1. Introduction
A new kind of logical algebra, known as
K-algebra, was introduced by Dar and Akram in [
1]. A
K-algebra is built on a group
G by adjoining the induced binary operation on
G. The group
G is particularly of the type in which each non-identity element is not of order 2. This algebraic structure is, in general, non-commutative and non-associative with right identity element [
1,
2,
3]. Akram et al. [
4] introduced fuzzy
K-algebras. They then developed fuzzy
K-algebras with other researchers worldwide. The concepts and results of
K-algebras have been broadened to the fuzzy setting frames by applying Zadeh’s fuzzy set theory and its generalizations, namely, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, bipolar fuzzy sets and vague sets [
5]. In handling information regarding various aspects of uncertainty, non-classical logic is considered to be a more powerful tool than the classical logic. It has become a strong mathematical tool in computer science, medical, engineering, information technology, etc. In 1998, Smarandache [
6] introduced neutrosophic set as a generalization of intuitionistic fuzzy set [
7]. A neutrosophic set is identified by three functions called truth-membership
, indeterminacy-membership
and falsity-membership
functions. To apply neutrosophic set in real-life problems more conveniently, Smarandache [
6] and Wang et al. [
8] defined single-valued neutrosophic sets which takes the value from the subset of [0, 1]. Thus, a single-valued neutrosophic set is an instance of neutrosophic set.
Algebraic structures have a vital place with vast applications in various areas of life. Algebraic structures provide a mathematical modeling of related study. Neutrosophic set theory has also been applied to many algebraic structures. Agboola and Davazz introduced the concept of neutrosophic
-algebras and discuss elementary properties in [
9]. Jun et al. introduced the notion of
neutrosophic subalgebra of a
-algebra [
10]. Jun et al. [
11] defined interval neutrosophic sets on
-algebra [
11]. Jun et al. [
12] proposed neutrosophic positive implicative
N-ideals and study their extension property [
12] Several set theories and their topological structures have been introduced by many researchers to deal with uncertainties. Chang [
13] was the first to introduce the notion of fuzzy topology. Later, Lowan [
14], Pu and Liu [
15], and Chattopadhyay and Samanta [
16] introduced other concepts related to fuzzy topology. Coker [
17] introduced the notion of intuitionistic fuzzy topology as a generalization of fuzzy topology. Salama and Alblowi [
18] defined the topological structure of neutrosophic set theory. Akram and Dar [
19] introduced the concept of fuzzy topological
K-algebras. They extended their work on intuitionistic fuzzy topological
K-algebras [
20]. In this paper, we introduce the notion of single-valued neutrosophic topological
K-algebras and investigate some of their properties. Further, we study certain properties, including
-connected, super connected, compact and Hausdorff, of single-valued neutrosophic topological
K-algebras. We also investigate the image and pre-image of single-valued neutrosophic topological
K-algebras under homomorphism.
3. Neutrosophic Topological K-algebras
Definition 5. Let Z be a nonempty set. A collection χ of single-valued neutrosophic sets (SNSs) in Z is called a single-valued neutrosophic topology (SNT) on Z if the following conditions hold:
- (a)
- (b)
If , then
- (c)
If , , then
The pair is called a single-valued neutrosophic topological space (SNTS). Each member of χ is said to be χ-open or single-valued neutrosophic open set (SNOS) and compliment of each open single-valued neutrosophic set is a single-valued neutrosophic closed set (SNCS). A discrete topology is a topology which contains all single-valued neutrosophic subsets of Z and indiscrete if its elements are only , .
Definition 6. Let be a single-valued neutrosophic set in . Then, is called a single-valued neutrosophic K-subalgebra of if following conditions hold for :
- (i)
- (ii)
∀
Example 1. Consider a K-algebra , where is the cyclic group of order 9 and Caley’s table for ⊙ is given as:
If we define a single-valued neutrosophic set in such that:
∀
According to Definition 5, the family of SNSs of K-algebra is a SNT on . We define a SNS in such that . Clearly, is a SN K-subalgebra of .
Definition 7. Let be a K-algebra and let be a topology on . Let be a SNS in and let be a topology on . Then, an induced single-valued neutrosophic topology on is a collection or family of single-valued neutrosophic subsets of which are the intersection with and single-valued neutrosophic open sets in defined as . Then, is called single-valued neutrosophic induced topology on or relative topology and the pair is called an induced topological space or single-valued neutrosophic subspace of .
Definition 8. Let and be two SNTSs and let . Then, f is called single-valued neutrosophic continuous if following conditions hold:
- (i)
For each SNS , .
- (ii)
For each SN K-subalgebra , is a SN K-subalgebra .
Definition 9. Let and be two SNTSs and let and be two single-valued neutrosophic subspaces over and . Let f be a mapping from into , then f is a mapping from to if .
Definition 10. Let f be a mapping from to . Then, f is relatively single-valued neutrosophic continuous if for every SNOS in , .
Definition 11. Let f be a mapping from to . Then, f is relatively single-valued neutrosophic open if for every SNOS in , the image .
Proposition 2. Let and be single-valued neutrosophic subspaces of and , where and are K-algebras. If f is a single-valued neutrosophic continuous function from to and . Then, f is relatively single-valued neutrosophic continuous function from into .
Definition 12. Let and be two SNTSs. A mapping is called a single-valued neutrosophic homomorphism if following conditions hold:
- (i)
f is a one-one and onto function.
- (ii)
f is a single-valued neutrosophic continuous function from to
- (iii)
is a single-valued neutrosophic continuous function from to
Theorem 1. Let be a SNTS and be an indiscrete SNTS on K-algebras and , respectively. Then, each function f defined as is a single-valued neutrosophic continuous function from to . If and be two discrete SNTSs and , respectively, then each homomorphism is a single values neutrosophic continuous function from to .
Proof. Let f be a mapping defined as . Let be SNT on and be SNT on , where . We show that is a single-valued neutrosophic K-subalgebra of , i.e., for each . Since , then for any consider such that .
Therefore, Likewise, Hence, f is a SN continuous function from to
Now, for the second part of the theorem, where both
and
are SNTSs on
and
, respectively, and
is a homomorphism. Therefore, for all
and
, where
f is not a usual inverse homomorphism. To prove that
is a single-valued neutrosophic
K-subalgebra in of
Let for
Hence, f is a single-valued neutrosophic continuous function from to □
Proposition 3. Let and be two SNTSs on . Then, each homomorphism is a single-valued neutrosophic continuous function.
Proof. Let and be two SNTSs, where is a K-algebra. To prove the above result, it is enough to show that result is false for a particular topology. Let and be two SNSs in . Take and . If , defined by , for all , then f is a homomorphism. Now, for , , ,
∀, i.e., . Therefore, Hence, f is not a single-valued neutrosophic continuous mapping. □
Definition 13. Let be a K-algebra and χ be a SNT on . Let be a single-valued neutrosophic K-algebra (K-subalgebra) of and be a SNT on . Then, is said to be a single-valued neutrosophic topological K-algebra (K-subalgebra) on if the self mapping defined as , , is a relatively single-valued neutrosophic continuous mapping.
Theorem 2. Let and be two SNTSs on and , respectively, and be a homomorphism such that . If is a single-valued neutrosophic topological K-algebra of then is a single-valued neutrosophic topological K-algebra of .
Proof. Let
be a single-valued neutrosophic topological
K-algebra of
. To prove that
be a single-valued neutrosophic topological
K-algebra of
. Let for any
Hence, is a single-valued neutrosophic K-algebra of .
Now, we prove that is single-valued neutrosophic topological K-algebra of . Since f is a single-valued neutrosophic continuous function, then by proposition , f is also a relatively single-valued neutrosophic continuous function which maps to .
Let
and
Y be a SNS in
and let
X be a SNS in
such that
We are to prove that
is relatively single-valued neutrosophic continuous mapping, then for any
we have
It concludes that . Thus, is a SNS in and a SNS in . Hence, and a single-valued neutrosophic topological K-algebra of . Hence, the proof. □
Theorem 3. Let and be two SNTSs on and , respectively, and let f be a bijective homomorphism of into such that . If is a single-valued neutrosophic topological K-algebra of , then is a single-valued neutrosophic topological K-algebra of .
Proof. Suppose that
is a SN topological K-algebra of
. To prove that
is a single-valued neutrosophic topological
K-algebra of
, let, for
,
Let
,
such that
Hence,
is a single-valued neutrosophic
K-subalgebra of
Now, we prove that the self mapping
defined by
, for all
, is a relatively single-valued neutrosophic continuous mapping. Let
be a SNS in
, there exists a SNS
in
such that
. We show that for a SNS in
,
Since f is an injective mapping, then is a SNS in which shows that f is relatively single-valued neutrosophic open. In addition, f is surjective, then for all , where .
This implies that . Since is relatively single-valued neutrosophic continuous mapping and f is relatively single-valued neutrosophic continues mapping from into , is a SNS in . Hence, is a SNS in , which completes the proof. □
Example 2. Let be a K-algebra, where is the cyclic group of order 9 and Caley’s table for ⊙
is given in Example 1. We define a SNS as:for all where . The collection of SNSs of is a SNT on and is a SNTS. Let be a SNS in , defined as: Clearly, is a single-valued neutrosophic K-subalgebra of . By direct calculations relative topology is obtained as . Then, the pair is a single-valued neutrosophic subspace of . We show that is a single-valued neutrosophic topological K-subalgebra of , i.e., the self mapping defined by is relatively single-valued neutrosophic continuous mapping, i.e., for a SNOS in , . Since is homomorphism, then . Therefore, is relatively single-valued neutrosophic continuous mapping. Hence, is a single-valued neutrosophic topological K-algebra of .
Example 3. Let be a K-algebra, where is the cyclic group of order 9 and Caley’s table for ⊙
is given in Example 3.1. We define a SNS as:for all where . The collection and of SNSs of are SNTs on and , be two SNTSs. Let be a SNS in , defined as: Now, Let be a homomorphism such that (we have not consider to be distinct), then, by Proposition 3, f is a single-valued neutrosophic continuous function and f is also relatively single-valued neutrosophic continues mapping from into . Since is a SNS in and with relative topology is also a single-valued neutrosophic topological K-algebra of . We prove that is a single-valued neutrosophic topological K-algebra in . Since f is a continuous function, then, by Definition 8, is a single-valued neutrosophic K-subalgebra in . To prove that is a single-valued neutrosophic topological K-algebra, then for takefor which shows that is a single-valued neutrosophic topological K-algebra in . Similarly, we can show that is a a single-valued neutrosophic topological K-algebra in by considering a bijective homomorphism. Definition 14. Let χ be a SNT on and be a SNTS. Then, is called single-valued neutrosophic -disconnected topological space if there exist a SNOS and SNCS such that and , otherwise is called single-valued neutrosophic -connected.
Example 4. Every indiscrete SNT space on is -connected.
Proposition 4. Let and be two SNTSs and be a surjective single-valued neutrosophic continuous mapping. If is a single-valued neutrosophic -connected space, then is also a single-valued neutrosophic -connected space.
Proof. Suppose on contrary that
is a single-valued neutrosophic
-disconnected space. Then, by Definition 14, there exist both SNOS and SNCS
be such that
and
. Since
f is a single-valued neutrosophic continuous and onto function, so
or
, where
is both SNOS and SNCS. Therefore,
and
a contradiction. Hence,
is a single-valued neutrosophic
-connected space. □
Corollary 1. Let χ be a SNT on . Then, is called a single-valued neutrosophic -connected space if and only if there does not exist a single-valued neutrosophic continuous map such that and
Definition 15. Let be a SNS in . Let χ be a SNT on . The interior and closure of in is defined as:
: The union of SNOSs which contained in
: The intersection of SNCSs for which is a subset of these SNCSs.
Remark 1. Being union of SNOS is a SNO and being intersection of SNCS is SNC.
Theorem 4. Let be a SNS in a SNTS . Then, is such an open set which is the largest open set of contained in .
Corollary 2. is a SNOS in if and only if and is a SNCS in if and only if .
Proposition 5. Let be a SNS in . Then, following results hold for :
- (i)
- (ii)
- (iii)
- (iv)
Definition 16. Let be a K-algebra and χ be a SNT on . A SNOS in is said to be single-valued neutrosophic regular open if Remark 2. Every SNOS which is regular is single-valued neutrosophic open and every single-valued neutrosophic closed and open set is a single-valued neutrosophic regular open.
Definition 17. A single-valued neutrosophic super connected K-algebra is such a K-algebra in which there does not exist a single-valued neutrosophic regular open set such that and . If there exists such a single-valued neutrosophic regular open set such that and , then K-algebra is said to be a single-valued neutrosophic super disconnected.
Example 5. Let be a K-algebra, where is the cyclic group of order 9 and Caley’s table for ⊙
is given in Example 1 We define a SNS as: Let be a SNT on and let be a SNS in . here Then, closure of is the intersection of closed sets which contain . Therefore, Now, interior of is the union of open sets which contain in . Therefore, Note that . Now, if we consider a SNS in a K-algebra and if is a SNT on . Then, and . Consequently,which shows that is a SN regular open set in K-algebra . Since is a SN regular open set in and then, by Definition 17, K-algebra is a single-valued neutrosophic supper disconnected K-algebra. Proposition 6. Let be a K-algebra and let be a SNOS. Then, the following statements are equivalent:
- (i)
A K-algebra is single-valued neutrosophic super connected.
- (ii)
, for each SNOS .
- (iii)
, for each SNCS .
- (iv)
There do not exist SNOSs such that and in K-algebra .
Definition 18. Let be a SNTS, where is a K-algebra. Let S be a collection of SNOSs in denoted by . Let be a SNOS in . Then, S is called a single-valued neutrosophic open covering of if .
Definition 19. Let be a K-algebra and be a SNTS. Let L be a finite sub-collection of S. If L is also a single-valued neutrosophic open covering of , then it is called a finite sub-covering of S and is called single-valued neutrosophic compact if each single-valued neutrosophic open covering S of has a finite sub-cover. Then, is called compact K-algebra.
Remark 3. If either is a finite K-algebra or χ is a finite topology on , i.e., consists of finite number of single-valued neutrosophic subsets of , then the SNT is a single-valued neutrosophic compact topological space.
Proposition 7. Let and be two SNTSs and f be a single-valued neutrosophic continuous mapping from into . Let be a SNS in . If is single-valued neutrosophic compact in , then is single-valued neutrosophic compact in
Proof. Let
be a single-valued neutrosophic continuous function. Let
be a single-valued neutrosophic open covering of
since
be a SNS in
. Let
be a single-valued neutrosophic open covering of
. Since
is compact, then there exists a single-valued neutrosophic finite sub-cover
such that
We have to prove that there also exists a finite sub-cover of
for
such that
Hence, is single-valued neutrosophic compact in □
Definition 20. A single-valued neutrosophic set in a K-algebra is called a single-valued neutrosophic point ifwith support u and value , denoted by . This single-valued neutrosophic point is said to “belong to" a SNS written as if and said to be “quasi-coincident with" a SNS , written as if Definition 21. Let be a K-algebra and let be a SNTS. Then, is called a single-valued neutrosophic Hausdorff space if and only if, for any two distinct single-valued neutrosophic points , there exist SNOSs such that , i.e.,and satisfy the condition that Then, is called single-valued neutrosophic Hausdorff space and K-algebra is said to ba a Hausdorff K-algebra. In fact, is a Hausdorff K-algebra. Example 6. Let be a K-algebra and let be a SNTS on , where is the cyclic group of order 9 and Caley’s table for ⊙
is given in Example 1. We define two SNSs as Consider a single-valued neutrosophic point for such that Then, is a single-valued neutrosophic point with support e and value . This single-valued neutrosophic point belongs to SNS but not SNS .
Then, is a single-valued neutrosophic point with support s and value . This single-valued neutrosophic point belongs to SNS but not SNS . Thus, and , and and . Thus, K-algebra is a Hausdorff K-algebra and is a Hausdorff topological space.
Theorem 5. Let , be two SNTSs. Let f be a single-valued neutrosophic homomorphism from into . Then, is a single-valued neutrosophic Hausdorff space if and only if is a single-valued neutrosophic Hausdorff K-algebra.
Proof. Let , be two SNTSs. Let be a single-valued neutrosophic Hausdorff space, then, according to the Definition 21, there exist two SNOSs X and Y for two distinct single-valued neutrosophic points also such that .
Now, for , consider , where if , otherwise 0. That is, . Therefore, we have . Similarly, . Now, since is a single-valued neutrosophic continuous mapping from into , there exist two SNOSs and of and , respectively, such that This implies that is a single-valued neutrosophic Hausdorff K-algebra. The converse part can be proved similarly. □
Theorem 6. Let f be a single-valued neutrosophic continuous function which is both one-one and onto, where f is a mapping from a single-valued neutrosophic compact K-algebra into a single-valued neutrosophic Hausdorff K-algebra . Then, f is a homomorphism.
Proof. Let be a single-valued neutrosophic continuous bijective function from single-valued neutrosophic compact K-algebra into a single-valued neutrosophic Hausdorff K-algebra . Since f is a single-valued neutrosophic continuous mapping from into , f is a homomorphism. Since f is bijective, we only prove that f is single-valued neutrosophic closed. Let be a single-valued neutrosophic closed in . If is single-valued neutrosophic closed in , then is single-valued neutrosophic closed in . However, if , then will be a single-valued neutrosophic compact, being subset of a single-valued neutrosophic compact K-algebra. Then, , being single-valued neutrosophic continuous image of a single-valued neutrosophic compact K-algebra, is also single-valued neutrosophic compact. Therefore, is closed, which implies that mapping f is closed. Thus, f is a homomorphism. □