super-compact open set if and only if there exists x ∈ N such that X = o ( x ) ;

3. for any closed set C in ( Prim ( N ) , τ ) , if ℬ is a subset of { o ( a ) : a ∈ N } satisfying (i) ℬ is closed under finite intersection; (ii) for any X ∈ ℬ , C ∩ X ≠ ∅ , then C ∩ ( ⋂ X ∈ ℬ X ) ≠ ∅ .

Proof

(1) Let x ∈ N . Suppose that { o ( a ) : a ∈ A } is a family of open sets with o ( x ) ⊆ ⋃ a ∈ A o ( a ) , then by Lemma 2 (3), c ( A ) = ⋂ a ∈ A c ( a ) ⊆ c ( x ) , and so by Lemma 2 (7), { P ∈ Prim ( N ) ∣ ⟨ A ⟩ ⊆ P } ⊆ { P ∈ Prim ( N ) ∣ ⟨ x ⟩ ⊆ P } . Thus, by Theorem 4 (3), ⟨ x ⟩ = ⋂ { P ∈ Prim ( N ) ∣ ⟨ x ⟩ ⊆ P } ⊆ ⋂ { P ∈ Prim ( N ) ∣ ⟨ A ⟩ ⊆ P } = ⟨ A ⟩ . Hence, by Proposition 5, x ∈ ⟨ x ⟩ ⊆ ⟨ A ⟩ = { neut ( s ) ∗ t − 1 : s ∈ A , t ∈ N } . Then there exist s ∈ A and t ∈ N such that x = neut ( s ) ∗ t − 1 . Since x = neut ( s ) ∗ t − 1 = ( s ∗ s − 1 ) ∗ t − 1 = s ∗ ( s − 1 ∗ t − 1 ) , by Lemma 2 (1), we have c ( s ) ⊆ c ( s ∗ ( s − 1 ∗ t − 1 ) ) = c ( x ) , so o ( x ) ⊆ o ( s ) . Therefore, o ( x ) is super-compact.

(2) Let X be a super-compact open set of ( Prim ( N ) , τ ) and I 0 the smallest ideal of N . If X = ∅ , by Lemma 2 (6), for every x ∈ I 0 , X = o ( x ) . If X ≠ ∅ , according to the fact that { o ( x ) : x ∈ N } is a base of topology ( Prim ( N ) , τ ) , there exists a subset Y of N such that X = ⋃ y ∈ Y o ( y ) , and then there exists y ∈ Y such that X ⊆ o ( y ) . Furthermore, o ( y ) ⊆ ⋃ y ∈ Y o ( y ) = X , so X = o ( y ) .

(3) Since C is a closed set in ( Prim ( N ) , τ ) , there exists an ideal I such that C = c ( I ) (i.e., because there exists B ⊆ N such that C = c ( B ) = c ( ⟨ B ⟩ ) ). Let A = { x ∈ N ∣ o ( x ) ∈ ℬ } , then ∀ x , y ∈ A , o ( x ) ∈ ℬ and o ( y ) ∈ ℬ . Since ℬ satisfies (i), by Lemma 2 (4), we have x ∗ y ∈ A . According to Lemma 2 (2), o ( x − 1 ) ∈ ℬ , that is, x − 1 ∈ A . Hence, A is a NT-subgroup. Now we can prove A ∩ I = ∅ . In fact, if there exists a ∈ A ∩ I , then c ( I ) ⊆ c ( a ) , and so c ( I ) ∩ o ( a ) = ∅ . However, o ( a ) ∈ ℬ and C ∩ o ( a ) ≠ ∅ . A contradiction arises. Therefore, by Theorem 4 (1), there exists a prime ideal P such that I ⊆ P and A ∩ P = ∅ . Then P ∈ c ( I ) = C and ∀ m ∈ A , P ∈ o ( m ) , that is, P ∈ C ∩ ( ⋂ X ∈ ℬ X ) , which means C ∩ ( ⋂ X ∈ ℬ X ) ≠ ∅ .□

Corollary 1

Prime spaces of weak commutative NETGs are SS-spaces.

Proof

Let ( N , ∗ ) be a weak commutative NETG. According to Lemma 2 (4) and Theorem 5 (2), we have SO ( Prim ( N ) ) is closed under finite intersection. Then by Lemma 2 (5), Theorem 5 (3) and Definition 9, we have Prim ( N ) is an S S -space.□

Lemma 3

Let ( N , ∗ ) be a semigroup, then ( N , ∗ ) is an idempotent weak commutative NETG if and only if ( N , ∗ ) is a commutative idempotent semigroup.

Proof

Suppose that ( N , ∗ ) is an idempotent weak commutative NETG. Let x , y ∈ N , by y 2 = y , we obtain neut ( y ) = y , so x ∗ y = x ∗ neut ( y ) = neut ( y ) ∗ x = y ∗ x . Hence, ( N , ∗ ) is commutative. Conversely, if ( N , ∗ ) is a commutative idempotent semigroup, for any a ∈ N , we obtain a 2 = a , by viewing neut ( a ) = a and a ∈ { anti ( a ) } , we can conclude ( N , ∗ ) is also a NETG, to be more precise, an idempotent weak commutative NETG.□

Remark 2

By Lemma 3, if ( N , ∗ ) is a commutative idempotent semigroup, we have ( N , ∗ ) is also an idempotent weak commutative NETG and Prim ( N ) = PS ( N ) .

Theorem 6

X is an SS-space if and only if X is homeomorphic to the prime space Prim ( N ) of some weak commutative NETG.

Proof

We only need to prove necessity. Since ( SO ( X ) , ∩ ) is closed under finite intersection, we have ( SO ( X ) , ∩ ) is a commutative idempotent semigroup with a zero element ∅ . By Theorem 2, X is homeomorphic to the space PS ( SO ( X ) ) . On the other hand, by Lemma 3, ( SO ( X ) , ∩ ) is an idempotent weak commutative NETG with the smallest ideal { ∅ } , then by Remark 2, PS ( SO ( X ) ) = Prim ( SO ( X ) ) . Therefore, X is homeomorphic to Prim ( SO ( X ) ) .□

Proposition 7

1. A function PR : NG O P → SST is defined as follows: PR assigns each weak commutative NETG to its prime space, and for any morphism N 1 → f N 2 in NG o p , PR ( f ) : Prim ( N 1 ) → Prim ( N 2 ) is given by PR ( f ) ( P ) = f − 1 ( P ) for every P ∈ Prim ( N 1 ) . Then PR is a contravariant functor from NG to SST .

2. A function SO : SST → NG O P is defined as follows: SO assigns each SS-space to its set of all super-compact open sets, and for any morphism X 1 → g X 2 in SST , SO ( g ) : SO ( X 1 ) → SO ( X 2 ) is given by SO ( g ) ( U ) = g − 1 ( U ) for every U ∈ SO ( X 2 ) . Then SO is a contravariant functor from SST to NG .