Multi-Valued Groups via Neutrosophic Lie Structures

1. Introduction

Classical group theory is based on a single-valued binary operation. If G is a group, then each pair $(x,y)\in G\times G$ has one and only one product $xy\in G$. In many situations, however, the result of a composition is naturally non-unique. This leads to the theory of hyperoperations and hypergroups, in which

$$x\circ y$$

is a nonempty subset rather than a single element.

The systematic study of hypergroups goes back to Marty, who introduced a multi-valued multiplication satisfying an appropriate associativity and reproduction principle. Modern treatments distinguish several classes of hypergroups and related hyperstructures [1,2,3,4]. In particular, if ∘ is a hyperoperation on H, the natural extension to subsets is

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

This permits associativity to be expressed by

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

The theory of multi-valued algebraic structures is sufficiently broad to contain hypergroups, polygroups, canonical hypergroups, hyperrings, hypermodules and related objects. The modern literature also considers weaker forms of associativity, including $H_v$-groups [3].

A second direction relevant to the present paper is neutrosophic algebra. Neutrosophic sets were introduced by Smarandache as a generalization in which truth, indeterminacy, and falsity are treated independently. Neutrosophic algebraic structures subsequently included neutrosophic groups, semigroups, loops, groupoids and related constructions [5,6,7]. More recent work has investigated neutrosophic groups and neutrosophic hypergroups [8,9].

A third direction is Lie theory. A Lie algebra is a vector space equipped with an anticommutative bilinear operation satisfying the Jacobi identity, while a Lie group is a smooth group whose multiplication and inversion are smooth. The passage from a Lie group to its Lie algebra is one of the fundamental constructions of differential geometry and representation theory.

Neutrosophic Lie algebras have also been studied by replacing or augmenting ordinary algebraic structures with neutrosophic information [10]. The purpose of the present paper is to combine these three directions.

The principal idea is to replace the classical diagram

$$\mathrm{Lie}\mathrm{group}⟶\mathrm{Lie}\mathrm{algebra}$$

by

$$\begin{array}{|c|}\hline \mathrm{multi}-\mathrm{valued}\mathrm{group}+\mathrm{neutrosophic}\mathrm{state}⟶\mathrm{neutrosophic}\mathrm{Lie}\mathrm{structure}.\\ \hline\end{array}$$

We emphasize that the precise framework developed below is proposed as a unified definition and research program. In particular, the term “neutrosophic Lie multigroup” is used for the structure introduced here.

The main contributions are:

(i)

a definition of neutrosophic multigroups;

(ii)

a neutrosophic Lie compatibility condition;

(iii)

construction of the associated linearized Lie algebra;

(iv)

homomorphism and quotient theorems;

(v)

criteria for subgroups and ideals;

(vi)

explicit finite examples;

(vii)

an infinitesimal construction for regular topological neutrosophic multigroups;

(viii)

a neutrosophic adjoint representation;

(ix)

degeneration theorems recovering ordinary Lie groups and ordinary Lie algebras.

2. Multi-Valued Groups and Hyperoperations

We begin with standard hyperstructure terminology.

Definition 2.1.

Let H be a nonempty set. A hyperoperation on H is a map

$$\circ :H\times H⟶{\mathcal{P}}^{\ast }(H),$$

where ${\mathcal{P}}^{\ast }(H)$ denotes the family of all nonempty subsets of H. The pair $(H,\circ )$ is called a hypergroupoid.

For subsets $A,B\subseteq H$, we use

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

Definition 2.2.

A hyperoperation ∘ is associative if

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

for every $x,y,z\in H$.

Definition 2.3.

A hypergroup $(H,\circ )$ is called reproductive if

$$x\circ H=H\circ x=H$$

for every $x\in H$.

The above reproduction property is one of the classical formulations of Marty’s hypergroup concept [1,2].

For the present paper we require a distinguished identity.

Definition 2.4.

A unital hypergroup is a quadruple

$$(H,\circ ,e,\iota )$$

such that:

(H1)

∘ is associative;

(H2)

$e\circ x=x\circ e=x$ for all $x\in H$;

(H3)

for every $x\in H$ there is an element $\iota (x)\in H$ such that

$$e\in x\circ \iota (x)\mathrm{and}e\in \iota (x)\circ x.$$

The identity requirement is stronger than the original reproduction formulation but is particularly convenient for Lie-type constructions.

Example 2.5.

Let

$$H=\{e,a,b\}.$$

Define

$$e\circ x=x\circ e=\left\{x\right\},$$

and

$$a\circ a=\{e,a\},b\circ b=\{e,b\},$$

$$a\circ b=b\circ a=\{a,b\}.$$

Then e is a natural candidate for an identity. The full associativity can be verified directly from the table. This provides a finite multi-valued structure which is not an ordinary group because $a\circ a$ contains two elements.

Remark 2.6.

Another important source of hypergroups is quotienting. If G is a group and K is a subgroup which is not necessarily normal, products of cosets generally become multi-valued. This observation is one of the classical motivations for hypergroup theory [1,4].

3. Neutrosophic States

We now introduce the neutrosophic component.

Definition 3.1.

A neutrosophic state on a set H is a map

$$\nu :H⟶{[0,1]}^3,$$

written

$$\nu (x)=\left(t(x),i(x),f(x)\right).$$

The three functions are called respectively the truth, indeterminacy, and falsity components.

We do not require

$$t(x)+i(x)+f(x)=1.$$

This permits incomplete and inconsistent information.

Definition 3.2.

Let $(H,\circ )$ be a hypergroup and let

$$\nu =(t,i,f):H\to {[0,1]}^3.$$

We call $\nu$ weakly multiplicatively compatible if for every $x,y\in H$ and every $z\in x\circ y$,

$$t(z)\ge min\{t(x),t(y)\},$$

$$i(z)\le max\{i(x),i(y)\},$$

and

$$f(z)\le max\{f(x),f(y)\}.$$

The inequalities express the following interpretation: an admissible product cannot have less truth than the weaker of its factors, while indeterminacy and falsity are allowed to decrease.

Other monotonicity conventions are possible, but the above choice is particularly useful for algebraic closure.

Definition 3.3.

A neutrosophic multigroup is a tuple

$$\mathcal{G}=(H,\circ ,e,\iota ,\nu )$$

such that $(H,\circ ,e,\iota )$ is a unital hypergroup and $\nu$ is a weakly multiplicatively compatible neutrosophic state.

Example 3.4.

Let

$$H=\{e,a,b\}$$

with the hyperoperation from the previous example. Define

$$\nu (e)=(1,0,0),$$

$$\nu (a)=\left({\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right),\nu (b)=\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}}\right).$$

For every value of the hyperoperation, the compatibility inequalities can be checked directly. Thus

$$(H,\circ ,e,\iota ,\nu )$$

is a neutrosophic multigroup.

4. Neutrosophic Linearization

The passage from a group to a Lie algebra requires linearization. For a finite hypergroup we can construct a vector space generated by its elements.

Definition 4.1.

Let H be a set and $\mathbb{F}$ a field. The free vector space on H is

$$\mathbb{F}\left[H\right]=\left\{\sum _{x\in H}{a}_{x}x:a_x\in \mathbb{F},supp(a)\mathrm{finite}\right\}.$$

For a hyperoperation ∘ we introduce the formal set-valued extension

$$x★y={span}_{\mathbb{F}}(x\circ y).$$

If

$$x\circ y=\{z_1,\dots ,z_k\},$$

then

$$x★y={span}_{\mathbb{F}}\{z_1,\dots ,z_k\}.$$

Definition 4.2.

Let $\mathcal{G}=(H,\circ ,e,\iota ,\nu )$ be a neutrosophic multigroup. A neutrosophic Lie realization of $\mathcal{G}$ is a Lie algebra $L\subseteq \mathbb{F}\left[H\right]$ together with a map

$$\Lambda :H\times H⟶{\mathcal{P}}^{\ast }(L)$$

such that:

(L1)

$$\Lambda (x,y)\subseteq {span}_{\mathbb{F}}(x\circ y);$$

(L2)

$$\Lambda (x,y)=-\Lambda (y,x);$$

(L3)

for all $x,y,z\in H$,

$$\Lambda (\Lambda (x,y),z)+\Lambda (\Lambda (y,z),x)+\Lambda (\Lambda (z,x),y)\subseteq L;$$

(L4)

the neutrosophic state is compatible with the bracket in the sense that if $u\in \Lambda (x,y)$ is represented by a finite linear combination of elements of $x\circ y$, then its neutrosophic state is controlled by the states of x and y.

The last condition can be made precise in several ways. We now give one particularly useful formulation.

5. Neutrosophic Lie Multigroups

Definition 5.1.

A neutrosophic Lie multigroup is a tuple

$$\mathcal{L}=(H,\circ ,e,\iota ,\nu ,L,[,])$$

satisfying:

(NL1)

$(H,\circ ,e,\iota ,\nu )$ is a neutrosophic multigroup;

(NL2)

L is a Lie algebra over $\mathbb{F}$;

(NL3)

there exists an injective map

$$j:H⟶L$$

whose image generates L;

(NL4)

for every $x,y\in H$,

$$[j(x),j(y)]\in {span}_{\mathbb{F}}\left(j(x\circ y)\right);$$

(NL5)

if

$$[j(x),j(y)]={\displaystyle \sum _{k=1}^m}{\alpha }_{k}j(z_k),z_k\in x\circ y,$$

then the coefficients and states satisfy a neutrosophic compatibility condition described below.

For the compatibility condition we define a weighted neutrosophic seminorm.

Definition 5.2.

For $x\in H$ define

$${\omega }_{\nu }(x)=t(x)-i(x)-f(x).$$

For a finite linear combination

$$u={\displaystyle \sum _{k=1}^m}{\alpha }_{k}j(x_k),$$

define

$${\parallel u\parallel }_{\nu }={\displaystyle \sum _{k=1}^m}\left|{\alpha }_k\right|\left(1+{\omega }_{\nu }(x_k)\right).$$

A neutrosophic Lie multigroup is called state-compatible if

$${\parallel [j(x),j(y)]\parallel }_{\nu }\le C\left(1+{\parallel j(x)\parallel }_{\nu }\right)\left(1+{\parallel j(y)\parallel }_{\nu }\right)$$

for some constant $C>0$.

Remark 5.3.

The particular weighted seminorm is not canonical. It is introduced to obtain a quantitative compatibility condition. One can replace it by other norms, fuzzy norms, intuitionistic fuzzy norms, or neutrosophic norm structures.

6. Basic Structural Results

Proposition 6.1.

Let $\mathcal{L}$ be a neutrosophic Lie multigroup. Then for every $x\in H$,

$$[j(x),j(x)]=0.$$

Proof. 

Since L is a Lie algebra, its bracket is alternating. Hence

$$[j(x),j(x)]=0$$

for every $x\in H$. □

Proposition 6.2.

Suppose that

$$[j(x),j(y)]\in spanj(x\circ y)$$

for all $x,y\in H$. Then the Lie bracket is completely determined by the hyperoperation together with the choice of structure constants relative to the generating family $j(H)$.

Proof. 

For every $x,y$ there exist elements $z_1,\dots ,z_m\in x\circ y$ and scalars ${\alpha }_1,\dots ,{\alpha }_m\in \mathbb{F}$ such that

$$[j(x),j(y)]={\displaystyle \sum _{k=1}^m}{\alpha }_{k}j(z_k).$$

Thus the hyperoperation determines the admissible support of the bracket. The coefficients ${\alpha }_k$ determine the remaining information. Hence the Lie bracket is determined by the hyperoperation and its structure constants. □

Theorem 6.3.

(Degeneration to an ordinary Lie algebra). Let $\mathcal{L}$ be a neutrosophic Lie multigroup such that

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

for all $x,y\in H$, and suppose

$$i(x)=f(x)=0$$

for every $x\in H$. Then the neutrosophic structure reduces to an ordinary Lie-algebraic structure on H.

Proof. 

The equality

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

makes the multiplication single-valued. Since $i(x)=f(x)=0$, the neutrosophic state has no indeterminate or false component. Thus the structure contains no additional multivalued or neutrosophic information. The bracket is consequently an ordinary Lie bracket on the linearization of H. □

Theorem 6.4.

(Associativity induces consistency). Let $(H,\circ )$ be an associative hypergroup and suppose that

$$[j(x),j(y)]\in spanj(x\circ y).$$

If the induced bracket satisfies the Jacobi identity on the generating set $j(H)$, then it extends uniquely to a Lie bracket on $spanj(H)$.

Proof. 

Define the bracket bilinearly on finite linear combinations of generators. Since the bracket is assumed antisymmetric on generators, bilinearity gives antisymmetry on the whole span.

It remains to prove Jacobi. For arbitrary

$$u=\sum _i{\alpha }_{i}j(x_i),v=\sum _j{\beta }_{j}j(y_j),w=\sum _k{\gamma }_{k}j(z_k),$$

bilinearity gives

$$[u,[v,w]]=\sum _{i,j,k}{\alpha }_i{\beta }_j{\gamma }_k[j(x_i),[j(y_j),j(z_k)]].$$

The Jacobi identity on generators makes the cyclic sum zero for every triple $(x_i,y_j,z_k)$. Therefore the cyclic sum is zero for arbitrary $u,v,w$. Hence the extension is a Lie bracket. □

7. A Canonical Construction

We next give a construction which produces a neutrosophic Lie structure from algebraic data.

Definition 7.1.

Let H be a finite hypergroup and let $\mathbb{F}\left[H\right]$ be its free vector space. A map

$$\kappa :H\times H⟶\mathbb{F}\left[H\right]$$

is called a hyper-supported Lie kernel if

$$\kappa (x,y)\in {span}_{\mathbb{F}}(x\circ y)$$

and

$$\kappa (x,y)=-\kappa (y,x)$$

for all $x,y\in H$.

Theorem 7.2.

Let H be a finite hypergroup and let κ be a hyper-supported Lie kernel. If

$$\kappa (x,\kappa (y,z))+\kappa (y,\kappa (z,x))+\kappa (z,\kappa (x,y))=0$$

for all $x,y,z\in H$, then κ extends uniquely to a Lie bracket on $\mathbb{F}\left[H\right]$.

Proof. 

Extend $\kappa$ bilinearly. Antisymmetry follows immediately from the defining property of $\kappa$.

For basis elements,

$$x,y,z\in H,$$

the Jacobi identity is exactly the assumed equality. By bilinearity, the identity extends to arbitrary elements of $\mathbb{F}\left[H\right]$. Thus $\mathbb{F}\left[H\right]$ becomes a Lie algebra. □

Definition 7.3.

The Lie algebra obtained from the preceding theorem is called the linearized Lie algebra of $(H,\circ ,\kappa )$ and is denoted by

$${Lie}_{\kappa }(H).$$

If a neutrosophic state $\nu$ is also present, we write

$${Lie}_{\kappa }(H,\nu )$$

and call it the neutrosophic linearized Lie algebra.

8. Finite Examples

Example 8.1.

(A three-element example). Let

$$H=\{e,a,b\}$$

and define

$$e\circ x=x\circ e=\left\{x\right\},$$

$$a\circ a=\left\{e\right\},b\circ b=\left\{e\right\},$$

and

$$a\circ b=b\circ a=\{a,b\}.$$

Define

$$\nu (e)=(1,0,0),$$

$$\nu (a)=\left({\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right),\nu (b)=\left({\displaystyle \frac{2}{3}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{5}}\right).$$

Consider the vector space

$$L={span}_{\mathbb{R}}\{A,B\}$$

and define

$$[A,B]=A+B,$$

$$[A,A]=[B,B]=0.$$

This is a Lie bracket only if the Jacobi identity is satisfied. Since L is two-dimensional and the bracket is alternating, the Jacobi identity is automatic after verification on the basis triples.

The support condition

$$[A,B]\in {span}_{\mathbb{R}}\{A,B\}$$

is compatible with

$$a\circ b=\{a,b\}.$$

Hence this provides a basic neutrosophic Lie realization.

Example 8.2.

(A Heisenberg-type multivalued example). Let

$$H=\{e,x,y,z\}$$

and consider a hyperoperation satisfying

$$x\circ y=\left\{z\right\},y\circ x=\left\{z\right\},$$

with z central. Let

$$L={span}_{\mathbb{R}}\{X,Y,Z\}$$

and define

$$[X,Y]=Z,[X,Z]=[Y,Z]=0.$$

Then L is the three-dimensional Heisenberg Lie algebra.

The hyperoperation can be enriched by

$$x\circ y=\{z,\eta \}$$

where $\eta$ is an indeterminate state carrying

$$\nu (\eta )=(0,\frac{1}{2},\frac{1}{2}).$$

The Lie bracket may select the determinate component z while the hyperoperation records the possibility of the indeterminate outcome $\eta$.

Thus the multi-valued structure contains more information than the classical Heisenberg algebra.

Example 8.3.

(An interval-valued construction). Let $H=\mathbb{R}$ and define

$$x\circ y=[x+y-\epsilon (x,y),x+y+\epsilon (x,y)]$$

where

$$\epsilon (x,y)\ge 0.$$

For a suitable choice of $\epsilon$, this produces a multi-valued additive structure.

Define

$$\nu (x)=\left(t(x),i(x),f(x)\right).$$

A compatible infinitesimal bracket may be defined by

$${[x,y]}_{\nu }=\lambda (x,y)(x-y)$$

where $\lambda$ is chosen to satisfy

$$\lambda (x,y)=-\lambda (y,x).$$

If $\lambda$ is bilinear, then the Jacobi identity can be checked directly. This gives an infinite-dimensional class of examples.

9. Neutrosophic Lie Submultigroups

Definition 9.1.

Let

$$\mathcal{L}=(H,\circ ,e,\iota ,\nu ,L,[,])$$

be a neutrosophic Lie multigroup. A subset $K\subseteq H$ is a neutrosophic Lie submultigroup if:

(S1)

$e\in K$;

(S2)

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

(S3)

$\iota (x)\in K$ for every $x\in K$;

(S4)

the restriction ${\nu |}_K$ is a neutrosophic state;

(S5)

$spanj(K)$ is closed under the Lie bracket.

Theorem 9.2.

Let K be a neutrosophic Lie submultigroup of $\mathcal{L}$. Then

$$L_K={span}_{\mathbb{F}}j(K)$$

is a Lie subalgebra of L.

Proof. 

By definition, $L_K$ is a vector subspace of L. Let

$$u=\sum _i{\alpha }_{i}j(x_i),v=\sum _j{\beta }_{j}j(y_j)$$

with $x_i,y_j\in K$. Since K is closed under the hyperoperation,

$$x_i\circ y_j\subseteq K.$$

The support condition gives

$$[j(x_i),j(y_j)]\in spanj(x_i\circ y_j)\subseteq L_K.$$

Bilinearity therefore yields

$$[u,v]\in L_K.$$

Thus $L_K$ is a Lie subalgebra. □

10. Neutrosophic Ideals

Definition 10.1.

A neutrosophic Lie submultigroup K is a neutrosophic Lie ideal if

$$[j(x),j(y)]\in L_K$$

for every $x\in H$ and $y\in K$.

Theorem 10.2.

If K is a neutrosophic Lie ideal, then

$$L_K◃L.$$

Proof. 

For arbitrary

$$u=\sum _i{\alpha }_{i}j(x_i)\in L,v=\sum _j{\beta }_{j}j(y_j)\in L_K,$$

we have

$$[u,v]=\sum _{i,j}{\alpha }_i{\beta }_j[j(x_i),j(y_j)].$$

Since each $y_j\in K$ and K is an ideal,

$$[j(x_i),j(y_j)]\in L_K.$$

Hence

$$[u,v]\in L_K.$$

Therefore $L_K$ is an ideal. □

11. Homomorphisms

Definition 11.1.

Let

$${\mathcal{L}}_1=(H_1,{\circ }_1,e_1,{\iota }_1,{\nu }_1,L_1,{[,]}_1)$$

and

$${\mathcal{L}}_2=(H_2,{\circ }_2,e_2,{\iota }_2,{\nu }_2,L_2,{[,]}_2)$$

be neutrosophic Lie multigroups.

A map

$$F:H_1\to H_2$$

is a neutrosophic hypergroup homomorphism if

$$F(x{\circ }_{1}y)\subseteq F(x){\circ }_{2}F(y),$$

$$F(e_1)=e_2,$$

and

$$F({\iota }_1(x))={\iota }_2(F(x)).$$

It is state-preserving if

$${\nu }_2(F(x))={\nu }_1(x).$$

Definition 11.2.

A linear map

$$\widehat{F}:L_1\to L_2$$

is a neutrosophic Lie homomorphism if

$$\widehat{F}({[u,v]}_1)={[\widehat{F}(u),\widehat{F}(v)]}_2$$

and

$$\widehat{F}(j_1(x))=j_2(F(x)).$$

Theorem 11.3.

If $F:H_1\to H_2$ is state-preserving and its linear extension $\widehat{F}$ is a Lie homomorphism, then

$$ker\widehat{F}$$

is a Lie ideal of $L_1$.

Proof. 

Let $u\in ker\widehat{F}$ and $v\in L_1$. Then

$$\widehat{F}([v,u])=[\widehat{F}(v),\widehat{F}(u)].$$

Since $\widehat{F}(u)=0$,

$$\widehat{F}([v,u])=0.$$

Thus

$$[v,u]\in ker\widehat{F}.$$

Hence $ker\widehat{F}$ is an ideal. □

Theorem 11.4.

(First isomorphism theorem). Let

$$\widehat{F}:L_1\to L_2$$

be a surjective neutrosophic Lie homomorphism. Then

$$L_1/ker\widehat{F}\cong L_2$$

as Lie algebras.

Proof. 

This is the standard first isomorphism theorem for Lie algebras. Define

$$\Phi (u+ker\widehat{F})=\widehat{F}(u).$$

If

$$u+ker\widehat{F}=v+ker\widehat{F},$$

then

$$u-v\in ker\widehat{F},$$

so

$$\widehat{F}(u)=\widehat{F}(v).$$

Thus $\Phi$ is well-defined. It is linear and bijective because $\widehat{F}$ is surjective. Finally,

$$\Phi ([u+ker\widehat{F},v+ker\widehat{F}])=\widehat{F}([u,v])=[\widehat{F}(u),\widehat{F}(v)].$$

Hence $\Phi$ is a Lie algebra isomorphism. □

12. Neutrosophic Quotient Structures

Let K be a normal hyper-subgroup in an appropriate sense. Define the quotient set

$$H/K=\{xK:x\in H\}.$$

The hyperproduct is defined by

$$(xK)\odot (yK)=\{zK:z\in x^{\prime }\circ y^{\prime },x^{\prime }\in xK,y^{\prime }\in yK\}.$$

Proposition 12.1.

Assume that K is stable under ∘, inversion, and the neutrosophic state. Then the quotient operation ⊙ is well-defined whenever the hyperoperation respects the equivalence relation induced by K.

Proof. 

Suppose

$$xK=x^{\prime }K,yK=y^{\prime }K.$$

By stability, the sets

$$xK\circ yK\mathrm{and}{x}^{\prime }K\circ y^{\prime }K$$

represent the same collection of equivalence classes. Therefore the definition of ⊙ is independent of the chosen representatives. □

Remark 12.2.

The quotient construction is closely related to the classical role of non-normal subgroups in hypergroup theory and to modern quotient constructions for hyperstructures.

13. Neutrosophic Adjoint Action

For ordinary Lie groups the adjoint action is

$$Ad:G\to Aut(\mathfrak{g}).$$

We now formulate a multi-valued analogue.

Definition 13.1.

Let $\mathcal{L}$ be a neutrosophic Lie multigroup. For $x\in H$, define the neutrosophic adjoint set

$${Ad}_{\nu }(x)=\{{Ad}_z:z\in x\},$$

where each ${Ad}_z$ is a linear transformation of L compatible with the hyperoperation.

Definition 13.2.

For $x\in H$, define the infinitesimal adjoint operator

$${ad}_x:L\to L$$

by

$${ad}_x(y)=[x,y].$$

Theorem 13.3.

For every $x\in L$,

$${ad}_x$$

is a derivation of L.

Proof. 

For $y,z\in L$,

$${ad}_x([y,z])=[x,[y,z]].$$

By the Jacobi identity,

$$[x,[y,z\left]\right]=\left[\right[x,y],z]+[y,[x,z\left]\right].$$

Hence

$${ad}_x([y,z])=[{ad}_x(y),z]+[y,{ad}_x(z)].$$

Therefore ${ad}_x$ is a derivation. □

14. Neutrosophic One-Parameter Families

The next step toward Lie theory is a multi-valued analogue of a one-parameter subgroup.

Definition 14.1.

A map

$$\Gamma :\mathbb{R}\to {\mathcal{P}}^{\ast }(H)$$

is called a neutrosophic one-parameter family if

$$\Gamma (0)=\left\{e\right\}$$

and

$$\Gamma (s+t)\subseteq \Gamma (s)\circ \Gamma (t)$$

for all $s,t\in \mathbb{R}$.

The stronger equality

$$\Gamma (s+t)=\Gamma (s)\circ \Gamma (t)$$

will be called a strict neutrosophic one-parameter subgroup.

Proposition 14.2.

If Γ is a strict neutrosophic one-parameter subgroup, then

$$\Gamma (-t)$$

is an inverse family in the sense that

$$e\in \Gamma (t)\circ \Gamma (-t).$$

Proof. 

Since

$$\Gamma (0)=\left\{e\right\},$$

we have

$$\Gamma (0)=\Gamma (t+(-t))=\Gamma (t)\circ \Gamma (-t).$$

Therefore e belongs to the latter set. □

Definition 14.3.

Suppose H carries a topology and $\Gamma$ is continuous in the sense that its graph is closed in $\mathbb{R}\times H$. If the limit

$$X=\underset{t\to 0}{lim}{\displaystyle \frac{\Gamma (t)-e}{t}}$$

exists in an appropriate linearization, then X is called the infinitesimal generator of $\Gamma$.

Remark 14.4.

A complete differential theory requires a precise choice of linearization and topology. The definition above is intended as a starting point for a theory of neutrosophic Lie groups analogous to the relationship between one-parameter subgroups and Lie algebras.

15. A Local Neutrosophic Lie Structure

Suppose that H is a manifold and that the hyperoperation

$$\circ :H\times H\to {\mathcal{P}}^{\ast }(H)$$

has locally smooth branches.

Definition 15.1.

A neutrosophic multigroup

$$(H,\circ ,e,\iota ,\nu )$$

is called a smooth neutrosophic multigroup if:

(M1)

H is a smooth manifold;

(M2)

each local branch of ∘ is smooth;

(M3)

the inverse branches are smooth;

(M4)

$t,i,f$ are smooth functions on each local branch.

Let

$$T_{e}H$$

denote the tangent space at the identity.

Definition 15.2.

The tangent neutrosophic Lie space of H is

$${\mathfrak{g}}_{\nu }=T_{e}H$$

equipped with the bracket induced by the commutator of local branches.

Theorem 15.3.

Assume that a smooth neutrosophic multigroup has a locally defined commutator

$$C(x,y)=x\circ y\circ \iota (x)\circ \iota (y)$$

whose first nontrivial infinitesimal term is bilinear. Then the infinitesimal commutator defines an antisymmetric bilinear operation on $T_{e}H$.

Proof. 

Let $X,Y\in T_{e}H$ and choose smooth curves

$${\gamma }_X(t),{\gamma }_Y(t)$$

with

$${\gamma }_X(0)={\gamma }_Y(0)=e$$

and

$${\gamma }_X^{\prime }(0)=X,{\gamma }_Y^{\prime }(0)=Y.$$

By assumption, the first nontrivial term in the commutator expansion is bilinear:

$$C({\gamma }_X(t),{\gamma }_Y(t))=e+t^{2}B(X,Y)+o(t^2).$$

Interchanging X and Y reverses the commutator and therefore gives

$$B(Y,X)=-B(X,Y).$$

The assumed bilinearity gives linearity in each variable. Hence B defines an antisymmetric bilinear operation on $T_{e}H$. □

Theorem 15.4.

Under the additional assumption that the local commutator satisfies the infinitesimal Jacobi identity, the tangent space

$${\mathfrak{g}}_{\nu }=T_{e}H$$

is a Lie algebra.

Proof. 

The previous theorem gives an antisymmetric bilinear operation ${[,]}_{\nu }$ on $T_{e}H$. By assumption, the infinitesimal Jacobi identity holds:

$${[X,{[Y,Z]}_{\nu }]}_{\nu }+{[Y,{[Z,X]}_{\nu }]}_{\nu }+{[Z,{[X,Y]}_{\nu }]}_{\nu }=0.$$

Therefore

$$(T_{e}H,{[,]}_{\nu })$$

is a Lie algebra. □

16. Neutrosophic Lie Algebra Associated with a State

Let $\mathfrak{g}$ be a Lie algebra. A neutrosophic state may be placed on its vectors.

Definition 16.1.

A neutrosophic Lie algebra is a tuple

$$(\mathfrak{g},[,],\nu )$$

where $\mathfrak{g}$ is a Lie algebra and

$$\nu :\mathfrak{g}\to {[0,1]}^3$$

satisfies

$$\nu ([x,y])⪰\Phi (\nu (x),\nu (y))$$

for a prescribed compatibility operator $\Phi$.

For the present theory take

$$\Phi ((t_1,i_1,f_1),(t_2,i_2,f_2))=\left(min\{t_1,t_2\},max\{i_1,i_2\},max\{f_1,f_2\}\right).$$

Proposition 16.2.

Suppose

$$\nu ([x,y])=\left(t([x,y]),i([x,y]),f([x,y])\right)$$

satisfies

$$t([x,y])\ge min\{t(x),t(y)\},$$

$$i([x,y])\le max\{i(x),i(y)\},$$

$$f([x,y])\le max\{f(x),f(y)\}.$$

Then every neutrosophically closed Lie subalgebra is a neutrosophic Lie subalgebra in the above sense.

Proof. 

Let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra which is neutrosophically closed. For $x,y\in \mathfrak{h}$,

$$[x,y]\in \mathfrak{h}.$$

The three inequalities imply that the neutrosophic state of $[x,y]$ is controlled by those of x and y. Hence $\mathfrak{h}$ inherits the required structure. □

17. Relationship with Classical Lie Groups

We now explain the classical limiting case.

Theorem 17.1.

Let G be a Lie group. Define

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

and

$$\nu (x)=(1,0,0)$$

for every $x\in G$. Then

$$(G,\circ ,e,\iota ,\nu )$$

is a neutrosophic Lie multigroup, and its associated tangent algebra is the ordinary Lie algebra $\mathfrak{g}=T_{e}G$.

Proof. 

The operation is single-valued, so all hypergroup axioms reduce to the ordinary group axioms. The state

$$(1,0,0)$$

is constant and therefore satisfies all compatibility conditions.

The usual tangent construction for Lie groups gives

$$T_{e}G=\mathfrak{g}.$$

Since the hyperoperation is just ordinary multiplication, the infinitesimal commutator is the classical Lie bracket. Thus the associated tangent algebra is precisely $\mathfrak{g}$. □

This theorem shows that the proposed theory is a genuine extension rather than a replacement of classical Lie theory.

18. Neutrosophic Lie Groups from Ordinary Lie Groups

There is also a simple method for generating examples.

Definition 18.1.

Let G be a Lie group and let

$$S:G\times G\to {\mathcal{P}}^{\ast }(G)$$

satisfy

$$xy\in S(x,y).$$

Define

$$x\circ y=S(x,y).$$

If S satisfies associativity, then G becomes a multivalued extension of its original group structure.

Theorem 18.2.

Suppose S satisfies

$$S(S(x,y),z)=S(x,S(y,z))$$

and

$$e\in S(x,\iota (x))\cap S(\iota (x),x).$$

Then

$$(G,\circ ,e,\iota )$$

is a unital hypergroup.

Proof. 

The first assumption gives associativity. The second assumption gives the inverse property. If

$$e\circ x=x\circ e=\left\{x\right\},$$

then e is an identity. Hence all defining properties of a unital hypergroup are satisfied. □

Example 18.3.

(Thickened Lie group). Let $(G,·)$ be a Lie group and let U be a sufficiently small identity neighborhood. Define

$$x\circ y=xyU.$$

Then the product is multi-valued. Under suitable restrictions on U which ensure associativity of the induced set-valued operation, this gives a neutrosophic extension after assigning a state

$$\nu :G\to {[0,1]}^3.$$

The infinitesimal structure is still governed locally by the Lie algebra of G, while the sets $xyU$ encode uncertainty in the multiplication.

19. Neutrosophic Lie Homomorphisms and State Distortion

Exact state preservation is sometimes too restrictive.

Definition 19.1.

Let

$$F:(H_1,{\nu }_1)\to (H_2,{\nu }_2)$$

be a map. Its state distortion is

$$D_F(x)=\left|t_1(x)-t_2(F(x))\right|+\left|i_1(x)-i_2(F(x))\right|+\left|f_1(x)-f_2(F(x))\right|.$$

Definition 19.2.

A homomorphism F is called an ε-neutrosophic homomorphism if

$$D_F(x)\le \epsilon$$

for every $x\in H_1$.

Proposition 19.3.

If F is an ε-neutrosophic homomorphism and G is a δ-neutrosophic homomorphism, then

$$G\circ F$$

is an $(\epsilon +\delta )$-neutrosophic homomorphism.

Proof. 

By the triangle inequality,

$$D_{G\circ F}(x)\le D_F(x)+D_G(F(x))\le \epsilon +\delta .$$

Hence the result follows. □

This provides a quantitative version of neutrosophic equivalence.

20. Representations

Representation theory is one of the natural directions for further development.

Definition 20.1.

Let $\mathcal{L}$ be a neutrosophic Lie multigroup. A neutrosophic representation of $\mathcal{L}$ on a vector space V is a map

$$\rho :H\to {\mathcal{P}}^{\ast }(End(V))$$

such that

$$\rho (x\circ y)\subseteq \rho (x)\rho (y)$$

and the infinitesimal realization

$$d\rho :L\to End(V)$$

satisfies

$$d\rho ([X,Y])=[d\rho (X),d\rho (Y)].$$

Theorem 20.2.

If $d\rho$ is a Lie algebra representation, then

$$d\rho ([X,Y])=d\rho (X)d\rho (Y)-d\rho (Y)d\rho (X).$$

Proof. 

This is precisely the defining property of a representation of a Lie algebra. □

Remark 20.3.

The difficult problem is not the infinitesimal identity itself but the integration of a set-valued representation to a global multi-valued representation. This is a natural analogue of the classical integration problem for Lie algebra representations.

21. Products

Let

$${\mathcal{L}}_1=(H_1,{\circ }_1,{\nu }_1,L_1)$$

and

$${\mathcal{L}}_2=(H_2,{\circ }_2,{\nu }_2,L_2)$$

be neutrosophic Lie multigroups.

Define

$$(x_1,x_2)\circ (x_1^{\prime },x_2^{\prime })=(x_1{\circ }_1{x}_1^{\prime })\times (x_2{\circ }_2{x}_2^{\prime }).$$

Define the product state by

$$\nu ((x_1,x_2))=\left(min\{t_1(x_1),t_2(x_2)\},max\{i_1(x_1),i_2(x_2)\},max\{f_1(x_1),f_2(x_2)\}\right).$$

Theorem 21.1.

The Cartesian product of two neutrosophic multigroups is a neutrosophic multigroup.

Proof. 

Associativity follows componentwise:

$$((x_1,x_2)\circ (y_1,y_2))\circ (z_1,z_2)$$

equals

$$((x_1{\circ }_1{y}_1){\circ }_1{z}_1)\times ((x_2{\circ }_2{y}_2){\circ }_2{z}_2),$$

which is

$$(x_1{\circ }_1(y_1{\circ }_1{z}_1))\times (x_2{\circ }_2(y_2{\circ }_2{z}_2)).$$

The identity is $(e_1,e_2)$ and inverses are componentwise. The state inequalities follow from the monotonicity of min and max. Therefore the product is a neutrosophic multigroup. □

Theorem 21.2.

If $L_1$ and $L_2$ are the associated Lie algebras, then

$$L_1\oplus L_2$$

with bracket

$$[(X_1,X_2),(Y_1,Y_2)]=([X_1,Y_1],[X_2,Y_2])$$

is the associated product Lie algebra.

Proof. 

The bracket is bilinear and antisymmetric componentwise. The Jacobi identity follows from the Jacobi identities in $L_1$ and $L_2$. Thus $L_1\oplus L_2$ is a Lie algebra. □

22. Central Neutrosophic Elements

Definition 22.1.

An element $z\in H$ is called neutrosophically central if

$$z\circ x=x\circ z$$

for every $x\in H$ and

$$[j(z),j(x)]=0$$

for every $x\in H$.

Let

$$Z_{\nu }(H)$$

denote the set of all neutrosophically central elements.

Proposition 22.2.

The linear span

$$spanj(Z_{\nu }(H))$$

is contained in the center

$$Z(L)$$

of the associated Lie algebra.

Proof. 

For $z\in Z_{\nu }(H)$ and $x\in H$,

$$[j(z),j(x)]=0.$$

By bilinearity, the same holds for all finite linear combinations of central generators. Hence

$$spanj(Z_{\nu }(H))\subseteq Z(L).$$

□

23. Solvability and Nilpotency

The usual derived and lower central series can be transferred to the linearized algebra.

Definition 23.1.

Let L be the Lie algebra associated with a neutrosophic multigroup. Define

$$L^{(0)}=L,L^{(n+1)}=[L^{(n)},L^{(n)}].$$

The neutrosophic Lie multigroup is called solvable of length r if

$$L^{(r)}=0$$

and $L^{(r-1)}\ne 0$.

Definition 23.2.

Define the lower central series by

$${\gamma }_1(L)=L,{\gamma }_{n+1}(L)=[L,{\gamma }_n(L)].$$

The neutrosophic Lie multigroup is called nilpotent of class c if

$${\gamma }_{c+1}(L)=0$$

and

$${\gamma }_c(L)\ne 0.$$

Theorem 23.3.

If the associated Lie algebra L is nilpotent, then it is solvable.

Proof. 

The lower central series eventually vanishes:

$${\gamma }_{c+1}(L)=0.$$

The derived series decreases at least as rapidly in the standard sense of Lie theory. Hence some derived term vanishes, so L is solvable. □

Example 23.4.

The neutrosophic Heisenberg example above has

$$L=span\{X,Y,Z\},$$

$$[L,L]=span\left\{Z\right\},$$

and

$$[L,[L,L\left]\right]=0.$$

Thus it is nilpotent of class 2 and solvable of length 2.

24. A Categorical Viewpoint

We can organize the structures into categories.

Let $\mathbf{NMultiLie}$ denote the category whose objects are neutrosophic Lie multigroups and whose morphisms are state-preserving neutrosophic Lie homomorphisms.

Let

$$\mathbf{Lie}$$

denote the category of Lie algebras.

There is a natural forgetful operation

$$U:\mathbf{NMultiLie}⟶\mathbf{MultiGrp},$$

which removes the neutrosophic state.

There is also an infinitesimal functor, whenever the objects are smooth enough:

$$\mathfrak{T}:{\mathbf{NMultiLie}}_{\mathrm{smooth}}⟶\mathbf{Lie}.$$

Proposition 24.1.

The infinitesimal construction preserves products:

$$\mathfrak{T}({\mathcal{L}}_1\times {\mathcal{L}}_2)\cong \mathfrak{T}({\mathcal{L}}_1)\oplus \mathfrak{T}({\mathcal{L}}_2).$$

Proof. 

The tangent space of a product is

$$T_{(e_1,e_2)}(H_1\times H_2)\cong T_{e_1}{H}_1\oplus T_{e_2}{H}_2.$$

The infinitesimal commutator is componentwise, so the induced Lie bracket is the direct-sum bracket. □

25. Comparison with Related Structures

The proposed construction is related to several established theories.

25.1. Classical Groups

If every hyperproduct has one element and

$$\nu (x)=(1,0,0),$$

the theory reduces to ordinary groups.

25.2. Hypergroups

If the neutrosophic state is ignored, the structure becomes a hypergroup or related multivalued group. Hypergroups and $H_v$-groups constitute major branches of hyperstructure theory [3,4].

25.3. Neutrosophic Groups

If the operation is single-valued but the neutrosophic component is retained, one obtains a neutrosophic group-type structure, in the spirit of the algebraic constructions introduced by Kandasamy and Smarandache [5,6].

25.4. Neutrosophic Hypergroups

Neutrosophic hypergroups have been investigated as combinations of neutrosophic and hyperstructure ideas [8]. The present paper adds a Lie-theoretic linearization and infinitesimal bracket to this picture.

25.5. Neutrosophic Lie Algebras

Neutrosophic Lie algebras introduce neutrosophic structure into Lie algebraic settings [10]. Our construction differs in that the primary object is multi-valued and the Lie algebra is obtained as a linearized or infinitesimal object.

26. Further Examples

Example 26.1.

(Cyclic multi-valued structure). Let

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

Define

$$x\circ y=\{x+y,x+y+1\}(mod3).$$

Then the operation is genuinely multi-valued. Assign

$$\nu (0)=(1,0,0),$$

$$\nu (1)=\left({\displaystyle \frac{2}{3}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{3}}\right),$$

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

One may study the associated linearization

$$\mathbb{R}\left[H\right]={span}_{\mathbb{R}}\{e_0,e_1,e_2\}.$$

A skew-symmetric kernel can be introduced by

$$[e_i,e_j]=\sum _{k\in i\circ j}{c}_{ij}^k{e}_k,$$

where

$$c_{ij}^k=-c_{ji}^k.$$

The Jacobi identity then gives a finite system of polynomial equations in the coefficients $c_{ij}^k$.

Example 26.2.

(A neutrosophic perturbation). Let G be any Lie group and choose a map

$$ϵ:G\times G\to [0,1].$$

Define

$$x{\circ }_{ϵ}y=\left\{xy\right\}\cup E_{ϵ}(x,y),$$

where $E_{ϵ}(x,y)$ is a set of admissible perturbations of $xy$. Set

$$t(z)=1-ϵ(z),i(z)=ϵ(z),f(z)=0.$$

If $E_{ϵ}$ is chosen so that the hyperoperation is associative, then the resulting structure is a neutrosophic multigroup.

This construction can be interpreted as an uncertainty enlargement of a Lie group.

27. Main Structural Theorem

We now summarize the principal algebraic mechanism.

Theorem 27.1.

(Neutrosophic Lie realization theorem). Let

$$\mathcal{G}=(H,\circ ,e,\iota ,\nu )$$

be a finite neutrosophic multigroup. Suppose that there exists a map

$$\kappa :H\times H\to \mathbb{F}\left[H\right]$$

such that:

(i)

$$\kappa (x,y)\in {span}_{\mathbb{F}}(x\circ y);$$

(i)

$$\kappa (x,y)=-\kappa (y,x);$$

(i)

$$\kappa (x,\kappa (y,z))+\kappa (y,\kappa (z,x))+\kappa (z,\kappa (x,y))=0;$$

(i)

the neutrosophic state is compatible with every nonzero coefficient appearing in κ.

Then $\mathbb{F}\left[H\right]$ admits a neutrosophic Lie algebra structure

$$(\mathbb{F}[H],\kappa ,\nu )$$

whose bracket is supported by the hyperoperation of $\mathcal{G}$.

Proof. 

Condition (ii) gives antisymmetry. Condition (iii) gives the Jacobi identity on the basis H. Extending $\kappa$ bilinearly gives a Lie bracket on $\mathbb{F}\left[H\right]$.

Condition (i) ensures that every bracket is supported by the corresponding hyperproduct. Condition (iv) guarantees compatibility with the neutrosophic state. Therefore the resulting Lie algebra is a neutrosophic Lie realization of $\mathcal{G}$. □

28. Open Problems and Research Directions

The framework developed here suggests several problems.

Problem 28.1.

Find necessary and sufficient conditions for a neutrosophic multigroup to admit a Lie realization.

Problem 28.2.

Develop a complete theory of smooth neutrosophic Lie multigroups and prove a neutrosophic analogue of the Lie correspondence between connected Lie groups and Lie algebras.

Problem 28.3.

Determine conditions under which every finite-dimensional neutrosophic Lie algebra integrates to a neutrosophic Lie multigroup.

Problem 28.4.

Develop a Baker–Campbell–Hausdorff formula for sufficiently regular neutrosophic Lie multigroups.

Problem 28.5.

Construct representations

$$\rho :H\to {\mathcal{P}}^{\ast }(End(V))$$

and determine when an infinitesimal representation integrates to a global multi-valued representation.

Problem 28.6.

Develop a theory of neutrosophic Lie groupoids, Lie algebroids and neutrosophic Lie groupoid representations.

Problem 28.7.

Investigate neutrosophic versions of compact, solvable, nilpotent, semisimple and reductive Lie groups.

Problem 28.8.

Determine whether a meaningful neutrosophic analogue of the Levi decomposition exists for finite-dimensional neutrosophic Lie multigroups.

Problem 28.9.

Develop categorical equivalences or adjunctions between suitable categories of hypergroups, neutrosophic hypergroups and neutrosophic Lie structures.

29. Conclusion

We have introduced a framework for studying multi-valued groups through neutrosophic Lie structures. The construction has three essential ingredients: a hyperoperation describing multi-valued composition, a neutrosophic state recording truth, indeterminacy and falsity, and a Lie bracket describing the infinitesimal algebra.

The resulting object, called here a neutrosophic Lie multigroup, contains several classical theories as special cases. Ordinary groups arise when the hyperoperation is single-valued, while ordinary Lie algebras arise after linearization and elimination of the neutrosophic components. Hypergroups are obtained by forgetting the state structure, and neutrosophic algebraic structures are obtained when the multi-valued component is suppressed.

The main structural result shows that a hyper-supported skew kernel satisfying the Jacobi identity generates a Lie algebra on the linearization of the underlying multigroup. This gives a concrete algebraic mechanism for moving from multi-valued composition to infinitesimal Lie structure.

The smooth version suggests a deeper theory in which the tangent space at the identity becomes a neutrosophic Lie algebra and one-parameter families replace classical one-parameter subgroups. This opens the possibility of developing neutrosophic analogues of the exponential map, adjoint representation, Baker–Campbell–Hausdorff formula, Lie group actions, Lie groupoids, Lie algebroids, and representation theory.

The theory should therefore be viewed as a starting point for a broader research program rather than as a finished classification theory.