Fuzzy Set Theory Applied on Autometrized Algebra

Gebrie Yeshiwas Tilahun

doi:10.12688/f1000research.161249.1

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Research Article

Fuzzy Set Theory Applied on Autometrized Algebra

[version 1; peer review: 2 approved, 1 approved with reservations]

Gebrie Yeshiwas Tilahun

PUBLISHED 04 Feb 2025

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Department of Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia

Gebrie Yeshiwas Tilahun
Roles: Writing – Review & Editing

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Abstract

This paper introduces fuzzy subalgebras of autometrized algebras and studies their properties. Also, we present fuzzy ideals of autometrized algebras and provide examples to illustrate our findings. We examine the homomorphisms of both the images and the inverse images of fuzzy subalgebras and ideals. Furthermore, we introduce fuzzy congruences on autometrized algebras. We prove that in normal autometrized algebra, the set of all fuzzy ideals is in one-to-one correspondence with the set of all fuzzy congruences.

Keywords

autometrized algebra, fuzzy subalgebra, fuzzy ideal, fuzzy congruence

Corresponding author: Gebrie Yeshiwas Tilahun

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2025 Tilahun GY. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Tilahun GY. Fuzzy Set Theory Applied on Autometrized Algebra [version 1; peer review: 2 approved, 1 approved with reservations]. F1000Research 2025, 14:159 (https://doi.org/10.12688/f1000research.161249.1) First published: 04 Feb 2025, 14:159 (https://doi.org/10.12688/f1000research.161249.1) Latest published: 04 Feb 2025, 14:159 (https://doi.org/10.12688/f1000research.161249.1)

1. Introduction

Ref. 1 introduced the notion of an autometrized algebra to obtain a unified theory of the then-known autometrized algebras: Boolean algebras,²^,³ Brouwerian algebras,⁴ Newman algebras,⁵ autometrized lattices⁴ and commutative lattice ordered groups or l-groups.⁶ Ideals and congruence in autometrized algebras were introduced and studied by Ref. 7. Further development of the autometrized algebra theory was done by Refs. 7–14. Moreover, the notion of representable autometrized algebras explored by Refs. 15–18.

In their research Refs. 19–22, established the theory of subalgebras, ideals, and homomorphisms. They analyzed the link between normal autometrized l-algebras and representable autometrized algebras. Additionally Ref. 23, introduced the homomorphism, isomorphism, and correspondence theorems of autometrized algebra by using congruency,²⁴ introduced fuzzy sets,²⁵ presented a fuzzy group,²⁶ investigated fuzzy groups and fuzzy relations.

The previous studies did not investigate fuzzy subalgebra, fuzzy ideals, and fuzzy congruence of autometrized algebra. This paper introduces the concept of fuzzy subalgebras and fuzzy ideals of autometrized algebras. It will also present and prove various theorems and facts related to fuzzy subalgebras and fuzzy ideals of autometrized algebras. Additionally, the study includes fuzzy congruence relations on autometrized algebras.

The paper will have the following structure. Section 2 will provide definitions and terms. Section 3 will introduce the concept of fuzzy subalgebras of autometrized algebras. In Section 4, we introduce fuzzy ideals of autometrized algebras. Section 5 will cover fuzzy congruences of autometrized algebras and present many basic facts related to the idea. Finally, in Section 6, we will conclude the paper.

In this paper, $A$ and $B$ denote autometrized algebras $(A, +, 0, \leq, \otimes)$ and $(B, +, 0, \leq, \otimes)$ , respectively.

2. Preliminaries

In this section, we’ll look at some essential concepts, definitions, and terms; that are important in other sections.

Definition 2.1

¹ A system $A = (A, +, 0, \leq, \otimes)$ is called an autometrized algebra if

(i) $(A, +, 0)$ is a commutative monoid.
(ii) $(A, \leq)$ is a partial ordered set, and $\leq$ is translation invariant, that is, $\forall γ_{1}, γ_{2}, γ_{3} \in A; γ_{1} \leq γ_{2} \Rightarrow γ_{1} + γ_{3} \leq γ_{2} + γ_{3}$ .
(iii) $\otimes : A \times A \to A$ is autometric on $A$ , that is, $\otimes$ satisfies metric operation axioms:
- (a) $\forall γ_{1}, γ_{2} \in A; γ_{1} \otimes γ_{2} \geq 0$ and, $γ_{1} \otimes γ_{2} = 0 \Leftrightarrow γ_{1} = γ_{2}$ ,
- (b) $\forall γ_{1}, γ_{2} \in A; γ_{1} \otimes γ_{2} = γ_{2} \otimes γ_{1}$ ,
- (c) $\forall γ_{1}, γ_{2}, γ_{3} \in A$ ; $γ_{1} \otimes γ_{3} \leq γ_{1} \otimes γ_{2} + γ_{2} \otimes γ_{3}$ .

Definition 2.2

⁷ $A$ is called normal if and only if

(i) $γ_{1} \leq γ_{1} \otimes 0 \forall γ_{1} \in A$ .
(ii) $(γ_{1} + γ_{3}) \otimes (γ_{2} + γ_{4}) \leq (γ_{1} \otimes γ_{2}) + (γ_{3} \otimes γ_{4}) \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ .
(iii) $(γ_{1} \otimes γ_{3}) \otimes (γ_{2} \otimes γ_{4}) \leq (γ_{1} \otimes γ_{2}) + (γ_{3} \otimes γ_{4}) \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ .
(iv) For any $γ_{1}$ and $γ_{2}$ in $A$ , $γ_{1} \leq γ_{2} \Rightarrow \exists γ_{3} \geq 0$ such that $γ_{1} + γ_{3} = γ_{2}$ .

Definition 2.3

¹⁹ Let $B \subseteq A$ . Then $B$ is said to be a subalgebra of $A$ if;

(i) $(B, +, 0)$ is a commutative monoid.
(ii) $(B, \leq)$ is a subposet, and $\leq$ is translation invariant, that is, $γ_{1} \leq γ_{2} \Rightarrow γ_{1} + γ_{3} \leq γ_{2} + γ_{3}$ for any $γ_{1}, γ_{2}, γ_{3} \in B$ .
(iii) $\otimes | B : B \times B \to B$ is metric.

Definition 2.4

¹⁹ A nonempty subset $I$ of $A$ is called an ideal if and only if

(i) $γ_{1}, γ_{2} \in I$ imply $γ_{1} + γ_{2} \in I$ .
(ii) $γ_{1} \in I, γ_{2} \in A and γ_{2} \otimes 0 \leq γ_{1} \otimes 0$ imply $γ_{2} \in I$ .

Definition 2.5

¹⁹ Every ideal of a normal autometrized algebra $A$ is a subalgebra of $A$ .

Lemma 2.6

⁷ In a normal autometrized algebra, the intersection of any nonempty collection of ideals in $A$ is again ideal.

Definition 2.7

⁷ $A$ is said to be semiregular if for any $γ \in A$ , $γ \geq 0 \Rightarrow γ \otimes 0 = γ$ .

Definition 2.8

⁷ Let $A$ is a normal autometrized algebra. An equivalence relation $Φ$ on $A$ is called a congruence relation if and only if

(i) $(γ_{1}, γ_{2}), (γ_{3}, γ_{4}) \in Φ \Rightarrow (γ_{1} + γ_{3}, γ_{2} + γ_{4}) \in Φ \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ ,
(ii) $(γ_{1}, γ_{2}), (γ_{3}, γ_{4}) \in Φ \Rightarrow (γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \in Φ \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ ,
(iii) $(γ_{3}, γ_{4}) \in Φ and γ_{1} \otimes γ_{2} \leq γ_{3} \otimes γ_{4} \Rightarrow (γ_{1}, γ_{2}) \in Φ \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ .

Definition 2.9

¹⁹ Let $f : A \to B$ be a map. Then $f$ is said to be a homomorphism from $A$ to $B$ if and only if

(i) (i) $f (γ_{1} + γ_{2}) = f (γ_{1}) + f (γ_{2}) \forall x, y \in A$ ,
(ii) $f (γ_{1} \otimes γ_{1}) = f (γ_{1}) \otimes f (γ_{2}) \forall γ_{1}, γ_{2} \in A$ and
(iii) $γ_{1} \leq γ_{2} \Rightarrow f (γ_{1}) \leq f (γ_{2}) \forall γ_{1}, γ_{2} \in A$ .

A homomorphism $f : A \to B$ is called an epimorphism if and only if $f$ is onto.

Theorem 2.10

¹⁹ Let $f : A \to B$ be a homomorphism. Let $I$ be an ideal of $B$ . Then, $N = f^{- 1} (I) = {γ \in A | f (γ) \in I}$ is an ideal of $A$ .

Theorem 2.11

¹⁹ Let $f : A \to B$ be an epimorphism and order-reversing. Let $I$ be an ideal of $A$ . Then, $L = f (I) = {f (γ) \in B | γ \in I}$ is an ideal of $B$ .

Theorem 2.12

¹⁹ Let $f : A \to B$ be a homomorphism. Then

(a) if $H$ is a subalgebra of $A \Rightarrow f (H)$ is a subalgebra of $B$ .
(b) if $H$ is a subalgebra of $B \Rightarrow f^{- 1} (H)$ is a subalgebra of $A$ .

Definition 2.13

²⁴ Let $A$ be a nonempty set, a fuzzy subset $ζ$ of $A$ is a mapping $ζ : A \to [0, 1]$ .

Definition 2.14

²⁴ Let $A$ be a nonempty set and $μ$ be a fuzzy subset of $A$ , for $π \in [0, 1]$ , the set $ζ_{π} = {γ \in A | ζ (γ) \geq π}$ is called a level subset of $ζ$ .

Definition 2.15

²⁵ A fuzzy subset $ζ$ of a set $A$ has $sup$ property if for any subset $T$ of $A$ , there exist $π_{0} \in T$ such that $ζ (π_{0}) = \sup {ζ (π) | π \in T}$ .

Definition 2.16

²⁵ Let $f : A \to B$ be a mapping nonempty sets $A$ and $B$ respectively. If $ζ$ is a fuzzy subset of $A$ , then the fuzzy subset $α$ of $B$ defined by:

f (ζ) (γ_{2}) = {\begin{cases} \sup {ζ (γ_{1}) | γ_{1} \in f^{- 1} (γ_{2})}, & {iff}^{- 1} (γ_{2}) = {γ_{1} \in A, f (γ_{1}) = γ_{2}} \neq \emptyset . \\ 0, & otherwise . \end{cases}

is said to be the image of

ζ

under

f

.

Similarly, if $α$ is a fuzzy subset of $B$ , then the fuzzy subset $ζ = (α ◦ f)$ of $A$ (that is, the fuzzy subset defined by $ζ (γ) = α (f (γ))$ for all $γ \in A$ ) is called the pre-image of $α$ under $f$ .

3. Fuzzy subalgebra of autometrized algebra

In this section, we will introduce the fuzzy subalgebras of autometrized algebras and examine their properties.

Definition 3.1

A fuzzy subset $ζ$ of $A$ is called a fuzzy subalgebra of $A$ if for all $γ_{1}, γ_{2} \in A$ ;

(i) $ζ (0) \geq ζ (γ_{1})$ .
(ii) $ζ (γ_{1} + γ_{2}) \geq \min {ζ (γ_{1}), ζ (γ_{2})}$ .
(iii) $ζ (γ_{1} \otimes γ_{2}) \geq \min {ζ (γ_{1}), ζ (γ_{2})}$ .

Theorem 3.2

Let $ζ$ be a fuzzy subset of $A$ . Then $ζ$ is a fuzzy subalgebra of $A$ if and only if for any $π \in [0, 1]$ , $ζ_{π}$ is a subalgebra of $A$ .

Proof.

Assume that $ζ$ is a fuzzy subalgebra of $A$ .

(i) Since $ζ (0) \geq ζ (γ)$ for all $γ \in A$ ; $ζ (0) \geq ζ (γ) \geq π$ for all $γ \in ζ_{π}$ . Therefore, $0 \in ζ_{π}$ .
(ii) Let $γ_{1}, γ_{2} \in A$ . Suppose that $γ_{1}, γ_{2} \in ζ_{π}$ . So, $ζ (γ_{1}) \geq π$ and $ζ (γ_{2}) \geq π$ . Since $ζ$ is a fuzzy subalgebra of $A$ ; $ζ (γ_{1} + γ_{2}) \geq min {ζ (γ_{1}), ζ (γ_{2})} \geq π$ . This implies that $γ_{1} + γ_{2} \in ζ_{π}$ .
(iii) Let $γ_{1}, γ_{2} \in A$ . Suppose that $γ_{1}, γ_{2} \in ζ_{π}$ . So, $ζ (γ_{1}) \geq π$ and $ζ (γ_{2}) \geq π$ . Since $ζ$ is a fuzzy subalgebra of $A$ ; $ζ (γ_{1} \otimes γ_{2}) \geq min {ζ (γ_{1}), ζ (γ_{2})} \geq π$ . This implies that $γ_{1} \otimes γ_{2} \in ζ_{π}$ . Hence $ζ_{π}$ is a subalgebra of $A$ .

Conversely, assume that $ζ_{π}$ is a subalgebra of $A$ . To show that $ζ$ is a fuzzy subalgebra of $A$ .

(i) To show that $ζ (0) \geq ζ (γ)$ for all $γ \in A$ . Assume that it is false. Therefore, there exists $γ^{'} \in A$ such that $ζ (0) < ζ (γ)$ . Take $π^{'} = \frac{ζ (γ^{'}) + ζ (0)}{2}$ . Then, $ζ (0) < π^{'}$ and $0 \leq π^{'} \leq ζ (γ^{'}) \leq 1$ . Therefore, $γ^{'} \in ζ_{π^{'}}$ . So, $ζ_{π^{'}} \neq \emptyset$ . As $ζ_{π^{'}}$ is a subalgebra of $A$ , we have $0 \in ζ_{π^{'}}$ . So, $ζ (0) \geq π^{'}$ . This is contradiction.
(ii) Let $γ_{1}, γ_{2} \in A$ . Take $π = min {ζ (γ_{1}), ζ (γ_{1})}$ . Therefore, $γ_{1} \in ζ_{π}$ and $γ_{2} \in ζ_{π}$ . Since $ζ_{π}$ is a subalgebra of $A$ ; $γ_{1} + γ_{2} \in ζ_{π}$ . As a result, $ζ (γ_{1} + γ_{2}) \geq π = min ζ (γ_{1}), ζ (γ_{2})$ .
(iii) Let $γ_{1}, γ_{2} \in A$ . Take $π = min {ζ (γ_{1}), ζ (γ_{2})}$ . Therefore, $γ_{1} \in ζ_{π}$ and $γ_{2} \in ζ_{π}$ . Since $ζ_{π}$ is a subalgebra of $A$ ; $γ_{1} \otimes γ_{2} \in ζ_{π}$ . As a result, $ζ (γ_{1} \otimes γ_{2}) \geq π = min {ζ (γ_{1}), ζ (γ_{2})}$ .

Hence $ζ$ is a fuzzy subalgebra of $A$ .

Theorem 3.3

The intersection of any set of fuzzy subalgebras of $A$ is also fuzzy subalgebra of $A$ .

Proof.

Let ${ζ_{i} | i \in I}$ be a family of fuzzy subalgebras of autometrized algebra $A$ . To show that $\cap_{i \in I} ζ_{i}$ is an ideal. Then for any $γ_{1}, γ_{2} \in A$ , $i \in I$ ,

(a)
$\begin{array}{l} (\cap_{i \in I} ζ_{i}) (0) = inf (ζ_{i} (0)) . \\ \geq inf (ζ_{i} (γ_{1})) . \\ = (\cap_{i \in I} ζ_{i}) (γ_{1}) . \end{array}$
(b)
$\begin{array}{l} (\cap_{i \in I} ζ_{i}) (γ_{1} + γ_{2}) = inf (ζ_{i} (γ_{1} + γ_{2})) . \\ \geq inf (min {ζ_{i} (γ_{1}), ζ_{i} (γ_{2})}) . \\ = min {inf (ζ_{i} (γ_{1})), inf (ζ_{i} (γ_{2}))} . \\ = min {(\cap_{i \in I} ζ_{i}) (γ_{1}), (\cap_{i \in I} ζ_{i}) (γ_{2})} . \end{array}$
(c)
$\begin{array}{l} (\cap_{i \in I} ζ_{i}) (γ_{1} \otimes γ_{2}) = inf (ζ_{i} (γ_{1} \otimes γ_{2})) . \\ \geq inf (min {ζ_{i} (γ_{1}), ζ_{i} (γ_{2})}) . \\ = min {inf (ζ_{i} (γ_{1})), inf (ζ_{i} (γ_{2}))} . \\ = min {(\cap_{i \in I} ζ_{i}) (γ_{1}), (\cap_{i \in I} ζ_{i}) (γ_{2})} . \end{array}$

Remark 3.4

The union of fuzzy subalgebras of $A$ is not a fuzzy subalgebra of $A$ .

Example 3.5

Let $A = {0, γ_{1}, γ_{2}, γ_{3}}$ with $0 \leq γ_{1}, γ_{2} \leq γ_{3}$ and elements $γ_{1}, γ_{2}$ are incomparable. Define $\otimes$ and $+$ by the following tables.

$\otimes$	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
0	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$γ_{1}$	$γ_{1}$	0	$γ_{3}$	$γ_{2}$
$γ_{2}$	$γ_{2}$	$γ_{3}$	0	$γ_{1}$
$γ_{3}$	$γ_{3}$	$γ_{2}$	$γ_{1}$	0

$+$	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
0	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$γ_{1}$	$γ_{1}$	$γ_{1}$	$γ_{3}$	$γ_{3}$
$γ_{2}$	$γ_{2}$	$γ_{3}$	$γ_{2}$	$γ_{3}$
$γ_{3}$	$γ_{3}$	$γ_{3}$	$γ_{3}$	$γ_{3}$

Then, $A$ is an autometrized algebra. Define a fuzzy subset $ζ$ and $α$ of $A$ by:

	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$ζ$	0.5	0.3	0.5	0.3
$α$	0.5	0.5	0.4	0.4
$ζ \cup α$	0.5	0.5	0.5	0.4

Then $ζ$ and $α$ are fuzzy subalgebras of $A$ . But the union $ζ \cup α$ is not a fuzzy subalgebras of $A$ . Since $ζ \cup α (γ_{1} \otimes γ_{2}) = ζ \cup α (γ_{3}) = 0.4 < 0.5 = min {ζ \cup α (γ_{1}), ζ \cup α (γ_{2})}$ .

Remark 3.6

The union of any chain set of fuzzy subalgebras of $A$ is also fuzzy subalgebra of $A$ .

Theorem 3.7

A homomorphic pre-image of a fuzzy subalgebra of $A$ is also a fuzzy subalgebra of $A$ .

Proof.

Let $f : A \to B$ be a homomorphism. Let $α$ a fuzzy subalgebra of $B$ and $ζ$ the pre-image of $α$ under $f$ , then $α (f (γ)) = ζ (γ)$ for all $γ \in A$ . To show that $ζ$ is a fuzzy subalgebra of $A$ .

(i) Since $f (γ) \in B$ and $α$ is a fuzzy subalgebra of $B$ ; $α (0^{'}) \geq α (f (γ)) = ζ (γ)$ , for every $γ \in A$ , where $0^{'}$ is the zero element of $B$ . But $α (0^{'}) = α (f (0)) = ζ (0)$ . Therefore, $ζ (0) \geq ζ (γ)$ for any $γ \in A$ .
(ii) Let $γ_{1}, γ_{2} \in A$ .

\begin{array}{l} ζ (γ_{1} + γ_{2}) = α (f (γ_{1} + γ_{2})) . \\ = α (f (γ_{1}) + f (γ_{2})) . \\ \geq min {α (f (γ_{1})), α (f (γ_{2}))} . \\ = min {ζ (γ_{1}), ζ (γ_{2})} . \end{array}

(iii) Let $γ_{1}, γ_{2} \in A$ .

\begin{array}{l} ζ (γ_{1} \otimes γ_{2}) = α (f (γ_{1} \otimes γ_{2})) . \\ = α (f (γ_{1}) \otimes f (γ_{2})) . \\ \geq min {α (f (γ_{1})), α (f (γ_{2}))} . \\ = min {ζ (γ_{1}), ζ (γ_{2})} . \end{array}

Hence, $ζ$ is a fuzzy subalgebra of $A$ .

Theorem 3.8

Let $f : A \to B$ be an epimorphism. For every fuzzy subalgebra $ζ$ of $A$ with sup property, $f (ζ)$ is a fuzzy subalgebra of $B$ .

Proof.

To show that $f (ζ)$ is a fuzzy subalgebra of $B$ . By definition $α (γ_{2}^{'}) = f (ζ) (γ_{2}^{'}) = sup {ζ (γ_{1}) | γ_{1} \in f^{- 1} (γ_{2}^{'})}$ , for all $γ_{2}^{'} \in B$ ( $sup \emptyset = 0$ ).

(i) Since $ζ$ is a fuzzy subalgebra of $A$ ; we have $ζ (0) \geq ζ (γ)$ , for every $γ \in A$ . We know that $0 \in f^{- 1} (0^{'})$ , where $0$ and $0^{'}$ are the zero elements of $A$ and $B$ respectively. Therefore, $α (0^{'}) = {sup}_{π \in f^{- 1} (0^{'})} ζ (π) = ζ (0) \geq ζ (γ)$ for all $γ \in A$ . Which implies that: $α (0^{'}) \geq {sup}_{π \in f^{- 1} (γ^{'})} ζ (π) = α (γ^{'})$ for any $γ^{'} \in B$ .
(ii) Let $γ^{'}, η^{'} \in B$ . Then, there exist $γ_{0}, η_{0} \in A$ such that $f (γ_{0}) = γ^{'}$ and $f (η_{0}) = η^{'}$ . So, $γ_{0} \in f^{- 1} (γ^{'})$ and $η_{0} \in f^{- 1} (η^{'})$ . Now, consider
$\begin{array}{l} ζ (γ_{0} + η_{0}) = α (f (γ_{0} + η_{0})) = α (f (γ_{0}) + f (η_{0})) = α (γ^{'} + η^{'}) = sup_{γ_{0} + η_{0} \in f^{- 1} (γ^{'} + η^{'})} ζ (γ_{0} + η_{0}) . \\ ζ (γ_{0}) = α (f (γ_{0})) = α (γ^{'}) = sup_{γ_{0} \in f^{- 1} (γ^{'})} ζ (γ_{0}) . \end{array}$
Also,
$\begin{array}{l} α (γ^{'} \otimes η^{'}) = sup_{π \in f^{- 1} (γ^{'} + η^{'})} ζ (π) . \\ \geq min {ζ (γ_{0}), ζ (η_{0})} . \\ = min {sup_{π \in f^{- 1} (γ^{'})} ζ (γ_{0}), sup_{π \in f^{- 1} (η^{'})} ζ (η_{0})} . \\ = min {α (γ^{'}), α (η^{'})} . \end{array}$
(iii) Let $γ^{'}, η^{'} \in B$ . Then, there exist $γ_{0}, η_{0} \in A$ such that $f (γ_{0}) = γ^{'}$ and $f (η_{0}) = η^{'}$ . So, $γ_{0} \in f^{- 1} (γ^{'})$ and $η_{0} \in f^{- 1} (η^{'})$ . Now, consider
$\begin{array}{l} ζ (γ_{0} \otimes η_{0}) = α (f (γ_{0} \otimes η_{0})) = α (f (γ_{0}) \otimes f (η_{0})) = α (γ^{'} \otimes η^{'}) = sup_{γ_{0} \otimes η_{0} \in f^{- 1} (γ^{'} \otimes η^{'})} ζ (γ_{0} \otimes η_{0}) . \\ ζ (γ_{0}) = α (f (γ_{0})) = α (γ^{'}) = sup_{γ_{0} \in f^{- 1} (γ^{'})} ζ (γ_{0}) . \end{array}$
Also,
$\begin{array}{l} α (γ^{'} \otimes η^{'}) = sup_{π \in f^{- 1} (γ^{'} \otimes η^{'})} ζ (π) . \\ \geq min {ζ (γ_{0}), ζ (η_{0})} . \\ = min {sup_{π \in f^{- 1} (γ^{'})} ζ (γ_{0}), sup_{π \in f^{- 1} (η^{'})} ζ (η_{0})} . \\ = min {α (γ^{'}), α (η^{'})} . \end{array}$

Hence, $α$ is a fuzzy subalgebra of $B$ .

4. Fuzzy ideals of autometrized algebra

This section introduces the notion of fuzzy ideals of autometrized algebras and examines several basic properties related to fuzzy ideals.

Definition 4.1

A fuzzy subset $ζ$ of $A$ is called a fuzzy ideal of $A$ if it satisfies the following conditions: for all $γ_{1}, γ_{2} \in A$ ;

(i) $ζ (γ_{1} + γ_{2}) \geq \min {ζ (γ_{1}), ζ (γ_{2})}$ .
(ii) if $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ , then $ζ (γ_{1}) \geq ζ (γ_{2})$ .

Example 4.2

Let $A = {0, γ_{1}, γ_{2}, γ_{3}}$ with $0 \leq γ_{1}, γ_{2} \leq γ_{3}$ and elements $γ_{1}, γ_{2}$ are incomparable. Define $\otimes$ and $+$ by the following tables.

$\otimes$	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
0	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$γ_{1}$	$γ_{1}$	0	$γ_{3}$	$γ_{2}$
$γ_{2}$	$γ_{2}$	$γ_{3}$	0	$γ_{1}$
$γ_{3}$	$γ_{3}$	$γ_{2}$	$γ_{1}$	0

$+$	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
0	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$γ_{1}$	$γ_{1}$	$γ_{1}$	$γ_{3}$	$γ_{3}$
$γ_{2}$	$γ_{2}$	$γ_{3}$	$γ_{2}$	$γ_{3}$
$γ_{3}$	$γ_{3}$	$γ_{3}$	$γ_{3}$	$γ_{3}$

Then, $A$ is an autometrized algebra. Define a fuzzy subset $ζ : A \to [0, 1]$ by, $ζ (0) = ζ (γ_{1}) = 0.5$ and $ζ (γ_{2}) = ζ (γ_{3}) = 0.2$ . Here, $ℐ = {0, γ_{1}}$ is an ideal of $A$ . It is clear that $ζ$ is a fuzzy ideal of $A$ .

Theorem 4.3

Let $ζ$ be a fuzzy subset of $A$ . Then $ζ$ is a fuzzy ideal of $A$ if and only if for any $π \in [0, 1]$ , $ζ_{π}$ is an ideal of $A$ .

Proof.

Assume that $ζ$ is a fuzzy ideal of $A$ .

(i) Since $0 \leq γ \otimes 0$ for any $γ \in A$ ; by properties of fuzzy ideal $ζ (0) \geq ζ (γ)$ for all $γ \in A$ . Therefore, $ζ (0) \geq ζ (γ) \geq π$ for $γ \in ζ_{π}$ . Therefore, $0 \in ζ_{π}$ .
(ii) Let $γ_{1}, γ_{2} \in A$ . Suppose that $γ_{1}, γ_{2} \in ζ_{π}$ . So, $ζ (γ_{1}) \geq π$ and $ζ (γ_{2}) \geq π$ . Since $ζ$ is a fuzzy ideal of $A$ ; $ζ (γ_{1} + γ_{2}) \geq min {ζ (γ_{1}), ζ (γ_{2})} \geq π$ . This implies that $γ_{1} + γ_{2} \in ζ_{π}$ .
(iii) Let $γ_{1}, γ_{2} \in A$ such that $γ_{1} \in ζ_{π}$ . suppose $γ_{2} \otimes 0 \leq γ_{1} \otimes 0$ . Clearly, $ζ (γ_{1}) \geq π$ . Since $ζ$ is a fuzzy ideal of $A$ ; $ζ (γ_{2}) \geq ζ (γ_{1}) \geq π$ . This implies that $γ_{2} \in ζ_{π}$ . Hence $ζ_{π}$ is a an ideal of $A$ .

Conversely, assume that $ζ_{π}$ is an ideal of $A$ . To show that $ζ$ is a fuzzy ideal of $A$ .

(i) Let $γ_{1}, γ_{2} \in A$ . Take $π = min {ζ (γ_{1}), ζ (γ_{1})}$ . Therefore, $γ_{1} \in ζ_{π}$ and $γ_{2} \in ζ_{π}$ . Since $ζ_{π}$ is an ideal of $A$ ; $γ_{1} + γ_{2} \in ζ_{π}$ . As a result, $ζ (γ_{1} + γ_{2}) \geq π = min ζ (γ_{1}), ζ (γ_{2})$ .
(ii) Let $γ_{1}, γ_{2} \in A$ . Suppose $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ . Take $π = ζ (γ_{2})$ . So, $γ_{2} \in ζ_{π}$ . Since $ζ_{π}$ is an ideal; $γ_{1} \in ζ_{π}$ . As a result, $ζ (γ_{1}) \geq π = ζ (γ_{2})$ .

Hence $ζ$ is a fuzzy ideal of $A$ .

Theorem 4.4

Let $ζ$ be a fuzzy set of a normal autometrized algebra $A$ . Define a fuzzy set $α$ as: $α (x) = sup {min {ζ (γ_{1}), \dots, ζ (γ_{n})} | x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0) for some positive integers m_{1}, \dots, m_{k} and γ_{1}, \dots, γ_{k} \in A}$ .

Then $α$ is the smallest fuzzy ideal of $A$ that contains $ζ$ .

Proof.

Let $x, y \in A$ . This implies

x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0)

and,

y \otimes 0 \leq n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0)

for

m_{1}, \dots, m_{k}, n_{1}, \dots, n_{l}

are positive integers and

γ_{i}, η_{i} \in A

. By property of normal; we get:

\begin{array}{l} (x + y) \otimes 0 \leq (x \otimes 0) + (y \otimes 0) \\ \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0) + n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0) \end{array}

for

m_{1}, \dots, m_{k}, n_{1}, \dots, n_{l}

are positive integers and

γ_{i}, η_{i} \in A

.

Therefore $α (x + y) \geq min {ζ (γ_{1}), \dots, ζ (γ_{n}), ζ (η_{1}), \dots, ζ (η_{l})}$ .

Denote by $H = {min {ζ (η_{1}), \dots, ζ (η_{l})} | y \otimes 0 \leq n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0) for some positive integers n_{1}, \dots, n_{l} and η_{1}, \dots, η_{l} \in A}$ .

And $J = {min {ζ (γ_{1}), \dots, ζ (γ_{n})} | x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + n_{k} (γ_{k} \otimes 0) for some positive integers m_{1}, \dots, m_{k}$ $and γ_{1}, \dots, γ_{k} \in A}$ .

We have $min {α (x), α (y)} = min {sup H, sup J} = sup min {ζ (η_{1}), \dots, ζ (η_{l}), ζ (γ_{1}), \dots, ζ (γ_{n})} | y \otimes 0 \leq n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0), x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + n_{k} (γ_{k} \otimes 0) .$ for some positive integers $m_{1}, \dots, m_{k}, n_{1}, \dots, n_{l} and η_{1}, \dots, η_{l}, γ_{1}, \dots, γ_{k} \in A$ . Hence, $α (x + y) \geq min {α (x), α (y)}$ .

Let $x, y \in A$ and $y \otimes 0 \leq x \otimes 0$ . Also,

x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0)

and,

y \otimes 0 \leq n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0)

for

m_{1}, \dots, m_{k}, n_{1}, \dots, n_{l}

are positive integers and

γ_{i}, η_{i} \in A

. Clearly,

y \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0) + n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0) .

Therefore, $α (y) = sup {min {ζ (η_{1}), \dots, ζ (η_{l}), ζ (γ_{1}), \dots, ζ (γ_{n})} | y \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0) + n_{1} (η_{1} \otimes 0) + \dots + n_{l} (η_{l} \otimes 0)$ for some positive integers $m_{1}, \dots, m_{k}, n_{1}, \dots, n_{l} and η_{1}, \dots, η_{l}, γ_{1}, \dots, γ_{k} \in A}$ . And hence $α (y) \geq α (x)$ . Therefore, $α$ is a fuzzy ideal.

Let $γ \in A$ . We know that $γ \otimes 0 \leq 1 (γ \otimes 0)$ . We have $α (γ) \geq min {ζ (γ), ζ (γ)} = ζ (γ)$ . That is $α$ contains $ζ$ .

Let $μ$ be a fuzzy ideal of $A$ containing $ζ$ . To show that $α (x) \leq μ (x)$ . Let $x \in A$ . Therefore,

$α (x) = sup min {ζ (γ_{1}), \dots, ζ (γ_{n})} | x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0)$ for some positive integers $m_{1}, \dots, m_{k} and γ_{1}, \dots, γ_{k} \in A \leq sup min {μ (γ_{1}), \dots, μ (γ_{n})} | x \otimes 0 \leq m_{1} (γ_{1} \otimes 0) + \dots + m_{k} (γ_{k} \otimes 0)$ for some positive integers $m_{1}, \dots, m_{k} and γ_{1}, \dots, γ_{k} \in A \leq μ (x)$ . Hence $α$ is the smallest fuzzy ideal of $A$ that containing $ζ$ .

Let $I$ be a nonempty subset of $A$ and $α, β \in [0, 1]$ such that $α > β$ . Now we define fuzzy set $ζ_{I}$ by

ζ_{I} (a) = {\begin{cases} α, & if a \in I . \\ β, & otherwise . \end{cases}

Theorem 4.5

Let $I$ be a nonempty subset of $A$ . Then $ζ_{ℐ}$ is a fuzzy ideal of $A$ if and only if $I$ is an ideal of $A$ .

Proof.

Assume that $ζ_{ℐ}$ is a fuzzy ideal of $A$ .

(i) Let $γ_{1}, γ_{2} \in A$ . If $γ_{1}, γ_{2} \in I$ , then $ζ_{ℐ} (γ_{1}) = ζ_{ℐ} (γ_{2}) = α$ . So, $ζ_{ℐ} (γ_{1} + γ_{2}) \geq min {ζ_{ℐ} (γ_{1}), ζ_{ℐ} (γ_{2})} = α$ . Therefore, $γ_{1} + γ_{2} \in I$ .

(iii) Let $γ_{1} \in I$ and $γ_{2} \in A$ . Suppose $γ_{2} \otimes 0 \leq γ_{1} \otimes 0$ . Clearly, $ζ_{ℐ} (γ_{1}) = α$ . Since $ζ_{ℐ} (γ_{2}) \geq ζ_{ℐ} (γ_{1}) \geq α$ . Clearly, $ζ_{ℐ} (γ_{2}) = α$ . Therefore, $γ_{2} \in I$ .

Conversely, let $I$ is an ideal of $A$ .

(a) Let $γ_{1}, γ_{2} \in A$ . Consider the following three cases.

(i) If $γ_{1}, γ_{2} \in I$ , then $γ_{1} + γ_{2} \in I$ . Clearly, $ζ_{ℐ} (γ_{1}) = ζ_{ℐ} (γ_{2}) = α$ . So, $ζ_{ℐ} (γ_{1} + γ_{2}) = α = min {ζ_{ℐ} (γ_{1}), ζ_{ℐ} (γ_{2})}$ .

(ii) If $γ_{1} \in I$ or $γ_{2} \notin I$ . Then $ζ_{ℐ} (γ_{1}) = α$ and $ζ_{ℐ} (γ_{2}) = β$ . Clearly, $ζ_{ℐ} (γ_{1} + γ_{2}) \geq β = min {ζ_{ℐ} (γ_{1}), ζ_{ℐ} (γ_{2})}$ .

(iii) If $γ_{1} \in I$ or $γ_{2} \in I$ . Then $ζ_{ℐ} (γ_{1}) = β$ or $ζ_{ℐ} (γ_{2}) = β$ . Clearly, $ζ_{ℐ} (γ_{1} + γ_{2}) \geq β = min {ζ_{ℐ} (γ_{1}), ζ_{ℐ} (γ_{2})}$ . Therefore, from Case - i, Case - ii and Case - iii, we get $ζ_{ℐ} (γ_{1} + γ_{2}) \geq β = min {ζ_{ℐ} (γ_{1}), ζ_{ℐ} (γ_{2})}$ for any $γ_{1}, γ_{2} \in A$ .

(b) Let $γ_{1}, γ_{2} \in A$ and $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ . Consider the following three cases.

(i) If $γ_{2} \in I$ , then $γ_{1} \in I$ . Therefore, $ζ_{ℐ} (γ_{2}) = ζ_{ℐ} (γ_{1}) = α$ .

(ii) If $γ_{2} \notin I$ , then $ζ_{ℐ} (γ_{2}) = β$ . Thus, $ζ_{ℐ} (γ_{1}) \geq ζ_{ℐ} (γ_{2}) = β$ .

(iii) If $γ_{1}, γ_{2} \notin I$ , then $ζ_{ℐ} (γ_{1}) = ζ_{ℐ} (γ_{2}) = β$ . Thus, $ζ_{ℐ} (γ_{1}) = ζ_{ℐ} (γ_{2}) = β$ . Consequently, for any $γ_{1}, γ_{2} \in A$ and $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ , we get $ζ_{ℐ} (γ_{1}) \geq ζ_{ℐ} (γ_{2})$ . So, $ζ_{ℐ}$ is a fuzzy ideal.

Theorem 4.6

Let $ζ$ be a fuzzy ideal of $A$ . Then the set $I_{0} = {γ \in A | ζ (γ) = ζ (0)}$ is an ideal of $A$ .

Proof.

(a) Let $γ_{1}, γ_{2} \in I_{0}$ . So, $ζ (γ_{1}) = ζ (0)$ and $ζ (γ_{2}) = ζ (0)$ . Since $ζ$ is a fuzzy ideal of $A$ ; $ζ (γ_{1} + γ_{2}) \geq min {ζ (γ_{1}), ζ (γ_{2})} \geq ζ (0)$ . Also, since $ζ (0) \geq ζ (γ)$ for any $γ \in A$ , we get $ζ (0) \geq ζ (γ_{1} + γ_{2})$ . Consequently, $ζ (γ_{1} + γ_{2}) = ζ (0)$ . Therefore, $γ_{1} + γ_{2} \in I_{0}$ .
(b) Let $γ_{1} \in I_{0}$ and $γ_{2} \in A$ . Clearly, $ζ (γ_{1}) = ζ (0)$ . Suppose $γ_{2} \otimes 0 \leq γ_{1} \otimes 0$ . Since $ζ$ is a fuzzy ideal of $A$ ; $ζ (γ_{2}) \geq ζ (γ_{1}) = ζ (0)$ . Also, since $ζ (0) \geq ζ (γ)$ for any $γ \in A$ , we get $ζ (0) \geq ζ (γ_{2})$ . As a result, $ζ (γ_{2}) = ζ (0)$ . Therefore, $γ_{2} \in I_{0}$ .

Hence, $I_{0}$ is an ideal of $A$ .

Theorem 4.7

Let $ζ$ be a fuzzy ideal of $A$ . Let $ζ_{α}$ , $ζ_{β}$ be level ideals of $ζ$ . Assume that $β < α$ . Then, $ζ_{α}$ = $ζ_{β}$ if and only if there is no $γ \in A$ such that $β \leq ζ (γ) < α$ .

Proof.

Assume that $ζ_{α}$ = $ζ_{β}$ . Suppose for $β \leq α$ , there exists $γ \in A$ such that $β \leq ζ (γ) < α$ . Then $ζ_{α}$ is a proper subset of $ζ_{β}$ . This is contradiction. Hence, there is no $γ \in A$ such that $β \leq ζ (γ) < α$ .

Conversely, suppose there is no $γ \in A$ such that $β \leq ζ (γ) < α$ . Clearly, $β \leq α$ . Therefore, $ζ_{α} \subseteq ζ_{β}$ . Let $η \in ζ_{β}$ . Then $ζ (η) \geq β$ . Since $ζ (η)$ doesnot lie between $β$ and $α$ ; $ζ (η) \geq α$ . Therefore, $η \in ζ_{α}$ . This implies that $ζ_{β} \subseteq ζ_{α}$ . Hence, $ζ_{α}$ = $ζ_{β}$ .

Theorem 4.8

The intersection of any set of fuzzy ideals of $A$ is also fuzzy ideal of $A$ .

Proof.

Let ${ζ_{i} | i \in I}$ be a family of fuzzy subalgebras of autometrized algebra $A$ . To show that $\cap_{i \in I} ζ_{i}$ is an ideal. Then for any $γ_{1}, γ_{2} \in A$ , $i \in I$ ,

(a)
$\begin{array}{l} (\cap_{i \in I} ζ_{i}) (γ_{1} + γ_{2}) = inf (ζ_{i} (γ_{1} + γ_{2})) . \\ \geq inf (min {ζ_{i} (γ_{1}), ζ_{i} (γ_{2})}) . \\ = min {inf (ζ_{i} (γ_{1})), inf (ζ_{i} (γ_{2}))} . \\ = min {(\cap_{i \in I} ζ_{i}) (γ_{1}), (\cap_{i \in I} ζ_{i}) (γ_{2})} . \end{array}$
(b) Let $γ_{1}, γ_{2} \in A$ and $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ .
$\begin{array}{l} \Rightarrow ζ_{i} (γ_{1}) \geq ζ_{i} (γ_{2}) . \\ \Rightarrow inf (ζ_{i} (γ_{1})) \geq inf (ζ_{i} (γ_{2})) . \\ \Rightarrow \cap_{i \in I} ζ_{i} (γ_{1}) \geq \cap_{i \in I} ζ_{i} (γ_{2}) . \end{array}$

Hence, $\cap_{i \in I} ζ_{i}$ is a fuzzy ideal of $A$ .

Remark 4.9

The union of fuzzy ideals of $A$ is not a fuzzy ideal of $A$ .

Example 4.10

Let $A = {0, γ_{1}, γ_{2}, γ_{3}}$ with $0 \leq γ_{1}, γ_{2} \leq γ_{3}$ and elements $γ_{1}, γ_{2}$ are incomparable. Define $\otimes$ and $+$ by the following tables.

$\otimes$	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
0	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$γ_{1}$	$γ_{1}$	0	$γ_{3}$	$γ_{2}$
$γ_{2}$	$γ_{2}$	$γ_{3}$	0	$γ_{1}$
$γ_{3}$	$γ_{3}$	$γ_{2}$	$γ_{1}$	0

$+$	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
0	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$γ_{1}$	$γ_{1}$	$γ_{1}$	$γ_{3}$	$γ_{3}$
$γ_{2}$	$γ_{2}$	$γ_{3}$	$γ_{2}$	$γ_{3}$
$γ_{3}$	$γ_{3}$	$γ_{3}$	$γ_{3}$	$γ_{3}$

Then, $A$ is an autometrized algebra. Define a fuzzy subset $ζ$ and $α$ of $A$ by:

	0	$γ_{1}$	$γ_{2}$	$γ_{3}$
$ζ$	0.5	0.3	0.5	0.3
$α$	0.5	0.5	0.4	0.4
$ζ \cup α$	0.5	0.5	0.5	0.4

Then $ζ$ and $α$ are fuzzy ideals of $A$ . But the union $ζ \cup α$ is not a fuzzy ideals of $A$ . Since $ζ \cup α (γ_{1} + γ_{2}) = ζ \cup α (γ_{3}) = 0.4 < 0.5 = min {ζ \cup α (γ_{1}), ζ \cup α (γ_{2})}$ .

Remark 4.11

The union of any chain set of fuzzy ideals of $A$ is also fuzzy ideal of $A$ .

Proof.

Let ${ζ_{i} | i \in I}$ be a family of fuzzy ideals of autometrized algebra $A$ . Since it is a chain $\cup_{i \in I} ζ_{i} = ζ_{k}$ , for some $k \in I$ . Then $\cup_{i \in I} ζ_{i}$ is a fuzzy ideal of $A$ .

Theorem 4.12

A homomorphic pre-image of a fuzzy ideal of $A$ is also a fuzzy ideal of $A$ .

Proof.

Let $f : A \to B$ be a homomorphism. Let $α$ a fuzzy ideal of $B$ and $ζ$ the pre-image of $α$ under $f$ , then $α (f (γ)) = ζ (γ)$ for all $γ \in A$ . To show that $ζ$ is a fuzzy ideal of $A$ .

(i) Let $γ_{1}, γ_{2} \in A$ .

\begin{array}{l} ζ (γ_{1} + γ_{2}) = α (f (γ_{1} + γ_{2})) . \\ = α (f (γ_{1}) + f (γ_{2})) . \\ \geq min {α (f (γ_{1})), α (f (γ_{2}))} . \\ = min {ζ (γ_{1}), ζ (γ_{2})} . \end{array}

(ii) Let $γ_{1}, γ_{2} \in A$ and $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ . Since $f$ is homomorphism; $f (γ_{1} \otimes 0) \leq f (γ_{2} \otimes 0)$ . So, $f (γ_{1}) \otimes 0 \leq f (γ_{2}) \otimes 0$ . Since $f (γ_{1}), f (γ_{2}) \in B$ ; $α (f (γ_{1})) \geq α (f (γ_{2}))$ . Therefore, $ζ (γ_{1}) \geq ζ (γ_{2})$ . Hence, $ζ$ is a fuzzy ideal of $A$ .

Theorem 4.13

Let $f : A \to B$ be an epimorphism and order reversing. For every fuzzy ideal $ζ$ of $A$ with sup property, $f (ζ)$ is a fuzzy ideal of $B$ .

Proof.

To show that $f (ζ)$ is a fuzzy ideal of $B$ . By definition $α (γ_{2}^{'}) = f (ζ) (γ_{2}^{'}) = sup {ζ (γ_{1}) | γ_{1} \in f^{- 1} (γ_{2}^{'})}$ , for all $γ_{2}^{'} \in B$ ( $sup \emptyset = 0$ ).

(i) Let $γ^{'}, η^{'} \in B$ . Then, there exist $γ_{0}, η_{0} \in A$ such that $f (γ_{0}) = γ^{'}$ and $f (η_{0}) = η^{'}$ . So, $γ_{0} \in f^{- 1} (γ^{'})$ and $η_{0} \in f^{- 1} (η^{'})$ . Now, consider

\begin{array}{l} ζ (γ_{0} + η_{0}) = α (f (γ_{0} + η_{0})) = α (f (γ_{0}) + f (η_{0})) = α (γ^{'} + η^{'}) = sup_{γ_{0} + η_{0} \in f^{- 1} (γ^{'} + η^{'})} ζ (γ_{0} + η_{0}) . \\ ζ (γ_{0}) = α (f (γ_{0})) = α (γ^{'}) = sup_{γ_{0} \in f^{- 1} (γ^{'})} ζ (γ_{0}) . \end{array}

Also,

\begin{array}{l} α (γ^{'} + η^{'}) = sup_{π \in f^{- 1} (γ^{'} + η^{'})} ζ (π) . \\ \geq min {ζ (γ_{0}), ζ (η_{0})} . \\ = min {sup_{π \in f^{- 1} (γ^{'})} ζ (γ_{0}), sup_{π \in f^{- 1} (η^{'})} ζ (η_{0})} . \\ = min {α (γ^{'}), α (η^{'})} . \end{array}

(ii) Let $γ^{'}, η^{'} \in ℬ$ and $γ^{'} \otimes 0 \leq η^{'} \otimes 0$ . To show that $α (γ^{'}) \geq α (η^{'})$ . Then, there exist $γ_{0}, η_{0} \in A$ such that $f (γ_{0}) = γ^{'}$ and $f (η_{0}) = η^{'}$ . So, $γ_{0} \in f^{- 1} (γ^{'})$ and $η_{0} \in f^{- 1} (η^{'})$ .
Since $f$ is order reversing;

\begin{array}{l} f (γ_{0}) \otimes 0 \leq f (η_{0}) \otimes 0 . \\ \Rightarrow f (γ_{0} \otimes 0) \leq f (η_{0} \otimes 0) . \\ \Rightarrow γ_{0} \otimes 0 \leq η_{0} \otimes 0 . \\ \Rightarrow ζ (γ_{0}) \geq ζ (η_{0}) . \\ \Rightarrow α (f (γ_{0})) \geq α (f (η_{0})) . \\ \Rightarrow α (γ^{'}) \geq α (η^{'}) . \end{array}

Hence, $α$ is a fuzzy ideal of $B$ .

5. Fuzzy congruence of autometrized algebra

This section provides an introduction to fuzzy congruence relations.

Let $A$ be an autometrized algebra. A fuzzy relation on $A$ is a mapping $Φ : A \times A \to [0, 1]$ .

Definition 5.1

If $Φ$ is a fuzzy equivalent relation on $A$ , then

(a) $Φ (γ_{1}, γ_{1}) = sup {Φ (γ_{2}, γ_{3}) | γ_{2}, γ_{3} \in A}$ .(reflexive)
(b) $Φ (γ_{1}, γ_{2}) = Φ (γ_{2}, γ_{1})$ .(symmetric)
(c) $Φ (γ_{1}, γ_{3}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{2}, γ_{3})}$ .(transitive)

Definition 5.2

A fuzzy equivalence relation $Φ$ on $A$ is called a fuzzy congruence relation on $A$ if

(a) $Φ (γ_{1} + γ_{3}, γ_{2} + γ_{4}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})} \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ ,
(b) $Φ (γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})} \forall γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ ,
(c) For any $γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ , if $γ_{3} \otimes γ_{4} \leq γ_{1} \otimes γ_{2}$ , then $Φ (γ_{3}, γ_{4}) \geq Φ (γ_{1}, γ_{2})$ .

Let $Φ$ be a fuzzy relation on $A$ . Consider $Φ_{π} = {(γ_{1}, γ_{2}) \in A \times A | Φ (γ_{1}, γ_{2}) \geq π}$ , $π \in [0, 1]$ .

Theorem 5.3

Let $Φ$ be a fuzzy relation on $A$ . $Φ$ is a fuzzy congruence relation on $A$ if and only if for all $π \in [0, 1]$ , $Φ_{π}$ is either empty or a congruence relation on $A$ .

Proof.

Assume that $Φ$ is a fuzzy congruence relation on $A$ .

(i) Since $Φ_{π}$ is a nonempty; let $(γ_{1}, γ_{2}) \in Φ_{π}$ . So, $Φ (γ_{1}, γ_{2}) \geq π$ . But $Φ (γ_{1}, γ_{1}) = sup {Φ (γ_{2}, γ_{3}) | γ_{2}, γ_{3} \in A} \geq Φ (γ_{1}, γ_{2}) \geq π$ . So, $Φ (γ_{1}, γ_{1}) \geq π$ . Therefore, $(γ_{1}, γ_{1}) \in Φ_{π}$ .
(ii) Let $(γ_{1}, γ_{2}) \in Φ_{π}$ . So, $Φ (γ_{1}, γ_{2}) \geq π$ . Clearly, $Φ (γ_{1}, γ_{2}) = Φ (γ_{2}, γ_{1}) \geq π$ . So, $Φ (γ_{2}, γ_{1}) \geq π$ . Therefore, $(γ_{2}, γ_{1}) \in Φ_{π}$ .
(iii) Let $(γ_{1}, γ_{2}), (γ_{2}, γ_{3}) \in Φ_{π}$ . Then, $Φ (γ_{1}, γ_{2}) \geq π$ and $Φ (γ_{2}, γ_{3}) \geq π$ . Clearly, $min {Φ (γ_{1}, γ_{2}), Φ (γ_{2}, γ_{3})} \geq π$ . Since $Φ$ is a fuzzy congruence; $Φ (γ_{1}, γ_{3}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{2}, γ_{3})} \geq π$ . Therefore, $(γ_{1}, γ_{3}) \in Φ_{π}$ .
Therefore, $Φ_{π}$ is an equivalence relation.
- (a) Let $(γ_{1}, γ_{2}), (γ_{3}, γ_{4}) \in Φ_{π}$ . So, $Φ (γ_{1}, γ_{2}) \geq π$ and $Φ (γ_{3}, γ_{4}) \geq π$ . Since $Φ$ is a fuzzy congruence relation on $A$ ; $Φ (γ_{1} + γ_{3}, γ_{2} + γ_{4}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})} \geq π$ . Therefore, $(γ_{1} + γ_{3}, γ_{2} + γ_{4}) \in Φ_{π}$ .
- (b) Let $(γ_{1}, γ_{2}), (γ_{3}, γ_{4}) \in Φ_{π}$ . So, $Φ (γ_{1}, γ_{2}) \geq π$ and $Φ (γ_{3}, γ_{4}) \geq π$ . Since $Φ$ is a fuzzy congruence relation on $A$ ; $Φ (γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})} \geq π$ . Therefore, $(γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \in Φ_{π}$ .
- (c) Let $(γ_{1}, γ_{2}) \in Φ_{π}$ and $γ_{3} \otimes γ_{4} \leq γ_{1} \otimes γ_{2}$ . Clearly, $(γ_{1}, γ_{2}) \geq π$ . Since $Φ$ is a fuzzy congruence relation on $A$ ; then $Φ (γ_{3}, γ_{4}) \geq Φ (γ_{1}, γ_{2}) \geq π$ . Therefore, $(γ_{3}, γ_{4}) \in Φ_{π}$ . Hence, $Φ_{π}$ is a congruence relation on $A$ .

Conversely, $Φ_{π}$ is a congruence relation on $A$ . To show that $Φ$ is a fuzzy congruence relation on $A$ .

(i) We know that for any $γ_{1}, γ_{2}, γ_{3} \in A$ , $γ_{3} \otimes γ_{3} = 0 \leq γ_{1} \otimes γ_{2}$ . Take $π = Φ (γ_{1}, γ_{2})$ . So, $(γ_{1}, γ_{2}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence relation; $(γ_{3}, γ_{3}) \in Φ_{π}$ . Then, $Φ (γ_{3}, γ_{3}) \geq π = Φ (γ_{1}, γ_{2})$ . Therefore, $Φ (γ_{3}, γ_{3}) = sup {Φ (γ_{1}, γ_{2}) | γ_{1}, γ_{2} \in A}$ .
(ii) Let $γ_{1}, γ_{2} \in A$ . Take $π = Φ (γ_{1}, γ_{2})$ . So, $(γ_{1}, γ_{2}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence; $(γ_{2}, γ_{1}) \in Φ_{π}$ . So, $Φ (γ_{2}, γ_{1}) \geq π$ . Clearly, $Φ (γ_{2}, γ_{1}) \geq Φ (γ_{1}, γ_{2})$ . Again, take $π = Φ (γ_{2}, γ_{1})$ . So, $(γ_{2}, γ_{1}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence; $(γ_{1}, γ_{2}) \in Φ_{π}$ . So, $Φ (γ_{1}, γ_{2}) \geq π$ . Clearly, $Φ (γ_{1}, γ_{2}) \geq Φ (γ_{2}, γ_{1})$ . Consequently, $Φ (γ_{1}, γ_{2}) = Φ (γ_{2}, γ_{1})$ .
(iii) Let $γ_{1}, γ_{2}, γ_{3} \in A$ . Take $π = min {Φ (γ_{1}, γ_{2}), Φ (γ_{2}, γ_{3})}$ . This implies that $(γ_{1}, γ_{2}), (γ_{2}, γ_{3}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence relation; $(γ_{1}, γ_{3}) \in Φ_{π}$ . Therefore, $Φ (γ_{1}, γ_{3}) \geq π$ . Thus, $Φ (γ_{1}, γ_{3}) \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{2}, γ_{3})}$ .
Therefore, $Φ$ is an equivalence relation.
- (a) Let $γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ . Take $π = min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})}$ . Therefore, $(γ_{1}, γ_{2}) \in Φ_{π}$ and $(γ_{3}, γ_{4}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence relation on $A$ ; $(γ_{1} + γ_{3}, γ_{2} + γ_{4}) \in Φ_{π}$ . As a result, $Φ (γ_{1} + γ_{3}, γ_{2} + γ_{4}) \geq π = min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})}$ .
- (b) Let $γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ . Take $π = min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})}$ . Therefore, $(γ_{1}, γ_{2}) \in Φ_{π}$ and $(γ_{3}, γ_{4}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence relation on $A$ ; $(γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \in Φ_{π}$ . As a result, $Φ (γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \geq π = min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})}$ .
- (c) Let $γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ and suppose that $γ_{3} \otimes γ_{4} \leq γ_{1} \otimes γ_{2}$ . Take $π = Φ (γ_{1}, γ_{2})$ . Therefore, $(γ_{1}, γ_{2}) \in Φ_{π}$ . Since $Φ_{π}$ is a congruence relation on $A$ ; $(γ_{3}, γ_{4}) \in Φ_{π}$ . Therefore, $Φ (γ_{3}, γ_{4}) \geq π = Φ (γ_{1}, γ_{2})$ . Hence, $Φ$ is a fuzzy congruence relation on $A$ .
For a fuzzy congruence relation $Φ$ , the fuzzy subset $Φ^{γ_{1}} : A \to [0, 1]$ , which is defined by $Φ^{γ_{1}} (γ_{2}) = Φ (γ_{1}, γ_{2})$ , is called the fuzzy congruence class containing $γ_{1}$ .

Theorem 5.4

For any fuzzy congruence relation $Φ$ on $A$ , $Φ^{0} = ζ_{Φ}$ is a fuzzy ideal of $A$ .

Proof.

Let $x \in A$ . Then,

(i) It is clear that, $Φ^{0} (0) = Φ (0, 0)$ . We know that $0 \otimes 0 \leq γ_{1} \otimes 0$ for all $γ_{1} \in A$ . Since $Φ$ is a fuzzy congruence relation; $Φ (0, 0) \geq Φ (0, γ_{1})$ . This implies that, $Φ^{0} (0) \geq Φ^{0} (γ_{1})$ .
(ii) $Φ^{0} (γ_{1} + γ_{2}) = Φ (0, γ_{1} + γ_{2}) = Φ (0 + 0, γ_{1} + γ_{2})$ . Since $Φ$ is a fuzzy congruence relation; $Φ^{0} (γ_{1} + γ_{2}) = Φ (0 + 0, γ_{1} + γ_{2}) \geq min {Φ (0, γ_{1}), Φ (0, γ_{2})} = min {Φ^{0} (γ_{1}), Φ^{0} (γ_{2})}$ .
(iii) Let $γ_{1}, γ_{2} \in A$ . Suppose $γ_{1} \otimes 0 \leq γ_{2} \otimes 0$ . Since $Φ$ is a fuzzy congruence relation; $Φ (γ_{1}, 0) \geq Φ (γ_{2}, 0)$ . Therefore, $Φ^{0} (γ_{1}) \geq Φ^{0} (γ_{2})$ . Hence, $Φ^{0}$ is a fuzzy ideal of $A$ .

Theorem 5.5

Let $A$ is a normal autometrized algebra. Let $ζ$ be a fuzzy ideal of $A$ and a fuzzy relation $Φ$ on $A$ defined by $Φ_{ζ} = Φ (γ_{1}, γ_{2}) = ζ (γ_{1} \otimes γ_{2})$ . Then, $Φ_{ζ}$ is a fuzzy congruence relation on $A$ .

Proof.

Let $ζ$ be a fuzzy ideal of $A$ . Therefore,

(i) It is clear that, $ζ (0) \geq ζ (γ_{1})$ for all $γ_{1} \in A$ . So, $Φ (γ_{1}, γ_{1}) = ζ (γ_{1} \otimes γ_{1}) = ζ (0) \geq ζ (γ_{1})$ for all $γ_{1} \in A$ . Therefore, $Φ (γ_{1}, γ_{1}) \geq ζ (γ_{2} \otimes γ_{3}) = Φ (γ_{2}, γ_{3})$ . Hence, $Φ (γ_{1}, γ_{1}) = sup {Φ (γ_{2}, γ_{3}) | γ_{2}, γ_{3} \in A}$ .
(ii) It is clear that; $Φ (γ_{1}, γ_{2}) = ζ (γ_{1} \otimes γ_{2}) = ζ (γ_{2} \otimes γ_{1}) = Φ (γ_{2}, γ_{1})$ .
(iii) By metric property; $γ_{1} \otimes γ_{2} \leq γ_{1} \otimes γ_{3} + γ_{3} \otimes γ_{2}$ . And also, $A$ is a normal autometrized algebra; implies that $(γ_{1} \otimes γ_{2}) \otimes 0 \leq (γ_{1} \otimes γ_{3} + γ_{3} \otimes γ_{2}) \otimes 0$ . Since $ζ$ is a fuzzy ideal; $ζ (γ_{1} \otimes γ_{2}) \geq ζ (γ_{1} \otimes γ_{3} + γ_{3} \otimes γ_{2})$ . Again, $ζ$ is a fuzzy ideal; $ζ (γ_{1} \otimes γ_{2}) \geq min {ζ (γ_{1} \otimes γ_{3}), ζ (γ_{2} \otimes γ_{3})}$ . Thus, $Φ (γ_{1}, γ_{2}) \geq min {Φ (γ_{1}, γ_{3}), Φ (γ_{2}, γ_{3})}$ .
Therefore, $Φ_{ζ}$ is an equivalence relation.
- (a) By property of normality; $[(γ_{1} + γ_{3}) \otimes (γ_{2} + γ_{4})] \otimes 0 \leq [(γ_{1} \otimes γ_{2}) + (γ_{3} \otimes γ_{4})] \otimes 0$ . Since $ζ$ is a fuzzy ideal;

\begin{array}{l} ζ ((γ_{1} + γ_{3}) \otimes (γ_{2} + γ_{4})) \geq ζ ((γ_{1} \otimes γ_{2}) + (γ_{3} \otimes γ_{4})) . \\ Φ (γ_{1} + γ_{3}, γ_{2} + γ_{4}) \geq min {ζ (γ_{1} \otimes γ_{2}), ζ (γ_{3} \otimes γ_{4})} . \\ \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})} . \end{array}

(b) By property of normality; $[(γ_{1} \otimes γ_{3}) \otimes (γ_{2} \otimes γ_{4})] \otimes 0 \leq [(γ_{1} \otimes γ_{2}) + (γ_{3} \otimes γ_{4})] \otimes 0$ . Since $ζ$ is a fuzzy ideal;

\begin{array}{l} ζ ((γ_{1} \otimes γ_{3}) \otimes (γ_{2} \otimes γ_{4})) \geq ζ ((γ_{1} \otimes γ_{2}) + (γ_{3} \otimes γ_{4})) . \\ Φ (γ_{1} \otimes γ_{3}, γ_{2} \otimes γ_{4}) \geq min {ζ (γ_{1} \otimes γ_{2}), ζ (γ_{3} \otimes γ_{4})} . \\ \geq min {Φ (γ_{1}, γ_{2}), Φ (γ_{3}, γ_{4})} . \end{array}

(c) Let $γ_{1}, γ_{2}, γ_{3}, γ_{4} \in A$ and suppose that $γ_{3} \otimes γ_{4} \leq γ_{1} \otimes γ_{2}$ . Since $A$ is a normal autometrized algebra; $(γ_{3} \otimes γ_{4}) \otimes 0 \leq (γ_{1} \otimes γ_{2}) \otimes 0$ . Since $ζ$ is a fuzzy ideal; $ζ (γ_{3} \otimes γ_{4}) \geq ζ (γ_{1} \otimes γ_{2})$ . Therefore, $Φ (γ_{3}, γ_{4}) \geq Φ (γ_{1}, γ_{2})$ . Hence, $Φ_{ζ}$ is a fuzzy congruence relation on $A$ .

Theorem 5.6

Let $A$ is a normal autometrized algebra. Then, $Φ_{ζ}^{γ_{1}} = Φ_{ζ}^{γ_{2}}$ if and only if $ζ (γ_{1} \otimes γ_{2}) = ζ (0)$ .

Proof.

Let $Φ_{ζ}^{γ_{1}} = Φ_{ζ}^{γ_{2}}$ for any $γ_{1}, γ_{2} \in A$ . Clearly, $Φ_{ζ}^{γ_{1}} (γ_{1}) = Φ_{ζ}^{γ_{2}} (γ_{1})$ . Therefore, $Φ_{ζ}^{γ_{1}} (γ_{1}) = Φ (γ_{1}, γ_{1}) = ζ (γ_{1} \otimes γ_{1}) = ζ (0)$ and $Φ_{ζ}^{γ_{1}} (γ_{2}) = Φ (γ_{1}, γ_{2}) = ζ (γ_{1} \otimes γ_{2})$ . This implies that; $ζ (0) = ζ (γ_{1} \otimes γ_{2})$ .

Conversely, for any $γ_{3} \in A$ ; $Φ_{ζ}^{γ_{1}} (γ_{3}) = Φ (γ_{1}, γ_{3}) = ζ (γ_{1} \otimes γ_{3})$ . By using the property of metric; $γ_{1} \otimes γ_{3} \leq γ_{1} \otimes γ_{2} + γ_{2} \otimes γ_{3}$ . And also, $A$ is a normal autometrized algebra; implies that $(γ_{1} \otimes γ_{3}) \otimes 0 \leq (γ_{1} \otimes γ_{2} + γ_{2} \otimes γ_{3}) \otimes 0$ . Since $ζ$ is fuzzy ideal;

(1)

\begin{array}{l} ζ (γ_{1} \otimes γ_{3}) \geq ζ (γ_{1} \otimes γ_{2} + γ_{3} \otimes γ_{2}) . \\ \geq min {ζ (γ_{1} \otimes γ_{2}), ζ (γ_{3} \otimes γ_{2})} . \\ \geq min {ζ (0), ζ (γ_{3} \otimes γ_{2})} . [Since ζ (γ_{1} \otimes γ_{2}) = ζ (0)] . \\ ζ (γ_{1} \otimes γ_{3}) \geq ζ (γ_{3} \otimes γ_{2}) . \end{array}

Similarly, by using the property of metric; $γ_{2} \otimes γ_{3} \leq γ_{1} \otimes γ_{2} + γ_{1} \otimes γ_{3}$ . And also, $A$ is a normal autometrized algebra; implies that $(γ_{2} \otimes γ_{3}) \otimes 0 \leq (γ_{1} \otimes γ_{2} + γ_{1} \otimes γ_{3}) \otimes 0$ . Since $ζ$ is fuzzy ideal;

(2)

\begin{array}{l} ζ (γ_{2} \otimes γ_{3}) \geq ζ (γ_{1} \otimes γ_{2} + γ_{3} \otimes γ_{1}) . \\ \geq min {ζ (γ_{1} \otimes γ_{2}), ζ (γ_{3} \otimes γ_{1})} . \\ \geq min {ζ (0), ζ (γ_{3} \otimes γ_{1})} . [Since ζ (γ_{1} \otimes γ_{2}) = ζ (0)] \\ ζ (γ_{2} \otimes γ_{3}) \geq ζ (γ_{1} \otimes γ_{3}) . \end{array}

By (1) and (2), $Φ_{ζ}^{γ_{1}} (γ_{3}) = ζ (γ_{1} \otimes γ_{3}) = ζ (γ_{3} \otimes γ_{2}) = Φ_{ζ}^{γ_{2}} (γ_{3})$ . Hence, $Φ_{ζ}^{γ_{1}} = Φ_{ζ}^{γ_{2}}$ .

Theorem 5.7

Let $A$ is a normal autometrized algebra. Let $Φ$ be a fuzzy congruence and $ζ$ be a fuzzy ideal of $A$ . Then

(i) $Φ_{ζ_{Φ}} = Φ$ .
(ii) $ζ_{Φ_{ζ}} = ζ$ .

Proof.

(i) $Φ_{ζ_{Φ}} (γ_{1}, γ_{2}) = ζ_{Φ} (γ_{1} \otimes γ_{2}) = Φ^{0} (γ_{1} \otimes γ_{2}) = Φ (0, γ_{1} \otimes γ_{2})$ for any $γ_{1}, γ_{2} \in A$ . Since $A$ is normal; $(γ_{1} \otimes γ_{2}) \otimes 0 = γ_{1} \otimes γ_{2}$ . This implies that $Φ (0, γ_{1} \otimes γ_{2}) = Φ (γ_{1}, γ_{2})$ . Hence, $Φ_{ζ_{Φ}} = Φ$ .
(ii) $ζ_{Φ_{ζ}} (γ_{1}) = Φ_{ζ}^{0} (γ_{1}) = Φ_{ζ} (0, γ_{1}) = ζ (γ_{1} \otimes 0)$ for any $γ_{1} \in A$ . Since $A$ is normal; $(γ_{1} \otimes 0) \otimes 0 = γ_{1} \otimes 0$ . This implies that $ζ (γ_{1} \otimes 0) = ζ (γ_{1})$ . Hence, $ζ_{Φ_{ζ}} = ζ$ .

Theorem 5.8

Let $A$ be a normal autometrized algebra.

Let $FI (A)$ = the set of all fuzzy ideals of $A$ .

Let $FC (A)$ = the set of all fuzzy congruences on $A$ .

Then $FI (A)$ and $FC (A)$ are in one to one correspondence.

Proof.

Define $ψ : FI (A) \to FC (A)$ by $ψ (ζ) = Φ_{ζ}$ .

Let $ζ, α \in FI (A)$ . Suppose $ζ = α$ . This implies $Φ_{ζ} = Φ_{α}$ . Then $ψ (ζ) = ψ (α)$ . Therefore $ψ$ is well-defined. Now to show that $ψ$ is onto. Let $Φ \in FC (A)$ . Then by theorem (5.4); $\exists ζ_{Φ} \in FI (A)$ such that:

\begin{array}{l} ψ (ζ_{Φ}) = Φ_{ζ_{Φ}} . \\ = Φ . [By (i) of the theorem (5.7)] \end{array}

Therefore $ψ$ is onto.

Finally let us show that $ψ$ is one to one. Suppose $ψ (ζ) = ψ (α)$ . This implies that $Φ_{ζ} = Φ_{α}$ . Thus, $ζ_{Φ_{ζ}} = ζ_{Φ_{α}}$ . Consider,

ζ_{Φ_{ζ}} = ζ . [By (ii) of the theorem (5.7)]

Whence, $ζ = α$ . Hence $ψ$ is one to one and onto. Then $ψ$ is one to one correspondence. Therefore $FI (A)$ and $FC (A)$ are in one to one correspondence.

6. Conclusion

This paper presented fuzzy subalgebras of autometrized algebras and examined their properties. We introduced fuzzy ideals of autometrized algebras and provided examples to illustrate our findings. Furthermore, we defined the notions of intersection and union for fuzzy subalgebras and fuzzy ideals in autometrized algebras. We also studied the homomorphisms of both the images and the inverse images of fuzzy subalgebras and ideals. Finally, we introduced fuzzy congruences on autometrized algebras. We showed that in normal autometrized algebra, the set of all fuzzy ideals is in one-to-one correspondence with the set of all fuzzy congruences.

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Department of Mathematics, Assosa University, Asosa, Benishangul-Gumuz, Ethiopia

Gebrie Yeshiwas Tilahun
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Tilahun GY. Fuzzy Set Theory Applied on Autometrized Algebra [version 1; peer review: 2 approved, 1 approved with reservations]. F1000Research 2025, 14:159 (https://doi.org/10.12688/f1000research.161249.1)

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Reviewer Report 28 Feb 2025

Sileshe Gone Korma, Hawassa University, Hawassa, Ethiopia

Approved

https://doi.org/10.5256/f1000research.177253.r365974

The author introduces the concept of fuzzy subalgebras in autometrized algebras and investigates their characteristics. Additionally, the paper examines fuzzy ideals, providing examples to illustrate key results. It also investigates homomorphisms of fuzzy subalgebras and ideals. Furthermore, the study introduces ... Continue reading

The author introduces the concept of fuzzy subalgebras in autometrized algebras and investigates their characteristics. Additionally, the paper examines fuzzy ideals, providing examples to illustrate key results. It also investigates homomorphisms of fuzzy subalgebras and ideals. Furthermore, the study introduces fuzzy congruences and establishes a one-to-one correspondence between fuzzy ideals and fuzzy congruences in normal autometrized algebras.

The author should review the points I have highlighted in my comments and address them accordingly to improve clarity and accuracy.

Definition 2.2 (line 1) It is better to say; An autometrized algebra A is called normal if and only if.
Definition 2.5 seems it is a result not a definition.
Definition 2.9 it is better to say “Let A and B are autometrized algebra and f:A ….”
Definition 2.16 (line 3) there must be a space between if and f-1(γ2)
Before Theorem 3.2, you need to state the equivalent condition for a subset to be a subalgebra, as you use this result in the proof of Theorem 3.2.
Theorem 3.2 (on converse part of the proof (i)) it seems typing error ζ0<ζ(γ-1). Also, I am not clear how ζ0<π-1.
Theorem 3.3 (proof part line 1) make it . . . to show a fuzzy subalgebra
Remark 3.4 : it is better to write The union of fuzzy subalgebras of A may not be a fuzzy subalgebra of A.
Theorem 4.3 (proof converse part (i) line 2) add { }
Theorem 4.5 (line 1) : Let I be a nonempty subset of A. Then ζI is a fuzzy ideal of A if and only if I is an ideal of A. Also, for converse part (iii) it seems ζ1 ∉I or ζ2∉I. Additionally, the second case does not seem necessary to me.
Theorem 4.7 (line 1) It seems β≤α
Remark 4.9 : The union of fuzzy ideals of A may not be a fuzzy ideal of A.
Definition 5.1 : It is better to write “ A fuzzy relation Φ is called a fuzzy equivalent relation on A”

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Algebra, lattice theory, universal algebra, and fuzzy algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Reviewer Report 13 Feb 2025

Ramazan Yasar, Ankara University, Ankara, Turkey

Approved with Reservations

https://doi.org/10.5256/f1000research.177253.r365973

Summary of the Article :

This paper investigates fuzzy subalgebras, fuzzy ideals, and fuzzy congruences within autometrized algebras, presenting rigorous mathematical definitions, theorems, and proofs. The study establishes a bijective correspondence between fuzzy ideals and fuzzy congruences ... Continue reading

Summary of the Article :

This paper investigates fuzzy subalgebras, fuzzy ideals, and fuzzy congruences within autometrized algebras, presenting rigorous mathematical definitions, theorems, and proofs. The study establishes a bijective correspondence between fuzzy ideals and fuzzy congruences in normal autometrized algebras, contributing significantly to the algebraic theory of fuzzy structures.

Key Strengths:
✅ Well-structured and logically organized.
✅ Clearly defined mathematical framework with precise notation.
✅ Theorems and proofs follow established techniques in fuzzy algebra.
✅ Relevant literature is well-referenced.

Areas for Improvement:

Clarification of Certain Proofs:
- The proof of Theorem 3.3 incorrectly refers to proving an ideal instead of a fuzzy subalgebra. This should be corrected.
- In Theorem 5.3, additional steps should be included to justify why Φπ\Phi_\piΦπ is a congruence relation.
- Definition 4.1 should better distinguish fuzzy subalgebras from fuzzy ideals to avoid confusion.
Enhancing Readability and Explanation:
- The definition of fuzzy congruence relations in Section 5 should include a brief discussion of its significance and applications.
- Example 3.5 should provide a counterexample demonstrating why the union of fuzzy subalgebras is not necessarily a fuzzy subalgebra.
Conclusion and Impact:
- The conclusion should explicitly state how the study extends prior research and suggest potential applications in soft computing or other domains.
Reproducibility:
- Some examples lack detailed computational verification. Expanding Example 4.2 with explicit calculations would improve transparency.
- The study could benefit from providing more concrete computational examples illustrating fuzzy congruences in practice.

Essential Revisions for Scientific Soundness:
✔ Correct the minor mathematical inconsistencies in proofs and definitions.
✔ Provide additional justifications for key theorems.
✔ Expand examples with detailed step-by-step calculations.
✔ Clarify the real-world significance of fuzzy congruences and their potential applications.
With these minor revisions, the paper will be more comprehensible and impactful. I recommend indexing after minor revisions.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Algebra, Fuzzy Set Theory, Soft Computing, Mathematical Logic, and Theoretical Computer ScienceExpertise in autometrized algebras and fuzzy logic-based mathematical structures, ensuring a thorough review of the algebraic and logical foundations of the paper.Specialized in fuzzy congruences, ideal theory, and homomorphisms within algebraic frameworks.Experience in soft computing applications related to fuzzy algebra and their implications in artificial intelligence and decision-making models.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

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Reviewer Report 13 Feb 2025

Jehad R Kider, University of Technology-Iraq, Baghdad, Iraq

Approved

https://doi.org/10.5256/f1000research.177253.r365978

“Fuzzy Set Theory Applied on Autometrized Algebra ”

Thank you for sending me to review this paper.
Generally, has a good structure and without mathematical mistakes.
This paper presented fuzzy subalgebras of autometrized algebras and ... Continue reading

“Fuzzy Set Theory Applied on Autometrized Algebra ”

Thank you for sending me to review this paper.
Generally, has a good structure and without mathematical mistakes.
This paper presented fuzzy subalgebras of autometrized algebras and examined their properties. They introduced fuzzy ideals of autometrized algebras and provided examples to illustrate our findings. Furthermore, they defined the notions of intersection and union for fuzzy subalgebras and fuzzy ideals in autometrized algebras. Also they studied the homomorphisms of both the images and the inverse images of fuzzy subalgebras and ideals. Finally, we introduced fuzzy congruences on autometrized algebras
I suggest for the authors to rewrite the abstract by determine the goals. The manuscript has a few minor typographical errors and grammatical mistakes that should be corrected before acceptance. Taking the above into consideration, I recommend the paper for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Fuzzy Metric spaces, Fuzzy Normed Spaces, Fuzzy inner product spaces.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Version 1

VERSION 1 PUBLISHED 04 Feb 2025

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Jehad R Kider, University of Technology-Iraq, Baghdad, Iraq
Ramazan Yasar, Ankara University, Ankara, Turkey
Sileshe Gone Korma, Hawassa University, Hawassa, Ethiopia

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Reviewer Report

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28 Feb 2025 | for Version 1

Sileshe Gone Korma, Hawassa University, Hawassa, Ethiopia

3 Views Cite this report Responses(0)

Approved

The author introduces the concept of fuzzy subalgebras in autometrized algebras and investigates their characteristics. Additionally, the paper examines fuzzy ideals, providing examples to illustrate key results. It also investigates homomorphisms of fuzzy subalgebras and ideals. Furthermore, the study introduces fuzzy congruences and establishes a one-to-one correspondence between fuzzy ideals and fuzzy congruences in normal autometrized algebras.

The author should review the points I have highlighted in my comments and address them accordingly to improve clarity and accuracy.

Definition 2.2 (line 1) It is better to say; An autometrized algebra A is called normal if and only if.
Definition 2.5 seems it is a result not a definition.
Definition 2.9 it is better to say “Let A and B are autometrized algebra and f:A ….”
Definition 2.16 (line 3) there must be a space between if and f-1(γ2)
Before Theorem 3.2, you need to state the equivalent condition for a subset to be a subalgebra, as you use this result in the proof of Theorem 3.2.
Theorem 3.2 (on converse part of the proof (i)) it seems typing error ζ0<ζ(γ-1). Also, I am not clear how ζ0<π-1.
Theorem 3.3 (proof part line 1) make it . . . to show a fuzzy subalgebra
Remark 3.4 : it is better to write The union of fuzzy subalgebras of A may not be a fuzzy subalgebra of A.
Theorem 4.3 (proof converse part (i) line 2) add { }
Theorem 4.5 (line 1) : Let I be a nonempty subset of A. Then ζI is a fuzzy ideal of A if and only if I is an ideal of A. Also, for converse part (iii) it seems ζ1 ∉I or ζ2∉I. Additionally, the second case does not seem necessary to me.
Theorem 4.7 (line 1) It seems β≤α
Remark 4.9 : The union of fuzzy ideals of A may not be a fuzzy ideal of A.
Definition 5.1 : It is better to write “ A fuzzy relation Φ is called a fuzzy equivalent relation on A”

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Algebra, lattice theory, universal algebra, and fuzzy algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

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Reviewer Report

10 Views

13 Feb 2025 | for Version 1

Ramazan Yasar, Ankara University, Ankara, Turkey

10 Views Cite this report Responses(0)

Approved With Reservations

Summary of the Article :

This paper investigates fuzzy subalgebras, fuzzy ideals, and fuzzy congruences within autometrized algebras, presenting rigorous mathematical definitions, theorems, and proofs. The study establishes a bijective correspondence between fuzzy ideals and fuzzy congruences in normal autometrized algebras, contributing significantly to the algebraic theory of fuzzy structures.

Key Strengths:
✅ Well-structured and logically organized.
✅ Clearly defined mathematical framework with precise notation.
✅ Theorems and proofs follow established techniques in fuzzy algebra.
✅ Relevant literature is well-referenced.

Areas for Improvement:

Clarification of Certain Proofs:
- The proof of Theorem 3.3 incorrectly refers to proving an ideal instead of a fuzzy subalgebra. This should be corrected.
- In Theorem 5.3, additional steps should be included to justify why Φπ\Phi_\piΦπ is a congruence relation.
- Definition 4.1 should better distinguish fuzzy subalgebras from fuzzy ideals to avoid confusion.
Enhancing Readability and Explanation:
- The definition of fuzzy congruence relations in Section 5 should include a brief discussion of its significance and applications.
- Example 3.5 should provide a counterexample demonstrating why the union of fuzzy subalgebras is not necessarily a fuzzy subalgebra.
Conclusion and Impact:
- The conclusion should explicitly state how the study extends prior research and suggest potential applications in soft computing or other domains.
Reproducibility:
- Some examples lack detailed computational verification. Expanding Example 4.2 with explicit calculations would improve transparency.
- The study could benefit from providing more concrete computational examples illustrating fuzzy congruences in practice.

Essential Revisions for Scientific Soundness:
✔ Correct the minor mathematical inconsistencies in proofs and definitions.
✔ Provide additional justifications for key theorems.
✔ Expand examples with detailed step-by-step calculations.
✔ Clarify the real-world significance of fuzzy congruences and their potential applications.
With these minor revisions, the paper will be more comprehensible and impactful. I recommend indexing after minor revisions.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Partly
If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable
Are all the source data underlying the results available to ensure full reproducibility?

Partly
Are the conclusions drawn adequately supported by the results?

Partly

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Algebra, Fuzzy Set Theory, Soft Computing, Mathematical Logic, and Theoretical Computer ScienceExpertise in autometrized algebras and fuzzy logic-based mathematical structures, ensuring a thorough review of the algebraic and logical foundations of the paper.Specialized in fuzzy congruences, ideal theory, and homomorphisms within algebraic frameworks.Experience in soft computing applications related to fuzzy algebra and their implications in artificial intelligence and decision-making models.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Respond to this report

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Reviewer Report

10 Views

13 Feb 2025 | for Version 1

Jehad R Kider, University of Technology-Iraq, Baghdad, Iraq

10 Views Cite this report Responses(0)

Approved

“Fuzzy Set Theory Applied on Autometrized Algebra ”

Thank you for sending me to review this paper.
Generally, has a good structure and without mathematical mistakes.
This paper presented fuzzy subalgebras of autometrized algebras and examined their properties. They introduced fuzzy ideals of autometrized algebras and provided examples to illustrate our findings. Furthermore, they defined the notions of intersection and union for fuzzy subalgebras and fuzzy ideals in autometrized algebras. Also they studied the homomorphisms of both the images and the inverse images of fuzzy subalgebras and ideals. Finally, we introduced fuzzy congruences on autometrized algebras
I suggest for the authors to rewrite the abstract by determine the goals. The manuscript has a few minor typographical errors and grammatical mistakes that should be corrected before acceptance. Taking the above into consideration, I recommend the paper for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Fuzzy Metric spaces, Fuzzy Normed Spaces, Fuzzy inner product spaces.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Respond to this report

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