GE-Algebras Advanced by Intuitionistic Fuzzy Points

Abstract

In this paper, we introduce the notion of intuitionistic fuzzy GE-algebra by combining the concepts of GE-algebras and intuitionistic fuzzy sets. We provide a necessary and sufficient condition for an intuitionistic fuzzy set to form an intuitionistic fuzzy GE-algebra. This study examines various properties and characterizations of intuitionistic fuzzy GE-algebra. In particular, we explore the roles of ${({\gimel }_A,t)}_{\in },{({ð}_A,s)}^{\in },{({\gimel }_A,t)}_q,{({ð}_A,s)}^q,{({\gimel }_A,t)}_{\in \vee q}$, and ${({ð}_A,s)}^{\in \vee q}$ sets in determining the subalgebra structures within GE-algebras. Examples illustrate the results, and counterexamples clarify the necessity of the conditions. These results not only enhance the theory of GE-algebras, but also contribute to the algebraic treatment of uncertainty using intuitionistic fuzzy logic.

1. Introduction

The study of algebraic structures under fuzzy environments has attracted substantial interest due to their applications in logic, artificial intelligence, and uncertainty modeling. Among the earliest algebraic structures to be explored in this direction are BCK-algebras, which were originally introduced to formalize implications in logic. The generalization of such structures through fuzzy set theory, and more notably through intuitionistic fuzzy set theory, has led to significant advancements in the field.

Atanassov introduced the concept of intuitionistic fuzzy sets (IFSs) as a generalization of Zadeh’s fuzzy sets [1,2]. Unlike traditional fuzzy sets that assign a single membership degree to each element, IFSs assign both a membership and a non-membership degree, constrained such that their sum does not exceed one. This additional dimension provides a more expressive framework for capturing uncertainty, hesitation, and incomplete information in mathematical models.

The fusion of IFSs with BCK-algebras has led to the emergence of several intuitionistic fuzzy generalizations. These include intuitionistic fuzzy ideals of BCK-algebras [3]. Further generalizations have been carried out for related structures such as BZ-algebras [4] and BE-algebras [5], enriching the study of logical algebras under fuzzy and intuitionistic fuzzy frameworks. Recent progress demonstrates the applicability of intuitionistic fuzzy sets in advanced mathematical models. For example, Shagari et al. [6] established common fixed-point results for intuitionistic fuzzy hybrid contractions in b-metric spaces, illustrating how intuitionistic fuzzy sets play a crucial role in fixed-point theory and its applications.

GE-algebras, a class of non-commutative algebras that generalize implication-based systems like BCK-algebras, have recently been introduced as an extension with applications to more flexible logical operations [7]. GE-algebras satisfy unique axioms different from those of classical implication algebras and provide a more general setting for studying algebraic operations in uncertain environments.

This paper focuses on applying intuitionistic fuzzy set theory to the structure of GE-algebras. We define the notion of an intuitionistic fuzzy GE-algebra and establish various characterizations, including necessary and sufficient conditions. The study explores how ${({\gimel }_A,t)}_{\in },{({ð}_A,s)}^{\in },{({\gimel }_A,t)}_q,{({ð}_A,s)}^q,{({\gimel }_A,t)}_{\in \vee q}$, and ${({ð}_A,s)}^{\in \vee q}$ sets associated with intuitionistic fuzzy points influence the substructure and properties of GE-algebras.

Illustrative examples are used to support the theoretical developments, while counterexamples are provided to demonstrate the importance and necessity of the assumptions. This research bridges the existing gap between the theory of GE-algebras and intuitionistic fuzzy logic, building on prior works in fuzzy topology [8] and Boolean systems [9], and contributes to the broader endeavor of extending logical algebraic systems through advanced fuzzy frameworks.

2. Preliminaries

Definition 1

([7]). A GE-algebra, denoted by $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$, is defined as a non-empty set G with a constant 1 and a binary operation ⋇ satisfying the following axioms:

(GE1) $\mathfrak{a}\divideontimes \mathfrak{a}=1$,

(GE2) $1\divideontimes \mathfrak{a}=\mathfrak{a}$,

(GE3) $\mathfrak{a}\divideontimes (\mathfrak{b}\divideontimes \mathfrak{c})=\mathfrak{a}\divideontimes (\mathfrak{b}\divideontimes (\mathfrak{a}\divideontimes \mathfrak{c}\left)\right)$

for all $\mathfrak{a},\mathfrak{b},\mathfrak{c}\in G$.

In a GE-algebra $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$, a binary relation ${\le }_G$ is defined by

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b}\in G)\left(\mathfrak{a}{\le }_G\mathfrak{b}\iff \mathfrak{a}\divideontimes \mathfrak{b}=1\right).\end{array}$$

(1)

Proposition 2

([7]). Every GE-algebra $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ satisfies the following items.

$$\begin{array}{cc}\hfill & (\forall \mathfrak{a}\in G)\left(\mathfrak{a}\divideontimes 1=1\right).\hfill \end{array}$$

(2)

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b}\in G)\left(\mathfrak{a}\divideontimes (\mathfrak{a}\divideontimes \mathfrak{b})=\mathfrak{a}\divideontimes \mathfrak{b}\right).\end{array}$$

(3)

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b}\in G)\left(\mathfrak{a}{\le }_G\mathfrak{b}\divideontimes \mathfrak{a}\right).\end{array}$$

(4)

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b},\mathfrak{c}\in G)\left(\mathfrak{a}\divideontimes (\mathfrak{b}\divideontimes \mathfrak{c}){\le }_G\mathfrak{b}\divideontimes (\mathfrak{a}\divideontimes \mathfrak{c})\right).\end{array}$$

(5)

$$\begin{array}{c}\hfill (\forall \mathfrak{a}\in G)\left(1{\le }_G\mathfrak{a}\Rightarrow \mathfrak{a}=1\right).\end{array}$$

(6)

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b}\in G)\left(\mathfrak{a}{\le }_G(\mathfrak{b}\divideontimes \mathfrak{a})\divideontimes \mathfrak{a}\right).\end{array}$$

(7)

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b}\in G)\left(\mathfrak{a}{\le }_G(\mathfrak{a}\divideontimes \mathfrak{b})\divideontimes \mathfrak{b}\right).\end{array}$$

(8)

$$\begin{array}{c}\hfill (\forall \mathfrak{a},\mathfrak{b},\mathfrak{c}\in G)\left(\mathfrak{a}{\le }_G\mathfrak{b}\divideontimes \mathfrak{c}\iff \mathfrak{b}{\le }_G\mathfrak{a}\divideontimes \mathfrak{c}\right).\end{array}$$

(9)

Definition 3

([7]). A subset F of a GE-algebra $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ is called a sub-GE-algebra of $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ if $\mathfrak{a}\divideontimes \mathfrak{b}\in F$ for all $\mathfrak{a},\mathfrak{b}\in F$.

Let G be a set. An intuitionistic fuzzy set $\tilde{A}$ in G (see [1]) is an object having the form

$$\begin{array}{c}\hfill \tilde{A}:=\{\langle \mathfrak{a},{\gimel }_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{a}\right)\rangle \mid {\gimel }_A\left(\mathfrak{a}\right)+{ð}_A\left(\mathfrak{a}\right)\le 1,\mathfrak{a}\in G\},\end{array}$$

which is simply denoted by $\tilde{A}:=({\gimel }_A,{ð}_A)$ where ${\gimel }_A$ and ${ð}_A$ are fuzzy sets in G,

The intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G can be represented as follows:

$$\begin{array}{c}\hfill \tilde{A}:=({\gimel }_A,{ð}_A):G\to [0,1]\times [0,1],\mathfrak{a}\mapsto ({\gimel }_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right))\end{array}$$

such that ${\gimel }_A\left(\mathfrak{a}\right)+{ð}_A\left(\mathfrak{b}\right)\le 1$.

An intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in a set G of the form

$$\begin{array}{c}\hfill \tilde{A}:=({\gimel }_A,{ð}_A):G\to [0,1]\times [0,1],\mathfrak{b}\mapsto \left\{\begin{array}{cc}(t,s)\in (0,1]\times [0,1)\hfill & \mathrm{if}\mathfrak{b}=\mathfrak{a},\hfill \\ (0,1)\hfill & \mathrm{if}\mathfrak{b}\ne \mathfrak{a},\hfill \end{array}\right.\end{array}$$

is said to be an intuitionistic fuzzy point with support $\mathfrak{a}$ and values $(t,s)$ such that $t+s\le 1$, and is denoted by $\langle {\mathfrak{a}}_{(t,s)}\rangle .$

Given an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ and intuitionistic fuzzy point $\langle {\mathfrak{a}}_{(t,s)}\rangle$ in G, we say

$$\begin{array}{cc}\hfill & \langle {\mathfrak{a}}_{(t,s)}\rangle \in \tilde{A}\mathrm{if}{\gimel }_A\left(\mathfrak{a}\right)\ge t\mathrm{and}{ð}_A\left(\mathfrak{a}\right)\le s.\hfill \end{array}$$

(10)

$$\begin{array}{c}\hfill \langle {\mathfrak{a}}_{(t,s)}\rangle q\tilde{A}\mathrm{if}{\gimel }_A\left(\mathfrak{a}\right)+t>1\mathrm{and}{ð}_A\left(\mathfrak{a}\right)+s<1.\end{array}$$

(11)

$$\begin{array}{c}\hfill \langle {\mathfrak{a}}_{(t,s)}\rangle \in \vee q\tilde{A}\mathrm{if}\langle {\mathfrak{a}}_{(t,s)}\rangle \in \tilde{A}\mathrm{or}\langle {\mathfrak{a}}_{(t,s)}\rangle q\tilde{A}.\end{array}$$

(12)

Given $(t,s)\in (0,1]\times [0,1)$ and an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G, consider the following sets:

$$\begin{array}{cc}\hfill & {({\gimel }_A,t)}_{\in }:=\{\mathfrak{a}\in G\mid {\gimel }_A\left(\mathfrak{a}\right)\ge t\},\hfill \\ & {({ð}_A,s)}^{\in }:=\{\mathfrak{a}\in G\mid {ð}_A\left(\mathfrak{a}\right)\le s\},\hfill \\ & {({\gimel }_A,t)}_q:=\{\mathfrak{a}\in G\mid {\gimel }_A\left(\mathfrak{a}\right)+t>1\},\hfill \\ & {({ð}_A,s)}^q:=\{\mathfrak{a}\in G\mid {ð}_A\left(\mathfrak{a}\right)+s<1\},\hfill \\ & {({\gimel }_A,t)}_{\in \vee q}:=\{\mathfrak{a}\in G\mid {\gimel }_A\left(\mathfrak{a}\right)\ge t\mathrm{or}{\gimel }_A\left(\mathfrak{a}\right)+t>1\},\hfill \\ & {({ð}_A,s)}^{\in \vee q}:=\{\mathfrak{a}\in G\mid {ð}_A\left(\mathfrak{a}\right)\le s\mathrm{or}{ð}_A\left(\mathfrak{a}\right)+s<1\},\hfill \end{array}$$

which are called lower ∈-set, upper ∈-set, lower q-set, upper q-set, lower $\in \vee q$-set, and upper $\in \vee q$-set of ${\gimel }_A$ and ${ð}_A$, respectively.

3. Intuitionistic Fuzzy GE-Algebras

In what follows, let $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ (or simply $\mathcal{G}$) denote a GE-algebra. Also, $(t,s)$ and $(t_i,s_i)$ are elements of $(0,1]\times [0,1)$ that satisfy $t+s\le 1$ and $t_i+s_i\le 1$, respectively, for $i=1,2,3,\cdots$.

First, we introduce a central concept that will be used throughout the paper.

Definition 4.

An intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G is called an intuitionistic fuzzy GE-algebra of $\mathcal{G}$ if it satisfies

$$\begin{array}{c}\hfill \begin{array}{c}x\in {({\gimel }_A,t_1)}_{\in }\cap {({ð}_A,s_1)}^{\in },y\in {({\gimel }_A,t_2)}_{\in }\cap {({ð}_A,s_2)}^{\in }\hfill \\ ⊢x\divideontimes y\in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\cap {({ð}_A,max\{s_1,s_2\})}^{\in }\hfill \end{array}\end{array}$$

(13)

for all $x,y\in G$.

Example 5.

Let $G=\{0,1,2,3,4,5\}$ be a set with a binary operation ⋇ given by

Table 1. Cayley table for the binary operation ⋇.

⋇	0	1	2	3	4	5
0	1	1	2	3	2	1
1	0	1	2	3	4	5
2	5	1	1	1	5	5
3	0	1	1	1	0	0
4	1	1	1	3	1	1
5	1	1	2	3	2	1

.

Then, it is routine to verify that $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ is a GE-algebra. Let $\tilde{A}:=({\gimel }_A,{ð}_A)$ be an intuitionistic fuzzy set in G given by

Table 2. Tabular representation of intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$.

G	0	1	2	3	4	5
${\gimel }_A\left(\mathfrak{a}\right)$	$0.6$	$0.8$	$0.6$	$0.6$	$0.8$	$0.6$
${ð}_A\left(\mathfrak{b}\right)$	$0.3$	$0.1$	$0.2$	$0.3$	$0.2$	$0.1$

.

Then, it is routine to verify that $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$.

Theorem 6.

An intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$ if and only if it satisfies

$$\begin{array}{c}\hfill (\forall x,y\in G)\left(\begin{array}{c}{\gimel }_A(x\divideontimes y)\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\},\hfill \\ {ð}_A(x\divideontimes y)\le max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}\hfill \end{array}\right).\end{array}$$

(14)

Proof. 

Assume that $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$. Let $x,y\in G$ be such that $(t_1,s_1)=({\gimel }_A\left(x\right),{ð}_A\left(x\right))$ and $(t_2,s_2)=({\gimel }_A\left(y\right),{ð}_A\left(y\right))$. Then $x\in {({\gimel }_A,t_1)}_{\in }\cap {({ð}_A,s_1)}^{\in }$ and $y\in {({\gimel }_A,t_2)}_{\in }\cap {({ð}_A,s_2)}^{\in }$. It follows from (13) that

$$\begin{array}{c}\hfill x\divideontimes y\in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\cap {({ð}_A,max\{s_1,s_2\})}^{\in },\end{array}$$

and so ${\gimel }_A(x\divideontimes y)\ge min\{t_1,t_2\}=min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}$ and

$${ð}_A(x\divideontimes y)\le max\{s_1,s_2\}=max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}.$$

Conversely, suppose that $\tilde{A}:=({\gimel }_A,{ð}_A)$ satisfies (14). Let $x,y\in G$ be such that $x\in {({\gimel }_A,t_1)}_{\in }\cap {({ð}_A,s_1)}^{\in }$ and $y\in {({\gimel }_A,t_2)}_{\in }\cap {({ð}_A,s_2)}^{\in }.$ Then ${\gimel }_A\left(x\right)\ge t_1$, ${ð}_A\left(x\right)\le s_1$, ${\gimel }_A\left(y\right)\ge t_2$, and ${ð}_A\left(y\right)\le s_2.$ It follows from (14) that

$${\gimel }_A(x\divideontimes y)\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}\ge min\{t_1,t_2\}$$

and ${ð}_A(x\divideontimes y)\le max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}\le max\{s_1,s_2\}$. Hence

$$x\divideontimes y\in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\cap {({ð}_A,max\{s_1,s_2\})}^{\in },$$

and therefore $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$. □

Corollary 7.

If $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$, then ${\gimel }_A\left(1\right)\ge {\gimel }_A\left(x\right)$ and ${ð}_A\left(1\right)\le {ð}_A\left(x\right)$ for all $x\in G$.

Theorem 8.

An intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$ if and only if the nonempty lower ∈-set ${({\gimel }_A,t)}_{\in }$ and the nonempty upper ∈-set ${({ð}_A,s)}^{\in }$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$.

Proof. 

Assume that $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$. Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$. Then, ${\gimel }_A\left(x\right)\ge t,$ ${\gimel }_A\left(y\right)\ge t,$ ${ð}_A\left(\mathfrak{a}\right)\le s,$ and ${ð}_A\left(\mathfrak{b}\right)\le s.$ It follows from Theorem 6 that

$$\begin{array}{cc}\hfill & {\gimel }_A(x\divideontimes y)\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}\ge t,\hfill \\ & {ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})\le max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}\le s.\hfill \end{array}$$

Hence, $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$, and, therefore, ${({\gimel }_A,t)}_{\in }$ and ${({ð}_A,s)}^{\in }$ are GE-subalgebras of $\mathcal{G}$.

Conversely, suppose that the nonempty lower ∈-set ${({\gimel }_A,t)}_{\in }$ and the nonempty upper ∈-set ${({ð}_A,s)}^{\in }$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$. If there exist $x,y\in G$, such that ${\gimel }_A(x\divideontimes y)<min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}$ or

$${ð}_A(x\divideontimes y)>max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\},$$

then $x,y\in {({\gimel }_A,t)}_{\in }\cap {({ð}_A,s)}^{\in }$, and $x\divideontimes y\notin {({\gimel }_A,t)}_{\in }$ or $x\divideontimes y\notin {({ð}_A,s)}^{\in }$ where $t=min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}$ and $s=max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}.$ This is a contradiction, and thus ${\gimel }_A(x\divideontimes y)\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}$ and ${ð}_A(x\divideontimes y)\le max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}.$ It follows from Theorem 6 that $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$. □

Theorem 9.

If $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$, then the sets

$$\begin{array}{c}\hfill G_0=\{x\in G\mid {\gimel }_A\left(x\right)>0\}\mathrm{and}{G}^1=\{x\in G\mid {ð}_A\left(x\right)<1\}\end{array}$$

are GE-subalgebras of $\mathcal{G}$.

Proof. 

Let $x,y\in G_0\cap G^1$. Then, ${\gimel }_A\left(x\right)>0,$ ${\gimel }_A\left(y\right)>0,$ ${ð}_A\left(x\right)<1,$ and ${ð}_A\left(y\right)<1.$ It follows from (14) that ${\gimel }_A(x\divideontimes y)\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}>0$ and ${ð}_A(x\divideontimes y)\le max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}<1.$ Hence, $x\divideontimes y\in G_0\cap G^1$, and thus $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$. □

The following example shows that the converse of Theorem 9 is generally not true.

Example 10.

Let $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ be a GE-algebra defined in Example 5. Let $\tilde{A}:=({\gimel }_A,{ð}_A)$ be an intuitionistic fuzzy set in G given by

Table 3. Tabular representation of intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$.

G	0	1	2	3	4	5
${\gimel }_A\left(\mathfrak{a}\right)$	0	$0.3$	$0.6$	$0.8$	0	0
${ð}_A\left(\mathfrak{b}\right)$	1	$0.3$	$0.2$	$0.1$	1	1

.

Clearly, $G_0=\{1,2,3\}$ and $G^1=\{1,2,3\}$ which are GE-subalgebras of $\mathcal{G}$. But $\tilde{A}:=({\gimel }_A,{ð}_A)$ is not an intuitionistic fuzzy GE-algebra of $\mathcal{G}$ since

$${\gimel }_A(2\divideontimes 3)={\gimel }_A\left(1\right)=0.3\ngeqq 0.6=min\{0.6,0.8\}=min\{{\gimel }_A\left(2\right),{\gimel }_A\left(3\right)\}$$

and/or ${ð}_A(3\divideontimes 2)={\gimel }_A\left(1\right)=0.3\nleqq 0.2=max\{0.1,0.2\}=max\{{\gimel }_A\left(3\right),{\gimel }_A\left(2\right)\}.$

We explore the conditions under which $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$.

Theorem 11.

If an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies

$$\begin{array}{c}\hfill \begin{array}{c}x\in {({\gimel }_A,t_1)}_{\in }\cap {({ð}_A,s_1)}^{\in },y\in {({\gimel }_A,t_2)}_{\in }\cap {({ð}_A,s_2)}^{\in }\hfill \\ ⊢x\divideontimes y\in {({\gimel }_A,max\{t_1,t_2\})}_q\cap {({ð}_A,min\{s_1,s_2\})}^q\hfill \end{array}\end{array}$$

(15)

for all $x,y\in G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$, then $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$.

Proof. 

We first note that the condition (15) means

$$\begin{array}{cc}\hfill & x\in {({\gimel }_A,t_1)}_{\in },y\in {({\gimel }_A,t_2)}_{\in }⊢x\divideontimes y\in {({\gimel }_A,max\{t_1,t_2\})}_q,\hfill \\ & \mathfrak{a}\in {({ð}_A,s_1)}^{\in },\mathfrak{b}\in {({ð}_A,s_2)}^{\in }⊢\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,min\{s_1,s_2\})}^q\hfill \end{array}$$

(16)

for all $x,y,\mathfrak{a},\mathfrak{b}\in G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$. Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in G_0\times G^1$. Note that

$$(x,y)\in {({\gimel }_A,{\gimel }_A\left(x\right))}_{\in }\times {({\gimel }_A,{\gimel }_A\left(y\right))}_{\in }$$

and $(\mathfrak{a},\mathfrak{b})\in {({ð}_A,{ð}_A\left(\mathfrak{a}\right))}^{\in }\times {({ð}_A,{ð}_A\left(\mathfrak{b}\right))}^{\in }$. It follows from (16) that

$$x\divideontimes y\in {({\gimel }_A,max\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_q$$

and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,min\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^q$. If $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\notin G_0\times G^1$, then ${\gimel }_A(x\divideontimes y)=0$ or ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})=1$. Hence

$${\gimel }_A(x\divideontimes y)+max\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}=max\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}\le 1$$

or ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})+min\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}=1+min\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}\ge 1$, that is,

$$x\divideontimes y\notin {({\gimel }_A,max\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_q$$

or $\mathfrak{a}\divideontimes \mathfrak{b}\notin {({ð}_A,min\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^q$. This is a contradiction, and so $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})$∈$G_0\times G^1$. Therefore, $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$. □

Theorem 12.

If an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies

$$\begin{array}{c}\hfill \begin{array}{c}(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_q\times {({ð}_A,s_1)}^q,(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q\hfill \\ ⊢(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,max\{t_1,t_2\})}_q\times {({ð}_A,min\{s_1,s_2\})}^q\hfill \end{array}\end{array}$$

(17)

for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$, then $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$.

Proof. 

Let $x,y,\mathfrak{a},\mathfrak{b}\in G$ be such that $(x,\mathfrak{a}),(y,\mathfrak{b})\in G_0\times G^1$. Then, ${\gimel }_A\left(x\right)\ne 0\ne {\gimel }_A\left(y\right)$ and ${ð}_A\left(\mathfrak{a}\right)\ne 1\ne {ð}_A\left(\mathfrak{b}\right)$. Hence, $(x,\mathfrak{a}),(y,\mathfrak{b})\in {({\gimel }_A,1)}_q\times {({ð}_A,0)}^q$, and so $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,1)}_q\times {({ð}_A,0)}^q$ by (17). If $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\notin G_0\times G^1$, then ${\gimel }_A(x\divideontimes y)=0$ or ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})=1$. Thus, $x\divideontimes y\notin {({\gimel }_A,1)}_q$ or $\mathfrak{a}\divideontimes \mathfrak{b}\notin {({ð}_A,0)}^q$, a contradiction. Hence, $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\notin G_0\times G^1$, and therefore, $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$. □

We look at the conditions under which an intuitionistic fuzzy set can be an intuitionistic fuzzy GE-algebra.

Theorem 13.

If an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies

$$\begin{array}{c}\hfill \begin{array}{c}(y,\mathfrak{b})\in {({\gimel }_A,t_1)}_{\in }\times {({ð}_A,s_1)}^{\in },(z,\mathfrak{c})\in {({\gimel }_A,t_2)}_{\in }\times {({ð}_A,s_2)}^{\in }\hfill \\ ⊢(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\times {({ð}_A,max\{s_1,s_2\})}^{\in }\hfill \end{array}\end{array}$$

(18)

for all $(x,\mathfrak{a}),(y,\mathfrak{b}),(z,\mathfrak{c})\in G\times G$ with $x{\le }_{G}z$ and $\mathfrak{a}{\le }_G\mathfrak{c}$, then $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$.

Proof. 

Let $x\in {({\gimel }_A,t_1)}_{\in }\cap {({ð}_A,s_1)}^{\in }$ and $y\in {({\gimel }_A,t_2)}_{\in }\cap {({ð}_A,s_2)}^{\in }$. Then, $(x,x)\in {({\gimel }_A,t_1)}_{\in }\times {({ð}_A,s_1)}^{\in }$ and $(y,y)\in {({\gimel }_A,t_2)}_{\in }\times {({ð}_A,s_2)}^{\in }$. Since $x{\le }_{G}x$ for all $x\in G$, it follows from (18) that $(x\divideontimes y,x\divideontimes y)\in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\times {({ð}_A,max\{s_1,s_2\})}^{\in }$. Thus $x\divideontimes y\in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\cap {({ð}_A,max\{s_1,s_2\})}^{\in }$; therefore, $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$. □

The following example shows that the converse of Theorem 13 may not be true, that is, there exists an intuitionistic fuzzy GE-algebra of $\mathcal{G}$ such that (18) does not hold for some $(x,\mathfrak{a}),(y,\mathfrak{b}),(z,\mathfrak{c})\in G\times G$ with $x{\le }_{G}z$ and $\mathfrak{a}{\le }_G\mathfrak{c}$.

Example 14.

Consider the intuitionistic fuzzy GE-algebra $\tilde{A}:=({\gimel }_A,{ð}_A)$ of $\mathcal{G}$ in Example 5. We can observe that $0{\le }_{G}1$, $3{\le }_{G}2$, $(4,5)\in {({\gimel }_A,0.78)}_{\in }\times {({ð}_A,0.18)}^{\in }$ and $(1,2)\in {({\gimel }_A,0.72)}_{\in }\times {({ð}_A,0.23)}^{\in }$. But

$$(0\divideontimes 4,3\divideontimes 5)=(2,0)\notin {({\gimel }_A,min\{0.78,0.72\})}_{\in }\times {({ð}_A,max\{0.18,0.23\})}^{\in }.$$

Given an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G, consider the following assertion:

$$\begin{array}{c}\hfill (\forall x,y,\mathfrak{a},\mathfrak{b}\in G)\left(\begin{array}{c}min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}\le max\{{\gimel }_A(x\divideontimes y),0.5\}\hfill \\ max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}\ge min\{{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b}),0.5\}\hfill \end{array}\right).\end{array}$$

(19)

Theorem 15.

An intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies the condition (19) if and only if the nonempty lower ∈-set ${({\gimel }_A,t)}_{\in }$ and the nonempty upper ∈-set ${({ð}_A,s)}^{\in }$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$ for all $(t,s)\in (0.5,1]\times [0,0.5)$.

Proof. 

Assume that $\tilde{A}:=({\gimel }_A,{ð}_A)$ satisfies the condition (19). Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$ for $(t,s)\in (0.5,1]\times [0,0.5)$. Then ${\gimel }_A\left(x\right)\ge t$, ${\gimel }_A\left(y\right)\ge t$, ${ð}_A\left(\mathfrak{a}\right)\le s$ and ${ð}_A\left(\mathfrak{b}\right)\le s$. It follows from (19) that

$$max\{{\gimel }_A(x\divideontimes y),0.5\}\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}\ge t>0.5$$

and $min\{{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b}),0.5\}\le max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}\le s<0.5.$ Hence, $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$, and thus ${({\gimel }_A,t)}_{\in }$ and ${({ð}_A,s)}^{\in }$ are GE-subalgebras of $\mathcal{G}$.

Conversely, suppose that the nonempty lower ∈-set ${({\gimel }_A,t)}_{\in }$ and the nonempty upper ∈-set ${({ð}_A,s)}^{\in }$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$ for all $(t,s)\in (0.5,1]\times [0,0.5)$. If the condition (19) is not valid, then

$$min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}>max\{{\gimel }_A(x\divideontimes y),0.5\}$$

or $max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}<min\{{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b}),0.5\}$ for some $x,y,\mathfrak{a},\mathfrak{b}\in G$. Taking $t=min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}$ and $s=max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}$ induces $(t,s)\in (0.5,1]\times [0,0.5)$ and $(x,\mathfrak{a}),(y,\mathfrak{b})\in {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$. Since ${({\gimel }_A,t)}_{\in }$ and ${({ð}_A,s)}^{\in }$ are GE-subalgebras of $\mathcal{G}$, it follows that $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$. On the other hand, since $max\{{\gimel }_A(x\divideontimes y),0.5\}<t$ or $min\{{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b}),0.5\}>s$, we have $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\notin {({\gimel }_A,t)}_{\in }\times {({ð}_A,s)}^{\in }$, which is a contradiction. Therefore, the condition (19) is valid. □

Theorem 16.

If $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$, then the nonempty lower q-set ${({\gimel }_A,t)}_q$ and the nonempty upper q-set ${({ð}_A,s)}^q$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$ for all $(t,s)\in (0,1]\times [0,1)$.

Proof. 

Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in {({\gimel }_A,t)}_q\times {({ð}_A,s)}^q$ for $(t,s)\in (0,1]\times [0,1)$. Then ${\gimel }_A\left(x\right)+t>1$, ${\gimel }_A\left(y\right)+t>1$, ${ð}_A\left(\mathfrak{a}\right)+t<1$ and ${ð}_A\left(\mathfrak{b}\right)+t<1$. It follows from Theorem 6 that ${\gimel }_A(x\divideontimes y)+t\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}+t=min\{{\gimel }_A\left(x\right)+t,{\gimel }_A\left(y\right)+t\}>1$ and ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})+s\le max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}+s=max\{{ð}_A\left(\mathfrak{a}\right)+s,{ð}_A\left(\mathfrak{b}\right)+s\}<1.$ Hence, $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t)}_q\times {({ð}_A,s)}^q,$ which shows that ${({\gimel }_A,t)}_q$ and ${({ð}_A,s)}^q$ are GE-subalgebras of $\mathcal{G}$ for all $(t,s)\in (0,1]\times [0,1)$ when ${({\gimel }_A,t)}_q\ne \varnothing$ and ${({ð}_A,s)}^q\ne \varnothing .$ □

Proposition 17.

Given an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G, if the nonempty lower q-set ${({\gimel }_A,t)}_q$ and the nonempty upper q-set ${({ð}_A,s)}^q$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$, then $\tilde{A}:=({\gimel }_A,{ð}_A)$ satisfies

$$\begin{array}{c}\hfill \begin{array}{c}(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_q\times {({ð}_A,s_1)}^q,(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q\hfill \\ ⊢(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,max\{t_1,t_2\})}_{\in }\times {({ð}_A,min\{s_1,s_2\})}^{\in }\hfill \end{array}\end{array}$$

(20)

for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,0.5]\times [0.5,1)$.

Proof. 

Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,0.5]\times [0.5,1)$ be such that $(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_q\times {({ð}_A,s_1)}^q$ and $(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q$. Then,

$$(x,\mathfrak{a}),(y,\mathfrak{b})\in {({\gimel }_A,max\{t_1,t_2\})}_q\times {({ð}_A,min\{s_1,s_2\})}^q,$$

which implies from the hypothesis that

$$\begin{array}{c}\hfill (x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,max\{t_1,t_2\})}_q\times {({ð}_A,min\{s_1,s_2\})}^q.\end{array}$$

Since $max\{t_1,t_2\}\le 0.5\le min\{s_1,s_2\}$, we have

$$\begin{array}{c}\hfill {\gimel }_A(x\divideontimes y)>1-max\{t_1,t_2\}\ge max\{t_1,t_2\}\end{array}$$

and

$$\begin{array}{c}\hfill {ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})<1-min\{s_1,s_2\}\le min\{s_1,s_2\}.\end{array}$$

Hence $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,max\{t_1,t_2\})}_{\in }\times {({ð}_A,min\{s_1,s_2\})}^{\in }$. □

The following example shows that the converse of Proposition 17 is generally not true, that is, the nonempty lower q-set ${({\gimel }_A,t)}_q$ of ${\gimel }_A$ or the nonempty upper q-set ${({ð}_A,s)}^q$ of ${ð}_A$ may not be a GE-subalgebra of $\mathcal{G}$ for an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfying (20) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),(t_2,s_2)\in (0,0.5]\times [0.5,1)$.

Example 18.

Consider a GE-algebra $\mathcal{G}$ $:=$ $(G,$ $\divideontimes ,$ $1)$ given in Example 5, and let $\tilde{A}:=({\gimel }_A,{ð}_A)$ be an intuitionistic fuzzy set in G given by

Table 4. Tabular representation of intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$.

G	0	1	2	3	4	5
${\gimel }_A\left(\mathfrak{a}\right)$	1	$0.5$	$0.3$	$0.3$	$0.3$	$0.3$
${ð}_A\left(\mathfrak{b}\right)$	0	$0.5$	$0.6$	$0.6$	$0.6$	$0.6$

.

Then, $\tilde{A}:=({\gimel }_A,{ð}_A)$ satisfies (20) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,0.5]\times [0.5,1)$. But ${({\gimel }_A,t)}_q={({ð}_A,s)}^q=\left\{0\right\}$, for $t\in (0,0.5]$ and $s\in [0.5,1)$, is not a GE-subalgebra of G since $0\divideontimes 0=1\notin {({ð}_A,s)}^q$ and $0\divideontimes 0=1\notin {({\gimel }_A,t)}_q$.

Theorem 19.

If an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies the condition (20) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$, then the sets $G_0$ and $G^1$, which are described in Theorem 9 are GE-subalgebras of $\mathcal{G}$.

Proof. 

Let $x,y,\mathfrak{a},\mathfrak{b}\in G$ be such that $(x,\mathfrak{a}),(y,\mathfrak{b})\in G_0\times G^1$. Since $(x,\mathfrak{a})\in {({\gimel }_A,1)}_q\times {({ð}_A,0)}^q$ and $(y,\mathfrak{b})\in {({\gimel }_A,1)}_q\times {({ð}_A,0)}^q$, we have $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,1)}_{\in }\times {({ð}_A,0)}^{\in }$ by (20). If $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\notin G_0\times G^1$, then $x\divideontimes y\notin G_0$ or $\mathfrak{a}\divideontimes \mathfrak{b}\notin G^1$. Hence, ${\gimel }_A(x\divideontimes y)=0$ or ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})=1$, which is a contradiction. Therefore, $G_0$ and $G^1$ are GE-subalgebras of $\mathcal{G}$. □

Proposition 20.

Given a GE-subalgebra F of $\mathcal{G}$, let $\tilde{A}:=({\gimel }_A,{ð}_A)$ be an intuitionistic fuzzy set in G such that

(a)

${\gimel }_A\left(x\right)=0$ and ${ð}_A\left(x\right)=1$ for all $x\in G\setminus F$,

(b)

${\gimel }_A\left(x\right)\ge 0.5$ and ${ð}_A\left(x\right)\le 0.5$ for all $x\in F$,

then $\tilde{A}:=({\gimel }_A,{ð}_A)$ satisfies

$$\begin{array}{c}\hfill \begin{array}{c}(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_q\times {({ð}_A,s_1)}^q,(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q\hfill \\ ⊢(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}\hfill \end{array}\end{array}$$

(21)

for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$.

Proof. 

Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$ be such that $(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_q\times {({ð}_A,s_1)}^q$ and $(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q$. Then ${\gimel }_A\left(x\right)+t_1>1,$ ${\gimel }_A\left(y\right)+t_2>1,$ ${ð}_A\left(\mathfrak{a}\right)+s_1<1$ and ${ð}_A\left(\mathfrak{b}\right)+s_2<1$. If $x\divideontimes y\notin F$, then $x\notin F$ or $y\notin F$. Hence ${\gimel }_A\left(x\right)=0$ or ${\gimel }_A\left(y\right)=0$, and so $t_1={\gimel }_A\left(x\right)+t_1>1$ or $t_2={\gimel }_A\left(y\right)+t_2>1$ which is a contradiction. Also, if $\mathfrak{a}\divideontimes \mathfrak{b}\notin F$, then $\mathfrak{a}\notin F$ or $\mathfrak{b}\notin F$. Thus ${ð}_A\left(\mathfrak{a}\right)=1$ or ${ð}_A\left(\mathfrak{b}\right)=1$, which implies that $1>{ð}_A\left(\mathfrak{a}\right)+s_1=1+s_1$ or $1>{ð}_A\left(\mathfrak{b}\right)+s_2=1+s_2$, a contradiction. Therefore, $x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b}\in F$, which implies from the second condition that ${\gimel }_A(x\divideontimes y)\ge 0.5\ge {ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})$. Now we have to consider the following cases:

(i)

$min\{t_1,t_2\}\le 0.5$ and $max\{s_1,s_2\}\ge 0.5$.

(ii)

$min\{t_1,t_2\}>0.5$ and $max\{s_1,s_2\}<0.5$.

(iii)

$min\{t_1,t_2\}\le 0.5$ and $max\{s_1,s_2\}<0.5$.

(iv)

$min\{t_1,t_2\}>0.5$ and $max\{s_1,s_2\}\ge 0.5$.

For the first case, we obtain ${\gimel }_A(x\divideontimes y)\ge 0.5\ge min\{t_1,t_2\}$ and $max\{s_1,s_2\}\ge 0.5\ge {ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})$. Hence

$$\begin{array}{cc}\hfill (x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})& \in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\times {({ð}_A,max\{s_1,s_2\})}^{\in }\hfill \\ & \subseteq {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}.\hfill \end{array}$$

For the second case, we have ${\gimel }_A(x\divideontimes y)+min\{t_1,t_2\}>1$ and ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})+max\{s_1,s_2\}<1$. Thus

$$\begin{array}{cc}\hfill (x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})& \in {({\gimel }_A,min\{t_1,t_2\})}_q\times {({ð}_A,max\{s_1,s_2\})}^q\hfill \\ & \subseteq {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}.\hfill \end{array}$$

Now, if the case (iii) is valid, then ${\gimel }_A(x\divideontimes y)\ge 0.5\ge min\{t_1,t_2\}$ and

$${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})+max\{s_1,s_2\}<1.$$

It follows that

$$\begin{array}{cc}\hfill (x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})& \in {({\gimel }_A,min\{t_1,t_2\})}_{\in }\times {({ð}_A,max\{s_1,s_2\})}^q\hfill \\ & \subseteq {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}.\hfill \end{array}$$

Lastly, if the case (iv) is valid, then ${\gimel }_A(x\divideontimes y)+min\{t_1,t_2\}>1$ and

$${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})\le 0.5\le max\{s_1,s_2\}.$$

Hence

$$\begin{array}{cc}\hfill (x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})& \in {({\gimel }_A,min\{t_1,t_2\})}_q\times {({ð}_A,max\{s_1,s_2\})}^{\in }\hfill \\ & \subseteq {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}.\hfill \end{array}$$

This completes the proof. □

Proposition 21.

If an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies (21) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$, then it also satisfies

$$\begin{array}{c}\hfill \begin{array}{c}(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_{\in }\times {({ð}_A,s_1)}^{\in },(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_{\in }\times {({ð}_A,s_2)}^{\in }\hfill \\ ⊢(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}\hfill \end{array}\end{array}$$

(22)

for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$.

Proof. 

Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$ be such that $(x,\mathfrak{a})\in {({\gimel }_A,t_1)}_{\in }\times {({ð}_A,s_1)}^{\in }$ and $(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_{\in }\times {({ð}_A,s_2)}^{\in }.$ Then, ${\gimel }_A\left(x\right)\ge t_1,$ ${ð}_A\left(\mathfrak{a}\right)\le s_1,$ ${\gimel }_A\left(y\right)\ge t_2,$ and ${ð}_A\left(\mathfrak{b}\right)\le s_2.$ If

$$(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\notin {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q},$$

then $x\divideontimes y\notin {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}$ or $\mathfrak{a}\divideontimes \mathfrak{b}\notin {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}$. It follows that

$$\begin{array}{c}\hfill {\gimel }_A(x\divideontimes y)<min\{t_1,t_2\}\mathrm{and}{\gimel }_A(x\divideontimes y)+min\{t_1,t_2\}\le 1\end{array}$$

or

$$\begin{array}{c}\hfill {ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})>max\{s_1,s_2\}\mathrm{and}{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})+max\{s_1,s_2\}\ge 1.\end{array}$$

Hence ${\gimel }_A(x\divideontimes y)<min\{t_1,t_2,0.5\}$ or ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})>max\{s_1,s_2,0.5\}.$ Therefore,

$$\begin{array}{cc}\hfill 1-{\gimel }_A(x\divideontimes y)& \ge t>1-min\{t_1,t_2,0.5\}=max\{1-t_1,1-t_2,0.5\}\hfill \\ & \ge max\{1-{\gimel }_A\left(x\right),1-{\gimel }_A\left(y\right),0.5\}\hfill \end{array}$$

for some $t\in (0,1]$, or

$$\begin{array}{cc}\hfill 1-{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})& \le s<1-max\{s_1,s_2,0.5\}=min\{1-s_1,1-s_2,0.5\}\hfill \\ & \le min\{1-{ð}_A\left(\mathfrak{a}\right),1-{ð}_A\left(\mathfrak{b}\right),0.5\}\hfill \end{array}$$

for some $s\in [0,1)$. It follows that

$$\begin{array}{c}\hfill {\gimel }_A\left(x\right)+t>1\mathrm{and}{\gimel }_A\left(y\right)+t>1,\mathrm{that}\mathrm{is},x,y\in {({\gimel }_A,t)}_q\end{array}$$

or

$$\begin{array}{c}\hfill {ð}_A\left(\mathfrak{a}\right)+s<1\mathrm{and}{ð}_A\left(\mathfrak{b}\right)+s<1,\mathrm{that}\mathrm{is},\mathfrak{a},\mathfrak{b}\in {({ð}_A,s)}^q.\end{array}$$

Using (21) derives to $x\divideontimes y\in {({\gimel }_A,t)}_{\in \vee q}$ or $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^{\in \vee q}$. Thus ${\gimel }_A(x\divideontimes y)\ge t$, ${\gimel }_A(x\divideontimes y)+t>1,$ ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})\le s$, or ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})+s<1.$ This is a contradiction, and so $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,min\{t_1,t_2\})}_{\in \vee q}\times {({ð}_A,max\{s_1,s_2\})}^{\in \vee q}$, that is, (22) is valid. □

The combination of Propositions 20 and 21 induces the following corollary.

Corollary 22.

Given a GE-subalgebra F of $\mathcal{G}$, if an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies $(x,\mathfrak{a})\in {({\gimel }_A,0.5)}_{\in }\times {({ð}_A,0.5)}^{\in }$ for $x,\mathfrak{a}\in F$ and $({\gimel }_A\left(x\right),{ð}_A\left(\mathfrak{a}\right))=(0,1)$ for $x,\mathfrak{a}\in G\setminus F$, then it satisfies the condition (22) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$.

Theorem 23.

Let $\tilde{A}:=({\gimel }_A,{ð}_A)$ be an intuitionistic fuzzy set in G that satisfies ${\gimel }_A\left(x\right)\le 0.5$ and ${ð}_A\left(x\right)\ge 0.5$ for all $x\in G$. If $\tilde{A}$ satisfies (22) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$, then it is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$.

Proof. 

Let $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$. Since $(x,\mathfrak{a})$∈${({\gimel }_A,{\gimel }_A\left(x\right))}_{\in }\times {({ð}_A,{ð}_A\left(\mathfrak{a}\right))}^{\in }$ and $(y,\mathfrak{b})$∈${({\gimel }_A,{\gimel }_A\left(y\right))}_{\in }\times {({ð}_A,{ð}_A\left(\mathfrak{b}\right))}^{\in }$, we have $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_{\in \vee q}\times {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^{\in \vee q}$ by (22). It follows that

$$\begin{array}{c}\hfill x\divideontimes y\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_{\in }\mathrm{or}x\divideontimes y\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_q,\end{array}$$

and

$$\begin{array}{c}\hfill \mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^{\in }\mathrm{or}\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^q.\end{array}$$

Therefore, we have to deal with the following four cases:

(i)

$x\divideontimes y\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_{\in }$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^{\in }$,

(ii)

$x\divideontimes y\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_{\in }$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^q$,

(iii)

$x\divideontimes y\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_q$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^{\in }$,

(iv)

$x\divideontimes y\in {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_q$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^q$.

The first case means that ${\gimel }_A(x\divideontimes y)\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}$ and

$${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})\le max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}.$$

Since ${\gimel }_A\left(x\right)\le 0.5$ and ${ð}_A\left(x\right)\ge 0.5$ for all $x\in G$, we know that

$${({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_q\subseteq {({\gimel }_A,min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\})}_{\in }$$

and ${({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^q\subseteq {({ð}_A,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\})}^{\in }$. Hence, all three cases (ii), (iii), and (iv) reach the first case. Using Theorem 6, we conclude that $\tilde{A}:=({\gimel }_A,{ð}_A)$ is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$. □

Corollary 24.

Let $\tilde{A}:=({\gimel }_A,{ð}_A)$ be an intuitionistic fuzzy set in G that satisfies ${\gimel }_A\left(x\right)\le 0.5$ and ${ð}_A\left(x\right)\ge 0.5$ for all $x\in G$. If $\tilde{A}$ satisfies (21) for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t_1,s_1),$ $(t_2,s_2)$∈$(0,1]\times [0,1)$, then it is an intuitionistic fuzzy GE-algebra of $\mathcal{G}$.

Proof. 

This is straightforward by the combination of Proposition 21 and Theorem 23. □

Theorem 25.

If an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G satisfies

$$\begin{array}{c}\hfill \begin{array}{c}min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}>1-t,max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}<1-s\hfill \\ ⊢(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t)}_{\in \vee q}\times {({ð}_A,s)}^{\in \vee q}\hfill \end{array}\end{array}$$

(23)

for all $(x,\mathfrak{a}),(y,\mathfrak{b})\in G\times G$ and $(t,s)$∈$(0.5,1]\times [0,0.5)$, then the nonempty lower q-set ${({\gimel }_A,t)}_q$ and the nonempty upper q-set ${({ð}_A,s)}^q$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$ for all $(t,s)$∈$(0.5,1]\times [0,0.5)$.

Proof. 

Let $x,y,\mathfrak{a},\mathfrak{b}\in G$ and $(t,s)$∈$(0.5,1]\times [0,0.5)$ be such that $(x,\mathfrak{a})\in {({\gimel }_A,t)}_q\times {({ð}_A,s)}^q$ and $(y,\mathfrak{b})\in {({\gimel }_A,t)}_q\times {({ð}_A,s)}^q$. Then, ${\gimel }_A\left(x\right)>1-t,$ ${ð}_A\left(\mathfrak{a}\right)<1-s,$ ${\gimel }_A\left(y\right)>1-t,$ and ${ð}_A\left(\mathfrak{b}\right)<1-s.$ Hence, $min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}>1-t$ and $max\{{ð}_A\left(\mathfrak{a}\right),{ð}_A\left(\mathfrak{b}\right)\}<1-s$, which imply from (23) that

$$(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t)}_{\in \vee q}\times {({ð}_A,s)}^{\in \vee q}.$$

It follows that $x\divideontimes y\in {({\gimel }_A,t)}_{\in \vee q}={({\gimel }_A,t)}_{\in }\cup {({\gimel }_A,t)}_q$ and

$$\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^{\in \vee q}={({ð}_A,s)}^{\in }\cup {({ð}_A,s)}^q.$$

We should consider the following cases:

(i)

$x\divideontimes y\in {({\gimel }_A,t)}_{\in }$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^{\in }$.

(ii)

$x\divideontimes y\in {({\gimel }_A,t)}_{\in }$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^q$.

(iii)

$x\divideontimes y\in {({\gimel }_A,t)}_q$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^{\in }$.

(iv)

$x\divideontimes y\in {({\gimel }_A,t)}_q$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^q$.

The last case means that ${({\gimel }_A,t)}_q$ and ${({ð}_A,s)}^q$ are GE-subalgebras of $\mathcal{G}$. If $x\divideontimes y\in {({\gimel }_A,t)}_{\in }$ and $\mathfrak{a}\divideontimes \mathfrak{b}\in {({ð}_A,s)}^{\in }$, then ${\gimel }_A(x\divideontimes y)\ge t>1-t$ and ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})\le s<1-s$ since $(t,s)$∈$(0.5,1]\times [0,0.5)$. Thus, all three cases (i), (ii), and (iii) reach the last case. This completes the proof. □

Theorem 26.

Given an intuitionistic fuzzy set $\tilde{A}:=({\gimel }_A,{ð}_A)$ in G, if the nonempty lower ∈-set ${({\gimel }_A,t_1)}_{\in }$ and the nonempty upper ∈-set ${({ð}_A,s_1)}^{\in }$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$ for all $(t_1,s_1)$∈$(0.5,1]\times [0,0.5)$, then the nonempty lower q-set ${({\gimel }_A,t_2)}_q$ and the nonempty upper q-set ${({ð}_A,s_2)}^q$ of ${\gimel }_A$ and ${ð}_A$, respectively, are GE-subalgebras of $\mathcal{G}$ for all $(t_2,s_2)$∈$(0,0.5]\times [0.5,1)$.

Proof. 

Let $x,y,\mathfrak{a},\mathfrak{b}\in G$ and $(t_2,s_2)$∈$(0,0.5]\times [0.5,1)$ be such that $(x,\mathfrak{a})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q$ and $(y,\mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q$. Then ${\gimel }_A\left(x\right)+t_2>1,$ ${ð}_A\left(\mathfrak{a}\right)+s_2<1,$ ${\gimel }_A\left(y\right)+t_2>1,$ and ${ð}_A\left(\mathfrak{b}\right)+s_2<1.$ It follows from Theorem 15 that $max\{{\gimel }_A(x\divideontimes y),0.5\}\ge min\{{\gimel }_A\left(x\right),{\gimel }_A\left(y\right)\}>1-t_2\ge 0.5$ and

$$min\{{ð}_A(\mathfrak{a}\divideontimes \mathfrak{b}),0.5\}\le max\{{ð}_A\left(x\right),{ð}_A\left(y\right)\}<1-s_2\le 0.5.$$

Hence ${\gimel }_A(x\divideontimes y)>1-t_2$ and ${ð}_A(\mathfrak{a}\divideontimes \mathfrak{b})<1-s_2$, that is, $(x\divideontimes y,\mathfrak{a}\divideontimes \mathfrak{b})\in {({\gimel }_A,t_2)}_q\times {({ð}_A,s_2)}^q$. Therefore, ${({\gimel }_A,t_2)}_q$ and ${({ð}_A,s_2)}^q$ are GE-subalgebras of $\mathcal{G}$ for all $(t_2,s_2)$∈$(0,0.5]\times [0.5,1)$ when ${({\gimel }_A,t_2)}_q\ne \varnothing$ and ${({ð}_A,s_2)}^q$ $\ne \varnothing$. □

4. Conclusions

In this work, we successfully defined and explored the structure of intuitionistic fuzzy GE-algebras. We established several characterizations of such algebras, showing how the ${({\gimel }_A,t)}_{\in },{({ð}_A,s)}^{\in },{({\gimel }_A,t)}_q,{({ð}_A,s)}^q,{({\gimel }_A,t)}_{\in \vee q}$ and ${({ð}_A,s)}^{\in \vee q}$ sets interact with the algebraic structure. The main results demonstrate that intuitionistic fuzzy GE-algebras preserve the GE-subalgebra structure under specific fuzzy constraints. Additionally, necessary and sufficient conditions have been provided for various generalized substructures. The examples and counterexamples further solidify the theoretical foundations laid out. These investigations may stimulate further research in the study of fuzzy algebraic systems, particularly those involving intuitionistic and type-2 fuzzy extensions of logical algebras. In this paper, we have concentrated only on the algebraic aspects of intuitionistic fuzzy GE-algebras. A natural continuation of this research would be to explore concrete applications of the developed theorems, for instance in uncertainty modeling, knowledge representation, and computational logic systems. We plan to address such applications in future work.

The notion of intuitionistic fuzzy GE-algebras could potentially contribute to fixed-point theory, especially in generalized metric spaces. For example, the fixed point results in intuitionistic fuzzy frameworks developed by Shagari et al. [6] suggest possible extensions of our results to BE-algebraic and GE-algebraic structures. Exploring such connections remains an interesting avenue for further research.