Hybrid near Algebra

Abstract

The objective of this paper is to study the hybrid near algebra. It has been summarized with the proper definitions and theorems of hybrid near algebra, hybrid near algebra homomorphism and direct product of hybrid near algebra. It has been proved that a homomorphic image of a hybrid near algebra is a hybrid near algebra. It also investigated the intersection of two hybrid near algebras is a hybrid near algebra.

1. Introduction

An algebraic system with two binary operations, that satisfies all of a ring axioms, with the possible exception of one distributive law, is recognized as a near ring. The notion of a near ring was promoted and popularized by G. Pilz [1]. A near ring that admits the field as a right operator domain is referred to as a near algebra. Near algebra was familiarized by H. Brown [2]. P. Narasimha Swamy [3,4] and Rakshitha Deshmukh were developed the concepts in near algebra. The operators form only a near algebra in a quantum mechanical formalism of P. Jordan. In view of this, the study of near algebra is interesting not only as an axiomatic question but also for physical reasons. A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function assigning a grade of membership to each object ranging between 0 and 1. The concept of fuzzy set was first established by Zadeh [5] in 1965. S. Nanda [6] and R. Biswas [7] were studied fuzzy field and fuzzy linear space. A hesitant fuzzy set is an excellent tool for expressing people’s hesitancy in daily life and for dealing with uncertainty, which can be precisely and perfectly described in terms of decision makers’ opinions. An extensive range of existing theories, including the theory of probability, fuzzy sets, vague sets, interval mathematics, rough sets, etc., are used to address a variety of problems in many different domains that require data with uncertainties. The concept of hesitant fuzzy sets was established by V. Torra [8]. All of these theories have their difficulties which are pointed out in soft set theory-first results [9]. To overcome these difficulties, Molodtosov [9] familiarized the soft set theory as a new mathematical tool for dealing with uncertainties that are free from problems. He successfully applied the soft set theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, theory of measurement, etc.

As a parallel circuit of fuzzy sets, soft sets and hesitant fuzzy sets advocate the notion of hybrid structures in a group of parameters over an initial universe set illustrating several properties according to Jun, Song and Muhiuddin [10]. Using the idea, they proposed the concept of a hybrid subalgebra, a hybrid field and a hybrid linear space. B. Elavarasan [11] has studied hybrid structures applied to ideals in near-rings. Saima Anis [12] has investigated hybrid ideals in semigroups. M. Himaya Jaleela Begum [13] were investigated has studied hybrid fuzzy bi-ideals in near-rings. Saima Anis [12] has investigated hybrid ideals in semigroups. M. Himaya Jaleela Begum [13] were investigated has studied hybrid fuzzy bi-ideals in near-rings. Srinivas et al. [14] introduced and studied the notion of fuzzy near algebra over a fuzzy filed.

In the present study, we introduce the notion of hybrid near algebra over a hybrid field and hybrid structure is used to analyze the structural statements of near algebras.

Throughout this paper, $Y$ means a (right) near algebra over a field L.

2. Preliminaries

Definition 1 ([10]). 

Let $U$ be a universal set, $P\left(U\right)$ be the power set, $L$ be the set of parameters, and $I$ be the unit interval. A mapping ${\tilde{\zeta }}_{\lambda }≔\left(\widetilde{\zeta },\lambda \right):L\to P\left(U\right)X I$, $z\to \left(\tilde{\zeta }\left(z\right),\lambda \left(z\right)\right),$ i.e., the image of $z$ is designated by $\left(\tilde{\zeta }\left(z\right),\lambda \left(z\right)\right)$ is entitled a hybrid structure in $L$ over $U,$ where $\tilde{\zeta }:L \to P\left(U\right)$ and $\lambda :L\to I$ are the mappings.

Definition 2 ([10]). 

Let ${\tilde{\zeta }}_{\lambda }$ be a hybrid structure in $L$ over $U$. Then, the sets

$$\begin{array}{l}\hspace{1em}\hspace{1em} {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right]=\left\{z\in L/\tilde{\zeta }\left(d\right)\supseteq \alpha , \lambda \left(d\right)\le t\right\};\\ \hspace{1em}\hspace{1em} {\tilde{\zeta }}_{\lambda }\left(\alpha ,t\right]=\left\{z\in L/\tilde{\zeta }\left(d\right)⊋\alpha , \lambda \left(d\right)\le t\right\};\\ \hspace{1em}\hspace{1em} {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right)=\left\{z\in L/\tilde{\zeta }\left(d\right)\supseteq \alpha , \lambda \left(d\right)<t\right\};\\ {\tilde{\zeta }}_{\lambda }\left(\alpha ,t\right)=\left\{z\in L/\tilde{f}\left(d\right)⊋\alpha , \lambda \left(d\right)<t\right\};\end{array}$$

are entitled the $\left[\alpha ,t\right]$ hybrid cut, $\left(\alpha ,t\right]$ hybrid cut, $\left[\alpha ,t\right)$ hybrid cut, and $\left(\alpha ,t\right)$ hybrid cut of ${\tilde{\zeta }}_{\lambda }$, respectively, where $\alpha \in P\left(U\right)$ and $t\in I.$ Obviously,

$${\tilde{\zeta }}_{\lambda }\left(\alpha ,t\right)\subseteq {\tilde{\zeta }}_{\lambda }\left(\alpha ,t\right]\subseteq {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right] and {\tilde{\zeta }}_{\lambda }\left(\alpha ,t\right)\subseteq {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right)\subseteq {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right].$$

Definition 3 ([10]). 

Let L be a field. A hybrid structure ${\tilde{\zeta }}_{\lambda }$ in $L$ over $U$ is entitled a hybrid field of $L$ over $U$ if

(i)

$\tilde{\zeta }\left(d+a\right)\supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\zeta }\left(a\right), \lambda \left(d+a\right)\le \bigvee \left\{\lambda \left(d\right),\lambda \left(a\right)\right\}, \forall d,a\in L$;

(ii)

$\tilde{\zeta }\left(-d\right)\supseteq \tilde{\zeta }\left(d\right), \lambda \left(-d\right)\le \lambda \left(d\right), \forall d\in L$;

(iii)

$\tilde{\zeta }\left(da\right)\supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\zeta }\left(a\right), \lambda \left(da\right)\le \bigvee \left\{\lambda \left(d\right),\lambda \left(a\right)\right\}, \forall d,a\in L$;

(iv)

If $d\ne 0\Rightarrow \tilde{\zeta }\left(d^{-1}\right)\supseteq \tilde{\zeta }\left(d\right), \lambda \left(d^{-1}\right)\le \lambda \left(d\right), \forall d\in L$.

Definition 4. 

Let ${\tilde{\zeta }}_{\lambda }$ be an H F of a field $L$ over $U,Z$ be an algebra over L. A hybrid structure ${\tilde{k}}_{\delta }$ in $Z$ over $U$ is entitled an H A over the H F $\left({\tilde{\zeta }}_{\lambda },L\right)$ if the following conditions hold:

(i)

$\tilde{k}\left(z+w\right)\supseteq \tilde{k}\left(z\right){\displaystyle \cap }\tilde{k}\left(w\right), \delta \left(z+w\right)\le \bigvee \left\{\delta \left(z\right),\delta \left(w\right)\right\}, \forall z,w\in Z$;

(ii)

$\tilde{k}\left(dz\right)\supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{k}\left(z\right), \delta \left(dz\right)\le \bigvee \left\{\lambda \left(d\right),\delta \left(z\right)\right\}, \forall z\in Z,d\in L;$

(iii)

$\tilde{k}\left(zw\right)\supseteq \tilde{k}\left(z\right){\displaystyle \cap }\tilde{k}\left(w\right), \delta \left(zw\right)\le \bigvee \left\{\delta \left(z\right),\delta \left(w\right)\right\}, \forall z,w\in Z;$

(iv)

$\tilde{\zeta }\left(1\right)\supseteq \tilde{k}\left(z\right), \lambda \left(1\right)\le \delta \left(z\right),\forall z\in Z.$

Example 1. 

Let $L={\left\{0,1\right\}}_{{⨁}_2, {⨁}_2}$ be a field. Then, the hybrid structure ${\tilde{\zeta }}_{\lambda }$ in $L$ over $U=\left\{u_1,u_2,u_3,u_4,u_5\right\},$ which is given as follows:

$\mathit{L}$	$\tilde{\mathsf{\zeta }}$	$\mathsf{\lambda }$
0	$\left\{u_1,u_2,u_3,u_4\right\}$	0.3
1	$\left\{u_2,u_3,u_4\right\}$	0.4

Therefore, ${\tilde{\zeta }}_{\lambda }$ is an H F in $L$ over $U$.

Let $Z=\left\{0,1,2,3,4\right\}$ be a set with two binary operations “${⨁}_5$”, “${⨁}_5$” by

${\mathsf{⨁}}_5$	0	1	2	3	4	${\mathsf{⨁}}_5$	0	1	2	3	4
0	0	1	2	3	4	0	0	0	0	0	0
1	1	2	3	4	0	1	0	1	2	3	4
2	2	3	4	0	1	2	0	2	4	1	3
3	3	4	0	1	2	3	0	3	1	4	2
4	4	0	1	2	3	4	0	4	3	2	1

Clearly, $Z$ forms an algebra over $L$. Then, the hybrid structure ${\tilde{k}}_{\delta }$ in $Z$ over $U=\left\{u_1,u_2,u_3,u_4,u_5\right\}$ is given by:

$\mathit{Z}$	$\tilde{\mathit{k}}$	$\mathsf{\delta }$
0	$\left\{u_2,u_3,u_5\right\}$	0.6
1	$\left\{u_3,u_4\right\}$	0.7
2	$\left\{u_3,u_5\right\}$	0.8
3	$\left\{u_1,u_3\right\}$	0.9
4	$\left\{u_2,u_3\right\}$	0.9

Therefore, $\left({\tilde{k}}_{\delta }, \mathrm{Z}\right)$ is an H A over $\left({\tilde{\zeta }}_{\lambda },L\right)$.

3. Hybrid near Algebra

In this section, we familiarize hybrid near algebra (H N A) and acquire some of the properties of hybrid near algebra over a hybrid field (H F).

Definition 5. 

Let ${\tilde{\zeta }}_{\lambda }$ be an H F of a field $L$ over $U,$ and let $Y$ be a near algebra over L. A hybrid structure ${\tilde{\varrho }}_{\gamma }$ in $Y$ over $U$ is entitled an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right)$ if the following conditions hold:

(i)

$\tilde{\varrho }\left(z+w\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right), \gamma \left(z+w\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}, \forall z,w\in Y$;

(ii)

$\tilde{\varrho }\left(dz\right)\supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right), \gamma \left(dz\right)\le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right)\right\}, \forall z\in Y, d\in L$;

(iii)

$\tilde{\varrho }\left(zw\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right), \gamma \left(zw\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}, \forall z, w\in Y$;

(iv)

$\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(1\right)\le \gamma \left(z\right),\forall z\in Y.$

Example 2. 

Let $L=Z_2={\left\{0,1\right\}}_{{⨁}_2, {⨁}_2}$ be a field. Then, the hybrid structure ${\tilde{\zeta }}_{\lambda }$ in $L$ over $U=\left\{u_1,u_2,u_3,u_4,u_5\right\}$ which is given below forms an H F in $L$ over $U$:

$\mathit{L}$	$\tilde{\mathsf{\zeta }}$	$\mathsf{\lambda }$
0	$\left\{u_1,u_2,u_3,u_4\right\}$	0.3
1	$\left\{u_2,u_3,u_4\right\}$	0.4

Let $Y=\left\{0,h,p,e\right\}$ be a set with two binary operations “+” and “.” By

+	0	h	p	e	.	0	h	p	e
0	0	h	p	e	0	0	0	0	0
h	h	0	e	p	h	h	h	h	h
p	p	e	0	h	p	p	p	p	p
e	e	p	h	0	e	e	e	e	e

Clearly, $Y$ forms a near algebra over $L$.

Then, the hybrid structure ${\tilde{\varrho }}_{\gamma }$ in $Y$ over $U=\left\{u_1,u_2,u_3,u_4,u_5\right\}$, which is given as follows:

$\mathit{Y}$	$\tilde{\mathsf{\varrho }}$	$\mathsf{\gamma }$
0	$\left\{u_2,u_3,u_4\right\}$	0.4
h	$\left\{u_2,u_4\right\}$	0.6
p	$\left\{u_2,u_3\right\}$	0.8
e	$\left\{u_2\right\}$	0.8

Clearly, $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right).$

Theorem 1. 

If $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right),$ then $\tilde{\zeta }\left(0\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(0\right)\le \gamma \left(z\right),\forall z\in Y$.

Proof. 

By the definition, $\tilde{\zeta }\left(0\right)\supseteq \tilde{\zeta }\left(1\right), \lambda \left(0\right)\le \lambda \left(1\right)$ and $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(1\right)\le \gamma \left(z\right),\forall z\in Y.$ Hence, $\tilde{\zeta }\left(0\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(0\right)\le \gamma \left(z\right),\forall z\in Y$. □

Theorem 2. 

If $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right),$ then $\tilde{\varrho }\left(0\right)\supseteq \tilde{\varrho }\left(z\right), \gamma \left(0\right)\le \gamma \left(z\right),\forall z\in Y$.

Proof. 

Let $z\in Y$. Then,

$$\begin{array}{ll}\tilde{\varrho }\left(0\right)& =\tilde{\varrho }\left(z-z\right)=\tilde{\varrho }\left(1.z-1.z\right)=\tilde{\varrho }\left(1.z+\left(-1\right).z\right);\\ & \supseteq \tilde{\varrho }\left(1.z\right){\displaystyle \cap }\tilde{\varrho }\left(\left(-1\right)z\right);\\ & \supseteq \left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(-1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right);\\ & \supseteq \left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right);\\ & \supseteq \left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right);\\ & \supseteq \left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right)=\tilde{\varrho }\left(z\right);\end{array}$$

and

$$\begin{array}{ll}\gamma \left(0\right)& =\gamma \left(z-z\right)=\gamma \left(1.z-1.z\right)=\gamma \left(1.z+\left(-1\right)z\right)\le \bigvee \left\{\gamma \left(1.z\right),\gamma \left(\left(-1\right).z\right)\right\};\\ & \le \bigvee \left\{\bigvee \left\{\lambda \left(1\right),\gamma \left(z\right)\right\},\bigvee \left\{\lambda (-1),\gamma \left(z\right)\right\}\right\};\\ & \le \bigvee \left\{\bigvee \left\{\lambda \left(1\right),\gamma \left(z\right)\right\},\bigvee \left\{\lambda \left(1\right),\gamma \left(z\right)\right\}\right\};\\ & \le \bigvee \left\{\bigvee \left\{\gamma \left(\mathrm{z}\right),\gamma \left(z\right)\right\},\bigvee \left\{\gamma \left(z\right),\gamma \left(z\right)\right\}\right\};\\ & \le \bigvee \left\{\gamma \left(z\right),\gamma \left(z\right)\right\}=\gamma \left(z\right).\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\square \end{array}$$

Theorem 3. 

Let ${\tilde{\zeta }}_{\lambda }$ be an H F of a field $L$ over $U$ and $Y$ be a near algebra over $L.$ A hybrid structure  ${\tilde{\varrho }}_{\gamma }$ in $Y$ over $U$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right)$ if and only if

(i)

$\widetilde{ \varrho }\left(dz+aw\right)\supseteq \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right)$

$\gamma \left(dz+aw\right)\le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\},\forall z,w\in Y;d,a\in L$;

(ii)

$\tilde{\varrho }\left(zw\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right), \gamma \left(zw\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\},\forall z,w\in Y$;

(iii)

$\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(1\right)\le \gamma \left(z\right),\forall z\in Y.$

Proof. 

Let $z,w\in Y \mathrm{and} d,a\in L$. Then,

$$\begin{array}{ll}\tilde{\varrho }\left(dz+aw\right)& \supseteq \tilde{\varrho }\left(dz\right){\displaystyle \cap }\tilde{\varrho }\left(aw\right)\supseteq \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right) \mathrm{and}\\ \gamma \left(dz+aw\right)& \le \bigvee \left\{\gamma \left(dz\right),\gamma \left(aw\right)\right\};\\ & \le \bigvee \left\{\bigvee \left\{\lambda \left(d\right),\gamma \left(z\right)\right\},\bigvee \left\{\lambda \left(a\right),\gamma \left(w\right)\right\}\right\};\\ & \le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}.\end{array}$$

(ii) and (iii) hold directly, as ${\tilde{\varrho }}_{\gamma }$ is an H N A of $Y$ over $U$.

Conversely, presume that the three conditions of the hypothesis hold.

Let $z,w\in Y \mathrm{and} d,a\in L$. Then,

$$\begin{array}{c}\tilde{\varrho }\left(z+w\right)=\tilde{\varrho }\left(1.z+1.w\right)\supseteq \left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right);\\ \supseteq \left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\varrho }\left(w\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right);\\ =\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right);\\ \gamma \left(z+w\right)=\gamma \left(1.z+1.w\right)\le \bigvee \left\{\lambda \left(1\right),\gamma \left(z\right),\lambda \left(1\right),\gamma \left(w\right)\right\};\\ \le \bigvee \left\{\gamma \left(z\right), \gamma \left(z\right), \gamma \left(w\right), \gamma \left(w\right)\right\};\\ =\bigvee \left\{\gamma \left(z\right), \gamma \left(w\right)\right\}.\end{array}$$

For any, $z\in Y,d\in L$,

$$\begin{array}{c}\tilde{\varrho }\left(dz\right)=\tilde{\varrho }\left(dz+0.z\right);\\ \supseteq \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(0\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right);\\ \supseteq \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right);\\ =\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right);\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\gamma \left(dz\right)=\gamma \left(dz+0.z\right)\le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(0\right),\gamma \left(z\right)\right\};\\ \le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(d\right),\gamma \left(z\right)\right\};\\ =\bigvee \left\{\lambda \left(d\right),\gamma \left(z\right)\right\}.\end{array}$$

From the hypothesis, $\tilde{\varrho }\left(zw\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right), \gamma \left(zw\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\},\forall z,w\in Y$ and $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(1\right)\le \gamma \left(z\right),\forall z\in Y$. Hence, $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right).$ □

Proposition 1. 

If $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right),$ then $\tilde{\varrho }\left(z-w\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right) and \gamma \left(z-w\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\},\forall z,w\in Y$.

Proof. 

Given, $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right).$

Let $z,w\in Y$ and $1$ be the identity element in $L.$

Then,

$$\begin{array}{c}\tilde{\varrho }\left(z-w\right)=\tilde{\varrho }\left(1.z+\left(-1\right).w\right);\\ \supseteq \tilde{\varrho }\left(1.z\right){\displaystyle \cap }\tilde{\varrho }\left(\left(-1\right)w\right);\\ \supseteq \left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(-1\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right);\\ \supseteq \left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(1\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right);\\ \supseteq \left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\varrho }\left(w\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right);\\ \supseteq \left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right);\end{array}$$

and

$$\begin{array}{l}\gamma \left(z-w\right)=\gamma \left(1.z+\left(-1\right).w\right);\\ \le \bigvee \left\{\gamma \left(1.z\right),\gamma \left(\left(-1\right).w\right)\right\};\\ \le \bigvee \left\{\bigvee \left\{\lambda \left(1\right),\gamma \left(z\right)\right\},\bigvee \left\{\lambda \left(-1\right),\gamma \left(w\right)\right\}\right\};\\ \le \bigvee \left\{\bigvee \left\{\lambda \left(1\right),\gamma \left(z\right)\right\},\bigvee \left\{\lambda \left(1\right),\gamma \left(w\right)\right\}\right\};\\ \le \bigvee \left\{\bigvee \left\{\gamma \left(z\right),\gamma \left(z\right)\right\},\bigvee \left\{\gamma \left(w\right),\gamma \left(w\right)\right\}\right\};\\ \le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}. \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\square \end{array}$$

Theorem 4. 

$\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over H F $\left({\tilde{\zeta }}_{\lambda },L\right)$ if and only if a nonempty set ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a sub near algebra over the field ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right] \forall t\in \left[0,1\right]$ and $\alpha \in P\left(U\right).$

Proof. 

Let $t\in \left[0,1\right]$ be such that ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]\ne \varnothing$ and $z,w\in {\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right];d\in {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right].$ Then, $z,w\in Y, d\in L$ and $\tilde{\varrho }\left(z\right)\supseteq \alpha , \gamma \left(z\right)\le t, \tilde{\varrho }\left(w\right)\supseteq \alpha , \gamma \left(w\right)\le t, \tilde{\zeta }\left(d\right)\supseteq \alpha , \lambda \left(d\right)\le t$. Thus, $z-w\in Y, dz\in Y.$ Now, $\tilde{\varrho }\left(z-w\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\supseteq \alpha {\displaystyle \cap }\alpha =\alpha , \gamma \left(z-w\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}\le \bigvee \left\{t,t\right\}=t$, $\tilde{\varrho }\left(dz\right)\supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\supseteq \alpha {\displaystyle \cap }\alpha =\alpha ,$ $\gamma \left(dz\right)\le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right)\right\}\le \bigvee \left\{t,t\right\}=t.$ Thus, $z-w\in {\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right] \mathrm{and} dz\in {\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$. Hence, ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a subspace of $Y$. Presume that ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right]\ne \varnothing .$ We know that ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right]$ is a subfield of $L.$ For $z,w\in {\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right], d\in {\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right]$, we have $z,w\in Y, d\in L, \tilde{\varrho }\left(z\right)\supseteq \alpha , \gamma \left(z\right)\le t, \tilde{\varrho }\left(w\right)\supseteq \alpha , \gamma \left(w\right)\le t, \tilde{\zeta }\left(d\right)\supseteq \alpha , \lambda \left(d\right)\le t.$ Thus, $z+w,zw,dz\in Y,$ $d\left(zw\right)=\left(dz\right)w.$ Since $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over H F $\left({\tilde{\zeta }}_{\lambda },L\right),$ we get $\tilde{\varrho }\left(z+w\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\supseteq \alpha {\displaystyle \cap }\alpha =\alpha$,$\gamma \left(z+w\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}\le \bigvee \left\{t,t\right\}=t$. And $\tilde{\varrho }\left(dz\right)\supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\supseteq \alpha {\displaystyle \cap }\alpha =\alpha ,$ $\gamma \left(dz\right)\le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right)\right\}\le \bigvee \left\{t,t\right\}=t,$ $\tilde{\varrho }\left(zw\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\supseteq \alpha {\displaystyle \cap }\alpha =\alpha$, $\left(zw\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}\le \bigvee \left\{t,t\right\}=t.$ $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right)\supseteq \alpha , \lambda \left(1\right)\le \gamma \left(z\right)\le t$ implies that $z+w,zw,dz\in {\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ and $d\left(zw\right)=\left(dz\right)w$. Hence, ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a sub near algebra of $Y$ over a field ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right]$.

Conversely, presume that ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]\ne \varnothing .$ Hence, ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a sub near algebra over a field ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right].$ If possible, presume that there exists $z,w\in Y; d,a\in L$ such that $\tilde{\varrho }\left(dz+aw\right)\subseteq \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right),$ $\gamma \left(dz+aw\right)\ge \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}$.

Put

$$\begin{array}{l}{u}_0=\frac{1}{2}\left\{ \tilde{\varrho }\left(dz+aw\right)+\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\zeta \left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right)\right\};\\ v_0=\frac{1}{2}\left\{ \gamma \left(dz+aw\right)+\bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}\right\}.\end{array}$$

• Then, we have $\tilde{\varrho }\left(dz+aw\right)<u_0<\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right)$
• and $\gamma \left(dz+aw\right)>v_0>$ $\bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}$.
• Hence, $\tilde{\varrho }\left(dz+aw\right)<u_0, \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right)>u_0$
• and $\gamma \left(dz+aw\right)>v_0, \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}<v_0$. So that
• $\tilde{\varrho }\left(dz+aw\right)<u_0, \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right)>u_0 \mathrm{and} \left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right)>u_0$
• and $\gamma \left(dz+aw\right)>v_0, \lambda \left(d\right)<v_0,\gamma \left(z\right)<v_0,\lambda \left(a\right)<v_0,\gamma \left(w\right)<v_0.$
• Thus, $\tilde{\varrho }\left(dz+aw\right)<u_0, \tilde{\zeta }\left(d\right)>u_0, \tilde{\varrho }\left(z\right)>u_0, \tilde{\zeta }\left(a\right)>u_0, \tilde{\varrho }\left(w\right)>u_0$
• and $\gamma \left(dz+aw\right)>v_0, \lambda \left(d\right)<v_0,\gamma \left(z\right)<v_0,\lambda \left(a\right)<v_0,\gamma \left(w\right)<v_0.$
• Since $z,w\in Y; d,a\in L, \mathrm{we} \mathrm{have}$ $dz+aw\in Y.$
• Hence, $\tilde{\varrho }\left(dz+aw\right)<u_0, \tilde{\zeta }\left(d\right)>u_0, \tilde{\varrho }\left(z\right)>u_0, \tilde{\zeta }\left(a\right)>u_0, \tilde{\varrho }\left(w\right)>u_0$ and
• $\gamma \left(dz+aw\right)>v_0, \lambda \left(d\right)<v_0,\gamma \left(z\right)<v_0,\lambda \left(a\right)<v_0,\gamma \left(w\right)<v_0$.
• Therefore, $dz+aw\notin {\tilde{\varrho }}_{\gamma }\left[u_0,t\right]{\displaystyle \cap }{\tilde{\varrho }}_{\gamma }\left[v_0,t\right]$, $d,a\in {\tilde{\zeta }}_{\lambda }\left[u_0,t\right]{\displaystyle \cap }{\tilde{\zeta }}_{\lambda }\left[v_0,t\right] \mathrm{and}$
• $z,w\in {\tilde{\varrho }}_{\gamma }\left[u_0,t\right]{\displaystyle \cap }{\tilde{\varrho }}_{\gamma }\left[v_0,t\right].$ Thus, $dz+aw\notin {\tilde{\varrho }}_{\gamma }\left[u_0,t\right]{\displaystyle \cap }{\tilde{\varrho }}_{\gamma }\left[v_0,t\right]$ and
• $dz,aw\in {\tilde{\varrho }}_{\gamma }\left[u_0,t\right]{\displaystyle \cap }{\tilde{\varrho }}_{\gamma }\left[v_0,t\right]$, which is a contradiction to the fact that ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a sub near algebra over a field ${\tilde{\zeta }}_{\gamma }\left[\alpha ,t\right]$. Hence, $\tilde{\varrho }\left(dz+aw\right)\supseteq \left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right),$ $\gamma \left(dz+aw\right)\le \bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}.$ If possible, presume that there exists $z,w\in Y$ such that $\tilde{\varrho }\left(zw\right)\subseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right), \gamma \left(zw\right)\ge \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}.$ Put $u_1=\frac{1}{2}\left\{ \tilde{\varrho }\left(zw\right)+\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right\}$, $v_1=\frac{1}{2}\left\{ \gamma \left(zw\right)+\bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}\right\}.$ Then, we have $\tilde{\varrho }\left(zw\right)<u_1<\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)$ and $\gamma \left(zw\right)>v_1>\bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}.$ Hence, $\tilde{\varrho }\left(zw\right)<u_1, \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)>u_1$,$\gamma \left(zw\right)>v_1, \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}<v_1.$
• So that $\tilde{\varrho }\left(zw\right)<u_1, \tilde{\varrho }\left(z\right)>u_1 \mathrm{and} \tilde{\varrho }\left(w\right)>u_1$ and $\gamma \left(zw\right)>v_1,\gamma \left(z\right)<v_1,\gamma \left(w\right)<v_1.$
• Since $z,w\in Y,$ we have $zw\in Y.$ Thus, $\tilde{\varrho }\left(zw\right)<u_1, \tilde{\varrho }\left(z\right)>u_1, \tilde{\varrho }\left(w\right)>u_1$ and $\gamma \left(zw\right)>v_1,\gamma \left(z\right)<v_1, \gamma \left(w\right)<v_1.$ Therefore, $zw\notin {\tilde{\varrho }}_{\gamma }\left[u_1,t\right]{\displaystyle \cap }{\tilde{\varrho }}_{\gamma }\left[v_1,t\right]$, $z,w\in {\tilde{\varrho }}_{\gamma }\left[u_1,t\right]{\displaystyle \cap }{\tilde{\varrho }}_{\gamma }\left[v_1,t\right]$, which is a contradiction to the fact that ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a sub near-algebra over a field ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right].$ Hence, $\tilde{\varrho }\left(zw\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right),$ $\gamma \left(zw\right)\le \bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}.$ If possible, presume that there exists $z\in Y$ such that $\tilde{\zeta }\left(1\right)\subseteq \tilde{\varrho }\left(z\right), \lambda \left(1\right)\ge \gamma \left(z\right).$ Put $u_2=\frac{1}{2}\left\{ \tilde{\zeta }\left(1\right)+\tilde{\varrho }\left(z\right)\right\}$ and $v_2=\frac{1}{2}\left\{ \lambda \left(1\right)+\gamma \left(z\right)\right\}.$ Then, $\tilde{\zeta }\left(1\right)<u_2<\tilde{\varrho }\left(z\right)$ and $\lambda \left(1\right)>v_2>\gamma \left(z\right).$ Hence, $\tilde{\zeta }\left(1\right)<u_2, \tilde{\varrho }\left(z\right)>u_2$ and $\lambda \left(1\right)>v_2,\gamma \left(z\right)<v_2$. So that $1\notin {\tilde{\zeta }}_{\lambda }\left[u_2,t\right], z\in {\tilde{\varrho }}_{\gamma }\left[u_2,t\right],$ $1\notin {\tilde{\zeta }}_{\lambda }\left[v_2,t\right], z\in {\tilde{\varrho }}_{\gamma }\left[v_2,t\right]$. Therefore, $1\notin {\tilde{\zeta }}_{\lambda }\left[u_2,t\right]{\displaystyle \cap }{\tilde{\zeta }}_{\lambda }\left[v_2,t\right]$
  and $z\in {\tilde{\varrho }}_{\gamma }\left[u_2,t\right]{\displaystyle \cap }z\in {\tilde{\varrho }}_{\gamma }\left[u_2,t\right]$, which is a contradiction to the fact that ${\tilde{\varrho }}_{\gamma }\left[\alpha ,t\right]$ is a sub near algebra over a field ${\tilde{\zeta }}_{\lambda }\left[\alpha ,t\right].$ Thus, $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right), \lambda \left(1\right)\le \gamma \left(z\right), \forall z\in Y$. Hence, $\left({\tilde{\varrho }}_{\gamma },Y\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right).$ □

Definition 6. 

Let ${\tilde{\varrho }}_{\gamma }$ and ${\tilde{h}}_{\mu }$ be hybrid structures in $L$ over $U.$ Then, the hybrid intersection of ${\tilde{\varrho }}_{\gamma }$ and ${\tilde{h}}_{\mu }$ is designated by ${\tilde{\varrho }}_{\gamma }⋒{\tilde{h}}_{\mu }$ and is demarcated to be a hybrid structure ${\tilde{\varrho }}_{\gamma }$ $⋒{\tilde{h}}_{\mu }:L\to P\left(U\right) X I, z\to \left(\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(z\right), \left(\gamma \vee \mu \right)\left(z\right)\right), for all z\in L$, i.e., the image of $z$ is $\left(\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(z\right), \left(\gamma \vee \mu \right)\left(z\right)\right)$, where $\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}:L\to P\left(U\right), z\to \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z\right),\gamma \bigvee \mu :L\to I,z\to \bigvee \left\{\gamma \right(z),\mu (z\left)\right\}$.

Theorem 5. 

Let ${\tilde{\varrho }}_{\gamma }$, ${\tilde{h}}_{\mu }$ be two H N As of $Y$ over an H F ${\tilde{\zeta }}_{\lambda }$ of $L$. Then, ${\tilde{\varrho }}_{\gamma }⋒{\tilde{h}}_{\mu }$ is an H N A of $Y$ over an H F ${\tilde{\zeta }}_{\lambda }$ of $L.$

Proof. 

Let ${\tilde{\varrho }}_{\gamma }$ and ${\tilde{h}}_{\mu }$ be two H N As of $Y$ over an H F ${\tilde{\zeta }}_{\lambda }$ of $L.$ For all $z,w\in Y$ and $d,a\in L$,

(i)

$\left({\tilde{\varrho }}_{\gamma }⋒{\tilde{h}}_{\mu }\right)\left(dz+aw\right)=\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(dz+aw\right)$

$$\begin{array}{l} =\tilde{\varrho }\left(dz+aw\right)\widetilde{{\displaystyle \cap }}\tilde{h}\left(dz+aw\right);\\ \supseteq \left(\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right)\right)\widetilde{{\displaystyle \cap }}\left(\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{h}\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{h}\left(w\right)\right)\right);\\ =\left(\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{h}\left(z\right)\right)\right)\widetilde{{\displaystyle \cap }}\left(\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{h}\left(w\right)\right)\right);\\ =\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\left(\tilde{\varrho }\left(z\right)\widetilde{{\displaystyle \cap }}\tilde{h}\left(z\right)\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\left(\tilde{\varrho }\left(w\right)\widetilde{{\displaystyle \cap }}\tilde{h}\left(w\right)\right)\right);\\ =\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(w\right)\right);\end{array}$$

and

$$\begin{array}{l}\left(\gamma \vee \mu \right)\left(dz+aw\right)=\bigvee \left\{\gamma \left(dz+aw\right),\mu \left(dz+aw\right)\right\};\\ \le \bigvee \left\{\bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(a\right),\gamma \left(w\right)\right\}, \bigvee \left\{\lambda \left(d\right),\mu \left(z\right),\lambda \left(a\right),\mu \left(w\right)\right\}\right\};\\ =\bigvee \left\{\bigvee \left\{\lambda \left(d\right),\gamma \left(z\right),\lambda \left(d\right),\mu \left(z\right)\right\}, \bigvee \left\{\lambda \left(a\right),\gamma \left(w\right),\lambda \left(a\right),\mu \left(w\right)\right\}\right\};\\ =\bigvee \left\{\right(\lambda \left(d\right),\bigvee \{\gamma \left(z\right),\mu \left(z\right)\}), (\lambda \left(a\right), \bigvee \left\{\gamma \left(w\right),\mu \left(w\right)\right\})\};\\ =\bigvee \left\{\left(\lambda \left(d\right),\left(\gamma \vee \mu \right)\left(z\right)\right), \left(\lambda \left(a\right),\left(\gamma \vee \mu \right)\left(w\right)\right)\right\}.\end{array}$$

(ii)

$$\begin{array}{l}\left({\tilde{\varrho }}_{\gamma }⋒{\tilde{h}}_{\mu }\right)\left(zw\right)=\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(zw\right)=\tilde{\varrho }\left(zw\right)\widetilde{{\displaystyle \cap }}\tilde{h}\left(zw\right)\supseteq \left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right)\widetilde{{\displaystyle \cap }}\left(\tilde{h}\left(z\right){\displaystyle \cap }\tilde{h}\left(w\right)\right);\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z\right)\right)\widetilde{{\displaystyle \cap }}\left(\tilde{\varrho }\left(w\right){\displaystyle \cap }\tilde{h}\left(w\right)\right)=\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(z\right){\displaystyle \cap }\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(w\right);\end{array}$$

and

$$\begin{array}{l}\left(\gamma \vee \mu \right)\left(zw\right) =\bigvee \left\{\gamma \left(zw\right),\mu \left(zw\right)\right\}\le \bigvee \left\{\bigvee \left\{\gamma \left(z\right),\gamma \left(w\right)\right\}, \bigvee \left\{\mu \left(z\right),\mu \left(w\right)\right\}\right\};\\ =\bigvee \left\{\bigvee \left\{\gamma \left(z\right),\mu \left(z\right)\right\}, \bigvee \left\{\gamma \left(w\right),\mu \left(w\right)\right\}\right\};\\ =\bigvee \left\{\left(\gamma \vee \mu \right)\left(z\right), \left(\gamma \vee \mu \right)\left(w\right)\right\}.\end{array}$$

We have $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right), \tilde{\zeta }\left(1\right)\supseteq \tilde{h}\left(z\right).$ Hence $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right)\widetilde{{\displaystyle \cap }}\tilde{h}\left(z\right)$.

Therefore, $\tilde{\zeta }\left(1\right)\supseteq$ $\left(\tilde{\varrho }\widetilde{{\displaystyle \cap }}\tilde{h}\right)\left(z\right),\forall z\in Y$ and $\lambda \left(1\right)\le \gamma \left(z\right), \lambda \left(1\right)\le \mu \left(z\right)$.

Thus, $\lambda \left(1\right)\le \gamma \left(z\right)\widetilde{{\displaystyle \cap }}$ $\mu \left(z\right)$. Hence $\lambda \left(1\right)\le (\gamma \widetilde{{\displaystyle \cap }}$ $\mu )\left(z\right) \forall z\in Y.$ □

Example 3. 

Let $L=Z_2={\left\{0,1\right\}}_{{⨁}_2, {⨁}_2}$ be a field. The hybrid structure ${\tilde{\zeta }}_{\lambda }$ in $L$ over $U=\left\{u_1,u_2,u_3,u_4,u_5\right\},$ which is given as follows:

$\mathit{L}$	$\tilde{\mathsf{\zeta }}$	$\mathsf{\lambda }$
0	$\left\{u_1,u_2,u_3,u_4\right\}$	0.3
1	$\left\{u_2,u_3,u_4\right\}$	0.4

Clearly, ${\tilde{\zeta }}_{\lambda }$ is an H F in $L$ over $U$. Let $Y=\left\{0,h,p,e\right\}$ be a set with two binary operations, “+” and “.” By

+	0	h	p	e	.	0	h	p	e
0	0	h	p	e	0	0	0	0	0
h	h	0	e	p	h	h	h	h	h
p	p	e	0	h	p	p	p	p	p
e	e	p	h	0	e	e	e	e	e

Clearly, $Y$ forms a near algebra over $L$. The hybrid structure ${\tilde{\varrho }}_{\gamma }, {\tilde{h}}_{\mu }$ in $Y$ over $U=\left\{u_1,u_2,u_3,u_4,u_5\right\}$, which is given as follows:

$\mathit{Y}$	$\tilde{\mathsf{\varrho }}$	$\mathsf{\gamma }$	$\tilde{\mathit{h}}$	$\mathsf{\mu }$
0	$\left\{u_2,u_3,u_4\right\}$	0.4	$\left\{u_1,u_3,u_4\right\}$	0.5
h	$\left\{u_2,u_4\right\}$	0.6	$\{u_1,u_4\}$	0.7
p	$\left\{u_2,u_3\right\}$	0.8	$\{u_1,u_5\}$	0.9
e	$\left\{u_2\right\}$	0.8	$\left\{u_1\right\}$	0.9

Hence, ${\tilde{\varrho }}_{\gamma }⋒{\tilde{h}}_{\mu }$ is an H N A of $Y$ over an H F ${\tilde{\zeta }}_{\lambda }$ of $L.$

Definition 7. 

Let Y and $Y^{\prime }$ be two near algebras. Let $\omega :Y\to Y^{\prime }$ be a mapping. Then,

(i)

If ${\tilde{h}}_{\mu }$ is a hybrid subset of $Y^{\prime }$, then the preimage of ${\tilde{h}}_{\mu }$ under $\omega$ is the hybrid subset in Y over $U$ defined by

$${\omega }^{-1}\left({\tilde{h}}_{\mu }\right)\left(z\right)=({\omega }^{-1}\left(\tilde{h}\right)\left(z\right), {\omega }^{-1}\left(\mu \right)\left(z\right))=\left(\tilde{h}\left(\omega \right)\left(z\right), \mu \left(\omega \right)\left(z\right)\right), \forall z\in Y.$$

(ii)

If ${\tilde{\varrho }}_{\gamma }$ is a hybrid subset of $Y$, then the image of ${\tilde{\varrho }}_{\gamma }$ under $\omega$ is the hybrid subset in $Y^{\prime }$ over $U$ defined by

$$\omega \left(\tilde{\varrho }\right)\left(w\right)=\{\begin{array}{c}\underset{z\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(z\right), if {\omega }^{-1}\left(w\right)\ne \varnothing \\ \varnothing , otherwise\end{array}$$

and

$$\omega \left(\gamma \right)\left(w\right)=\{\begin{array}{c}\underset{z\in {\omega }^{-1}\left(w\right)}{{\displaystyle \wedge }}\gamma \left(z\right), if {\omega }^{-1}\left(w\right)\ne \varnothing \\ 1, otherwise,\end{array}$$

for every $w\in Y.$

Theorem 6. 

Let Y and $Y^{\prime }$ be two near algebras over a field $L$ and $\omega :Y\to Y^{\prime }$ be onto near algebra homomorphism. If $({\tilde{\varrho }}_{\gamma },Y)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right)$, then $\left(\omega \left({\tilde{\varrho }}_{\gamma }\right),Y^{\prime }\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right).$

Proof. 

Let $z,w\in Y^{\prime }$ and $d,a\in L.$ Then, we have $\left\{q:q\in {\omega }^{-1}\left(dz+aw\right)\right\}\supseteq \left\{ds+at:s\in {\omega }^{-1}\left(z\right) \mathrm{and} t\in {\omega }^{-1}\left(w\right)\right\}$ and $\left\{q:q\in {\omega }^{-1}\left(zw\right)\right\}\supseteq \left\{st:s\in {\omega }^{-1}\left(z\right) \mathrm{and} t\in {\omega }^{-1}\left(w\right)\right\}$.

If ${\omega }^{-1}\left(z\right)\ne \varnothing ,{\omega }^{-1}\left(w\right)\ne \varnothing ,$ then ${\omega }^{-1}\left(zw\right)\ne \varnothing .$ Now

(i)

For all $z,w\in Y^{\prime }, \exists s,t\in Y$ such that $z=\omega \left(s\right), w=\omega \left(t\right)$

$$\begin{array}{l}\omega \left(\tilde{\varrho }\right)\left(dz+aw\right)=\underset{q\in {\omega }^{-1}\left(dz+aw\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(q\right);\\ \supseteq \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(ds+at\right);\\ \supseteq \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(s\right){\displaystyle \cap }\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{g}\left(t\right);\\ =\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\underset{s\in {\omega }^{-1}\left(z\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(s\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\underset{t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(t\right)\right);\\ =\left(\tilde{\zeta }\left(d\right){\displaystyle \cap }\omega \left(\tilde{\varrho }\right)\left(z\right)\right){\displaystyle \cap }\left(\tilde{\zeta }\left(a\right){\displaystyle \cap }\omega \left(\tilde{\varrho }\right)\left(w\right)\right);\\ =\tilde{\zeta }\left(d\right){\displaystyle \cap }\omega \left(\tilde{\varrho }\right)\left(z\right){\displaystyle \cap }\tilde{\zeta }\left(a\right){\displaystyle \cap }\omega \left(\tilde{\varrho }\right)\left(y\right);\end{array}$$

and

$$\begin{array}{l}\omega \left(\gamma \right)\left(dz+aw\right)=\underset{q\in {\omega }^{-1}\left(dz+aw\right)}{{\displaystyle \wedge }}\gamma \left(q\right);\\ \le \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \wedge }}\gamma \left(ds+at\right);\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le {{\displaystyle \wedge }}_{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)} \left(\vee \left\{\lambda \left(d\right),\gamma \left(s\right),\lambda \left(a\right),\gamma \left(t\right)\right\}\right);\\ =\vee \left\{\vee \left\{\lambda \left(d\right),\underset{s\in {\omega }^{-1}\left(z\right)}{{\displaystyle \wedge }}\gamma \left(s\right)\right\},\vee \left\{\lambda \left(a\right),\underset{t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \wedge }}\gamma \left(t\right)\right\}\right\};\\ =\vee \left\{\lambda \left(a\right),\omega \left(\gamma \right)\left(z\right),\lambda \left(b\right),\omega \left(\gamma \right)\left(w\right)\right\}.\end{array}$$

(ii)

$$\begin{array}{l}\omega \left(\tilde{\varrho }\right)\left(zw\right)=\underset{q\in {\omega }^{-1}\left(zw\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(q\right);\\ \supseteq \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(st\right);\\ \supseteq \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(s\right){\displaystyle \cap }\tilde{\varrho }\left(t\right);\\ =\left(\underset{s\in {\omega }^{-1}\left(z\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(s\right)\right){\displaystyle \cap }\left(\underset{t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \cup }}\tilde{\varrho }\left(t\right)\right);\\ =\omega \left(\tilde{\varrho }\right)\left(z\right){\displaystyle \cap } \omega \left(\tilde{\varrho }\right)\left(w\right);\end{array}$$

and

$$\begin{array}{l}\omega \left(\gamma \right)\left(zw\right)=\underset{q\in {\omega }^{-1}\left(zw\right)}{{\displaystyle \wedge }}\gamma \left(q\right); \\ \le \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \wedge }}\gamma \left(st\right);\\ \le \underset{s\in {\omega }^{-1}\left(z\right),t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \wedge }}\left(\vee \left\{\gamma \left(s\right),\gamma \left(t\right)\right\}\right);\\ =\vee \left\{\underset{s\in {\omega }^{-1}\left(z\right)}{{\displaystyle \wedge }}\gamma \left(s\right), \underset{t\in {\omega }^{-1}\left(w\right)}{{\displaystyle \wedge }}\gamma \left(t\right) \right\};\\ =\vee \left\{\omega \left(\gamma \right)\left(z\right), \omega \left(\gamma \right)\left(w\right)\right\}.\end{array}$$

(iii)

$\tilde{\zeta }\left(1\right)\supseteq \omega \left(\tilde{\varrho }\right)\left(z\right)$ and $\lambda \left(1\right)\le \omega \left(\gamma \right)\left(z\right), \forall z\in Y^{\prime }$.

Therefore, $\left(\omega \left({\tilde{\varrho }}_{\gamma }\right),Y^{\prime }\right)$ is an H N A over $\left({\tilde{\zeta }}_{\lambda },L\right)$. □

Theorem 7. 

Let $Y,$ $Y^{\prime }$ be two near algebras over a field $L$ and $\omega : Y\to Y^{\prime }$ be onto near algebra homomorphism. If ${\tilde{h}}_{\mu }$ is a hybrid near algebra in $Y^{\prime }$ over $\left({\tilde{\zeta }}_{\lambda },L\right),$ then ${\omega }^{-1}\left({\tilde{h}}_{\mu }\right):=\left({\omega }^{-1}\left(\tilde{h}\right),{\omega }^{-1}\left(\mu \right)\right)$ is an H N A in $Y$ over $\left({\tilde{\zeta }}_{\lambda },L\right).$

Proof. 

Let $z,w\in Y$ and $d\in L.$

(i)

$$\begin{array}{c}{\omega }^{-1}\left(\tilde{h}\right)\left(dz+aw\right)=\tilde{h}\left(\omega \left(dz+aw\right)\right);\\ =\tilde{h}\left(d \omega \left(z\right)+a \omega \left(w\right)\right);\\ \supseteq \tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{h}\left(\omega \left(z\right)\right){\displaystyle \cap }\tilde{\zeta }\left(a\right){\displaystyle \cap }\tilde{h}\left(\omega \left(w\right)\right);\\ =\tilde{\zeta }\left(d\right){\displaystyle \cap } {\omega }^{-1}\left(\tilde{h}\right)\left(z\right){\displaystyle \cap }\tilde{\zeta }\left(a\right){\displaystyle \cap } {\omega }^{-1}\left(\tilde{h}\right)\left(w\right);\end{array}$$

and

$$\begin{array}{l}{\omega }^{-1}\left(\mu \right)\left(dz+aw\right)=\mu \left(\omega \left(dz+aw\right)\right);\\ = \mu \left(d \omega \left(z\right) + a\omega \left(w\right)\right);\\ \le \vee \left\{\lambda \left(d\right),\mu \left(\omega \right(z\left)\right), \lambda \left(a\right),\mu \left(\omega \right(w\left)\right)\right\};\\ =\vee \left\{\lambda \left(d\right),{\omega }^{-1}\left(\mu \right)\left(z\right), \lambda \left(a\right),{\omega }^{-1}\left(\mu \right)\left(w\right)\right\}.\end{array}$$

(ii)

$$\begin{array}{c} {\omega }^{-1}\left(\tilde{h}\right)\left(zw\right)=\tilde{h}\left(\omega \left(zw\right)\right)=\tilde{h}\left(\omega \left(z\right)\omega \left(w\right)\right)\supseteq \tilde{h}\left(\omega \left(z\right)\right){\displaystyle \cap }\tilde{h}\left(\omega \left(w\right)\right); \\ ={\omega }^{-1}\left(\tilde{h}\right)\left(z\right){\displaystyle \cap }{\omega }^{-1}\left(\tilde{h}\right)\left(w\right);\end{array}$$

and

$$\begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\omega }^{-1}\left(\mu \right)\left(zw\right)=\mu \left(\omega \left(zw\right)\right)=\mu \left( \omega \left(z\right)\omega \left(w\right)\right)\le \vee \left\{\mu \left(\omega \left(z\right)\right),\mu \left(\omega \left(w\right)\right)\right\};\\ =\vee \left\{{\omega }^{-1}\left(\mu \right)\left(z\right),{\omega }^{-1}\left(\mu \right)\left(w\right)\right\}.\end{array}$$

(iii)

$$\tilde{\zeta }\left(1\right)\supseteq {\omega }^{-1}\left(\tilde{h}\right)\left(z\right) \mathrm{and} \lambda \left(1\right)\le {\omega }^{-1}\left(\mu \right)\left(z\right), \forall z\in Y.$$

Therefore, ${\omega }^{-1}\left({\tilde{h}}_{\mu }\right)=\left({\omega }^{-1}\left(\tilde{h}\right),{\omega }^{-1}\left(\mu \right)\right)$ is an H N A in $Y$ over $\left({\tilde{\zeta }}_{\lambda },L\right).$ □

Definition 8. 

Let ${\tilde{\varrho }}_{\gamma }$, ${\tilde{h}}_{\mu }$ be two hybrid near algebras of $Y,$ $Y^{\prime }$, respectively. Then, the direct product of ${\tilde{\varrho }}_{\gamma }$ and ${\tilde{h}}_{\mu }$ is denoted by $\tilde{\varrho }X\tilde{h}$ and $\gamma X \mu$ is the function demarcated by

$$\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z,z^{\prime }\right)=\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right) \mathrm{and} \left(\gamma X \mu \right)\left(z,z^{\prime }\right)=\vee \left\{\gamma \left(z\right), \mu \left(z^{\prime }\right)\right\} \forall z\in Y and z^{\prime }\in Y^{\prime }.$$

Theorem 8. 

Let $Y$,$Y^{\prime }$ be two near algebras over a field $L$ and ${\tilde{\varrho }}_{\gamma }$, ${\tilde{h}}_{\mu }$ be two hybrid near algebras of $Y$, $Y^{\prime },$ respectively, over an H F $\left({\tilde{\zeta }}_{\lambda },L\right).$ Then, ${\tilde{\varrho }}_{\gamma }X{\tilde{h}}_{\mu }$ is an H N A of $YX{Y}^{\prime }$ over an H F $\left({\tilde{\zeta }}_{\lambda },L\right).$

Proof. 

We have $\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z,z^{\prime }\right)=\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right),$ $\left(\gamma X \mu \right)\left(z,z^{\prime }\right)= \vee \left\{\gamma \left(z\right), \mu \left(z^{\prime }\right)\right\}, \forall z\in Y, z^{\prime }\in Y^{\prime }.$ Let $\left(z,z^{\prime }\right),\left(w,w^{\prime }\right)\in YX{Y}^{\prime }$ and $d\in L.$ Then,

(i)

$$\begin{array}{l}\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left[\left(z,z^{\prime }\right)+\left(w,w^{\prime }\right)\right]=\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z+w, z^{\prime }+w^{\prime }\right);\\ =\tilde{\varrho }\left(z+w\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }+w^{\prime }\right);\\ \supseteq \left[\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right]{\displaystyle \cap }\left[\tilde{h}\left(z^{\prime }\right){\displaystyle \cap }\tilde{h}\left(w^{\prime }\right)\right];\\ =\left[\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right)\right]{\displaystyle \cap }\left[\tilde{\varrho }\left(w\right){\displaystyle \cap }\tilde{h}\left(w^{\prime }\right)\right];\\ =\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z,z^{\prime }\right){\displaystyle \cap }\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(w,w^{\prime }\right);\\ \left(\gamma X \mu \right)\left[\left(z,z^{\prime }\right)+\left(w,w^{\prime }\right)\right]=\left(\gamma X \mu \right)\left(z+w, z^{\prime }+w^{\prime }\right);\\ =\bigvee \left\{\gamma \left(z+w\right), \mu \left(z^{\prime }+w^{\prime }\right)\right\};\\ \le \bigvee \left\{\bigvee \left\{\gamma \left(z\right), \gamma \left(w\right)\right\}, \bigvee \left\{\mu \left(z^{\prime }\right),\mu \left(w^{\prime }\right)\right\}\right\};\\ \le \bigvee \left\{\bigvee \left\{\gamma \left(z\right), \mu \left(z^{\prime }\right)\right\}, \bigvee \left\{\gamma \left(w\right),\mu \left(w^{\prime }\right)\right\}\right\};\\ =\bigvee \left\{\left(\gamma X \mu \right)\left(z,z^{\prime }\right),\left(\gamma X \mu \right)\left(w,w^{\prime }\right)\right\}.\end{array}$$

(ii)

$$\begin{array}{l}\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(d\left(z,z^{\prime }\right)\right)=\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(dz,d{z}^{\prime }\right);\\ =\tilde{\varrho }\left(dz\right){\displaystyle \cap }\tilde{h}\left(d{z}^{\prime }\right);\\ \supseteq \left[\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{\varrho }\left(z\right)\right]{\displaystyle \cap }\left[\tilde{\zeta }\left(d\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right)\right];\\ =\tilde{\zeta }\left(d\right){\displaystyle \cap }\left(\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right)\right);\\ =\tilde{\zeta }\left(d\right){\displaystyle \cap }\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z,z^{\prime }\right);\\ \left(\gamma X \mu \right)\left[d\left(z,z^{\prime }\right)\right]=\left(\gamma X \mu \right)\left(dz,d{z}^{\prime }\right);\\ =\vee \left\{\gamma \left(dz\right), \mu \left(d{z}^{\prime }\right)\right\};\\ \le \bigvee \left\{\bigvee \left\{\lambda \left(d\right), \gamma \left(z\right)\right\}, \bigvee \left\{\lambda \left(d\right),\mu \left(z^{\prime }\right)\right\}\right\};\\ \le \bigvee \left\{\lambda \left(d\right), \bigvee \left\{\gamma \left(z\right),\mu \left(z^{\prime }\right)\right\}\right\};\\ =\bigvee \left\{\lambda \left(d\right),\left(\gamma X \mu \right)\left(z,z^{\prime }\right)\right\}.\end{array}$$

(iii)

$$\begin{array}{l} \left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left[\left(z,z^{\prime }\right)\left(w,w^{\prime }\right)\right]=\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(zw, z^{\prime }{w}^{\prime }\right); \\ =\tilde{\varrho }\left(zw\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }{w}^{\prime }\right);\\ \supseteq \left[\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{\varrho }\left(w\right)\right]{\displaystyle \cap }\left[\tilde{h}\left(z^{\prime }\right){\displaystyle \cap }\tilde{h}\left(w^{\prime }\right)\right];\\ =\left[\tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right)\right]{\displaystyle \cap }\left[\tilde{\varrho }\left(w\right){\displaystyle \cap }\tilde{h}\left(w^{\prime }\right)\right];\\ =\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z,z^{\prime }\right){\displaystyle \cap }\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(w,w^{\prime }\right);\end{array}$$

and

$$\begin{array}{l}\left(\gamma X \mu \right)\left[\left(z,z^{\prime }\right)\left(w,w^{\prime }\right)\right]=\left(\gamma X \mu \right)\left(zw, z^{\prime }{w}^{\prime }\right);\\ = \bigvee \left\{\gamma \left(zw\right), \mu \left(z^{\prime }{W}^{\prime }\right)\right\};\\ \le \bigvee \left\{\bigvee \left\{\gamma \left(z\right), \gamma \left(w\right)\right\}, \bigvee \left\{\mu \left(z^{\prime }\right),\mu \left(w^{\prime }\right)\right\}\right\};\\ \le \bigvee \left\{\bigvee \left\{\gamma \left(z\right), \mu \left(z^{\prime }\right)\right\}, \bigvee \left\{\gamma \left(w\right),\mu \left(w^{\prime }\right)\right\}\right\};\\ =\bigvee \left\{\left(\gamma X \mu \right)\left(z,z^{\prime }\right),\left(\gamma X \mu \right)\left(w,w^{\prime }\right)\right\}.\end{array}$$

(iv)

Let ‘1’ be the unity in $L$. Since $\left({\tilde{\varrho }}_{\gamma },Y\right)$ and $\left({\tilde{h}}_{\mu },Y^{\prime }\right)$ are H N A over $\left({\tilde{\zeta }}_{\lambda },L\right),$

$$\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right),\forall z\in Y, \lambda \left(1\right)\le \gamma \left(z\right), \tilde{\zeta }\left(1\right)\supseteq \tilde{h}\left(z^{\prime }\right),\forall z^{\prime }\in Y^{\prime } \mathrm{and} \lambda \left(1\right)\le \mu \left(z\prime \right).$$

Then, $\tilde{\zeta }\left(1\right)\supseteq \tilde{\varrho }\left(z\right){\displaystyle \cap }\tilde{h}\left(z^{\prime }\right)=\left(\tilde{\varrho }\mathrm{X}\tilde{h}\right)\left(z,z^{\prime }\right)$ and $\lambda \left(1\right)\le \bigvee \left\{\gamma \left(z\right),\mu \left(z^{\prime }\right)\right\}=\left(\gamma X \mu \right)\left(z,z^{\prime }\right).$

Hence, ${\tilde{\varrho }}_{\gamma }X{\tilde{h}}_{\mu }$ is an H N A of $YX{Y}^{\prime }$ over an H F $\left({\tilde{\zeta }}_{\lambda },L\right).$ □

4. Conclusions

In this paper, we introduced the notion of H N A over an H F and investigated several properties. Using these notions, we introduced the concepts of H N A homomorphism and the direct product of H N A. Research can be extended to the ideal of an H N A and hybrid gamma near algebra.