Fuzzy Hom–Lie Ideals of Hom–Lie Algebras

Abstract

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras.

1. Introduction

Algebraic structures are an indispensable part of the domain of mathematics and beyond, possessing an extensive array of its implications for numerous disciplinary fields (i.e., engineering and computer sciences [1,2,3,4,5], topological spaces [6,7,8], theoretical physics [9], coding theory [10,11], etc.). Such a wide-reaching impact of algebra forms the basis upon which to further extend this notion by investigating its distinct categories, sub-categories, and forms in order to shed light on the underlying ideas that research scholars seek to address in attempt to push forward the knowledge boundary in this paradigm. The same idea also encourages mathematicians to scrutinize varying concepts and the resulting factors from within the context of abstract algebra in the widely stretched-out framework of the fuzzy setting [12]. The current work intended to perform the same.

The inception of the concept of Lie algebras was performed by one of the renowned Norwegian mathematicians of the 19th century, Sophus Lie (1842–1899) [13,14]. After him, the idea has undergone many developments, modifications, and additions, by multiple authors who incorporated further advanced and novel types and factors with this concept, one of which is the notion of Hom–Lie algebras. The base of the idea of Hom–Lie algebras was initially formulated by Hartwig, Larsson, and Silvestrov in 2006 [15]. It lies among the generalities of the notion of classical Lie algebras. In recent years, Hom–Lie algebras have turned into a stimulating area of physics and mathematics. For further deliberations concerning Hom–Lie algebras, we refer the reader to [15,16,17,18,19,20,21].

In terms of the case of fuzzy sets, Zadeh [22] was the first one to present this notion. On a non-empty set X, a fuzzy set denotes a map termed as the membership function $\mu :X\to [0,1]$. It should also be noted in the classical set theory, which is written as $\mu \left(x\right)=1$ if $x\in X$, and $\mu \left(x\right)=0$ if x is not in X. The application of fuzzy set theory extends to various fields including decision theory [23,24], logic [25], soil science [26], computer science [27], social life [28,29], artificial intelligence [30,31,32], management science [33,34], and so forth. Over the years, fuzzy algebraic structures, including fuzzy groups, fuzzy (gamma-)rings, and fuzzy modules, fuzzy (color-)Lie (super-)algebras, fuzzy co-algebras, and fuzzy bi-algebras have been extensively studied and applied in various disciplines.

Originally, the study of fuzzy Lie subalgebras of Lie algebras was first introduced by Yehia [35] in the year 1996. Afterwards, numerous authors [36,37,38,39,40,41,42,43,44], as well as the references within) have made contributions to the concept of fuzzy sets (in more general, complex fuzzy sets and intuitionistic fuzzy sets) by applying them in a variety of directions in Lie algebras.

The current study revolves around the investigation of fuzzy Hom–Lie algebras. Our obtained results in the theory of fuzzy Hom–Lie subalgebras (and ideals, respectively) are of significant importance for several reasons. Firstly, they contribute to our understanding of fuzzy algebraic structures and their applications. Fuzzy Hom–Lie subalgebras (and ideals, respectively) are a generalization of traditional Hom–Lie subalgebras (and ideals, respectively), where the notion of exact membership is replaced by degrees of membership. These findings establish a framework that enables the examination and characterization of fuzzy Hom–Lie subalgebras (or ideals), facilitating a more extensive exploration of their structural properties and dynamic behavior. By extending the concepts and theorems from classical (Hom–)Lie subalgebras to the fuzzy setting, these results enhance our ability to model and reason about uncertain or imprecise information in mathematical, physical and real-world scenarios. The study of fuzzy Hom–Lie subalgebras (and ideals, respectively) offer a valuable tool for investigating uncertainties, fluctuations, and emergent properties in physical theories, leading to deeper insights into the fundamental laws of nature.

In [45], the authors introduced and studied complex neutrosophic Lie algebras. The author welcomes to study complex neutrosophic Hom–Lie algebras as a generalization of the previous paper. Integrating the representations of Hom–Lie algebras and (complex) (neutrophic) (fuzzy) (Hom–)Lie algebras can be approached by identifying the common themes and connections between the two topics. Here is a possible way to integrate the topics:

The study of representations in the context of Hom–Lie algebras provides insights into the algebraic properties of these non-associative structures. Investigating the representations of Hom–Lie algebras involves understanding how these algebras act on vector spaces and how the Hom structure influences their properties.

Building upon the theory of representations in (Hom–)Lie algebras, the second topic introduces the concept of (complex) (neutrophic) (fuzzy) (Hom-)Lie algebras. Complex (neutrophic) (fuzzy) (Hom–)Lie algebras extend the notion of fuzzy algebra to Hom–Lie algebras, allowing for fuzzy membership degrees and complex-valued elements. This extension provides a framework to handle uncertainty and imprecise information in the context of Hom–Lie algebras. By integrating these topics, we can explore the representation theory of complex (neutrophic) (fuzzy) (Hom–)Lie algebras. This entails studying how complex (neutrophic) (fuzzy) (Hom–)Lie algebras act on vector spaces while considering the fuzzy membership degrees and the influence of the (Hom–)Lie structure. This integrated approach enables a more comprehensive understanding of the algebraic properties and representation theory of complex (neutrophic) (fuzzy) (Hom–)Lie algebras.

The integration of these topics facilitates a broader exploration of the connections between representation theory, (Hom-)Lie algebras, and complex (neutrophic) (fuzzy) algebra. It offers a unified framework for investigating the interplay between algebraic structures, representation theory, and the handling of uncertainty in the context of (Hom–)Lie algebras extended with fuzzy and complex-valued elements.

With the aspiration of facilitating further investigation into this subject, and for the convenience of readers, we endeavored to make this paper self-contained and accessible to individuals with minimal knowledge in linear and abstract algebras.

2. Preliminaries

Let F be a ground field. Consider the vector spaces V, U, and W over F, with $V\times U$ representing the Cartesian product of sets V and U. A function $f:V\times U\to W$ is said to be bilinear if it exhibits linearity with respect to each variable separately. In other words, for any $v,v^{\prime }\in V$, $u,u^{\prime }\in U$, and $a,b\in F$, the following conditions hold:

$$f(av+b{v}^{\prime },u)=af(v,u)+bf(v^{\prime },u)$$

and

$$f(v,au+b{u}^{\prime })=af(v,u)+bf(v,u^{\prime }).$$

A Lie algebra can be defined as a pair $(L,[,\left]\right)$, where L is a vector space over F, and $[,]:L\times L\to L$ is a bilinear map known as the Lie bracket. The Lie algebra must satisfy the two following properties:

(i)

Skew-symmetry: for any elements x and y in L, the Lie bracket demonstrates skew-symmetry:

$$[x,y]=-[y,x].$$

(ii)

Jacobi identity: for any elements x, y, and z in L, the Lie bracket satisfies the Jacobi identity:

$$[x,[y,z\left]\right]+[y,[z,x\left]\right]+[z,[x,y\left]\right]=0.$$

A Hom–Lie algebra over F is defined as a triple $(L,[,],\alpha )$, where L is a vector space over F, $\alpha :L\to L$ is a linear map, and $[,]:L\times L\to L$ is a bilinear map, referred to as a bracket. The Hom–Lie algebra satisfies the following properties:

(i)

For any elements x and y in L, the bracket exhibits skew-symmetry:

$$[x,y]=-[y,x].$$

(ii)

For any elements x, y, and z in L, the bracket satisfies the Hom–Jacobi identity:

$$\left[\alpha \right(x),[y,z\left]\right]+\left[\alpha \right(y),[z,x\left]\right]+\left[\alpha \right(z),[x,y\left]\right]=0.$$

It is noteworthy that each instance of a Lie algebra can be viewed as a Hom–Lie algebra by setting $\alpha$ as the identity map, as denoted by ${\mathrm{id}}_L$.

Recall that a characteristic of a field F refers to a mathematical concept that describes the behavior of the field’s elements with respect to addition. The characteristic of a field refers to the minimum positive integer n for which the sum of n copies of an element $x\in F$, represented as n · x, yields the additive identity element denoted by 0. In simpler terms, one can prove that it represents the number of times you need to add the number 1 to itself in order to obtain 0. Moreover, it is well known that the characteristic of a field can be either a prime number or zero. In the case of a Hom–Lie algebra L over a field F having characteristic $\ne 2$ as in the scenario of Lie algebras, it can be shown that $[x,x]=0$ for every $x\in L$. Additionally, for an arbitrary Hom–Lie algebra L, for every $x\in L$, we have $[x,0]=[0,x]=0$.

Example 1. 

Consider a vector space L over F and a skew-symmetric-bilinear map $[,]:L\times L\to L$. If the map $\alpha :L\to L$ is defined as the zero map (i.e., $\alpha \left(x\right)=0$ for all $x\in L$), then the triple $(L,[,],\alpha )$ forms a Hom–Lie-algebra.

Example 2. 

Consider a vector space L and an arbitrary linear map α. In this context, the triple $(L,[,],\alpha )$ forms a Hom–Lie algebra, where the Lie bracket $[x,y]$ is equal to zero for all $c,d\in L$. Hom–Lie algebras of this type are commonly referred to as abelian or commutative Hom–Lie algebras.

Hom–Lie algebra $(L,[,],\alpha )$ can be classified as multiplicative if

$$\alpha \left(\right[c,d\left]\right)=\left[\alpha \right(c),\alpha (d\left)\right]$$

for all $c,d\in L$. If a multiplicative Hom–Lie algebra has the additional property of $\alpha$ being a bijective map, it is referred to as a regular multiplicative Hom–Lie algebra.

Example 3

([46]). Consider a vector space L over F with a basis $\{c_1,c_2,c_3\}$. We define the linear map $\alpha :L\to L$ by setting $\alpha \left(c_1\right)=c_2$ and $\alpha \left(c_2\right)=\alpha \left(c_3\right)=0$. Consider the skew-bilinear map $[,]:L\times L\to L$ such that

$$[c_1,c_2]=[c_2,c_3]=0,[c_1,c_3]=c_1$$

and $[c_i,c_i]=0$ for all $i=1,2,3$.

In this context, we can observe that $(L,[,],\alpha )$ forms a Hom–Lie algebra. It is worth noting that, for any elements $x,y\in L$, the value $[x,y]$ is a scalar multiple of $c_1$. Similarly, for each $x\in L$, the value $\alpha \left(x\right)$ is a scalar multiple of $c_2$ for each $x\in L$. Consequently, $\left[\alpha \right(x),[y,z\left]\right]=0$ for each $x,y,z\in L$. This observation leads us to the conclusion that the Hom–Jacobi identity holds in this context.

Example 4. 

Consider L as a vector space over F with a basis $\{c_1,c_2,c_3\}$. Define the linear map $\alpha :L\to L$ by setting $\alpha \left(c_1\right)=c_1$, $\alpha \left(c_2\right)=2c_2,$ and $\alpha \left(c_3\right)=2c_3$. Let $[,]:L\times L\to L$ be the skew symmetric bilinear map such that

$$[c_1,c_2]=c_1,[c_1,c_3]=2c_2,[c_2,c_3]=2c_3,$$

and for all $i=1,2,3$, we have $[c_i,c_i]=0$. Then, $(L,[,],\alpha )$ becomes a Hom–Lie algebra, but the bracket $[,]$ does not define a Lie algebra on L.

Consider a Hom–Lie algebra $(L,[,],\alpha )$. A subspace H of L is considered a Hom–Lie subalgebra if it satisfies the following conditions: $\alpha \left(H\right)\subseteq H$ and $[x,y]\in H$ for all $x,y\in H$. If a Hom–Lie subalgebra H also fulfills the property that $[x,y]\in H$ for all $x\in H$ and $y\in L$, it is referred to as a Hom–Lie ideal.

Consider the Hom–Lie algebras $(L_1,{[,]}_1,{\alpha }_1)$ and $(L_2,{[,]}_2,{\alpha }_2)$. A linear map $\phi :L_1\to L_2$ is referred to as a morphism of Hom–Lie algebras if it satisfies the two following conditions:

(i)

$\phi \left({[x,y]}_1\right)={[\phi \left(x\right),\phi \left(y\right)]}_2$ for all $x,y\in L_1$.

(ii)

$\phi \circ {\alpha }_1={\alpha }_2\circ \phi$.

Thus, all through the present work, L will be considered a Hom–Lie algebra over F.

3. Results

3.1. Fuzzy Hom–Lie Subalgebras and Fuzzy Hom–Lie Ideals

Let $a,b\in [0,1]$. Intended for the ease of understanding, the expressions $a\wedge b$ and $a\vee b$ were used to indicate $\min \{a,b\}$ and $\max \{a,b\}$, respectively.

Definition 1. 

A fuzzy set μ defined on L is referred to as a “fuzzy Hom–Lie subalgebra” if it satisfies the following identities for all $c,d\in L$ and $\gamma \in F$:

(i) 

$\mu (c+d)\ge \mu \left(c\right)\wedge \mu \left(d\right)$,

(ii) 

$\mu (\gamma$·$c)\ge \mu (c)$,

(iii) 

$\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\wedge \mu \left(d\right)$,

(iv) 

$\mu \left(\alpha \right(c\left)\right)\ge \mu \left(c\right)$.

When the condition $\left(iii\right)$ is modified to $\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\vee \mu \left(d\right)$, the fuzzy set $\mu$ is referred to as a “fuzzy Hom–Lie ideal” of L. It is important to note that the second condition implies $\mu \left(c\right)\le \mu \left(0\right)$ and $\mu (-c)\le \mu \left(c\right)$ for all $c\in L$. Furthermore, if $\alpha ={\mathrm{I}}_L$, which denotes the identity map on L, the resulting fuzzy sets correspond to fuzzy Lie subalgebras (and fuzzy ideals, respectively).

If $\gamma \ne 0$, the condition $\left(ii\right)$ where $\mu \left(\gamma c\right)\ge \mu \left(c\right)$ in a fuzzy Hom–Lie subalgebra (and ideal, respectively) implies that $\mu (\gamma$·$c)=\mu (c)$. Indeed, for $\gamma \ne 0$, we find

$$\mu \left(c\right)=\mu (({\gamma }^{-1}·\gamma )·c)=\mu ({\gamma }^{-1}·(\gamma ·c))\ge \mu (\gamma ·c).$$

Now, it is apparent that as long as $\mu$ is a fuzzy Hom–Lie ideal of L, then it is a fuzzy Hom–Lie subalgebra of L.

Example 5. 

Consider a vector space L with a basis $c_1,c_2,c_3$. We define a linear map $\alpha :L\to L$ by assigning $\alpha \left(c_1\right)=c_2$ and $\alpha \left(c_2\right)=\alpha \left(c_3\right)=0$. Furthermore, let $[,]:L\times L\to L$ be the skew-symmetric bilinear map defined as follows:

$$[c_1,c_2]=0,[c_2,c_3]=0,\mathrm{and}[c_1,c_3]=c_1$$

Additionally, we set $[c_i,c_i]=0$ for all $i=1,2,3$. With these definitions, it can be observed that $(L,[,],\alpha )$ forms a Hom–Lie algebra. Notably, for any $x,y\in L$, the element $[x,y]$ can be expressed as a scalar multiple of $c_1$. Similarly, $\alpha \left(x\right)$ is a scalar multiple of $c_2$ for all $x\in L$. Consequently, the expression $\left[\alpha \right(x),[y,z\left]\right]=0$ holds true for every $x,y,z\in L$, indicating that the Hom–Jacobi identity is satisfied.

We define μ as follows:

$$\mu \left(x\right)=\left\{\begin{array}{cc}0.9\hfill & :ifx=0\hfill \\ 0.4\hfill & :x\in span\{c_1,c_2\}-\left\{0\right\}\hfill \\ 0.2\hfill & :\mathrm{otherwise}.\hfill \end{array}\right.$$

Subsequently, one can obtain that L exhibits μ as a fuzzy Hom–Lie ideal.

3.2. Relations between Fuzzy Hom–Lie Ideals and Hom–Lie Ideals

Let V be a vector space equipped with a fuzzy set $\mu$. We define the set $U(\mu ,t)$ as the upper level of $\mu$ for $t\in [0,1]$, denoted by $U(\mu ,t)=\{x\in V|\mu \left(x\right)\ge t\}$. The subsequent theorem will demonstrate the connection between fuzzy Hom–Lie subalgebras of L and Hom–Lie subalgebras of L.

Theorem 1. 

Consider a fuzzy subset μ of L. In this scenario, the following statements are equivalent:

(i) 

μ is a fuzzy-Hom–Lie subalgebra of L.

(ii) 

For every $t\in \mathrm{Im}\left(\mu \right)$, the non-empty set $U(\mu ,t)$ is a Hom–Lie subalgebra of L.

Proof. 

Let $\mu$ be a fuzzy Hom–Lie subalgebra of L. Consider $c,d\in U(\mu ,t)$ and $\gamma \in F$, where $\mu \left(c\right)\ge t$ and $\mu \left(d\right)\ge t$. Referring to Definition 1, we can observe the following:

(i)

$\mu (c+d)\ge \mu \left(c\right)\wedge \mu \left(d\right)\ge t$;

(ii)

$\mu (\gamma .c)\ge \mu \left(c\right)\ge t$;

(iii)

$\mu \left(\alpha \right(c\left)\right)\ge \mu \left(c\right)\ge t$;

(iv)

$\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\wedge \mu \left(d\right)\ge t$.

Consequently, $c+d,\gamma$·$c,\alpha \left(c\right)$, and $[c,d]$ belong to $U(\mu ,t)$.

In reverse, suppose that $U(\mu ,t)$ is a Hom–Lie subalgebra of L for every $t\in \mathrm{Im}\left(\mu \right)$. Take $c,d\in L$, assuming that $\mu \left(d\right)\ge \mu \left(c\right)=t_1$, which implies $c,d\in U(\mu ,t_1)$. Since $U(\mu ,t_1)$ is a subspace of L, the following hold:

(i)

$\gamma$·$c\in U(\mu ,t_1)$;

(ii)

$c+d\in U(\mu ,t_1)$.

Consequently, we can deduce:

(i’)

$\mu (\gamma$·$c)\ge t_1=\mu \left(c\right)$;

(ii’)

$\mu (c+d)\ge t_1=\mu \left(c\right)\wedge \mu \left(d\right)$.

Since $U(\mu ,t_1)$ is a Hom–Lie subalgebra of L, the following conditions are satisfied:

(i)

$[c,d]\in U(\mu ,t_1)$,

(ii)

$\alpha \left(c\right)\in U(\mu ,t_1)$.

Consequently, we have:

(i’)

$\mu \left([c,d]\right)\ge t_1=\mu \left(c\right)\wedge \mu \left(d\right)$;

(ii’)

$\mu \left(\alpha \left(c\right)\right)\ge t_1=\mu \left(c\right)$.

The proof now is complete. □

Theorem 2. 

The equivalence between the following statements holds for a fuzzy subset μ of L:

(i) 

μ is a fuzzy Hom–Lie ideal of L;

(ii) 

The nonempty-set $U(\mu ,t)$ is a Hom–Lie ideal of L for every $t\in \mathrm{Im}\left(\mu \right)$.

Proof. 

If $\mu$ is a fuzzy Hom–Lie ideal of L, it is also a fuzzy Hom–Lie subalgebra of L. As shown in the previous theorem, for any $c,d\in U(\mu ,t)$ and $\gamma \in F$, we have $c+d$, $\gamma$ · c, and $\alpha \left(c\right)$ in $U(\mu ,t)$. Additionally, for $c\in L$ and $d\in U(\mu ,t)$, we find that $\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\vee \mu \left(d\right)\ge \mu \left(c\right)\ge t$. Therefore, $[c,d]\in U(\mu ,t)$.

In reverse, suppose that, for every non-empty $U(\mu ,t)$, it is a Hom–Lie ideal of L. Therefore, $U(\mu ,t)$ is also a Hom–Lie subalgebra. The proof follows a similar approach as in the previous theorem, with the only difference appearing in the proof of the following statement:

$$\mu \left(\right[c,d]\ge \mu (c)\vee \mu (d)\forall c,d\in L.$$

Let $c,d\in L$. Without loss of generality, let us assume that $\mu \left(c\right)\ge \mu \left(d\right)$. Set $t_0=\mu \left(c\right)$, which means that $c\in U(\mu ,t_0)$. Since $U(\mu ,t_0)$ is a Hom–Lie ideal of L, we have $[c,d]\in L$. Consequently, $\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\vee \mu \left(d\right)$. □

Consider V to be a vector space. For $t\in [0,1]$ and a fuzzy set $\mu$ on V, the set $U({\mu }^{>},t)=\left\{c\in V\left|\mu \right(c)>t\right\}$ is termed a strong-upper level of $\mu$. The following results are presented.

Theorem 3. 

The equivalence of the following statements holds for a fuzzy subset μ of L:

(i) 

μ is a fuzzy Hom–Lie subalgebra of L;

(ii) 

The strong upper-level $U({\mu }^{>},t)$ forms a subalgebra of L for every t in the image of μ.

Proof. 

For each t in the image of $\mu$, let $c,d\in U({\mu }^{>},t)$, and $\gamma \in F$. Since $\mu$ is a fuzzy Hom–Lie subalgebra of L, the following conditions hold:

• $\mu (c+d)\ge \mu \left(c\right)\wedge \mu \left(d\right)>t$;
• $\mu (\gamma$·$c)>\mu (c)\ge t$;
• $\mu \left(\alpha \right(c\left)\right)>\mu \left(c\right)$;
• and $\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\wedge \mu \left(d\right)>t$.

As a result, $\alpha \left(c\right)$, $c+d$, $[c,d]$, and $\gamma .c$ are elements of $U({\mu }^{>},t)$.

Conversely, suppose that for each t in the image of $\mu$, $U({\mu }^{>},t)$ is a Hom–Lie subalgebra of L. Now, consider $c,d\in L$ and $\gamma \in F$. We need to show that the conditions from Definition 1 are satisfied. If $\mu \left(c\right)=0$ or $\mu \left(d\right)=0$, then $\mu (c+d)\ge 0=\mu \left(c\right)\wedge \mu \left(d\right)$.

Assume that $\mu \left(c\right)\ne 0$ and $\mu \left(d\right)\ne 0$. Furthermore, assume that $\mu (c+d)$ and $\mu \left(\right[c,d\left]\right)$ are both less than $\mu \left(c\right)\wedge \mu \left(d\right)$. Let $t_0$ be the greatest lower bound of the set $\{t\mid t<\mu (c)\wedge \mu (d\left)\right\}$. Since $c,d\in U({\mu }^{>},t_0)$, we have $c+d,[c,d]\in U({\mu }^{>},t_0)$, and hence $\mu (c+d),\mu \left([c,d]\right)>t_0$. This contradicts the assumption that there is no element $x\in L$ such that $t_0<\mu \left(x\right)<\mu \left(c\right)\wedge \mu \left(d\right)$. Therefore, we conclude that $\mu (c+d),\mu \left(\right[c,d\left]\right)\ge \mu \left(c\right)\wedge \mu \left(d\right)$. Yet again, suppose $t_0$ is the largest number of $[0,1]$ so as $t_0<\mu \left(c\right)$ and we cannot find an element $x\in L$ with $t_0<\mu \left(x\right)<\mu \left(c\right)$. Since $U({\mu }^{>},t_0)$ is a Hom–Lie subalgebra, we have $\gamma$·$c,\alpha \left(c\right)$ in $U({\mu }^{>},t_0)$, and so $\mu (\gamma .c)>t_0$ and $\mu \left(\alpha \left(c\right)\right)>t_0$. Therefore, $\mu (\gamma$·$c)$ and $\mu \left(\alpha \right(c\left)\right)$ are $\ge \mu \left(a\right)$. □

Furthermore, by using nearly the same argument, the given results can be shown.

Theorem 4. 

The following statements are equivalent for a fuzzy subset μ of L:

(i) 

μ is a fuzzy Hom–Lie ideal of L;

(ii) 

All strong-upper-levels $U({\mu }^{>},t)$ are Hom–Lie ideals of L for every t in the image of μ.

3.3. Direct Sum of Fuzzy Hom–Lie Subalgebras

Recalling the case of n Hom–Lie algebras $(L_i,{[,]}_i,{\alpha }_i)$ for $i=1,\dots ,n$, we have the Hom–Lie algebra structure defined as

$$(L_1\oplus L_2\oplus \dots \oplus L_n,[,],{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n)$$

where the binaryperation $[,]$ is given by

$$[,]:(L_1\oplus L_2\oplus \dots \oplus L_n)\times (L_1\oplus L_2\oplus \dots \oplus L_n)\to (L_1\oplus L_2\oplus \dots \oplus L_n)$$

$$((x_1,x_2,\dots ,x_n),(y_1,y_2,\dots ,y_n))\mapsto ({[x_1,y_2]}_1,{[x_1,y_2]}_2,\dots ,{[x_n,y_n]}_n),$$

and the linear map ${\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n$ is defined as

$$({\alpha }_1+{\alpha }_2+\dots +{\alpha }_n):(L_1\oplus L_2\oplus \dots \oplus L_n)\to (L_1\oplus L_2\oplus \dots \oplus L_n)$$

$$(x_1,x_2,\dots ,x_n)\mapsto ({\alpha }_1\left(x_1\right),{\alpha }_2\left(x_2\right),\dots ,{\alpha }_n\left(x_n\right)).$$

In the special case where $n=2$, we obtain ([15], [Proposition 2.2]) (see [20]).

Let $(L_1,{[,]}_1,{\alpha }_1),(L_2,{[,]}_2,{\alpha }_2),\dots ,(L_n,{[,]}_n,{\alpha }_n)$ be Hom–Lie algebras. Assume that ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ are fuzzy subsets of $L_1,L_2,\dots ,L_n$, respectively. Then, the generalized Cartesian sum of fuzzy sets induced by ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ on $L_1\oplus L_2\oplus \cdots \oplus L_n$ is

$${\mu }_1\oplus {\mu }_2\oplus \cdots \oplus {\mu }_n:L_1\oplus L_2\cdots \oplus L_2\to [0,1];(x_1,x_2,\dots ,x_n)\mapsto {\mu }_1\left(x_1\right)\wedge {\mu }_2\left(x_2\right)\wedge {\mu }_n\left(x_n\right).$$

Theorem 5. 

Let $(L_1,{[,]}_1,{\alpha }_1),(L_2,{[,]}_2,{\alpha }_2),\dots ,(L_n,{[,]}_n,{\alpha }_n)$ be Hom–Lie algebras. Let ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ be fuzzy Hom–Lie subalgebras of $L_1,L_2,\dots ,L_n$, respectively. Then, ${\mu }_1\oplus {\mu }_2\oplus \cdots \oplus {\mu }_n$ is a fuzzy Hom–Lie subalgebra of $L_1\oplus L_2\oplus \cdots \oplus L_n$.

Proof. 

Let $(x_1,x_2,\dots ,x_n),(y_1,y_2,\dots ,y_n)\in L_1\oplus L_2\oplus \cdots \oplus L_n$. Then

$$\begin{array}{ccc}\hfill {\mu }_1\left([(x_1,\dots ,x_n),(y_1,\dots ,y_n)]\right)& =& ({\mu }_1\oplus {\mu }_2\oplus \cdots \oplus {\mu }_n)({[x_1,y_1]}_1,{[x_2,y_2]}_2,\dots ,{[x_n,y_n]}_n)\hfill \\ & =& {\mu }_1\left({[x_1,y_1]}_1\right)\wedge {\mu }_2\left({[x_2,y_2]}_2\right)\wedge \cdots \wedge {\mu }_n\left({[x_n,y_n]}_n\right)\hfill \\ & \ge & ({\mu }_1\left(x_1\right)\wedge {\mu }_1\left(y_1\right)\wedge {\mu }_2\left(x_2\right)\wedge {\mu }_2\left(y_2\right)\cdots \wedge {\mu }_n\left(x_n\right)\wedge {\mu }_n\left(y_n\right)\hfill \\ & =& ({\mu }_1\oplus \cdots \oplus {\mu }_n)\left((x_1,\dots ,x_2)\right)\wedge ({\mu }_1\oplus \cdots \oplus {\mu }_n)\left((y_1,\dots ,y_n)\right).\hfill \end{array}$$

Additionally,

$$\begin{array}{ccc}\hfill ({\mu }_1\oplus \cdots \oplus {\mu }_n)({\alpha }_1+\cdots +{\alpha }_n)(x_1,\dots ,x_n)& =& ({\mu }_1\oplus \cdots \oplus {\mu }_n)({\alpha }_1\left(x_1\right),{\alpha }_2\left(x_2\right),\dots ,{\alpha }_n\left(x_n\right))\hfill \\ & =& {\mu }_1\left({\alpha }_1\left(x_1\right)\right)\wedge {\mu }_2\left({\alpha }_2\left(x_2\right)\right)\wedge \cdots \wedge {\mu }_n\left({\alpha }_n\left(x_n\right)\right)\hfill \\ & \ge & {\mu }_1\left(x_1\right)\wedge {\mu }_2\left(x_2\right)\wedge \cdots \wedge {\mu }_n\left(x_n\right)\hfill \\ & =& ({\mu }_1\oplus {\mu }_2\oplus \cdots {\mu }_n)\ast x_1,x_2,\dots ,x_n).\hfill \end{array}$$

The remaining proof is identical to ([16], [Theorem 5.2]); hence, it has been omitted. □

It has been noted that the direct sum of the fuzzy Hom–Lie ideals of Hom–Lie algebras $L_1$ and $L_2$ does not have to be a fuzzy Hom–Lie ideal of the Hom–Lie algebra $L_1\oplus L_2$. In our prior work [46], the infinite direct product of Hom–Lie algebras has been introduced and examined. Thus, the fuzzy Hom–Lie subalgebras of such Hom–Lie algebras can be considered.

3.4. On Fuzzy Hom–Lie Algebras and Hom–Lie Algebra Morphisms

Let $f:X\to Y$ be a function. If ${\mu }_B$ is a fuzzy set defined on Y, a corresponding fuzzy set can be defined on X induced by f and ${\mu }_A$ as follows:

$${\mu }_{f^{-1}\left(B\right)}\left(x\right)={\mu }_B\left(f\left(x\right)\right)$$

for any $x\in X$.

Similarly, if ${\mu }_A$ is a fuzzy set on X, the fuzzy set ${\mu }_{f\left(A\right)}\left(y\right)$ on Y induced by f and ${\mu }_A$ can be defined as:

$${\mu }_{f\left(A\right)}\left(y\right)=\left\{\begin{array}{cc}{\sup }_{x\in f^{-1}\left(y\right)}\left\{{\mu }_A\left(x\right)\right\}\hfill & \hfill :y\in f\left(X\right)\\ 0\hfill & \hfill :y\notin f\left(X\right)\end{array}\right.$$

is a fuzzy set on Y induced by f and ${\mu }_A$ (as an instance, refer to [42]. In the setting of Lie algebras, the subsequent theorem was obtained by Kim and Lee in [47]. Thus, in the given setting, it is being extended for the case of Hom–Lie algebras.

Theorem 6. 

Let $f:(L_1,{[,]}_1,{\alpha }_1)\to (L_2,{[,]}_2,{\alpha }_1)$ be a morphism of Hom–Lie algebras. If $B={\mu }_B$ is a fuzzy Hom–Lie subalgebra (and ideal, respectively) of $L_2$, then the fuzzy set $f^{-1}\left(B\right)$ is also a fuzzy Hom–Lie-subalgebra (and ideal, respectively) of $L_1$.

Proof. 

Let $a_1,a_2\in L_1$. Then

$$\begin{array}{ccc}\hfill {\mu }_{f^{-1}\left(B\right)}(a_1+a_2)& =& {\mu }_B\left(f(a_1+a_2)\right)\hfill \\ & =& {\mu }_B(f\left(a_1\right)+f\left(a_2\right))\left(f\mathrm{is}\mathrm{linear}\right)\hfill \\ & \ge & {\mu }_B\left(f\left(a_1\right)\right)\wedge {\mu }_B\left(f\left(a_2\right)\right)({\mu }_B\mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{Hom}--\mathrm{Lie}-\mathrm{subalgebra})\hfill \\ & =& {\mu }_{f^{-1}\left(B\right)}\left(a_1\right)\wedge {\mu }_{f^{-1}\left(B\right)}\left(a_2\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mu }_{f^{-1}\left(B\right)}\left([a_1,a_2]\right)& =& {\mu }_B(f\left([a_1,a_2]\right)\hfill \\ & =& {\mu }_B\left([f\left(a_1\right),f\left(a_2\right)]\right)\left(f\mathrm{is}\mathrm{morphism}\right)\hfill \\ & \ge & {\mu }_B\left(f\left(a_1\right)\right)\wedge {\mu }_B\left(f\left(a_2\right)\right)({\mu }_B\mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{Hom}--\mathrm{Lie}-\mathrm{subalgebra})\hfill \\ & =& {\mu }_{f^{-1}\left(B\right)}\left(a_1\right)\wedge {\mu }_{f^{-1}\left(B\right)}\left(a_2\right).\hfill \end{array}$$

Let $c\in L_1$ and $\gamma \in F$. Then

$$\begin{array}{ccc}\hfill {\mu }_{f^{-1}\left(B\right)}(\gamma .c)& =& {\mu }_B\left(f(\gamma .c)\right)\hfill \\ & =& {\mu }_B(\gamma .f\left(c\right))\left(f\mathrm{is}\mathrm{linear}\right)\hfill \\ & \ge & {\mu }_B\left(f\left(c\right)\right)({\mu }_B\mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{Hom}--\mathrm{Lie}-\mathrm{subalgebra})\hfill \\ & =& {\mu }_{f^{-1}\left(B\right)}\left(c\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mu }_{f^{-1}\left(B\right)}\left({\alpha }_1\left(c\right)\right)& =& {\mu }_B\left(f\left({\alpha }_1\left(c\right)\right)\right)\hfill \\ & =& {\mu }_B\left({\alpha }_2\left(f\left(c\right)\right)\right)(f\mathrm{is}\mathrm{a}\mathrm{morphism}\mathrm{of}\mathrm{Hom}--\mathrm{Lie}\mathrm{algebras})\hfill \\ & \ge & {\mu }_B\left(f\left(c\right)\right)({\mu }_B\mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{Hom}--\mathrm{Lie}-\mathrm{subalgebra})\hfill \\ & =& {\mu }_{f^{-1}\left(B\right)}\left(c\right),\hfill \end{array}$$

Similarly, the case of the fuzzy Hom–Lie ideal is also identical to the present. □

Suppose that $f:L_1\to L_2$ is a Lie algebra homomorphism and $A={\mu }_A$ is a fuzzy-subalgebra of $L_1$. Now, the image of A, $f\left(A\right)$ is a fuzzy-subalgebra of $f\left(L_1\right)$ [47]. The given theorem offers the case where an analogue result for the case of Hom–Lie algebras has been established.

Theorem 7. 

Let $f:(L_1,{[,]}_1,{\alpha }_1)\to (L_2,{[,]}_2,{\alpha }_2)$ be a morphism from $L_1$ onto $L_2$. Provided that $A={\mu }_A$ is a fuzzy Hom–Lie subalgebra of $L_1$, the fuzzy Hom–Lie subalgebra of $L_2$ is $f\left(A\right)$.

Proof. 

Let $b_1,b_2\in L_2$. As f is onto, there are $a_1,a_2\in L_1$ such that $f\left(a_1\right)=b_1$ and $f\left(a_2\right)=b_2$. Then, we have

$$\{a_1+a_2|a_1\in f^{-1}\left(b_1\right)\mathrm{and}{a}_2\in f^{-1}\left(b_2\right)\}\subseteq \left\{x\right|x\in f^{-1}(b_1+b_2)\},$$

and

$$\left\{{[a_1,a_2]}_1\right|a_1\in f^{-1}\left(b_1\right)\mathrm{and}{a}_2\in f^{-1}\left(b_2\right)\}\subseteq \left\{x\right|x\in f^{-1}\left({[b_1,b_2]}_2\right)\}.$$

Now, it is found that

$$\begin{array}{ccc}\hfill {\mu }_{f\left(A\right)}(b_1+b_2)& =& {\sup }_{x\in f^{-1}(b_1+b_2)}\left\{{\mu }_A\left(x\right)\right\}\hfill \\ & \ge & \left\{{\mu }_A(a_1+a_2)\right|a_1\in f^{-1}\left(b_1\right)\mathrm{and}{a}_2\in f^{-1}\left(b_2\right)\}\hfill \\ & \ge & \sup \{{\mu }_A\left(a_1\right)\wedge {\mu }_A\left(a_2\right)|a_1\in f^{-1}\left(b_1\right)\mathrm{and}{a}_2\in f^{-1}\left(b_2\right)\}\hfill \\ & =& {\sup }_{a_1\in f^{-1}\left(b_1\right)}\left\{{\mu }_A\left(a_1\right)\right\}\wedge {\sup }_{a_2\in f^{-1}\left(b_2\right)}\left\{{\mu }_A\left(a_2\right)\right\}\hfill \\ & =& {\mu }_{f\left(A\right)}\left(b_1\right)\wedge {\mu }_{f\left(A\right)}\left(b_2\right).\hfill \end{array}$$

Additionally,

$$\begin{array}{ccc}\hfill {\mu }_{f\left(A\right)}\left({[b_1,b_2]}_2\right)& =& {\sup }_{x\in f^{-1}\left({[b_1,b_2]}_2\right)}\left\{{\mu }_A\left(x\right)\right\}\hfill \\ & \ge & \left\{{\mu }_A\left({[a_1,a_2]}_1\right)\right|a_1\in f^{-1}\left(y_1\right)\mathrm{and}{a}_2\in f^{-1}\left(y_2\right)\}\hfill \\ & \ge & \sup \{{\mu }_A\left(a_1\right)\wedge {\mu }_A\left(a_2\right)|a_1\in f^{-1}\left(b_1\right)\mathrm{and}{a}_2\in f^{-1}\left(b_2\right)\}\hfill \\ & =& {\sup }_{a_1\in f^{-1}\left(b_1\right)}\left\{{\mu }_A\left(a_1\right)\right\}\wedge {\sup }_{a_2\in f^{-1}\left(b_2\right)}\left\{{\mu }_A\left(a_2\right)\right\}\hfill \\ & =& {\mu }_{f\left(A\right)}\left(b_1\right)\wedge {\mu }_{f\left(A\right)}\left(b_2\right).\hfill \end{array}$$

For $b\in L_2$ and $\gamma \in F$, it is found

$$\{\gamma ·a|a\in f^{-1}\left(b\right)\}\subseteq \left\{a\right|a\in f^{-1}(\gamma ·b)\},$$

and

$$\left\{{\alpha }_1\left(a\right)\right|a\in f^{-1}\left(b\right)\}\subseteq \left\{a\right|a\in f^{-1}\left({\alpha }_2\left(b\right)\right)\}.$$

and so

$$\begin{array}{ccc}\hfill {\mu }_{f\left(A\right)}(\gamma ·b)& =& {\sup }_{a\in f^{-1}(\gamma ·b)}\left\{{\mu }_A\left(a\right)\right\}\hfill \\ & \ge & \left\{{\mu }_A(\gamma ·a)\right|a\in f^{-1}(\gamma ·b)\}\hfill \\ & \ge & \left\{{\mu }_A\left(a\right)\right|a\in f^{-1}\left(b\right)\}\hfill \\ & =& {\mu }_{f\left(A\right)}\left(b\right),\hfill \end{array}$$

furthermore,

$$\begin{array}{ccc}\hfill {\mu }_{f\left(A\right)}\left({\alpha }_2\left(b\right)\right)& =& {\sup }_{a\in f^{-1}\left({\alpha }_2\left(b\right)\right)}\left\{{\mu }_A\left(a\right)\right\}\hfill \\ & \ge & \left\{{\mu }_A\left({\alpha }_1\left(a\right)\right)\right|a\in f^{-1}\left({\alpha }_2\left(b\right)\right)\}\hfill \\ & \ge & \left\{{\mu }_A\left(a\right)\right|a\in f^{-1}\left(b\right)\}\hfill \\ & =& {\mu }_{f\left(A\right)}\left(b\right).\hfill \end{array}$$

□

In their work [47], Kim and Lee demonstrated that, if $\phi :L\to L^{\prime }$ is a surjective Lie algebra homomorphism and $A={\mu }_A$ is a fuzzy ideal of L, then $\phi \left(A\right)$ is a fuzzy ideal of $L^{\prime }$. Building upon this concept, we can extend these results to the case of fuzzy Hom–Lie algebras.

Theorem 8. 

Consider the onto morphism $f:(L_1,{[,]}_1,{\alpha }_1)\to (L_2,{[,]}_2,{\alpha }_2)$ between Hom–Lie algebras. Assuming that $A={\mu }_A$ is a fuzzy Hom–Lie ideal of $L_1$, it follows that $f\left(A\right)$ is likewise a fuzzy Hom–Lie ideal of $L_2$.

Proof. 

The proof for this case is identical to that of the previous theorem. Therefore, we only need to establish the inequality ${\mu }_{f\left(A\right)}\left([y_1,y_2]2\right)\ge \mu f\left(A\right)\left(y_1\right)\vee {\mu }_{f\left(A\right)}\left(y_2\right)$ for every $y_1,y_2\in L_2$.

Let $y_1,y_2\in L_2$. Suppose, by contradiction, that ${\mu }_{f\left(A\right)}\left([y_1,y_2]2\right)<\mu f\left(A\right)\left(y_1\right)\vee {\mu }_{f\left(A\right)}\left(y_2\right)$. This implies that ${\mu }_{f\left(A\right)}\left([y_1,y_2]2\right)<\mu f\left(A\right)\left(y_1\right)$ or ${\mu }_{f\left(A\right)}\left([y_1,y_2]2\right)<\mu f\left(A\right)\left(y_2\right)$. Without loss of generality, let us assume that ${\mu }_{f\left(A\right)}\left([y_1,y_2]2\right)<\mu f\left(A\right)\left(y_1\right)$.

Choose any number $t\in [0,1]$ such that ${\mu }_{f\left(A\right)}\left([y_1,y_2]2\right)<t<\mu f\left(A\right)\left(y_2\right)$. There exists $a\in f^{-1}\left(y_1\right)$ such that ${\mu }_A\left(a\right)>t$. Since f is onto, there exists $b\in f^{-1}\left(y_2\right)$. It is worth noting that:

$$f\left({[a,b]}_1\right)={[f\left(a\right),f\left(b\right)]}_2={[y_1,y_2]}_2.$$

Hence, we find,

$${\mu }_{f\left(A\right)}\left({[y_1,y_2]}_2\right)\ge {\mu }_A\left({[a,b]}_1\right)\ge {\mu }_A\left(a\right)\vee {\mu }_A\left(b\right)>t>{\mu }_{f\left(A\right)}\left({[y_1,y_1]}_2\right).$$

This leads to a contradiction. □

4. Conclusions and Further Recommendations

In our previous work [46], we verified that the isomorphism theorems for Lie algebras are also satisfied for Hom–Lie algebras. In my other work [48], an analog of the isomorphism theorems has been established from $\Gamma$-rings out of complex fuzzy $\Gamma$-rings. Thus, as a future research recommendation, it has been proposed that attempts should be made for the purpose of also attaining identical results for Hom–Lie algebras. The author aims and welcomes other researchers in this field to study for the same results in the case of Hom algebras, Hom co-algebras, Hom bi-algebras, Hom–Hopf algebras, and Hom (n-)(color) Lie (super-)algebras.

The concept of intuitionistic fuzzy sets was first introduced by Atanassov in 1986 as a generalization of classical fuzzy sets [49]. Atanassov’s idea was to allow for the possibility that an element might not only be partially but also not belong to the set at all. This concept gained widespread attention in the field of mathematics, leading to the development of intuitionistic fuzzy set theory. Complex intuitionistic fuzzy sets (CIFSs) are an extension of intuitionistic fuzzy sets that incorporate complex numbers to represent uncertainty and ambiguity in a more expressive way. CIFS were introduced by Alkouri and Salleh in 2012 [50]. The author aims and welcomes other researchers to study for the same results in the case of (complex) (intuitionistic)-fuzzy Hom–Lie algebras.

Here are some other potential research questions you can explore:

• How can fuzzy Hom–Lie ideals be effectively characterized and classified within different classes of Hom–Lie algebras?
• Can fuzzy Hom–Lie ideals be extended to fuzzy Hom associative algebras or other related algebraic structures? What are the potential applications and significance of such extensions?
• What are the properties and structures of fuzzy Hom–Lie ideals compared to classical (non-fuzzy) Hom–Lie ideals?
• How can fuzzy Hom–Lie ideals be utilized in the study of fuzzy Hom–Lie algebras, and what implications do they have for the theory of fuzzy algebras?
• How can the concept of fuzzy Hom–Lie ideals be employed to investigate fuzzy representations and module theory within the framework of Hom–Lie algebras?
• How can the theory of fuzzy Hom–Lie ideals be extended to more general frameworks, such as fuzzy Hom–Lie (color) superalgebras or fuzzy Hom–Lie bi-algebras?
• Are there any connections or relationships between fuzzy Hom–Lie ideals and other areas of mathematics, such as fuzzy logic, fuzzy set theory, or fuzzy topology?