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Neutrosophic Soft Topological K-Algebras.

1 Introduction

A K-algebra (G, x [dot circle], e) is a new class of logical algebra, introduced by Dar and Akram [1] in 2003. A Kalgebra is constructed on a group (G, x, e) by adjoining an induced binary operation [C] on G and attached to an abstract K-algebra (G, x, [dot circle], e). This system is, in general, non-commutative and non-associative with a right identity e. If the given group G is not an elementary abelian 2-group, then the K-algebra is proper. Therefore, a K-algebra K = (G, x, [dot circle], e) is abelian and non-abelian, proper and improper purely depends upon the base group G. In 2004, a K-algebra renamed as K(G)-algebra due to its structural basis G and characterized by left and right mappings when the group G is abelian and non-abelian by Dar and Akram in [2, 3]. In 2007, Dar and Akram [4] investigated the K-homomorphisms of K-algebras.

Non-classical logic leads to classical logic due to various aspects of uncertainty. It has become a conventional tool for computer science and engineering to deal with fuzzy information and indeterminate data and executions. In our daily life, the most frequently encountered uncertainty is incomparability. Zadeh's fuzzy set theory [5] revolutionized the systems, accomplished with vagueness and uncertainty. A number of researchers extended the conception of Zadeh and presented different theories regarding uncertainty which includes intuitionistic fuzzy set theory, interval-valued intuitionistic fuzzy set theory [6] and so on. In addition, Smarandache [7] generalized intuitionistic fuzzy set by introducing the concept of neutrosophic set in 1998. It is such a branch of philosophy which studies the origin, nature, and scope of neutralities as well as their interactions with different ideational spectra. To have real life applications of neutrosophic sets such as in engineering and science, Wang et al. [8] introduced the single-valued neutrosophic set in 2010. In 1999, Molodtsov [9] introduced another mathematical approach to deal with ambiguous data, called soft set theory. Soft set theory gives a parameterized outlook to uncertainty. Maji [10] defined the notion of neutrosophic soft set by unifying the fundamental theories of neutrosophic set and soft set to deal with inconsistent data in a much-unified mode. A large number of theories regarding uncertainty with their respective topological structures have been introduced. In 1968, Chang [11] introduced the concept of fuzzy topology. Chattopadhyay and Samanta [12], Pu and Liu [13] and Lowan [14] defined some certain notions related to fuzzy topology. Recently, Tahan et al. [15] presented the notion of topological hypergroupoids. Onasanya and Hoskova-Mayerova [16] discussed some topological and algebraic properties of [alpha]--level subsets of fuzzy subsets. Coker [17] considered the notion of an intuitionistic fuzzy topology. Salama and Alblowi [18] studied the notion of neutrosophic topological spaces. In 2017, Bera and Mahapatra [19] described neutrosophic soft topological spaces. Akram and Dar [20, 21] considered fuzzy topological K-algebras and intuitionistic topological K-algebras. Recently, Akram et al. [22, 23, 24, 25] presented some notions, including single-valued neutrosophic K-algebras, single-valued neutrosophic topological K-algebras and single-valued neutrosophic Lie algebras. In this research article, In this paper, we propose the notion of single-valued neutrosophic soft topological K-algebras. We discuss certain concepts, including interior, closure, [C.sub.5]-connected, super connected, Compactness and Hausdorff in single-valued neutrosophic soft topological K-algebras. We illustrate these concepts with examples and investigate some of their related properties. We also study image and pre-image of single-valued neutrosophic soft topological K-algebras.

The rest of the paper is organized as follows: In Section 2, we review some elementary concepts related to K-algebras, single-valued neutrosophic soft sets and their topological structures. In Section 3, we define the concept of single-valued neutrosophic soft topological K-algebras and discuss certain concepts with some numerical examples. In Section 4, we present concluding remarks.

2 Preliminaries

This section consists of some basic definitions and concepts, which will be used in the next sections.

Definition 2.1. [1] A K-algebra K = (G, x, [dot circle], e) is an algebra of the type (2, 2, 0) defined on the group (G, x, e) in which each non-identity element is not of order 2 with the following [dot circle]--axioms:

([K.sub.1]) (x [dot circle] y) [dot circle] (x [dot circle] z) = (x [dot circle] ([z.sup.-1] [dot circle] [y.sup.-1])) [dot circle] x = (x [dot circle] ((e [dot circle] z) [dot circle] (e [dot circle] y))) [dot circle] x,

([K.sub.2]) x [dot circle] (x [dot circle] y) = (x [dot circle] [y.sup.-1]) [dot circle] x = (x [dot circle] (e [dot circle] y)) [dot circle] x,

(K3) (x [dot circle] x) = e,

(K4) (x [dot circle] e) = x,

(K5) (e [dot circle] x) = [x.sup.-1]

for all x, y, z [member of] G.

Definition 2.2. [1] A nonempty set S in a K-algebra K is called a subalgebra of K if for all x, y [member of] S, x [dot circle] y [member of] S.

Definition 2.3. [1] Let [K.sub.1] and [K.sub.2] be two K-algebras. A mapping f : [K.sub.1] [right arrow] [K.sub.2] is called a homomorphism if

f(x [dot circle] y) = f (x) [dot circle] f (y) for x, [dot circle] y [member of] K.

Definition 2.4. [7] Let Z be a nonempty set of objects. A single-valued neutrosophic set H in Z is of the form

H = {s [member of] Z : [T.sub.H](s), [I.sub.H](s), [F.sub.H](s)}, where T, I, F : Z [right arrow] [0, 1] for all s [member of] Z with 0 [less than or equal to] [T.sub.H](s) + [I.sub.H](s) + [F.sub.H](s) [less than or equal to] 3.

Definition 2.5. [22] Let H = ([T.sub.H], [I.sub.H], [F.sub.H]) be a single-valued neutrosophic set in K, then H is said to be a single-valued neutrosophic K-subalgebra of K if it possess the following properties:

(a) [T.sub.H](s [dot circle] t) > min{[T.sub.H](s), [T.sub.H](t)},

(b) [I.sub.H](s [dot circle] t) > min{[I.sub.H] (s), [I.sub.H](t)},

(c) [F.sub.H](s [dot circle] t) < max{[F.sub.H](s), [F.sub.H](t)} for all s [not equal to] [member of] K.

A K-subalgebra also satisfies the following conditions:

[T.sub.H](e) [greater than or equal to] [T.sub.H](s), [I.sub.H](e) [greater than or equal to] [I.sub.H](s), [F.sub.H](e) [less than or equal to] [F.sub.H](s) for all s [not equal to] e [member of] K.

Definition 2.6. [26] A t-norm is a two-valued function defined by a binary operation *, where * : [0,1] x [0,1] [right arrow] [0,1]. A t-norm is an associative, monotonic and commutative function possess the following properties, for all a,b,c,d G [0,1],

(i) * is a commutative binary operation.

(ii) * is an associative binary operation.

(iii) *(0, 0) = 0 and *(a, 1) = *(1, a) = a.

(iv) If a [less than or equal to] c and b [less than or equal to] d, then *(a, b) [less than or equal to] *(c, d).

Definition 2.7. [26] A t-conorm (s-norm) is a two-valued function defined by a binary operation o such that o : [0,1] x [0,1] [right arrow] [0,1]. A t-conorm is an associative, monotonic and commutative two-valued function, possess the following properties, for all a,b,c,d [member of] [0,1],

(i) o is a commutative binary operation.

(ii) o is an associative binary operation.

(iii) o(1,1) = 1 and o(a, 0) = o(0, a) = a.

(iv) If a [less than or equal to] c and b [less than or equal to] d, then o(a, b) [less than or equal to] o(c, d).

Definition 2.8. [23] Let [[chi].sub.K] be a single-valued neutrosophic topology over K. Let H be a single-valued neutrosophic K-algebra of K and [[chi].sub.H] be a single-valued neutrosophic topology on H. Then H is called a single-valued neutrosophic topological K-algebra over K if the self map pa : (H, [[chi].sub.H]) [[rho].sub.a] (H, [[chi].sub.H]) for all a [member of] K, defined as [[rho].sub.a](s) = s [dot circle] a, is relatively single-valued neutrosophic continuous.

Definition 2.9. [9] Let Z be a universe of discourse and E be a universe of parameters. Let P(Z) denotes the set of all subsets of Z and A [subset not equal to] E. Then a soft set [F.sub.A] over Z is represented by a set-valued function [[zeta].sub.A], where [[zeta].sub.A] : E [right arrow] P(Z) such that [[zeta].sub.A]([theta]) = 0 if [theta] [member of] E - A. In other words, [F.sub.A] can be represented in the form of a collection of parameterized subsets of Z such as [F.sub.A] = {([theta], [[zeta].sub.A]([theta])) : [theta] [member of] E, [[zeta].sub.A]([theta]) = 0 if [theta] [member of] E - A}.

Definition 2.10. [27] Let Z be a universe of discourse and E be a universe of parameters. A single-valued neutrosophic soft set H in Z is defined by a set-valued function [[zeta].sub.H], where [[zeta].sub.H] : E [right arrow] P(Z) and P(Z) denotes the power set set of Z. In other words, a single-valued neutrosophic soft set is a parameterized family of single-valued neutrosophic sets in Z and therefore can be written as:

[mathematical expression not reproducible] are called truth, indeterminacy and falsity membership functions of [[zeta].sub.H]([theta]), respectively.

Definition 2.11. [27] Let H be a single-valued neutrosophic soft set. The compliment of H, denoted by [H.sup.c], is defined as follows:

[mathematical expression not reproducible].

Definition 2.12. [27] Let H and T be two single-valued neutrosophic soft sets over (Z, E). Then H is called a neutrosophic soft subset of J, denoted by H [subset not equal to] J, if the following conditions hold:

(i) [mathematical expression not reproducible],

(ii) [mathematical expression not reproducible],

(iii) [mathematical expression not reproducible].

Throughout this article, we take the t-norm (*) as min(a, b) and t-conorm (o) as max(a, b) for intersection of two single-valued neutrosophic soft sets and (*) as max(a, b) and t-conorm (o) as min(a, b) for union of two single-valued neutrosophic soft sets. The union and the intersection for two single-valued neutrosophic soft sets are defined as follows.

Definition 2.13. [27] Let H and T be two single-valued neutrosophic soft sets over (Z, E). Then the union of H and T is denoted by H [union] T = L and defined as:

[mathematical expression not reproducible]

where

[mathematical expression not reproducible].

Definition 2.14. [27] Let H and T be two single-valued neutrosophic soft sets over (Z, E). Then their intersection is denoted by H [intersection] T = L and defined as:

[mathematical expression not reproducible]

where

[mathematical expression not reproducible].

Definition 2.15. [27] A single-valued neutrosophic soft set H over the universe Z is termed to be an empty or null single-valued neutrosophic soft set with respect to the parametric set [mathematical expression not reproducible] and can be written as:

[mathematical expression not reproducible].

Definition 2.16. [27] A single-valued neutrosophic soft set H over the universe Z is called an absolute or a whole single-valued neutrosophic soft set if [mathematical expression not reproducible] and can be written as:

[mathematical expression not reproducible].

Definition 2.17. [10] Let ([Z.sub.1], E) and ([Z.sub.2], E) be two initial universes. Then a pair ([phi], [rho]) is called a single-valued neutrosophic soft function from ([Z.sub.1], E) into ([Z.sub.2], E), where [phi] : [Z.sub.1] [right arrow] [Z.sub.2] and [rho] : E [right arrow] E, and E is a parametric set of [Z.sub.1] and [Z.sub.2].

Definition 2.18. [10] Let (H, E) and (J, E) be two single-valued neutrosophic soft sets over [G.sub.1] and [G.sub.2], respectively. If ([phi], [rho]) is a single-valued neutrosophic soft function from ([G.sub.1], E) into ([G.sub.2], E), then under this single-valued neutrosophic soft function ([phi], [rho]), image of (H, E) is a single-valued neutrosophic soft set on [K.sub.2], denoted by ([phi], [rho])(H, E) and defined as follows:

for all m [member of] [rho](E) and y [member of] [G.sub.2], ([phi], [rho])(H, E) = ([phi](H), [rho](E)), where

[mathematical expression not reproducible].

The preimage of (J, E),denoted by [([phi], [rho]).sup.-1](J,E),is defined as [for all] l [member of] [[rho].sup.-1](E) andforall x [member of] [G.sub.1], [([phi], [rho]).sup.-1](J, E) = ([[phi].sup.-1](J), [[rho].sup.-1](E)), where

[mathematical expression not reproducible].

Proposition 2.19. Let [Z.sub.1] and [Z.sub.2] be two initial universes with parametric set [E.sub.1] and [E.sub.2], respectively. Let H, ([H.sub.i], i [member of] I) be a single-valued neutrosophic soft set in [Z.sub.1] and T be a single-valued neutrosophic soft set in [Z.sub.2].

Let f : [Z.sub.1] [right arrow] [Z.sub.2] be a function. Then

(i) [mathematical expression not reproducible], if f is a surjective function.

(ii) [mathematical expression not reproducible].

(iii) [mathematical expression not reproducible].

(iv) [mathematical expression not reproducible].

(v) [mathematical expression not reproducible].

Through out this article, Z is considered as initial universe, E is a parametric set and [theta] [member of] E an arbitrary parameter.

3 Single-Valued Neutrosophic Soft Topological K-Algebras

Definition 3.1. Let Z be a nonempty set and E be a universe of parameters. A collection x of single-valued neutrosophic soft sets is called a single-valued neutrosophic soft topology if the following properties hold:

(1) [0.sub.E], [1.sub.E] [member of] x.

(2) The intersection of any two single-valued neutrosophic soft sets of [chi] belongs to [chi].

(3) The union of any collection of single-valued neutrosophic soft sets of [chi] belongs to [chi].

The triplet (Z, E, [chi]) is called a single-valued neutrosophic soft topological space over (Z, E). Each element of [chi] is called a single-valued neutrosophic soft open set and compliment of each single-valued neutrosophic soft open set is a single-valued neutrosophic soft closed set in [chi]. A single-valued neutrosophic soft topology which contains all single-valued neutrosophic soft subsets of Z is called a discrete single-valued neutrosophic soft topology and indiscrete single-valued neutrosophic soft topology if it consists of [0.sub.E] and [1.sub.E].

Definition 3.2. Let H be a single-valued neutrosophic soft set over a K-algebras K. Then H is called a single-valued neutrosophic soft K-subalgebra of K if the following conditions hold:

(i) [mathematical expression not reproducible],

(ii) [mathematical expression not reproducible],

(iii) [mathematical expression not reproducible].

Note that

[mathematical expression not reproducible].

Example 3.3. Consider a K-algebra K = (G, x, [dot circle], e) on a group (G, x), where G = {e, x, [x.sup.2], [x.sup.3], [x.sup.4], [x.sup.5], [x.sup.6], [x.sup.7]} is the cyclic group of order 8 and [C] is given by the following Cayley's table as:

[mathematical expression not reproducible]

Let E be a set of parameters defined as E = {[l.sub.1], [l.sub.2]}. We define single-valued neutrosophic soft sets H, T and L in K as:

[[zeta].sub.H]([l.sub.1]) = {(e, 0.8, 0.7, 0.2), (h, 0.6, 0.5, 0.4)}, [[zeta].sub.H]([l.sub.2]) = {(e, 0.7, 0.7, 0.2), (h, 0.4, 0.1, 0.5)},

[[zeta].sub.J]([l.sub.1]) = {(e, 0.4, 0.6, 0.6), (h, 0.3, 0.5, 0.7)}, [[zeta].sub.J]([l.sub.2]) = {(e, 0.9, 0.8, 0.1), (h, 0.7, 0.6, 0.4)},

[[zeta].sub.L]([l.sub.1]) = {(e, 0.9, 0.7, 0.1), (h, 0.7, 0.6, 0.4)}, [[zeta].sub.L]([l.sub.2]) = {(e, 0.7, 0.7, 0.2), (h, 0.6, 0.6, 0.5)}

for all h [not equal to] e [member of] G.

The collection [[chi].sub.K] = {[0.sub.E], [1.sub.E], H, J, L} is a single-valued neutrosophic soft topology on K and the triplet (K, E, [[chi].sub.K]) is a single-valued neutrosophic soft topological space over K. It is interesting to note that corresponding to each parameter [theta] [member of] E, we get a single-valued neutrosophic topology over K which means that a single-valued neutrosophic soft topological space gives a parameterized family of single-valued neutrosophic topological space on K. Now, we define a single-valued neutrosophic soft set Q in K as:

[[zeta].sub.Q]([l.sub.1]) = {(e, 0.8, 0.5, 0.1), (h, 0.6, 0.4, 0.3)},

[[zeta].sub.Q]([l.sub.2]) = {(e, 0.5, 0.6, 0.5), (h, 0.3, 0.4, 0.6)}.

Clearly, by Definition 3.2, Q is a single-valued neutrosophic soft K-subalgebra over K.

Proposition 3.4. Let (K, E, [[chi]'.sub.K]) and (K, E, [[chi]".sub.K]) be two single-valued neutrosophic topological spaces over K. If [[chi]'.sub.K] [intersection] [[chi]".sub.K] = M', where M' is a single-valued neutrosophic soft set from the set of all single-valued neutrosophic soft sets in K, then [[chi]'.sub.K] [intersection] [[chi]".sub.K] is also a single-valued neutrosophic soft topology on K.

Remark 3.5. The union of two single-valued neutrosophic soft topologies over K may not be a single-valued neutrosophic soft topology over K.

Example 3.6. Consider a K-algebra K = (G, X, [dot circle], e), where G = {e, x, [x.sup.2], [x.sup.3], [x.sup.4], [x.sup.5], [x.sup.6], [x.sup.7]} is the cyclic group of order 8 and Cayley's table for [dot circle] is given in Example 3.3. We take E = {[l.sub.1], [l.sub.2]} and two single-valued neutrosophic soft topological spaces [[chi]'.sub.K] = {[0.sub.E], [1.sub.E], H, J}, [[chi]".sub.K] = {[0.sub.E], [1.sub.E], R, S} on K, where R = H and single-valued neutrosophic soft set S is defined as:

[[zeta].sub.S]([l.sub.1]) = {(e, 0.7, 0.6, 0.2), (h, 0.5, 0.5, 0.6)},

[[zeta].sub.S]([l.sub.2]) = {(e, 0.9, 0.8, 0.2), (h, 0.7, 0.7, 0.3)}.

Suppose that [[chi]'".sub.K] = [[chi]'.sub.K] [union] [[chi]".sub.K] = {[0.sub.E], [1.sub.E], H, J, S}. We see that [[chi]'".sub.K] is not a single-valued neutrosophic soft topology over K since S [intersection] T [not member of] [[chi]'".sub.K].

Definition 3.7. Let (K, E, [[chi]'".sub.K]) be a single-valued neutrosophic soft topological space over K, where [[chi].sub.K] is a single-valued neutrosophic soft topology over K. Let F be a single-valued neutrosophic soft set in K, then [[chi].sub.F] = {F [intersection] H : H [member of] [[chi].sub.K]} is called a single-valued neutrosophic soft topology on F and (F, E, [[chi].sub.F]) is called a single-valued neutrosophic soft subspace of (K, E, [[chi].sub.K]).

Definition 3.8. Let ([K.sub.1], E, [[chi].sub.1]) and ([K.sub.2], E, [[chi].sub.2]) be two single-valued neutrosophic soft topological spaces, where [K.sub.1] and [K.sub.2] are two K-algebras. Then, a mapping f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) is called single-valued neutrosophic soft continuous mapping of single-valued neutrosophic soft topological spaces if it the following properties hold:

(i) For each single-valued neutrosophic soft set H [member of] [[chi].sub.2], [f.sup.-1](H) [member of] [[chi].sub.1].

(ii) For each single-valued neutrosophic soft K-subalgebra H [member of] [[chi].sub.2], [f.sup.-1](H) is a single-valued neutrosophic soft K-subalgebra [member of] [[chi].sub.1].

Definition 3.9. Let H and T be two single-valued neutrosophic soft sets in a K-algebra K and f : (H, E, [[chi].sub.H]) [right arrow] (J, E, [[chi].sub.J]). Then, f is called a relatively single-valued neutrosophic soft open function if for every singlevalued neutrosophic soft open set V in [[chi].sub.H], the image f (V) [member of] [[chi].sub.J].

Definition 3.10. If f is a mapping such that f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]). Then f is a mapping from (H, E, [[chi].sub.H]) into (J, E, [[chi].sub.J]) if f (H) [subset] J, where (H, E, [[chi].sub.H]) and (J, E, [[chi].sub.J]) are two single-valued neutrosophic soft subspaces of ([K.sub.1], E, [[chi].sub.1]) and ([K.sub.2], E, [[chi].sub.2]), respectively.

Definition 3.11. A mapping f such that f : (H, E, [[chi].sub.H]) [right arrow] (J, E, [[chi].sub.J]) is called relatively single-valued neutrosophic soft continuous if for every single-valued neutrosophic soft open set [Y.sub.J] [member of] [[chi].sub.J], [f.sup.1]([y.sub.j]) [intersection] H [member of] [[chi].sub.H].

Definition 3.12. Let ([K.sub.1], E, [[chi].sub.1]) and ([K.sub.2], E, [[chi].sub.2]) be two single-valued neutrosophic soft topological spaces. Then, a function f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) is called a single-valued neutrosophic soft homomorphism if it satisfies the following properties:

(i) f is a bijective function.

(ii) Both f and [f.sup.-1] are single-valued neutrosophic soft continuous functions.

Proposition 3.13. Let f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) be a single-valued neutrosophic soft continues mapping and (H, E, [[chi].sub.H]) and (J, E, [[chi].sub.J]) two single-valued neutrosophic soft topological subspaces of ([K.sub.1], E, [[chi].sub.1]) and ([K.sub.2], E, [[chi].sub.2]), respectively. If f (H) [subset not equal to] J, then f is a relatively single-valued neutrosophic soft continuous mapping from (H, E, [[chi].sub.H]) into (J, E, [[chi].sub.J]).

Proposition 3.14. Let ([K.sub.1], E, [[chi].sub.1]) and ([K.sub.2], E, [[chi].sub.2]) be two single-valued neutrosophic soft topological spaces, where [[chi].sub.1] is a single-valued neutrosophic soft topology on [K.sub.1] and [[chi].sub.2] is an indiscrete single-valued neutrosophic soft topology on [K.sub.2]. Then for each [theta] [member of] E, every function f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) is a single-valued neutrosophic soft continues function.

Proof. Let [[chi].sub.1] be a single-valued neutrosophic soft topology on [K.sub.1] and [[chi].sub.2] an indiscrete single-valued neutrosophic soft topology on [K.sub.2] such that [[chi].sub.2] = {[0.sub.E], [1.sub.E]}. Let f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) be any function. Now, to prove that f is a single-valued neutrosophic soft continues function for each [theta] [member of] E, we show that f satisfies both conditions of Definition 3.8. Clearly, every member of [[chi].sub.2] is a single-valued neutrosophic soft K-subalgebra of [K.sub.2] for each [theta] [member of] E. Now, there is only need to show that for all H [member of] [[chi].sub.2] and for each [there exists] [member of] E, [f.sup.-1](H) [member of] [x.sub.1]. For this purpose, let us assume that [0.sub.[theta]] [member of] [[chi].sub.2], for any u [member of] [K.sub.1] and [theta] [member of] E, we have [f.sup.-1]([0.sub.[theta]])(u) = [0.sub.[theta]](f(u)) = [0.sub.[theta]](u) [??] [0.sub.[theta]] [member of] [[chi].sub.1]. Similarly, [f.sup.-1]([1.sub.[theta]])(u) = [1.sub.[theta]](f(u)) = [1.sub.[theta]](u) [??][1.sub.[theta]] [member of] [[chi].sub.1]. For an arbitrary choice of [theta], result holds for each [theta] [member of] E. This shows that f is a single-valued neutrosophic soft continues function.

Proposition 3.15. Let [[chi].sub.1] and [[chi].sub.2] be any two discrete single-valued neutrosophic soft topological spaces on [K.sub.1] and [K.sub.2], respectively and ([K.sub.1], E, [[chi].sub.1]) and ([K.sub.2], E, [[chi].sub.2]) two discrete single-valued neutrosophic soft topological spaces. Then for each [theta] [member of] E, every homomorphism f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) is a single-valued neutrosophic soft continuous function.

Proof. Let [mathematical expression not reproducible] be a single-valued neutrosophic soft set in [K.sub.2] defined by a set-valued function [[zeta].sub.H]. Let f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) be a homomorphism (not a usual inverse homomorphism). Since [[chi].sub.1] and [[chi].sub.2] be two discrete single-valued neutrosophic soft topologies, then for every H [member of] [[chi].sub.2], [f.sup.-1](H) [member of] [[chi].sub.1]. Now, we show that for each [theta] [member of] E, the mapping [f.sup.-1](H) is a single-valued neutrosophic soft K-subalgebra of K-algebra [K.sub.1]. Then for any s, t [member of] [K.sub.1] and [theta] [member of] E, we have

[mathematical expression not reproducible]

Therefore, [f.sup.-1](H) is single-valued neutrosophic soft K-subalgebra of [K.sub.1]. Hence [f.sup.-1](H) [member of] [[chi].sub.2] which shows that f is a single-valued neutrosophic soft continuous function from ([K.sub.1], E, [[chi].sub.1]) into ([K.sub.2], E, [[chi].sub.2]).

Proposition 3.16. Let [[chi].sub.1] and [[chi].sub.2] be any two single-valued neutrosophic soft topological spaces on K and (K, E, [[chi].sub.1]) and (K, E, [[chi].sub.2]) be two single-valued neutrosophic soft topological spaces. Then for each [theta] [member of] E, every homomorphism f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) is a single-valued neutrosophic soft continuous function.

Definition 3.17. Let [chi] be a single-valued neutrosophic soft topology on K-algebra K. Let [mathematical expression not reproducible] be a single-valued neutrosophic soft K-algebra (K-subalgebra) of K and [[chi].sub.H] a single-valued neutrosophic soft topology over H. Then H is called a single-valued neutrosophic soft topological K-algebra of K if the self mapping [[rho].sub.a] : (H, E, [[chi].sub.H]) [right arrow] (H, E, [[chi].sub.H]) defined as [[rho].sub.a](u) = u [dot circle] a, [for all] a [member of] K, is a relatively single-valued neutrosophic soft continuous mapping.

Theorem 3.18. Let [[chi].sub.1] and [[chi].sub.2] be two single-valued neutrosophic soft topological spaces on [K.sub.1] and [K.sub.2], respectively. Let f : [K.sub.1] [right arrow] [K.sub.2] be a homomorphism of K-algebras such that [f.sup.-1]([[chi].sub.2]) = [[chi].sub.1]. If for each [theta] [member of] E, [mathematical expression not reproducible] is a single-valued neutrosophic soft topological K-algebra of [K.sub.2], then for each [theta] [member of] E, [f.sup.-1](H) is a single-valued neutrosophic soft topological K-algebra of [K.sub.1].

Proof. In order to prove that [f.sup.-1](H) is a single-valued neutrosophic soft topological K-algebra of K-algebra [K.sub.1]. Firstly, we show that [f.sup.-1](H) is a single-valued neutrosophic soft K-algebra of [K.sub.1]. One can easily show that for all [mathematical expression not reproducible]. Let for any s, t [member of] [K.sub.1] and [theta] [member of] E,

[mathematical expression not reproducible]

This shows that [f.sup.-1](H) is a single-valued neutrosophic soft K-algebra of [K.sub.1].

Since f is a homomorphism and also a single-valued neutrosophic soft continuous mapping, then clearly, f is relatively single-valued neutrosophic soft continuous mapping from [mathematical expression not reproducible] such that for a single-valued neutrosophic soft set V in [[chi].sub.H], and a single-valued neutrosophic soft set U in [mathematical expression not reproducible],

[f.sup.-1](V) = U. (1)

Now, we prove that the self mapping [mathematical expression not reproducible] is relatively single-valued neutrosophic soft continuous mapping. Now, for any a [member of] [K.sub.1] and [theta] [member of] E, we have

[mathematical expression not reproducible]

This implies that [mathematical expression not reproducible] is a single-valued neutrosophic soft set in [f.sup.-1](H) and a single-valued neutrosophic soft set in [mathematical expression not reproducible]. Hence [f.sup.-1](H) is a single-valued neutrosophic soft topological K-algebra of [K.sub.1]. This completes the proof.

Theorem 3.19. Let [[chi].sub.1] and [[chi].sub.2] be two single-valued neutrosophic soft topologies on [K.sub.1] and [K.sub.2], respectively and f : [K.sub.1] [right arrow] [K.sub.2] an isomorphism of K-algebras such that f ([[chi].sub.1]) = [[chi].sub.2]. If for each [mathematical expression not reproducible] is a single-valued neutrosophic soft topological K-algebra of Kalgebra [K.sub.1], then for each [theta] [member of] E, f(H) is a single-valued neutrosophic soft topological K-algebra of [K.sub.2].

Proof. Let H be a single-valued neutrosophic soft topological K-algebra of [K.sub.1]. For u, v [member of] [K.sub.2].

Let [t.sub.o] [member of] [f.sup.-1](u), [s.sub.o] [member of] [f.sup.-1](v) such that

[mathematical expression not reproducible]

We now have,

[mathematical expression not reproducible]

Hence f(H) is a single-valued neutrosophic soft K-subalgebra of [K.sub.2]. To show that f(H) is a single-valued neutrosophic soft topological K-algebra of [K.sub.2], i.e., the self map [[rho].sub.b] : (f(H), [[chi].sub.f(H)]) [right arrow] (f(H), [[chi].sub.f(H)]), defined as [[rho].sub.b](v) = v [dot circle] b, [for all] b [member of] [K.sub.2] is a relatively single-valued neutrosophic soft continuous mapping. Let [Y.sub.H] be a single-valued neutrosophic soft set in [[chi].sub.H], then there exists a single-valued neutrosophic soft set Y in [[chi].sub.1] be such that [Y.sub.H] = Y [intersection] H.

[[rho].sup.-1.sub.b] ([Y.sub.f(H)]) [intersection] f(H) [member of] [[chi].sub.f(H)]

Then f ([Y.sub.H]) = f (Y [intersection] H) = f(Y) [intersection] f(H) is a single-valued neutrosophic soft set in [[chi].sub.f](H) since f is an injective function. Thus, f is relatively single-valued neutrosophic soft open. Since f is also an onto function, then for all b [member of] [K.sub.2] and a [member of] [K.sub.1], a = f(b), we have

[mathematical expression not reproducible]

This shows that [f.sup.-1]([[rho].sup.-1.sub.(b)](([Y.sub.f(h)]))) = [[rho].sup.-1.sub.(a)]([f.sup.-1] ([Y.sub.(H)])). Since [[rho].sub.(a)] : (H, [[chi].sub.H]) [right arrow] (H, [[chi].sub.H]) is relatively single valued neutrosophic soft continuous mapping and f is also relatively single-valued neutrosophic soft continues function. Therefore, [f.sup.-1]([[rho].sup.-1.sub.(b)]))) [intersection] H = [[rho].sup.-1.sub.(a)] ([f.sup.1]([Y.sub.(H)])) [intersection] H is a single-valued neutrosophic soft set in [[chi].sub.H]. Thus, f([f.sup.-1]([[rho].sub.(b)](([Y.sub.f(H)]))) [intersection] A) = [[rho].sup.-1.sub.(b)] (([Y.sub.f(A)]) [intersection] f(A) is a single-valued neutrosophic soft set in [[chi].sub.A].

Example 3.20. Consider a K-algebra K on a cyclic group of order 8 and Cayley's table for [dot circle] is given Example 3.3, where G = {e,x,[x.sup.2],[x.sup.3],[x.sup.4],[x.sup.5],[x.sup.6],[x.sup.7]}. Consider a set of parameters E = {[l.sub.1], [l.sub.2]} and single-valued neutrosophic soft sets H, J, L defined as:

[[zeta].sub.H]([l.sub.1]) = {(e, 0.8, 0.7, 0.2), (h, 0.6, 0.5, 0.4)}, [[zeta].sub.H]([l.sub.2]) = {(e, 0.7, 0.7, 0.2), (h, 0.6, 0.6, 0.5)},

[[zeta].sub.J]([l.sub.1]) = {(e, 0.7, 0.7, 0.2), (h, 0.4, 0.1, 0.5)}, [[zeta].sub.J]([l.sub.2]) = {(e, 0.4, 0.6, 0.6), (h, 0.3, 0.5, 0.7)},

[[zeta].sub.L]([l.sub.1]) = {(e, 0.9, 0.8, 0.1), (h, 0.7, 0.6, 0.4)}, [[zeta].sub.L]([l.sub.2]) = {(e, 0.9, 0.7, 0.1), (h, 0.7, 0.6, 0.4)}

for all h [not equal to] e [member of] G. Then the family [[chi].sub.K] = {[0.sub.E], [1.sub.E], H, J, L} is a singlevalued neutrosophic soft topology on K and (K, E, [[chi].sub.K]) is a single-valued neutrosophic soft topological space over K. We define another single-valued neutrosophic soft set Q in K as:

[[zeta].sub.Q]([l.sub.1]) = {(e, 0.8, 0.5, 0.1), (h, 0.6, 0.4, 0.3)}, [[zeta].sub.Q]([l.sub.2]) = {(e, 0.5, 0.6, 0.5), (h, 0.3, 0.4, 0.6)}.

It is obvious that Q is a single-valued neutrosophic soft K-algebra of K.

Now, we prove that the self map [[rho].sub.a] : (Q, E, [[chi].sub.Q]) [right arrow] (Q, E, [[chi].sub.Q]), defined as [[rho].sub.a](s) = s [dot circle] a for all a [member of] K, is a relatively single-valued neutrosophic soft continuous mapping.

We get Q [intersection] [0.sub.E] = [0.sub.E], Q [intersection] [1.sub.E] = [1.sub.E], Q [intersection] H = [R.sub.1], Q [intersection] J = [R.sub.2], Q [intersection] L = [R.sub.3], where [R.sub.1], [R.sub.2], [R.sub.3] are as follows:

[mathematical expression not reproducible]

Thus, [[chi].sub.Q] = {[0.sub.E], [1.sub.E], [R.sub.1], [R.sub.2], [R.sub.3]} is a relatively topology of Q and (Q, E, [[chi].sub.Q]) is a single-valued neutrosophic soft subspace of (K, E, [[chi].sub.K]). Since [[rho].sub.a] is a homomorphism, then for a single-valued neutrosophic soft set R [member of] [[chi].sub.Q], [[rho].sup.-1.sub.a])(R) [intersection] Q [member of] [[chi].sub.Q]. Which shows that [[rho].sub.a] : (Q, E, [[chi].sub.Q]) [right arrow] (Q, E, [[chi].sub.Q]) is relatively single-valued neutrosophic soft continuous mapping. Therefore, Q is a single-valued neutrosophic soft topological K-algebra.

4 Single-Valued Neutrosophic Soft [C.sub.5]-connected K-Algebras

In this section, we discuss single-valued neutrosophic soft [C.sub.5]-connected K-algebras.

Definition 4.1. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space over K. A single-valued neutrosophic soft separation of (K, E, [[chi].sub.K]) is a pair of nonempty single-valued neutrosophic soft open sets H, T if the following conditions hold:

(i) H [union] T = [1.sub.E].

(ii) H [intersection] T = [0.sub.E].

Definition 4.2. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space over K. Then (K, E, [[chi].sub.K]) is called a single-valued neutrosophic soft [C.sub.5]-disconnected if there exists a single-valued neutrosophic soft separation of (K, E, [[chi].sub.K]), otherwise [C.sub.5]-connected.

Definition 4.2 can be written as:

Definition 4.3. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space over K. If there exists a single-valued neutrosophic soft open set and single-valued neutrosophic soft closed set L such that L [not equal to] [1.sub.E] and L [not equal to] [0.sub.E], then (K, E, [[chi].sub.K]) is called a single-valued neutrosophic soft [C.sub.5]disconnected, otherwise (K, E, [[chi].sub.K]) is called a single-valued neutrosophic soft [C.sub.5]-connected.

Example 4.4. By considering Example 3.3, we consider a single-valued neutrosophic soft topological space [[chi].sub.K] = {[0.sub.E], [1.sub.E], H, J, L}. Since H [intersection] J [not equal to] [0.sub.E], H [intersection] L [not equal to] [0.sub.E], T [intersection] L [not equal to] [0.sub.E] and H [union] J [not equal to] [1.sub.E], H [intersection] L [not equal to] [1.sub.E], T [union] L [not equal to] [1.sub.E]. Thus, [[chi].sub.K] is a single-valued neutrosophic soft [C.sub.5]-connected.

Example 4.5. Every indiscrete single-valued neutrosophic soft space is [C.sub.5]-connected since the only single-valued neutrosophic soft sets in single-valued neutrosophic soft indiscrete space that are both single-valued neutrosophic soft open and single-valued neutrosophic soft closed are [0.sub.E] and [1.sub.E].

Theorem 4.6. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space on K-algebra K. Then (K, E, [[chi].sub.K]) is a single-valued neutrosophic soft [C.sub.5]-connected if and only if [[chi].sub.K] contains only [0.sub.E] and [1.sub.E] which are both single-valued neutrosophic soft open and single-valued neutrosophic soft closed. Proof. Straightforward.

Proposition 4.7. Let [K.sub.1] and [K.sub.2] be two K-algebras and [mathematical expression not reproducible] two single-valued neutrosophic soft topological spaces on [K.sub.1] and [K.sub.2], respectively. Let f : [K.sub.1] [right arrow] [K.sub.2] be a singlevalued neutrosophic soft continuous surjective function. If [mathematical expression not reproducible] is a single-valued neutrosophic soft [C.sub.5]-connected space, then [mathematical expression not reproducible] is also single-valued neutrosophic soft [C.sub.5]-connected.

Proof. Let [mathematical expression not reproducible] be two single-valued neutrosophic soft topological spaces and [mathematical expression not reproducible] be a single-valued neutrosophic soft [C.sub.5]-connected space. We prove that [mathematical expression not reproducible] is also single-valued neutrosophic soft [C.sub.5]-connected. Let us suppose on contrary that ([K.sub.2], [[chi].sub.2]) be a single-valued neutrosophic soft [C.sub.5]-disconnected space. According to Definition 4.3, we have both single-valued neutrosophic soft open set and single-valued neutrosophic soft closed set L such that L [not equal to] [1.sub.SN] and L [not equal to] [0.sub.SN]. Then [f.sup.-1](L) = [1.sub.SN] or [f.sup.-1](L) = [0.sub.SN] since f is a single-valued neutrosophic soft continuous surjective mapping, where [f.sup.1](L) is both single-valued neutrosophic soft open set and single-valued neutrosophic soft closed set. Therefore, L = f([f.sup.-1](L)) = f([1.sub.SN]) = [1.sub.SN] and L = f([f.sup.-1](L)) = f([0.sub.SN]) = [0.sub.SN], a contradiction. Hence ([K.sub.2], E, [[chi].sub.2]) is a single-valued neutrosophic soft [C.sub.5]-connected space.

5 Single-Valued Neutrosophic Soft Super Connected K-Algebras

Definition 5.1. Let ([K.sub.2], E, [[chi].sub.2]) be a single-valued neutrosophic soft topological space over [mathematical expression not reproducible] a single-valued neutrosophic soft set in K. Then the interior and closure of H in a K-algebra K is defines as:

[H.sup.Int] = [union]{O : O is a single-valued neutrosophic soft open set in K and O [subset not equal to] H},

[H.sup.Clo = [intersection] {C : C is a single-valued neutrosophic soft closed set in K and H [subset not equal to] C}.

It is interesting to note that [H.sup.Int], being union of single-valued neutrosophic soft open sets is single-valued neutrosophic soft open and [H.sup.Clo, being intersection of single-valued neutrosophic soft closed set is single-valued neutrosophic soft closed.

Theorem 5.2. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space on K. Let [mathematical expression not reproducible] be a single-valued neutrosophic soft set in [[chi].sub.K]. Then [H.sup.Int] is the largest single-valued neutrosophic soft open set contained in H.

Proof. Obvious.

Proposition 5.3. Let H be a single-valued neutrosophic soft set in K. Then the following properties hold:

(i) [([1.sub.E]).sup.Int] = [1.sub.E]

(ii) [([0.sub.E]).sup.Clo] = [0.sub.E]

(iii) [[bar.(H)].sup.Int] = [[bar.(H)].sup.Clo]

(iv) [[bar.(H)].sup.Clo] = [[bar.(H)].sup.Int]

Corollary 5.4. If H is a single-valued neutrosophic soft set in K, then H is single-valued neutrosophic soft open if and only if [H.sup.Int] = H and H is a single-valued neutrosophic soft closed if and only if [H.sup.Clo] = H.

Definition 5.5. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space on K and [[chi].sub.K] be a single-valued neutrosophic soft topology on K. Let [mathematical expression not reproducible] be a single-valued neutrosophic soft open set in K. Then H is called a single-valued neutrosophic soft regular open if

H = [([H.sup.Clo]).sup.Int].

Remark 5.6. (1) Every single-valued neutrosophic soft regular is single-valued neutrosophic soft open.

(2) Every single-valued neutrosophic soft clopen set is single-valued neutrosophic soft regular open.

Definition 5.7. Let [[chi].sub.K] be a single-valued neutrosophic soft topology on K. Then K is called a single-valued neutrosophic soft super disconnected if there exists a single-valued neutrosophic soft regular open set [mathematical expression not reproducible] such that [1.sub.E] [not equal to] H and [0.sub.E] [not equal to] H. But if there does not exist such a single-valued neutrosophic soft regular open set H such that [1.sub.E] [not equal to] H and [0.sub.E] [not equal to] H, then K is called singlevalued neutrosophic soft super connected.

Example 5.8. Consider a K-algebra on a cyclic group of order 8 and Cayley's table for [dot circle] is given in Example 3.3, where G {e, x, [x.sup.2], [x.sup.3], [x.sup.4], [x.sup.5], [x.sup.6], [x.sup.7]}. We have a single-valued neutrosophic soft topology [[chi].sub.K] = {[0.sub.E], [1.sub.E], H, J}, where H, T with a parametric set E = {[l.sub.1], [l.sub.2]} are given as:

[[zeta].sub.H]([l.sub.1]) = {(e, 0.8, 0.7, 0.2), (h, 0.6, 0.5, 0.4)}, [[zeta].sub.H]([l.sub.2]) = {(e, 0.7, 0.7, 0.2), (h, 0.6, 0.6, 0.5)},

[[zeta].sub.J]([l.sub.1]) = {(e, 0.7, 0.7, 0.2), (h, 0.4, 0.1, 0.5)}, [[zeta].sub.J]([l.sub.2]) = {(e, 0.4, 0.6, 0.6), (h, 0.3, 0.5, 0.7)},

for all h [not equal to] e [member of] G.

Let L be a single-valued neutrosophic soft set in K, defined by:

[[zeta].sub.L]([l.sub.1]) = {(e, 0.9, 0.8, 0.1), (h, 0.7, 0.6, 0.4)}, [[zeta].sub.L]([l.sub.2]) = {(e, 0.9, 0.7, 0.1), (h, 0.7, 0.6, 0.4)}.

Now, we have single-valued neutrosophic soft open sets: [0.sub.E], [1.sub.E], H, J. single-valued neutrosophic soft closed sets : [([0.sub.E]).sup.c] = [1.sub.E], [([1.sub.E]).sup.c] = [0.sub.E], [(H).sup.c] = H', [(J).sup.c] = J', where H', J' are obtained as:

[[zeta].sub.H']([l.sub.1]) = {(e, 0.2, 0.7, 0.8), (h, 0.4, 0.5, 0.6)}, [[zeta].sub.H']([l.sub.2]) = {(e, 0.2, 0.7, 0.7), (h, 0.5, 0.6, 0.6)},

[[zeta].sub.J']([l.sub.1]) = {(e, 0.2; 0.7, 0.7), (h, 0.5, 0.1, 0.4)}, [[zeta].sub.J']([l.sub.2]) = {(e, 0.6; 0.6, 0.4), (h, 0.7, 0.5, 0.3)},

for all h [not equal to] e [member of] G. Then, interior and closure of a single-valued neutrosophic soft set L is obtained as:

[L.sup.Int] = H,

[L.sup.Clo] = [1.sub.E].

For L to be a single-valued neutrosophic soft regular open, then L = [([L.sup.Clo]).sup.Int]. But since L = [([1.sub.E]).sup.Int] = [1.sub.E] [not equal to] L. This shows that [1.sub.E] [not equal to] L [not equal to] [0.sub.E] is not a single-valued neutrosophic soft regular open set. By Definition 5.7, defined K-algebra is a single-valued neutrosophic soft super connected K-algebra.

6 Single-Valued Neutrosophic Soft Compactness K-Algebras

Definition 6.1. Let [[chi].sub.K] be a single-valued neutrosophic soft topology on K. Let H be a single-valued neutrosophic soft set in K. A collection [mathematical expression not reproducible] of single-valued neutrosophic soft sets in K is called a single-valued neutrosophic soft open covering of H if H [subset not equal to] [union] [OMEGA]. A finite sub-collection of [OMEGA] say ([OMEGA]') is also a single-valued neutrosophic soft open covering of H, called a finite subcovering of H.

Definition 6.2. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space of K. Let H be a single-valued neutrosophic soft set in K. Then H is called a single-valued neutrosophic soft compact if every single-valued neutrosophic soft open covering [OMEGA] of H has a finite sub-covering ([OMEGA]).

Example 6.3. A single-valued neutrosophic soft topological space (K, E, [[chi].sub.K]) is single-valued neutrosophic soft compact if either K is finite or [[chi].sub.K] is a finite single-valued neutrosophic soft topology on K.

Proposition 6.4. Let [mathematical expression not reproducible] be a single-valued neutrosophic soft continuous mapping, where [mathematical expression not reproducible] are two single-valued neutrosophic soft topological spaces of [K.sub.1] and [K.sub.2], respectively. If H is a single-valued neutrosophic soft compact in [mathematical expression not reproducible] is single-valued neutrosophic soft compact in [mathematical expression not reproducible].

Proof. Let f be a single-valued neutrosophic soft continuous map from [K.sub.1] into [K.sub.2]. Let [OMEGA] = {[f.sup.1]([H.sub.i] : i [member of] I)} be a single-valued neutrosophic soft open covering of H and [DELTA] = {[H.sub.i] : i [member of] I} a single-valued neutrosophic soft open covering of f(H). Then there exists a single-valued neutrosophic soft finite sub-covering [mathematical expression not reproducible] such that

[mathematical expression not reproducible]

Thus,

[mathematical expression not reproducible]

This shows that there exists a single-valued neutrosophic soft finite sub-covering of f(H). Therefore, f(H) is single-valued neutrosophic soft compact in [mathematical expression not reproducible].

7 Single-Valued Neutrosophic Soft Hausdorff K-Algebras

Definition 7.1. Let [mathematical expression not reproducible] be a single-valued neutrosophic soft set in a K. Then H is called a single-valued neutrosophic soft point if, for [theta] [member of] E

[[zeta].sub.H]([theta]) [not equal to] [0.sub.E],

and

[[zeta].sub.H]([theta]') = [0.sub.E],

for all [theta]' [member of] E - {[theta]}. A single-valued neutrosophic soft point in H is denoted by [[theta].sub.H].

Definition 7.2. A single-valued neutrosophic soft point [[theta].sub.H] is said to belong to a single-valued neutrosophic soft set J, i.e., [[theta].sub.H] [member of] T if, for [theta] [member of] E

[[zeta].sub.H]([theta]) [less than or equal to] [[zeta].sub.J]([theta]).

Definition 7.3. Let (K, E, [[chi].sub.K]) be a single-valued neutrosophic soft topological space over K and [[theta].sub.L], [[theta].sub.Q] be two single-valued neutrosophic soft points in K. If for these two single-valued neutrosophic soft points, there exist two disjoint single-valued neutrosophic soft open sets H, T such that [[theta].sub.L] [member of] H and [[theta].sub.Q] [member of] J. Then (K, E, [[chi].sub.K]) is called a single-valued neutrosophic soft Hausdorff topological space over K and K is called a single-valued neutrosophic soft Hausdorff K-algebra.

Example 7.4. Consider a K-algebra K on a cyclic group of order 8 and Cayley's table for [dot circle] is given in Example 3.3, where G = {e,x,[x.sup.2],[x.sup.3],[x.sup.4],[x.sup.5],[x.sup.6],[x.sup.7]}. Let E = {l} and [[chi].sub.K] = {[0.sub.E], [1.sub.E], H, J} be a single-valued neutrosophic soft topological space over K. We define two single-valued neutrosophic soft points [l.sub.L], [l.sub.Q] such that

[l.sub.L] = {(e, 1, 0, 1), (h, 0, 0, 1)}, [l.sub.Q] = {(e, 0, 0, 1), (h, 0, 1, 0)}.

Since for l [member of] E, [[zeta].sub.L](l) [not equal to] [0.sub.E], [[zeta].sub.Q](l) [not equal to] [0.sub.E], and [l.sub.L] [not equal to] [l.sub.Q], then clearly [l.sub.L] and [l.sub.Q] are two single-valuedneutrosophic soft points. Now, consider two single-valued neutrosophic soft open sets H and T defined as:

[[zeta].sub.H](l) = {(e, 1, 1, 0), (h, 0, 0, 1)}, [[zeta].sub.H](l) = {(e, 0, 0, 1), (h, 1, 1, 0)},

for all h [not equal to] e [member of] G. Since [[zeta].sub.L](l) [less than or equal to] [[zeta].sub.H](l) and [[zeta].sub.Q](l) [less than or equal to] J(l), i.e., [l.sub.L] [member of] H and [l.sub.q] [member of] T and H [intersection] T = [0.sub.E]. Thus, (K, E, [[chi].sub.K]) is a single-valued neutrosophic soft Hausdorff space and K is a single-valued neutrosophic soft Hausdorff K-algebra.

Theorem 7.5. Let f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) be a single-valued neutrosophic soft homomorphism. Then [K.sub.1] is a single-valued neutrosophic soft Hausdorff space if and only if [K.sub.2] is a single-valued neutrosophic soft Hausdorff K-algebra.

Proof. Let f : ([K.sub.1], E, [[chi].sub.1]) [right arrow] ([K.sub.2], E, [[chi].sub.2]) be a single-valued neutrosophic soft homomorphism and [K.sub.1], [K.sub.2] be two single-valued neutrosophic soft topologies on [K.sub.1] and [K.sub.2], respectively. Suppose that [K.sub.1] is a single-valued neutrosophic soft Hausdorff space. To prove that [K.sub.2] is a single-valued neutrosophic soft Hausdorff K-algebra, Let for l [member of] E, [l.sub.L] and [l.sub.Q] be two single-valued neutrosophic soft points in [[chi].sub.2] such that [l.sub.L] = [l.sub.Q] with u, v [member of] [K.sub.1], u [not equal to] v. Then for these two distinct single-valued neutrosophic soft points, there exist two single-valued neutrosophic soft open sets H and T such that [l.sub.L] [member of] H, [l.sub.Q] [member of] J with H [intersection] J = [0.sub.E]. For x [member of] [K.sub.1], we consider

[mathematical expression not reproducible]

Therefore, [f.sup.-1]([l.sub.L]) = ([f.sup.-1](l))L. Likewise, [f.sup.-1]([l.sub.Q]) = [([f.sup.-1](l)).sub.Q]. Since f is a single-valued neutrosophic soft continuous function from [K.sub.1] into [K.sub.2] and also [f.sup.-1] is a single-valued neutrosophic soft continuous function from [K.sub.2] into [K.sub.1], then there exist two disjoint single-valued neutrosophic soft open sets f(H) and f(J) of single-valued neutrosophic soft points [l.sub.L] and [l.sub.Q], respectively be such that f(H) [intersection] f(J) = f ([0.sub.E]) = [0.sub.E]. This shows that [K.sub.2] is a single-valued neutrosophic soft Hausdorff K-algebra. The proof of converse part is straightforward.

Theorem 7.6. let f : [K.sub.1] [right arrow] [K.sub.2] be a bijective single-valued neutrosophic soft continuous function, where [K.sub.1] is a single-valued neutrosophic soft compact K-algebra and [K.sub.2] is a single-valued neutrosophic soft Hausdorff K-algebra. Then mapping f is a [K.sub.1] is a single-valued neutrosophic soft homomorphism.

Proof. Let f be a bijective single-valued neutrosophic soft mapping from a single-valued neutrosophic soft compact K-algebra into a single-valued neutrosophic soft Hausdorff K-algebra. Then clearly, f is a single-valued neutrosophic soft homomorphism. We only prove that f is single-valued neutrosophic soft closed since f is a bijective mapping. Let a single-valued neutrosophic soft set [mathematical expression not reproducible] be closed in Kalgebra [K.sub.1]. Now if Q = [0.sub.E], then f(Q) = [0.sub.E] is single-valued neutrosophic soft closed in [K.sub.2]. But if Q [not equal to] [0.sub.E], then being a subset of a single-valued neutrosophic soft compact K-algebra, Q is single-valued neutrosophic soft compact. Also f(Q) is single-valued neutrosophic soft compact, being a single-valued neutrosophic soft continuous image of a single-valued neutrosophic soft compact K-algebra. Hence f is closed thus, f is a single-valued neutrosophic soft homomorphism.

8 Conclusions

In 1998, Smarandache originally considered the concept of neutrosophic set from philosophical point of view. The notion of a single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view, and an extension of intuitionistic fuzzy sets [32]. In 1999, Molodtsov introduced the idea of soft set theory as another powerful mathematical tool to handle indeterminate and inconsistent data. This theory fixes the problem of establishing the membership function for each specific case by giving a parameterized outlook to indeterminacy. By using a hybrid model of these two mathematical techniques with a topological structure, we have developed the concept of single-valued neutrosophic soft topological K-algebras to analyze the element of indeterminacy in K-algebras. We have defined some certain concepts such as the interior, closure, [C.sub.5]-connected, super connected, compactness and Hausdorff of single-valued neutrosophic soft topological K-algebras. In future, we aim to extend our notions to (1) Rough neutrosophic K-algebras, (2) Soft rough neutrosophic K-algebras, (3) Bipolar neutrosophic soft K-algebras, and (4) Rough neutrosophic K-algebras.

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Received: December 15, 2018. Accepted: March 30, 2019.

Muhammad Akram (1), Hina Gulzar (1), Florentin Smarandache (2)

(1) Department of Mathematics, University of the Punjab, New Campus, Lahore- 54590, Pakistan. E-mail: m.akram@pucit.edu.pk, E-mail: hinagulzar5@gmail.com

(2) University of New Mexico Mathematics & Science Department 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: fsmarandache@gmail.com
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Date:May 22, 2019
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