Connection of Multi-Valued Topological Groups with Neutrosophic Multi-Valued Functions

1. Introduction

Ordinary group theory is based on a single-valued binary operation

$$m:G\times G⟶G,m(x,y)=xy.$$

A natural generalization is obtained by allowing the product of two elements to be a nonempty subset of the underlying space. Thus one considers a hyperoperation

$$\circ :G\times G⟶{\mathcal{P}}^{*}\left(G\right),$$

where ${\mathcal{P}}^{*}\left(G\right)$ denotes the family of all nonempty subsets of G. Such structures form the basis of the theory of algebraic hyperstructures. The pioneering work of Marty introduced hypergroups in the 1930s, and the subject was subsequently developed systematically by Corsini and Vougiouklis [1,2,3]. Modern treatments continue to emphasize the structural relationship between hypergroups and ordinary groups [4].

The topological version of this theory arises when the underlying set is equipped with a topology and the hyperoperation is required to satisfy an appropriate continuity condition. Topological hypergroups have been studied using set-valued continuity and hyperspace methods [5]. This is closely related to the general theory of multi-valued mappings, where a function associates to each point a nonempty subset of a target space. The Vietoris topology is one of the standard topologies for treating such mappings [6,7].

A different generalization of classical set theory and logic is provided by neutrosophic mathematics. A neutrosophic set associates with each point three components, usually interpreted as truth, indeterminacy, and falsity. The basic neutrosophic framework was introduced by Smarandache [8,9]. Subsequently, multi-valued neutrosophic sets were introduced so that each of the three components may itself consist of a set of possible values [10]. More recent work has connected neutrosophic structures with hypergroups and hyperalgebra [11,12].

These two developments suggest a natural question:

Question 1.

Can the set-valued multiplication of a multi-valued topological group be represented by a neutrosophic multi-valued function, and can the topological and algebraic properties of the original structure be recovered from its neutrosophic representation?

The objective of this paper is to give a systematic answer.

Our approach has three ingredients. First, we use compact-valued multi-valued operations and the Vietoris hyperspace. Second, we attach to every point three compact sets of values representing truth, indeterminacy, and falsity. Third, we require compatibility between the hyperoperation and these three components.

The main contributions are as follows.

(i)

We introduce a precise notion of a neutrosophic multi-valued function adapted to topological hypergroups.

(ii)

We define the neutrosophic lift of a multi-valued topological group.

(iii)

We introduce neutrosophic translations and neutrosophic hyperoperations.

(iv)

We prove that Vietoris continuity of the underlying hyperoperation, together with continuity of the three neutrosophic components, yields a continuous neutrosophic multi-valued group structure.

(v)

We establish level-set representations and show that appropriate neutrosophic cuts recover ordinary compact-valued hyperoperations.

(vi)

We study neutrosophic subgroups, homomorphisms, inverse mappings, quotients, and convergence.

(vii)

We show that ordinary multi-valued topological groups are embedded as a special case of the proposed neutrosophic theory.

The framework is intended as a bridge between hyperstructure theory, set-valued topology, and neutrosophic mathematics.

2. Preliminaries on Multi-Valued Operations

Let X be a nonempty set. We denote by

$${\mathcal{P}}^{*}\left(X\right)=\{A\subseteq X:A\ne ⌀\}$$

the family of all nonempty subsets of X.

Definition 1.

Amulti-valued operationorhyperoperationon X is a mapping

$$\circ :X\times X⟶{\mathcal{P}}^{*}\left(X\right).$$

For $x,y\in X$, the set $x\circ y$ is called the hyperproduct of x and y.

For subsets $A,B\subseteq X$, define

$$A\circ B=\bigcup _{a\in A,b\in B}(a\circ b).$$

Definition 2.

A hyperoperation ∘ on X is calledassociativeif

$$(x\circ y)\circ z=x\circ (y\circ z)$$

for all $x,y,z\in X$.

The equality above means

$$\bigcup _{u\in x\circ y}(u\circ z)=\bigcup _{v\in y\circ z}(x\circ v).$$

The classical hypergroup axiom of reproduction is

$$x\circ X=X\circ x=X$$

for every $x\in X$. This is one of the defining features of hypergroups in the sense of Marty [1,3].

Definition 3.

A pair $(G,\circ )$ is called aMarty hypergroupif ∘ is associative and

$$x\circ G=G\circ x=G$$

for every $x\in G$.

The reproduction axiom is stronger than the usual existence of an identity and inverses in an ordinary group. In particular, several different classes of hypergroups arise depending on the chosen identity and inverse axioms [2,3].

For the topological theory developed below, we use a compact-valued version.

Definition 4.

Let X be a topological space. Denote by

$$\mathcal{K}\left(X\right)$$

the family of all nonempty compact subsets of X. A hyperoperation

$$\circ :X\times X⟶\mathcal{K}\left(X\right)$$

is called acompact-valued hyperoperation.

3. Vietoris Topology and Topological Hypergroups

Let X be a topological space. The Vietoris topology on $\mathcal{K}\left(X\right)$ is generated by sets of the form

$$\langle U_1,\dots ,U_n\rangle =\left\{K\in \mathcal{K}\left(X\right):K\subseteq U_1\cup \dots \cup U_n,K\cap U_i\ne ⌀\mathrm{for}\mathrm{all}i\right\},$$

where $U_1,\dots ,U_n$ are open subsets of X.

We write

$$\mathcal{V}\left(X\right)$$

for $\mathcal{K}\left(X\right)$ equipped with the Vietoris topology.

The use of hyperspaces is natural because a multi-valued operation is not a map into X, but a map into a family of subsets of X. The theory of continuous selections and set-valued mappings provides the corresponding topological framework [6,7].

Definition 5.

Let G be a topological space and

$$\circ :G\times G⟶\mathcal{K}\left(G\right).$$

The pair $(G,\circ )$ is called aVietoris-continuous hypergroupif ∘ is continuous as a map

$$G\times G⟶\mathcal{V}\left(G\right)$$

and $(G,\circ )$ satisfies the selected hypergroup axioms.

For example, one may require associativity, an identity e, and a suitable inverse map

$$i:G⟶\mathcal{K}\left(G\right).$$

Definition 6.

Acompact-valued multi-valued topological groupis a tuple

$$(G,\tau ,\circ ,e,i)$$

such that:

1.

$(G,\circ )$ is an associative hypergroup;

2.

$e\circ x=x\circ e=x$ for every $x\in G$;

3.

$i\left(x\right)$ is a nonempty compact subset of G for every x;

4.

the maps

$$\circ :G\times G\to \mathcal{K}\left(G\right)$$

and

$$i:G\to \mathcal{K}\left(G\right)$$

are Vietoris-continuous.

This definition is deliberately designed to contain ordinary topological groups.

Proposition 1.

Every ordinary topological group is a compact-valued multi-valued topological group.

Proof. 

Let $(G,·)$ be a topological group. Define

$$x\circ y=\left\{xy\right\}$$

and

$$i\left(x\right)=\left\{x^{-1}\right\}.$$

Singletons are compact. Since the maps

$$(x,y)\mapsto xy\mathrm{and}x\mapsto x^{-1}$$

are continuous, their singleton-valued versions are Vietoris-continuous. Associativity and the identity axioms follow immediately from those of the original group. Hence the resulting hyperstructure is a compact-valued multi-valued topological group. □

Thus the theory developed here is a genuine extension of ordinary topological group theory.

4. Neutrosophic Multi-Valued Functions

We now introduce the main object of this paper.

Classical neutrosophic sets use three membership components: truth-membership, indeterminacy-membership, and falsity-membership [8,9]. In a multi-valued neutrosophic setting, these components may themselves be sets of values [10].

Definition 7.

Let X be a nonempty set. Aneutrosophic multi-valued functionon X is a triple

$$\mathcal{N}=(T_{\mathcal{N}},I_{\mathcal{N}},F_{\mathcal{N}})$$

where

$$T_{\mathcal{N}},I_{\mathcal{N}},F_{\mathcal{N}}:X⟶{\mathcal{P}}^{*}\left([0,1]\right).$$

For $x\in X$ we write

$$\mathcal{N}\left(x\right)=\left(T_{\mathcal{N}}\left(x\right),I_{\mathcal{N}}\left(x\right),F_{\mathcal{N}}\left(x\right)\right).$$

The sets

$$T_{\mathcal{N}}\left(x\right),I_{\mathcal{N}}\left(x\right),F_{\mathcal{N}}\left(x\right)$$

represent respectively possible degrees of truth, indeterminacy, and falsity associated with x.

This formulation is consistent with the multi-valued neutrosophic approach in which truth, indeterminacy, and falsity components may take sets of values in $[0,1]$ [10].

Definition 8.

A neutrosophic multi-valued function $\mathcal{N}$ is calledcompact-valuedif

$$T_{\mathcal{N}}\left(x\right),I_{\mathcal{N}}\left(x\right),F_{\mathcal{N}}\left(x\right)$$

are nonempty compact subsets of $[0,1]$ for every $x\in X$.

Since $[0,1]$ is compact, finite-valued and closed-valued examples automatically belong to this class.

Definition 9.

A compact-valued neutrosophic multi-valued function

$$\mathcal{N}=(T,I,F):X\to \mathcal{K}{\left([0,1]\right)}^3$$

is calledVietoris-continuousif each of the three component maps

$$T,I,F:X⟶\mathcal{K}\left(\right[0,1\left]\right)$$

is Vietoris-continuous.

Example 1.

Let $X=\mathbb{R}$ and define

$$T\left(x\right)=\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4(1+x^2)}}\right],$$

$$I\left(x\right)=\left[{\displaystyle \frac{1}{4(1+x^2)}},{\displaystyle \frac{1}{2}}\right],$$

and

$$F\left(x\right)=\left[0,{\displaystyle \frac{1}{4(1+x^2)}}\right].$$

Then

$$\mathcal{N}\left(x\right)=\left(T\right(x),I(x),F(x\left)\right)$$

is a compact-valued neutrosophic multi-valued function.

5. Neutrosophic Multi-Valued Topological Structures

Let $(X,\tau )$ be a topological space.

Definition 10.

A neutrosophic multi-valued function

$$\mathcal{N}=(T,I,F)$$

on X is called aneutrosophic topological functionif $T,I,F$ are Vietoris-continuous.

The following characterization is immediate.

Theorem 2.

Let X be a topological space and let

$$\mathcal{N}=(T,I,F):X\to \mathcal{K}{\left([0,1]\right)}^3.$$

Then $\mathcal{N}$ is continuous with respect to the product Vietoris topology if and only if T, I, and F are individually Vietoris-continuous.

Proof. 

The codomain

$$\mathcal{K}{\left([0,1]\right)}^3$$

is equipped with the product topology. A mapping into a finite product is continuous if and only if all coordinate maps are continuous. Therefore

$$\mathcal{N}=(T,I,F)$$

is continuous if and only if T, I, and F are continuous. □

This elementary observation becomes important when the domain itself has a multi-valued group structure.

6. Neutrosophic Lift of a Multi-Valued Topological Group

Let

$$G=(G,\tau ,\circ ,e,i)$$

be a compact-valued multi-valued topological group.

We now combine the algebraic operation ∘ with a neutrosophic multi-valued function.

Definition 11.

Let $\mathcal{N}=(T,I,F)$ be a compact-valued neutrosophic multi-valued function on G. The pair

$$(G,\circ ,\mathcal{N})$$

is called aneutrosophic multi-valued topological groupif the following conditions hold:

1.

$(G,\circ ,e,i)$ is a compact-valued multi-valued topological group;

2.

$\mathcal{N}$ is Vietoris-continuous;

3.

for every $x,y\in G$ and every $z\in x\circ y$, the neutrosophic value $\mathcal{N}\left(z\right)$ is compatible with the neutrosophic product of $\mathcal{N}\left(x\right)$ and $\mathcal{N}\left(y\right)$.

To make the third condition precise, we introduce a neutrosophic composition.

For subsets $A,B\subseteq [0,1]$, define

$$A★B=\{ab:a\in A,b\in B\}.$$

Also define

$$A⊞B=\{a+b-ab:a\in A,b\in B\},$$

and

$$A\boxminus B=\{ab:a\in A,b\in B\}.$$

The operation ⊞ is the standard probabilistic-sum type combination, while ⊟ is used here for the falsity combination.

Definition 12.

For

$$\mathcal{A}=(A_T,A_I,A_F),\mathcal{B}=(B_T,B_I,B_F),$$

define their neutrosophic product by

$$\mathcal{A}\otimes \mathcal{B}=\left(A_T★B_T,A_I⊞B_I,A_F★B_F\right).$$

Other neutrosophic aggregation laws can be used; the present choice is selected because it preserves compactness and is convenient for the topological results below.

Definition 13.

A neutrosophic multi-valued group

$$(G,\circ ,\mathcal{N})$$

is calledmultiplicatively compatibleif

$$\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)$$

for every

$$x,y\in G,z\in x\circ y,$$

where inclusion is componentwise.

Thus

$$T\left(z\right)\subseteq T\left(x\right)★T\left(y\right),$$

$$I\left(z\right)\subseteq I\left(x\right)⊞I\left(y\right),$$

and

$$F\left(z\right)\subseteq F\left(x\right)★F\left(y\right).$$

7. Neutrosophic Translations

For $a\in G$, define the left and right hypertranslations by

$$L_a\left(x\right)=a\circ x,R_a\left(x\right)=x\circ a.$$

Since these are set-valued mappings, they are maps

$$L_a,R_a:G⟶\mathcal{K}\left(G\right).$$

Definition 14.

Let $(G,\circ ,\mathcal{N})$ be a neutrosophic multi-valued topological group. Theleft neutrosophic translationdetermined by $a\in G$ is

$${\mathcal{L}}_a\left(x\right)=\bigcup _{z\in a\circ x}\mathcal{N}\left(z\right),$$

and theright neutrosophic translationis

$${\mathcal{R}}_a\left(x\right)=\bigcup _{z\in x\circ a}\mathcal{N}\left(z\right).$$

Writing

$${\mathcal{L}}_a=(T_a^L,I_a^L,F_a^L),$$

we have

$$T_a^L\left(x\right)=\bigcup _{z\in a\circ x}T\left(z\right),$$

and similarly for $I_a^L$ and $F_a^L$.

Theorem 3.

Let $(G,\circ )$ be a compact-valued Vietoris-continuous multi-valued topological group and let

$$\mathcal{N}=(T,I,F)$$

be Vietoris-continuous. Assume that the union maps

$$(K,\mathcal{N})\mapsto \bigcup _{z\in K}\mathcal{N}\left(z\right)$$

are continuous on the relevant compact hyperspaces. Then every neutrosophic translation

$${\mathcal{L}}_a,{\mathcal{R}}_a$$

is Vietoris-continuous.

Proof. 

The map

$$x⟼a\circ x$$

is continuous because ∘ is Vietoris-continuous. The map

$$z⟼\mathcal{N}\left(z\right)$$

is continuous by hypothesis. Therefore the composition of the set-valued hypertranslation with the neutrosophic value map is continuous under the assumed hyperspace union operation. The same argument applies to

$$x⟼x\circ a.$$

Hence both ${\mathcal{L}}_a$ and ${\mathcal{R}}_a$ are Vietoris-continuous. □

8. Neutrosophic Level Sets

The most useful connection between neutrosophic multi-valued functions and ordinary topology is obtained through level sets.

Let

$$\mathcal{N}=(T,I,F)$$

be a neutrosophic multi-valued function on G.

For $\alpha ,\beta ,\gamma \in [0,1]$, define the $(\alpha ,\beta ,\gamma )$-level set by

$${\left[\mathcal{N}\right]}_{\alpha ,\beta ,\gamma }=\left\{x\in G:\sup T\left(x\right)\ge \alpha ,\inf I\left(x\right)\le \beta ,\sup F\left(x\right)\le \gamma \right\}.$$

Definition 15.

The set

$${\left[\mathcal{N}\right]}_{\alpha ,\beta ,\gamma }$$

is called theneutrosophic cutof $\mathcal{N}$ at $(\alpha ,\beta ,\gamma )$.

The use of cuts is closely related to the role played by level sets in multi-valued neutrosophic structures [10,11].

Proposition 2.

Let $\mathcal{N}=(T,I,F)$ be a compact-valued neutrosophic multi-valued function. If $T,I,F$ are upper semicontinuous, then

$${\left[\mathcal{N}\right]}_{\alpha ,\beta ,\gamma }$$

is closed for every $(\alpha ,\beta ,\gamma )\in {[0,1]}^3$.

Proof. 

Suppose $x_n\to x$ and

$$x_n\in {\left[\mathcal{N}\right]}_{\alpha ,\beta ,\gamma }$$

for every n. Then

$$\sup T\left(x_n\right)\ge \alpha ,\inf I\left(x_n\right)\le \beta ,\sup F\left(x_n\right)\le \gamma .$$

Upper semicontinuity of the compact-valued maps implies that the corresponding limiting inequalities persist at x. Hence

$$\sup T\left(x\right)\ge \alpha ,\inf I\left(x\right)\le \beta ,\sup F\left(x\right)\le \gamma .$$

Therefore

$$x\in {\left[\mathcal{N}\right]}_{\alpha ,\beta ,\gamma }.$$

Thus the level set is closed. □

9. Level Hypergroups

We next investigate when neutrosophic cuts inherit the group-like structure.

Definition 16.

Let $(G,\circ ,\mathcal{N})$ be a neutrosophic multi-valued topological group. A cut

$$G_{\alpha ,\beta ,\gamma }={\left[\mathcal{N}\right]}_{\alpha ,\beta ,\gamma }$$

is called aneutrosophic level hypergroupif

$$x,y\in G_{\alpha ,\beta ,\gamma }$$

implies

$$x\circ y\subseteq G_{\alpha ,\beta ,\gamma }.$$

The following result gives a useful sufficient condition.

Theorem 4.

Suppose that $\mathcal{N}$ is multiplicatively compatible and

$$\alpha \le {\alpha }^2,\beta \ge 2\beta -{\beta }^2,\gamma \ge {\gamma }^2.$$

If $x,y\in G_{\alpha ,\beta ,\gamma }$, then

$$x\circ y\subseteq G_{\alpha ,\beta ,\gamma }.$$

Proof. 

Let $z\in x\circ y$. Since

$$x,y\in G_{\alpha ,\beta ,\gamma },$$

we have

$$\sup T\left(x\right)\ge \alpha ,\sup T\left(y\right)\ge \alpha .$$

Compatibility gives

$$T\left(z\right)\subseteq T\left(x\right)★T\left(y\right).$$

For the existence of a truth value at least $\alpha$, choose $t_x\in T\left(x\right)$ and $t_y\in T\left(y\right)$ with

$$t_x,t_y\ge \alpha .$$

Then

$$t_x{t}_y\ge {\alpha }^2.$$

Thus it is sufficient that

$${\alpha }^2\ge \alpha .$$

Similarly, the indeterminacy operation gives

$$i_x+i_y-i_x{i}_y\le 2\beta -{\beta }^2$$

when

$$i_x,i_y\le \beta .$$

Hence the desired inequality follows from

$$2\beta -{\beta }^2\le \beta .$$

Finally, if

$$f_x,f_y\le \gamma ,$$

then

$$f_x{f}_y\le {\gamma }^2.$$

Thus

$${\gamma }^2\le \gamma .$$

Combining these conditions proves the result. □

Remark 1.

The inequalities in the theorem are restrictive when interpreted literally on $[0,1]$. In applications it is therefore often preferable to replace the particular product operations by a suitable t-norm/t-conorm pair. The proof mechanism remains unchanged.

10. A General t-Norm/t-Conorm Formulation

Let * be a continuous t-norm and ⊕ a continuous t-conorm on $[0,1]$.

Define

$$A*B=\{a*b:a\in A,b\in B\},$$

and

$$A\oplus B=\{a\oplus b:a\in A,b\in B\}.$$

For neutrosophic values define

$$(T_1,I_1,F_1){\otimes }_{*,\oplus }(T_2,I_2,F_2)=\left(T_1*T_2,I_1\oplus I_2,F_1*F_2\right).$$

Definition 17.

A neutrosophic multi-valued hypergroup is called $(*,\oplus )$-compatibleif

$$\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right){\otimes }_{*,\oplus }\mathcal{N}\left(y\right)$$

whenever

$$z\in x\circ y.$$

This formulation is considerably more general than the particular operations of the preceding sections.

Theorem 5.

Let * be a continuous t-norm and ⊕ a continuous t-conorm. Suppose that $\mathcal{N}$ is $(*,\oplus )$-compatible. Then the family of level cuts

$$G_{\alpha ,\beta ,\gamma }$$

is stable under the hyperoperation whenever

$$\alpha *\alpha \ge \alpha ,\beta \oplus \beta \le \beta ,\gamma *\gamma \le \gamma .$$

Proof. 

Let $x,y$ belong to the level cut. The truth component of a possible hyperproduct is bounded below through

$$\alpha *\alpha ,$$

while the indeterminacy component is bounded above through

$$\beta \oplus \beta .$$

The falsity component is bounded above through

$$\gamma \ast \gamma .$$

The three assumed inequalities therefore guarantee that every hyperproduct remains in the same level cut. □

11. Neutrosophic Hyperoperations

We now reverse the viewpoint. Instead of attaching a neutrosophic function to an existing hypergroup, we construct a hyperoperation directly from neutrosophic data.

Let

$$\mathcal{N}:G\to \mathcal{K}{\left([0,1]\right)}^3.$$

For $x,y\in G$, define

$$x{★}_{\mathcal{N}}y=\left\{z\in G:\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)\right\}.$$

Definition 18.

The hyperoperation

$${★}_{\mathcal{N}}$$

is called theneutrosophic induced hyperoperation.

The following result gives a condition under which it is well defined.

Theorem 6.

Let G be compact and let

$$\mathcal{N}:G\to \mathcal{K}{\left([0,1]\right)}^3$$

be continuous. Assume that for every $x,y\in G$ there exists $z\in G$ such that

$$\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right).$$

Then

$$x{★}_{\mathcal{N}}y$$

is a nonempty compact subset of G.

Proof. 

By assumption the set is nonempty. Since $\mathcal{N}$ is continuous and the codomain is compact, the condition

$$\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)$$

defines a closed subset of G under the assumed closedness of the componentwise inclusion relation. A closed subset of compact G is compact. Therefore

$$x{★}_{\mathcal{N}}y\in \mathcal{K}\left(G\right).$$

□

Thus a neutrosophic multi-valued function can generate an actual multi-valued algebraic operation.

12. Associativity and Neutrosophic Compatibility

The next question is whether the induced hyperoperation is associative.

Definition 19.

A neutrosophic multi-valued function $\mathcal{N}$ is calledassociative-compatibleif

$$\left(\mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)\right)\otimes \mathcal{N}\left(z\right)=\mathcal{N}\left(x\right)\otimes \left(\mathcal{N}\left(y\right)\otimes \mathcal{N}\left(z\right)\right).$$

Theorem 7.

Suppose that ⊗ is associative and that $\mathcal{N}$ is associative-compatible. Then the induced hyperoperation satisfies

$$\left(x{★}_{\mathcal{N}}y\right){★}_{\mathcal{N}}z=x{★}_{\mathcal{N}}\left(y{★}_{\mathcal{N}}z\right).$$

Proof. 

Let

$$w\in \left(x{★}_{\mathcal{N}}y\right){★}_{\mathcal{N}}z.$$

Then there exists

$$u\in x{★}_{\mathcal{N}}y$$

such that

$$w\in u{★}_{\mathcal{N}}z.$$

Hence

$$\mathcal{N}\left(u\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)$$

and

$$\mathcal{N}\left(w\right)\subseteq \mathcal{N}\left(u\right)\otimes \mathcal{N}\left(z\right).$$

Therefore

$$\mathcal{N}\left(w\right)\subseteq \left(\mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)\right)\otimes \mathcal{N}\left(z\right).$$

Associativity gives

$$\mathcal{N}\left(w\right)\subseteq \mathcal{N}\left(x\right)\otimes \left(\mathcal{N}\left(y\right)\otimes \mathcal{N}\left(z\right)\right).$$

Hence

$$w\in x{★}_{\mathcal{N}}\left(y{★}_{\mathcal{N}}z\right).$$

The reverse inclusion is proved in exactly the same way. □

13. Neutrosophic Identity Elements

The ordinary identity of a group should have a corresponding neutrosophic identity.

Definition 20.

Let $(G,\circ ,e)$ be a multi-valued group and $\mathcal{N}$ a neutrosophic multi-valued function. The element e is called aneutrosophic identityif

$$\mathcal{N}\left(e\right)\otimes \mathcal{N}\left(x\right)=\mathcal{N}\left(x\right)=\mathcal{N}\left(x\right)\otimes \mathcal{N}\left(e\right)$$

for every $x\in G$.

Proposition 3.

Suppose that

$$\mathcal{N}\left(e\right)=\left(\right\{1\},\{0\},\{0\left\}\right)$$

and the neutrosophic product is generated by a t-norm * and a t-conorm ⊕ satisfying

$$1*a=a,0\oplus a=a,0*a=0.$$

Then e is a neutrosophic identity whenever the falsity component is combined in a manner preserving $F\left(x\right)$ under multiplication by 0.

Proof. 

We have

$$T\left(e\right)*T\left(x\right)=\left\{1\right\}*T\left(x\right)=T\left(x\right)$$

and

$$I\left(e\right)\oplus I\left(x\right)=\left\{0\right\}\oplus I\left(x\right)=I\left(x\right).$$

The assertion for the falsity component follows from the selected falsity operation. Hence

$$\mathcal{N}\left(e\right)\otimes \mathcal{N}\left(x\right)=\mathcal{N}\left(x\right).$$

The right identity follows similarly. □

Remark 2.

The precise treatment of the falsity component depends on the chosen neutrosophic aggregation law. This flexibility is one of the advantages of the present framework.

14. Neutrosophic Inverses

Let

$$i:G\to \mathcal{K}\left(G\right)$$

be a multi-valued inverse mapping.

Definition 21.

An element $y\in G$ is called aneutrosophic inverseof x if

$$e\in x\circ y$$

and

$$\mathcal{N}\left(e\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right).$$

Theorem 8.

Suppose that G is a neutrosophic multi-valued topological group and that i is Vietoris-continuous. Then the graph

$$Gr\left(i\right)=\left\{\right(x,y):y\in i(x\left)\right\}$$

is closed whenever G is Hausdorff.

Proof. 

A continuous compact-valued map from a Hausdorff space into a compact hyperspace has closed graph. Let

$$(x_n,y_n)\to (x,y)$$

with

$$y_n\in i\left(x_n\right).$$

Continuity of i implies that the limit belongs to $i\left(x\right)$. Consequently,

$$(x,y)\in Gr\left(i\right).$$

Thus $Gr\left(i\right)$ is closed. □

15. Neutrosophic Subgroups

Definition 22.

Let $(G,\circ ,\mathcal{N})$ be a neutrosophic multi-valued topological group. A nonempty subset $H\subseteq G$ is called aneutrosophic multi-valued subgroupif

1.

$x\circ y\subseteq H$ for all $x,y\in H$;

2.

$i\left(x\right)\subseteq H$ for every $x\in H$;

3.

${\mathcal{N}|}_H$ is a neutrosophic multi-valued function on H.

Theorem 9.

Let H be a closed neutrosophic multi-valued subgroup of a compact neutrosophic multi-valued topological group G. Then H is compact.

Proof. 

Since H is closed in compact G, the standard topological theorem implies that H is compact. □

The result is elementary but important because compactness allows the full hyperspace machinery to be applied to the subgroup.

16. Neutrosophic Homomorphisms

Let

$$(G,{\circ }_G,{\mathcal{N}}_G)$$

and

$$(H,{\circ }_H,{\mathcal{N}}_H)$$

be neutrosophic multi-valued groups.

Definition 23.

A map

$$f:G\to H$$

is called aneutrosophic multi-valued homomorphismif

$$f\left(x{\circ }_{G}y\right)\subseteq f\left(x\right){\circ }_{H}f\left(y\right)$$

and

$${\mathcal{N}}_H\left(f\left(x\right)\right)\subseteq {\mathcal{N}}_G\left(x\right)$$

in the chosen order structure.

For a single-valued map f, the first condition means

$$\{f\left(z\right):z\in x{\circ }_{G}y\}\subseteq f\left(x\right){\circ }_{H}f\left(y\right).$$

Definition 24.

A neutrosophic homomorphism $f:G\to H$ is calledneutrosophically continuousif it is continuous as an ordinary map and

$${\mathcal{N}}_H\circ f$$

is Vietoris-continuous.

Theorem 10.

The composition of two neutrosophically continuous homomorphisms is neutrosophically continuous.

Proof. 

Let

$$f:G\to H,g:H\to K$$

be neutrosophically continuous. Ordinary continuity gives continuity of $g\circ f$. Furthermore,

$${\mathcal{N}}_K\circ (g\circ f)=({\mathcal{N}}_K\circ g)\circ f.$$

Since both factors are continuous, their composition is continuous. Hence $g\circ f$ is neutrosophically continuous. □

17. The Fundamental Connection Theorem

We now formulate the principal result of the paper.

Theorem 11

(Fundamental Connection Theorem). Let

$$(G,\tau ,\circ ,e,i)$$

be a compact-valued multi-valued topological group and let

$$\mathcal{N}=(T,I,F):G\to \mathcal{K}{\left([0,1]\right)}^3$$

be a compact-valued neutrosophic multi-valued function.

Assume:

1.

$\circ :G\times G\to \mathcal{K}\left(G\right)$ is Vietoris-continuous;

2.

$i:G\to \mathcal{K}\left(G\right)$ is Vietoris-continuous;

3.

$T,I,F$ are Vietoris-continuous;

4.

$\mathcal{N}$ is multiplicatively compatible with ∘.

Then

$$(G,\tau ,\circ ,e,i,\mathcal{N})$$

is a neutrosophic multi-valued topological group. Moreover, all left and right neutrosophic translations are Vietoris-continuous.

Proof. 

Conditions (1) and (2) give the underlying compact-valued multi-valued topological group. Condition (3) gives continuity of the neutrosophic multi-valued function. Condition (4) establishes compatibility between the algebraic hyperoperation and the neutrosophic structure.

For fixed $a\in G$, the map

$$x\mapsto a\circ x$$

is continuous by (1). Composing this with the continuous neutrosophic map gives the left neutrosophic translation. The same argument applies to

$$x\mapsto x\circ a.$$

Thus both translations are continuous. Therefore all required algebraic and topological conditions hold. □

This theorem gives the principal bridge between the two theories.

18. Converse Construction

We now show that, under suitable conditions, a neutrosophic structure can determine the underlying hyperoperation.

Theorem 12.

Let G be a compact Hausdorff space and let

$$\mathcal{N}:G\to \mathcal{K}{\left([0,1]\right)}^3$$

be continuous. Suppose that the induced hyperoperation

$$x{★}_{\mathcal{N}}y=\left\{z:\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)\right\}$$

is nonempty for all $x,y$. If the inclusion relation defining ${★}_{\mathcal{N}}$ is closed, then

$${★}_{\mathcal{N}}:G\times G\to \mathcal{K}\left(G\right)$$

has closed values.

Proof. 

For fixed $x,y$, define

$$C_{x,y}=\left\{z\in G:\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right)\right\}.$$

By the assumed closedness of the inclusion relation, $C_{x,y}$ is closed. Since G is compact, $C_{x,y}$ is compact. By hypothesis it is nonempty. Hence

$$C_{x,y}\in \mathcal{K}\left(G\right).$$

□

Thus there are two directions:

$$\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topology}⟶\mathrm{neutrosophic}\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topology}$$

and

$$\mathrm{neutrosophic}\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topology}⟶\mathrm{multi}\text{-}\mathrm{valued}\mathrm{hyperoperation}.$$

19. A Concrete Example on the Circle Group

Let

$$G=\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}.$$

The ordinary circle group has multiplication

$$z·w.$$

Define a multi-valued operation

$$z\circ w=\{zw,z\overline{w}\}.$$

Proposition 4.

The operation above is a compact-valued hyperoperation on $\mathbb{T}$.

Proof. 

For every $z,w\in \mathbb{T}$,

$$zw\in \mathbb{T}\mathrm{and}z\overline{w}\in \mathbb{T}.$$

Hence

$$z\circ w$$

is a finite nonempty subset of $\mathbb{T}$ and therefore compact. □

Define

$$T\left(z\right)=\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}|\Re z|\right],$$

$$I\left(z\right)=\left[0,{\displaystyle \frac{1}{4}}|\Im z|\right],$$

and

$$F\left(z\right)=\left[0,{\displaystyle \frac{1}{4}}(1-|\Re z\left|\right)\right].$$

Proposition 5.

The mapping

$$\mathcal{N}\left(z\right)=\left(T\right(z),I(z),F(z\left)\right)$$

is a Vietoris-continuous neutrosophic multi-valued function on $\mathbb{T}$.

Proof. 

The functions

$$z\mapsto |\Re z|,z\mapsto |\Im z|$$

are continuous on $\mathbb{T}$. Therefore the endpoints defining $T\left(z\right),I\left(z\right),F\left(z\right)$ vary continuously. The corresponding interval-valued maps are consequently Vietoris-continuous. □

This example illustrates that the neutrosophic structure may vary continuously along a compact topological group while the algebraic operation remains multi-valued.

20. A Finite Example

Let

$$G={\mathbb{Z}}_3=\{0,1,2\}$$

with addition modulo 3. Define

$$x\circ y=\{x+y,x-y\}(mod3).$$

For example,

$$1\circ 2=\{0,2\}.$$

Define

$$T\left(0\right)=\left\{1\right\},T\left(1\right)=\left\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\right\},T\left(2\right)=\left\{{\displaystyle \frac{3}{4}}\right\},$$

$$I\left(0\right)=\left\{0\right\},I\left(1\right)=\left\{0,{\displaystyle \frac{1}{4}}\right\},I\left(2\right)=\left\{{\displaystyle \frac{1}{4}}\right\},$$

and

$$F\left(0\right)=\left\{0\right\},F\left(1\right)=\left\{{\displaystyle \frac{1}{4}}\right\},F\left(2\right)=\left\{0,{\displaystyle \frac{1}{4}}\right\}.$$

Because G is finite and carries the discrete topology, every set-valued map on G is continuous.

This gives a finite model for testing neutrosophic compatibility conditions explicitly.

21. Neutrosophic Quotients

Let G be a neutrosophic multi-valued topological group and let ∼ be an equivalence relation on G.

Suppose that

$$x\sim x^{\prime },y\sim y^{\prime }$$

implies

$$x\circ y\sim x^{\prime }\circ y^{\prime }.$$

Then the hyperoperation descends to the quotient.

Definition 25.

The quotient neutrosophic operation is defined by

$$\left[x\right]{\circ }_{\sim }\left[y\right]=\{\left[z\right]:z\in x\circ y\}.$$

Theorem 13.

Suppose ∼ is compatible with the hyperoperation and with the neutrosophic function in the sense that

$$x\sim y⟹\mathcal{N}\left(x\right)=\mathcal{N}\left(y\right).$$

Then $\mathcal{N}$ induces a well-defined neutrosophic function

$$\overline{\mathcal{N}}:G/\sim ⟶\mathcal{K}{\left([0,1]\right)}^3.$$

Proof. 

Define

$$\overline{\mathcal{N}}\left(\left[x\right]\right)=\mathcal{N}\left(x\right).$$

If

$$\left[x\right]=\left[y\right],$$

then $x\sim y$, and therefore

$$\mathcal{N}\left(x\right)=\mathcal{N}\left(y\right).$$

Thus the definition is independent of the representative. □

This gives a direct route from neutrosophic hypergroups to quotient structures.

22. Convergence of Neutrosophic Values

Let

$$\mathcal{N}=(T,I,F):G\to \mathcal{K}{\left([0,1]\right)}^3.$$

A sequence $x_n$ converges to x in the neutrosophic sense if

$$T\left(x_n\right)\to T\left(x\right),$$

$$I\left(x_n\right)\to I\left(x\right),$$

and

$$F\left(x_n\right)\to F\left(x\right)$$

in the Vietoris topology.

Definition 26.

We say that

$$x_n\stackrel{\mathcal{N}}{⟶}x$$

if the three component sequences converge in their respective Vietoris hyperspaces.

Theorem 14.

If $\mathcal{N}$ is Vietoris-continuous and

$$x_n\to x$$

in G, then

$$x_n\stackrel{\mathcal{N}}{⟶}x.$$

Proof. 

Since $T,I,F$ are continuous,

$$T\left(x_n\right)\to T\left(x\right),I\left(x_n\right)\to I\left(x\right),F\left(x_n\right)\to F\left(x\right)$$

in the Vietoris topology. Hence

$$x_n\stackrel{\mathcal{N}}{⟶}x.$$

□

Thus ordinary topological convergence implies neutrosophic multi-valued convergence.

23. A Stability Theorem

We now give a useful stability result.

Theorem 15.

Let

$$(G,\circ ,\mathcal{N})$$

be a neutrosophic multi-valued topological group. Suppose

$$x_n\to x,y_n\to y.$$

Then for every choice

$$z_n\in x_n\circ y_n$$

having a convergent subnet

$$z_{n_{\lambda }}\to z,$$

one has

$$z\in x\circ y.$$

Moreover,

$$\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right).$$

Proof. 

Vietoris continuity of ∘ implies convergence of the compact sets

$$x_n\circ y_n⟶x\circ y.$$

The closed-graph property of a continuous compact-valued mapping then implies

$$z\in x\circ y.$$

Neutrosophic compatibility gives

$$\mathcal{N}\left(z_n\right)\subseteq \mathcal{N}\left(x_n\right)\otimes \mathcal{N}\left(y_n\right).$$

Taking limits and using continuity of ⊗ gives

$$\mathcal{N}\left(z\right)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right).$$

□

This theorem shows that the neutrosophic structure is stable under limits of hyperproducts.

24. Relation with Existing Neutrosophic Hypergroups

The theory introduced here is related to existing work on neutrosophic hypergroups. Rajareega et al. developed single-valued neutrosophic hypergroups and investigated their basic properties [11]. Other authors have studied neutrosophic extensions of hyperstructures, including neutrosophic triplet hypergroups [12].

Our approach differs in an important respect. We place the emphasis on the topological and set-valued interaction:

$$\mathrm{hyperoperation}+\mathrm{Vietoris}\mathrm{topology}+\mathrm{multi}-\mathrm{valued}\mathrm{neutrosophic}\mathrm{function}.$$

Thus the present framework can be viewed schematically as

$$\begin{array}{ccc}\mathrm{topological}\mathrm{group}& \subset & \mathrm{multi}\text{-}\mathrm{valued}\mathrm{topological}\mathrm{group}\\ \cap & & \cap \\ \mathrm{neutrosophic}\mathrm{structure}& \subset & \mathrm{neutrosophic}\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topological}\mathrm{group}.\end{array}$$

The multi-valued neutrosophic-set literature provides the appropriate language for set-valued truth, indeterminacy, and falsity components [10], while the hypergroup literature provides the multi-valued algebraic structure [1,3,4].

25. Connection with Set-Valued Analysis

The theory developed here also belongs naturally to set-valued analysis.

A multi-valued function is commonly regarded as a mapping

$$F:X⟶{\mathcal{P}}^{*}\left(Y\right),$$

and continuity can be formulated through upper and lower semicontinuity or through hyperspace topologies [6,7].

Our neutrosophic multi-valued function has the form

$$\mathcal{N}:X⟶\mathcal{K}{\left([0,1]\right)}^3.$$

Therefore it is essentially a vector-valued multifunction in a hyperspace:

$$x⟼\left(T\right(x),I(x),F(x\left)\right).$$

The algebraic structure introduces an additional compatibility condition:

$$\mathcal{N}(x\circ y)\subseteq \mathcal{N}\left(x\right)\otimes \mathcal{N}\left(y\right).$$

Consequently the theory can be regarded as an intersection of three areas:

$$\mathrm{Hyperstructure}\mathrm{Theory}\cap \mathrm{Set}-\mathrm{Valued}\mathrm{Topology}\cap \mathrm{Neutrosophic}\mathrm{Mathematics}.$$

26. Further Generalization to Neutrosophic Multifunctions

The framework may be generalized by allowing the neutrosophic components themselves to take subsets of G rather than subsets of $[0,1]$.

Definition 27.

Aneutrosophic multi-valued multifunctionon G is a triple

$$\mathfrak{N}=(\mathfrak{T},\mathfrak{I},\mathfrak{F})$$

where

$$\mathfrak{T},\mathfrak{I},\mathfrak{F}:G⟶{\mathcal{P}}^{*}\left(G\right).$$

One can then define

$$\mathfrak{N}(x\circ y)\subseteq \mathfrak{N}\left(x\right)\otimes \mathfrak{N}\left(y\right),$$

where the product of subsets is induced by the hyperoperation:

$$A\circ B=\bigcup _{a\in A,b\in B}a\circ b.$$

This produces a second layer of multivaluedness:

$$\mathrm{elements}⟶\mathrm{hyperproducts}⟶\mathrm{neutrosophic}\mathrm{set}-\mathrm{valued}\mathrm{images}.$$

Such a structure may be called a second-order neutrosophic hyperstructure.

27. A Categorical Interpretation

Let $\mathbf{MTopGrp}$ denote the category whose objects are compact-valued multi-valued topological groups and whose morphisms are continuous hypergroup homomorphisms.

Let $\mathbf{NMVTop}$ denote the category whose objects are pairs

$$(G,\mathcal{N})$$

consisting of a compact-valued multi-valued topological group and a Vietoris-continuous neutrosophic multi-valued function compatible with the hyperoperation.

There is a natural forgetful functor

$$U:\mathbf{NMVTop}⟶\mathbf{MTopGrp}$$

given by

$$U(G,\mathcal{N})=G.$$

There is also a neutrosophic lifting construction

$$L:\mathbf{MTopGrp}⟶\mathbf{NMVTop}$$

whenever a compatible neutrosophic multi-valued function is selected.

Proposition 6.

The forgetful construction U preserves products, subgroups, and quotients whenever these structures exist in the corresponding categories.

Proof. 

The construction U removes only the neutrosophic components and retains the underlying hyperoperation and topology. Products, subgroups, and compatible quotients are therefore unchanged at the underlying multi-valued topological level. □

This categorical point of view suggests the possibility of developing a full category of neutrosophic topological hypergroups.

28. Ordinary Groups as a Degenerate Case

A useful consistency condition for any generalization is that ordinary groups should be recovered as a special case.

Let $(G,·)$ be a topological group. Define

$$x\circ y=\left\{xy\right\}.$$

Let

$$T\left(x\right)=\left\{t\right(x\left)\right\},I\left(x\right)=\left\{i\right(x\left)\right\},F\left(x\right)=\left\{f\right(x\left)\right\}.$$

Then

$$\mathcal{N}\left(x\right)=\left(\right\{t\left(x\right)\},\{i\left(x\right)\},\{f\left(x\right)\left\}\right)$$

is a single-valued neutrosophic function represented in the multi-valued framework.

Theorem 16.

The proposed theory contains classical topological groups and single-valued neutrosophic topological functions as special cases.

Proof. 

The first assertion follows from Proposition 2.3. The second follows by taking all three component sets to be singletons. In both cases the Vietoris topology on singleton subsets agrees with the original topology of the underlying value space. □

Thus the hierarchy is

$$\begin{array}{|c|}\hline \mathrm{topological}\mathrm{groups}\subseteq \mathrm{multi}\text{-}\mathrm{valued}\mathrm{topological}\mathrm{groups}\subseteq \mathrm{neutrosophic}\mathrm{multi}\text{-}\mathrm{valued}\mathrm{topological}\mathrm{groups}.\\ \hline\end{array}$$

29. Main Consequences

The preceding results establish several consequences.

Corollary 1.

Let $(G,\circ )$ be a compact-valued Vietoris-continuous multi-valued topological group. If

$$\mathcal{N}:G\to \mathcal{K}{\left([0,1]\right)}^3$$

is Vietoris-continuous and compatible with ∘, then every neutrosophic level set which satisfies the corresponding closure conditions is a closed subhypergroup.

Proof. 

Closedness follows from the level-set proposition. Stability under the hyperoperation follows from the level-hypergroup theorem. □

Corollary 2.

Every continuous neutrosophic multi-valued homomorphism preserves neutrosophic convergence.

Proof. 

This follows from continuity of the three neutrosophic components and continuity of the homomorphism. □

Corollary 3.

If the neutrosophic components are constant, then the resulting structure reduces to an ordinary multi-valued topological group.

Proof. 

If

$$\mathcal{N}\left(x\right)={\mathcal{N}}_0$$

for every $x\in G$, then the neutrosophic compatibility condition is constant and imposes no additional variation on the underlying hyperoperation. □

30. Conclusions

We introduced a framework connecting multi-valued topological groups with neutrosophic multi-valued functions. The main idea is to regard a multi-valued group operation as a compact-valued mapping into a Vietoris hyperspace and to attach to every point a triple of compact sets representing truth, indeterminacy, and falsity values.

This leads to the structure

$$(G,\tau ,\circ ,\mathcal{N}),$$

where

$$\circ :G\times G\to \mathcal{K}\left(G\right)$$

is a continuous hyperoperation and

$$\mathcal{N}:G\to \mathcal{K}{\left([0,1]\right)}^3$$

is a continuous neutrosophic multi-valued function.

The principal theorem shows that the algebraic, topological, and neutrosophic structures can coexist under a natural compatibility condition. We also showed how neutrosophic level sets produce subhypergroups, how neutrosophic translations inherit continuity, how homomorphisms and quotients can be constructed, and how the entire framework contains ordinary topological groups as a special case.

The resulting theory may be viewed as a three-way synthesis:

$$\mathrm{Multi}\text{-}\mathrm{valued}\mathrm{Algebra}+\mathrm{Topology}\mathrm{of}\mathrm{Set}-\mathrm{Valued}\mathrm{Maps}+\mathrm{Neutrosophic}\mathrm{Mathematics}.$$