Fuzzy PTM subalgebra of PTM - algebra

Alemayehu Girum Melaku; Berhanu Assaye Alaba; Bekalu Tarekegn Bitew; Beza Lamesgin Derseh

doi:10.12688/f1000research.160239.1

Home Browse Fuzzy PTM subalgebra of PTM - algebra

ALL Metrics

Views

Downloads

Get PDF

Get XML

Export

▬

✚

Research Article

Fuzzy PTM subalgebra of PTM - algebra

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

Alemayehu Girum Melaku ¹, Berhanu Assaye Alaba¹, Bekalu Tarekegn Bitew¹, Beza Lamesgin Derseh²

PUBLISHED 06 Jan 2025

Author details Author details

¹ Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia
² Debre Markos University College of Natural and Computational Science, Debre Markos, Amhara, Ethiopia

Alemayehu Girum Melaku
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Berhanu Assaye Alaba
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Bekalu Tarekegn Bitew
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Beza Lamesgin Derseh
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Software, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background

This article introduces the concept of fuzzy pseudo-TM subalgebra (FPTMSA) of pseudo-TM algebra (PTMA). This study explores the relationship between FPTMSAs and their level sets within the context of PTMA. It further investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby providing a foundation for further exploration of the algebraic structure of fuzzy pseudo-ideals within PTMAs.

Methods

The concept of level sets for FPTMSAs is defined and examined, highlighting their role in characterizing these subalgebras within a PTMA. This study explores the behavior of FPTMSAs under homomorphisms, considering both homomorphic images (HI) and inverse images (II) of FPTMSAs in PTMAs. In addition, the composition of epimorphic images and inverse images of FPTMSAs is discussed. The study also considers the properties of FPTMSAs under Cartesian products (CPs), demonstrating that the Cartesian product of two FPTMSAs remains an FPTMSA.

Results

This study establishes several key properties of FPTMSAs in PTMAs, particularly focusing on their behavior under homomorphisms, epimorphisms, and Cartesian products. Specifically, this study demonstrates that the Cartesian product of two FPTMSAs results in another FPTMSA. Additionally, this study examines the relationships between the homomorphic image and inverse image of FPTMSAs and presents further algebraic results related to these concepts.

Conclusions

The article concludes by highlighting the significant role of FPTMSAs in the structure of PTMAs, particularly in their interaction with homomorphisms, epimorphisms, and Cartesian products. The results suggest that the FPTMSAs are robust under various algebraic operations. Future work will extend these concepts to intuitionistic fuzzy pseudo-TM subalgebra within PTMAs with the goal of uncovering new algebraic insights and results.

Keywords

PTMA, FPTMSA, Level set of a PTMA, CP of FPTMSA

Corresponding author: Alemayehu Girum Melaku

Competing interests: No competing interests were disclosed.

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

Copyright: © 2025 Melaku AG et al. 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: Melaku AG, Alaba BA, Bitew BT and Derseh BL. Fuzzy PTM subalgebra of PTM - algebra [version 1; peer review: 1 approved with reservations]. F1000Research 2025, 14:32 (https://doi.org/10.12688/f1000research.160239.1) First published: 06 Jan 2025, 14:32 (https://doi.org/10.12688/f1000research.160239.1) Latest published: 06 Jan 2025, 14:32 (https://doi.org/10.12688/f1000research.160239.1)

Introduction

Iski and Imai examined two types of algebras: $BCI$ -algebras and $BCK$ -algebras.^5,4,8 It is established that $BCK$ -algebras form a proper subclass within the broader class of $BCI$ -algebras. Iorgulescu and Georgescu introduced the concept of pseudo- $BCI$ algebra as a generalization of $BCK$ -algebra.^1,3 Jun examined the idea of a pseudo-ideal in pseudo-BCK-algebra.⁹ H. S. Kim and J. Neggers explained the notion of d-algebras, which is another useful generalization of BCK-algebras, and investigated several relations between d-algebras and oriented diagraphs.¹² Roh, Jun, and Kim introduced a new concept known as BH algebra, which is a generalization of BCI and BCK-algebras. The FS $f$ of a set $S$ is defined as a mapping element of $S$ to values within the closed interval $[0, 1]$ . In 1965, Zadeh introduced the idea of an FS for a set that was initially introduced.¹⁹ Kim¹⁰ introduced the concept of BG-algebra, which generalizes B-algebra. Somjanta et al.¹⁸ introduced and explored various properties of fuzzy UP-subalgebras and fuzzy UP-ideals within UP algebras. Prabpayak et al.¹⁵ examined various results related to fuzzy pseudo-KU algebras. Y. B. H. S., Jun, Kim, and S.S. Ahn⁷ introduced pseudo-Q-algebras as a generalization of Q-algebras.¹³ Jawad⁶ defined the concept of fuzzy pseudo-ideals (FPI) in pseudo Q-algebras. Megalai and Tamilarasi¹¹ introduced TMA, which generalizes BCK, BCH, Q, and BCI algebras. Chanwit and Prabpayak¹⁶ presented the concept of homomorphisms in FTMA, and established several properties. Mostafa et al.¹² studied the idea of FTMA within TMA and investigated its related properties.

In this article, we present the concept and characteristics of FPTMSAs within PTMA We explain the concept of level subsets in FPTMSA of PTMAs. Various properties of these FPTMSAs have been investigated with respect to homomorphisms and CP leading to several noteworthy findings. We denote if and only if by iff, $\land$ by minimum, $\lor$ by maximum, $\lor$ by sup, TM algebra by TMA, pseudo TM-algebra by PTMA, fuzzy subset by FS, fuzzy pseudo TM-subalgebra by FPTMSA, pseudo TM homomorphism by PTMH, homomorphic image by HI, inverse image by II, and cartesian product by CP.

Preliminaries

In this section, we review the basic results and definitions essential for the analysis presented in this article.

Definition 2.1.

(Ref. 14) A PTMA is defined as an algebra $(Σ; ⋉, ⋊, 0)$ that satisfies the following axioms: For every $ρ, ς, ϱ \in Σ$

1 . ρ ⋉ 0 = ρ ⋊ 0 = ρ 2 . (ρ ⋉ ς) ⋊ (ρ ⋉ ϱ) = ϱ ⋉ ς a n d (ρ ⋊ ς) ⋉ (ρ ⋊ ϱ) = ϱ ⋊ ς .

In a PTMA $Σ$ , define a relation $\leq such$ that $ρ \leq ς$ iff $ρ ⋉ ς = 0$ and $ρ ⋊ ς = 0$ . Furthermore, any PTMA $Σ$ in which $ρ ⋉ ς = ρ ⋊ ς$ for every $ρ, ς \in Σ$ is classified as TMA.

Lemma 2.2.

(Ref. 14) In PTMA $Σ$ the next holds for every $ρ, ς \in Σ$ .

(i) $ρ ⋉ ρ = ρ ⋊ ρ = 0$
(ii) $ς ⋉ (ρ ⋊ ς) = ς = ς ⋊ (ρ ⋉ ς)$
(iii) $(ρ ⋊ ς) ⋉ ρ = 0 ⋊ ς$ and $(ρ ⋉ ς) ⋊ ρ = 0 ⋉ ς$
(iv) $ρ ⋉ (ρ ⋊ (ρ ⋊ ς)) = ρ ⋊ ς, ρ ⋊ (ρ ⋉ (ρ ⋉ ς)) = ρ ⋉ ς$
(v) $0 ⋉ (ρ ⋊ ς) = ς, 0 ⋊ (ρ ⋉ ς) = ς$
(vi) $0 ⋉ (ς ⋊ ρ) = ρ ⋊ ς, 0 ⋊ (ς ⋉ ρ) = ρ ⋉ ς$
(vii) $(ρ ⋉ (ρ ⋊ ς)) ⋉ ς = 0, (ρ ⋊ (ρ ⋉ ς)) ⋊ ς = 0$
(viii) $ρ ⋉ (ρ ⋊ ς) = ς, ρ ⋊ (ρ ⋉ ς) = ς$
(ix) $ρ ⋉ (ς ⋊ ρ) = ρ, ρ ⋊ (ς ⋉ ρ) = ρ$
(x) $ς ⋉ (ς ⋊ ρ) = ρ, ς ⋊ (ς ⋉ ρ) = ρ$

Definition 2.3.

(Ref. 19) Let $Σ$ be a non-void. The mapping $n : Σ \to [0, 1]$ is called the FS of $Σ$ . The complement of $n$ , denoted by $n^{c}$ , is given by $n^{c} (ρ) = 1 - n (ρ)$ for each $ρ \in Σ$ .

Definition 2.4.

(Ref. 2) Let $n$ be the FS in $Σ$ . For a fixed $e \in [0, 1]$ , the set $v (n : e) = n_{e} = {ρ \in$ $Σ | n (ρ) \geq e}$ is the upper level of $n$ . Similarly, $ι (n : e) = n_{e} = {ρ \in Σ | n (ρ) \leq e}$ is the lower level of $n$ .

In addition, we can define the upper-strong and lower-strong level subsets of $n$ as follows:

The upper-strong-level subset is $v^{+} (n : e) = n_{e} = {ρ \in Σ | n (ρ) > e}$ .

ii. The lower-strong-level subset is $ι^{-} (n : e) = n_{e} = {ρ \in ρ | n (ρ) < e}$ .

Remark 2.5.

(Ref. 19) If $e_{1} \leq e_{2}$ , then $v (n : e_{2}) \subseteq v (n : e_{1})$ and $ι (n : e_{1}) \subseteq ι (n : e_{2})$ for every $e_{1}, e_{2} \in [0, 1]$ .

Definition 2.6.

(Ref. 17) A FS $n$ of a nonvoid $Σ$ , with a membership map $n : Σ \to [0, 1]$ , is said to be the supremum property if, for any subset $W$ of $Σ$ , there exists a $ρ \in W$ in which $n (ρ) = {sup}_{w \in W} n (w)$ .

Definition 2.7.

(Ref. 17) Let $h : W \to Z$ be a mapping, where $W$ and $Z$ are nonempty subsets of $Σ$ . If $n$ and $r$ are FSs of $W$ and $Z$ , respectively, then the image of $n$ under $h$ , denoted by $h (n)$ , is an FS of $Z$ defined as follows:

h (n) (ς) = {\begin{cases} ⋁ {n (ρ) | ρ \in h^{- 1} (ς)} & if h^{- 1} (ς) \neq \emptyset, \\ 0 & otherwise \end{cases}

where

ς \in Z

. The II of

r

under

h

, denoted by

h^{- 1} (r)

, is an FS of

W

defined by

h^{- 1} (r) (ρ) =

r (h (ρ))

for every

ρ \in W

Definition 2.8.

(Ref. 19) Let $n$ and $r$ be any two FSs of $Σ$ . Then th statements hereunder hold:

1. $n \subseteq r$ iff $n (ρ) \leq r (ρ)$ for every $ρ \in Σ$ .
2. $n = r$ iff $n (ρ) = r (ρ)$ for every $ρ \in Σ$ .
3. The product of $n$ and $r$ , denoted $(n \times r) (ρ, ς)$ , is given by $\land {n (ρ), r (ς)}$ for every $(ρ, ς) \in ρ \times ς$ .
4. The intersection of $n$ and $r$ is given by $n \cap r = \land {n (ρ), r (ρ)}$ for every $ρ \in Σ$ .
5. The union of $n$ and $r$ is $n \cup r = \lor {n (ρ), r (ρ)}$ for every $ρ \in Σ$ .

Lemma 2.9.

(Ref. 2) Let $n$ be any $FS$ of $Σ$ . Then, the axioms hold for any $, ς \in Σ$ :

1. $1 - \land {n (ρ), n (ς)} = \lor {1 - n (ρ), 1 - n (ς)}$ .
2. $1 - \lor {n (ρ), n (ς)} = \land {1 - n (ρ), 1 - n (ς)}$ .

Fuzzy pseudo TM-Subalgebra of pseudo TM-algebra

In this section, we examine the FPTMSAs within PTMAs and explore some fundamental properties of these subalgebras. Unless specified otherwise, $Σ$ and $ψ$ represent the PTMAs throughout this section and the subsequent sections.

Definition 3.1.

An FS $n$ of a PTMA $Σ$ is called an FPTMSA of $Σ$ if the following statements hold.

1. $n (ρ ⋉ ς) \geq \land {n (ρ), n (ς)}$ for every $ρ, ς \in Σ$ .
2. $n (ρ ⋊ ς) \geq \land {n (ρ), n (ς)}$ for every $ρ, ς \in Σ$ .

Example 3.2.

Let $Σ = {0, g, s, t}$ be a set equipped with two binary compositions, $⋉$ and $⋊$ , defined in the tables below.

$⋉$	0	g	s	t
0	0	0	0	0
g	g	0	0	0
s	s	s	0	0
t	t	s	s	0

$⋊$	0	g	s	t
0	0	0	0	0
g	g	0	0	0
s	s	s	0	0
t	t	t	g	0

Thus, $(Σ, ⋉, ⋊, 0)$ forms PTMA. Let $n$ and $r$ be FSs of $Σ$ defined hereunder.

n (ρ) = {\begin{cases} 1, & if ρ = 0; \\ \frac{3}{10}, & if ρ = t; \\ 0 . & otherwise . \end{cases} and r (ρ) = {\begin{cases} \frac{7}{10}, & if ρ = 0 \\ \frac{1}{2}, & if ρ = t; \\ 0 . & otherwise . \end{cases}

It is evident that $n$ and $r$ are FPTMSAs of $Σ$ .

Lemma 3.3.

If $n$ is an FPTMSA of $Σ$ , then $n (0) \geq n (ρ)$ , for each $ρ \in Σ$ .

Proof.

Assuming $n$ is an FPTMSA of $Σ$ , we can apply Lemma 2.2 to obtain

n (0) = n (ρ ⋉ ρ) \geq \land {n (ρ), n (ρ)} = n (ρ)

and

n (0) = n (ρ ⋊ ρ) \geq \land {n (ρ), n (ρ)} = n (ρ)

Therefore, $n (0) \geq n (ρ)$ for each $ρ \in Σ$ .

Theorem 3.4.

Let $n$ be a FPTMSA of $Σ$ . Then $n (ρ ⋉ ς) = n (ς)$ iff $n (ρ) = n (0)$ for every $ρ, ς \in Σ$ .

Proof.

Assume $n (ρ ⋉ ς) = n (ς)$ for every $ρ, ς \in Σ$ . Claim: $n (ρ) = n (0)$ for every $ρ \in Σ$ .

By a PTMA, we have $ρ ⋉ 0 = 0$ and $ρ ⋊ 0 = 0$ for every $ρ \in Σ$ . Thus

(ρ ⋉ 0) ⋉ 0 = ρ ⋉ 0 = ρ

and

(ρ ⋊ 0) ⋊ 0 = ρ ⋊ 0 = ρ

Therefore:

\begin{array}{l} n (ρ) = n ((ρ ⋉ 0) ⋉ 0) \geq \land {n (ρ ⋉ 0), n (0)} = \land {n (0), n (0)} = n (0) \\ n (ρ) = n ((ρ ⋊ 0) ⋊ 0) \geq \land {n (ρ ⋊ 0), n (0)} = \land {n (0), n (0)} = n (0) \end{array}

From Lemma 3.3, we also have $n (0) \geq n (ρ)$ . Therefore, $n (ρ) = n (0)$ for each $ρ \in Σ$ .

Conversely, it is assumed $n (ρ) = n (0)$ for every $ρ \in Σ$ . Claim: $n (ρ ⋉ ς) = n (ς)$ and $n (ρ ⋊ ς) = n (ς)$ .

Using Lemma 3.3, $n (0) \geq n (ς)$ for any $ς \in ρ$ , so $n (ρ) \geq n (ς)$ . Also:

\begin{array}{l} n (ρ ⋉ ς) \geq \land {n (ρ), n (ς)} = n (ς) \\ n (ρ ⋊ ς) \geq \land {n (ρ), n (ς)} = n (ς) \end{array}

Additionally:

\begin{array}{l} n (ς) = n (0 ⋉ (ρ ⋊ ς)) \geq \land {n (0), n (ρ ⋊ ς)} = \land {n (ρ), n (ρ ⋊ ς)} = n (ρ ⋊ ς) \\ n (ς) = n (0 ⋊ (ρ ⋉ ς)) \geq \land {n (0), n (ρ ⋉ ς)} = \land {n (x), n (x ⋉ ς)} = n (x ⋉ ς) \end{array}

Thus, $n (x ⋉ ς) = n (ς)$ and $n (x ⋊ ς) = n (ς)$ .

Theorem 3.5.

If $n$ and $r$ are two FPTMSAs of $Σ$ , then their intersection $n \cap r$ is also $a$ FPTMSA of $X$ .

Proof.

Let $n$ and $r$ be the two FPTMSAs of $Σ$ . Claim: $n \cap r$ is an FPTMSA of $Σ$ .

Consider $ρ, ς \in Σ : (n \cap r) (ρ ⋉ ς) = \land {n (ρ ⋉ ς), r (ρ ⋉ ς)} \geq \land {\land {n (ρ), n (ς)}, \land {r (ρ), r (ς)}} = \land {\land {n (ρ), r (ρ)}, \land {n (ς), r (ς)}}$ . This implies that $(n \cap r) (ρ ⋉ ς) \geq \land {(n \cap r) (ρ), (n \cap r) (ς)}$ .

Similarly, $(n \cap r) (ρ ⋊ ς) = \land {n (ρ ⋊ ς), r (ρ ⋊ ς)} \geq \land {\land {n (ρ), n (ς)}, \land {r (ρ), r (ς)}} = \land {\land {n (ρ), r (ρ)}, \land {n (ς), r (ς)}}$ . Thus, $(n \cap r) (ρ ⋊ ς) \geq \land {(n \cap r) (ρ), (n \cap r) (ς)}$ .

Corollary 3.6.

Let ${n_{i} | i \in I}$ be a family of FPTMSA with $Σ$ . Their intersection $\cap_{i \in I} n_{i}$ is also an FPTMSA of $Σ$ .

Remark 3.7.

The union of the two FPTMSAs of $X$ is not necessarily an FPTMSA of $Σ$ .

Example 3.8.

Let $Q$ be a set of rational numbers. Define two binary operations $⋉$ and $⋊$ on $Q$ as $ρ ⋉ ς = ρ - ς$ and $x ⋊ ς = ρ - ς$ for every $ρ, ς \in Q$ , where – denotes the usual subtraction in $Q$ . Thus, $(Q, ⋉, ⋊, 0)$ is a PTMA.

Let $n$ and $r$ be FSs of $Q$ by:

n (ρ) = {\begin{cases} \frac{9}{10} & if ρ \in ⟨ 5 ⟩ \\ \frac{1}{5} & otherwise \end{cases}

and

r (ρ) = {\begin{cases} \frac{4}{5} & if ρ \in ⟨ 4 ⟩ \\ \frac{1}{10} & otherwise \end{cases}

Clearly, $n$ and $r$ are the FPTMSAs of $Q$ .

Consider the union ( $n \cup r$ ) defined by

(n \cup r) (ρ) = {\begin{cases} \frac{9}{10} & if ρ \in ⟨ 5 ⟩ \\ \frac{4}{5} & if ρ \in ⟨ 4 ⟩ \ ⟨ 5 ⟩ \\ \frac{1}{10} & otherwise \end{cases}

For $ρ = 5$ and $= 4$ :

\begin{array}{l} (n \cup r) (5) = 0.9 \\ (n \cup r) (4) = 0.8 \end{array}

Then:

(n \cup r) (5 ⋉ 4) = \lor {n (5 - 4), r (5 - 4)} = \lor {0.2,0.1} = 0.2

(n \cup r) (5 ⋊ 4) = \lor {n (5 - 4), r (5 - 4)} = \lor {0.2,0.1} = 0.2

So:

(n \cup r) (5 ⋉ 4) = 0.2 = (n \cup r) (5 ⋊ 4)

However:

\land {(n \cup r) (5), (n \cup r) (4)} = \land {0.9,0.8} = 0.8

Thus, $0.2 \geq 0.8$ , which contradicts the definition of an FPTMSA. This indicates that the union of the two FPTMSAs may not be an FPTMSA of $Σ$ .

Proposition 3.9.

If $n$ is an FPTMSA of $Σ$ , then the following results hold.

1. $n (0 ⋉ ρ) \geq n (ρ)$ and $n (0 ⋊ ρ) \geq n (ρ)$ .
2. If there exists a sequence $ρ_{k}$ in $Σ$ such that ${lim}_{k \to \infty} n (ρ_{k}) = 1$ , then $n (0) = 1$ .

Proof.

Suppose that $n$ is an FPTMSA of $Σ$ . Claim: $n (0 ⋉ ρ) \geq n (ρ)$ and $n (0 ⋊ ρ) \geq n (ρ)$ .

i. Since $n (0 ⋉ ρ) \geq \land {n (0), n (ρ)}$ . It implies that $n (0 ⋉ ρ) \geq n (ρ)$ and $n (0 ⋊ ρ) \geq \land {n (0), n (ρ)} \Rightarrow n (0 ⋊ ρ) \geq n (ρ)$ .
ii. Suppose that $n$ is an FPTMSA of $Σ$ . From Lemma $3.3 n (0) \geq n (ρ)$ , for every $ρ \in Σ$ . If $ρ_{k}$ is a sequence in $Σ$ , then $1 \geq n (0) = n (ρ_{k} ⋉ ρ_{n}) \geq \land {n (ρ_{k}), n (ρ_{k})} = n (ρ_{k})$ . As $k \to \infty$ , we have ${lim}_{k \to \infty} 1 \geq {lim}_{k \to \infty} n (0) \geq {lim}_{k \to \infty} n (ρ_{k})$ .

Hence, $n (0) = 1$ .

Similarly, $1 \geq n (0) = n (ρ_{n} ⋊ ρ_{n}) \geq \land {n (ρ_{k}), n (k_{k})} = n (ρ_{k})$ . As $k \to \infty$ . Thus, ${lim}_{k \to \infty} 1 \geq {lim}_{k \to \infty} n (0) \geq {lim}_{k \to \infty} n (ρ_{k})$ .

Hence, $n (0) = 1$ .

Definition 3.10.

Let $v (n : e)$ be non-void for every $e \in [0, 1]$ . An FS $n$ of a PTMA $Σ$ is called the upper level of $n$ if $v (n : e) = n_{e} = {ρ \in Σ | n (ρ) \geq e}$ . Similarly, $ι (n : e) = n_{e} = {ρ \in Σ |$ $n (ρ) \leq e}$ is the lower level of $n$ .

Theorem 3.11.

Let $n$ be a FPTMSA of $Σ$ . Then the set $G_{n} = {ρ \in Σ | n (ρ) = n (0)}$ forms a FPTMSA of $Σ$ .

Proof.

Suppose that $n$ is an FPTMSA of $Σ$ . Let $ρ, ς \in G_{n}$ . Then $n (ρ) = n (0) = n (ς)$ . Now, $n (ρ ⋉ ς) \geq \land {n (ρ), n (ς)} = n (0)$ . This implies that $n (ρ ⋉ ς) \geq n (0)$ . By Lemma 3.3, $n (0) \geq n (ρ ⋉ ς)$ for every $ρ, ς \in Σ$ . It follows that $n (ρ ⋉ ς) = n (0)$ . This implies that $ρ ⋉ ς \in G_{n}$ . Similarly, $n (ρ ⋊ ς) \geq \land {n (ρ), n (ς)} = n (0)$ . It implies $n (ρ ⋊ ς) \geq n (0)$ . By Lemma 3.3, $n (0) \geq n (ρ ⋊ ς)$ for every $ρ, ς \in Σ$ . It follows that $n (ρ ⋊ ς) = n (0)$ . This implies that $ρ ⋊ ς \in G_{n}$ . Hence, $G_{n}$ is an FPTMSA with $Σ$ .

Theorem 3.12.

An FS n of a PTMA $Σ$ is an FPTMSA iff for every $e \in [0, 1]$ the level set $v (n, e)$ is either empty or a pseudo TM subalgebra of $Σ$ .

Proof.

We assume that $n$ is an FPTMSA of $Σ$ . Claim: The level subset $v (n, e)$ is either empty or PTMA of $Σ$ . Suppose that the level subset $v (n, e) \neq \emptyset$ . For any $ρ, ς \in v (n, e)$ we have $n (ρ) \geq e$ and $n (ς) \geq e$ . Then:

\begin{array}{l} n (ρ ⋉ ς) \geq \land {n (ρ), n (ς)} \geq \land {e, e} = e and \\ n (ρ ⋊ ς) \geq \land {n (ρ), n (ς)} \geq \land {e, e} = e \end{array}

Hence, $ρ ⋉ ς, ρ ⋊ ς \in v (n, e)$ . Therefore, $v (n, e)$ represents the PTMSA of $Σ$ .

Conversely, we assume that $v (n, e)$ is a PTMA of $Σ$ . Claim $n$ is the FPTMSA of $Σ$ .

For any $ρ, ς \in Σ$ . Take $e = \land {n (ρ), n (ς)}$ . Then $ρ ⋉ ς, ρ ⋊ ς \in v (n, e)$ which implies that $n (ρ ⋉ ς) \geq e = \land {n (ρ), n (ς)}$ and $n (ρ ⋊ y) \geq e = \land {n (ρ), n (ς)}$ .

Hence, $ρ ⋉ ς, ρ ⋊ ς \in n$ . Therefore, $n$ is a PTMSA with $Σ$ .

Corollary 3.13.

Let $n$ be a PTMSA of $Σ$ and $e \in [0, 1]$ . If $e = 1$ , then the upper level set $v (n, e)$ is either empty or a PTMSA of $Σ$ .

Theorem 3.14.

Let $W$ be any non-void subset of a PTMA $Σ$ , and let $n$ be an FS of $Σ$ defined by

n (x) = {\begin{cases} u & if ρ \in W \\ v & if ρ \notin W \end{cases}

where

u, v \in [0, 1]

and

u \geq v

Then, $n$ is an FPTMSA of $Σ$ if $W$ is a PTMSA of $Σ$ .

Proof.

We assume that $n$ is an FPTMSA of $Σ$ . Claim: $W$ is PTMSA of $Σ$ .

Let $ρ, ς \in W$ . Because $n$ is an FPTMSA of $Σ$ , we have

\begin{array}{l} n (ρ ⋉ ς) \geq \land {n (ρ), n (ς)} = \land {u, u} = u and \\ n (ρ ⋊ ς) \geq \land {n (ρ), n (ς)} = \land {u, u} = u . \end{array}

Hence, 0, ρ ⋉ ς, ρ ⋊ ς ∈ W.

Hence, $W$ is the FPTMSA of $Σ$ .

Conversely, we assume that $n$ is the PTMA of $Σ$ . Claim $n$ is the FPTMSA of $Σ$ . Consider Case 1. If $ρ, ς \in W$ , then $ρ ⋉ ς, ρ ⋊ ς \in W$ .

\begin{array}{l} n (ρ ⋉ ς) = u \geq \land {n (ρ), n (ς)} and \\ n (ρ ⋊ ς) = u \geq \land {n (ρ), n (ς)} . \end{array}

Hence, ρ ⋉ς, ρ ⋊ ς ∈ n.

Case 2. If $ρ \in W$ and $ς \notin W$ , then $n (ρ) = u$ and $n (ς) = v$ .Thus

\begin{array}{l} n (ρ ⋉ ς) \geq \land {u, v} = \land {n (ρ), n (ς)} and \\ n (ρ ⋊ ς) \geq \land {u, v} = \land {n (ρ), n (y)} . \end{array}

Hence, ρ ⋉ς, ρ ⋊ ς ∈ n.

Case 3. If $ρ \notin W$ and $ς \in W$ , then swapping the roles of $ρ$ and $ς$ in case (2) produces similar results.

\begin{array}{l} n (ρ ⋉ ς) \geq \land {v, u} = \land {n (ς), n (ρ)} = \land {n (ρ), n (ς)} and \\ n (ρ ⋊ ς) \geq \land {v, u} = \land {n (ς), n (ρ)} = \land {n (ς), n (ρ)} . \end{array}

Hence, ρ ⋉ς, ρ ⋊ ς ∈ n.

Case 4. If $ρ \notin W, ς \notin W$ , then $n (ρ) = v$ and $n (ς) = v$ .Thus

\begin{array}{l} n (ρ ⋉ ς) \geq \land {v, v} = \land {n (ρ), n (ς)} and \\ n (ρ ⋊ ς) \geq \land {v, v} = \land {n (ρ), n (ς)} . \end{array}

Hence, ρ ⋉ς, ρ ⋊ ς ∈ n.

Therefore, for each case, $n$ is the FPTMSA of $Σ$ .

Lemma 3.15.

Any PTMSA of a PTMA $Σ$ can be represented as a level PTMSA of an FPTMSA of $Σ$ .

Proof.

Let $n$ be a PTMSA of a PTMA $Σ$ . For $α \in [0, 1]$ , let $n$ be an FS of $Σ$ defined as:

n (ρ) = {\begin{cases} α, & if ρ \in n \\ 0, & otherwise \end{cases}

If $ρ, ς \in n$ , then $ρ ⋉ ς, ρ ⋊ ς \in n$ . Then by definition $n (x) = n (ς) = n (ρ ⋉ ς) = n (ρ ⋊ ς) = α$ .

\begin{array}{l} n (ρ ⋉ ς) = α \geq \land {n (ρ), n (ς)} and \\ n (ρ ⋊ ς) = α \geq \land {n (ρ), n (ς)} . \end{array}

Hence, ρ ⋉ ς, ρ ⋊ ς ∈ n.

Similarly, if $ρ, ς \notin n$ , then $n (ρ) = n (ς) = 0$ .

\begin{array}{l} n (ρ ⋉ ς) = 0 \geq \land {n (ρ), n (ς)} & and \\ n (ρ ⋊ ς) = 0 \geq \land {n (ρ), n (ς)} \end{array}

Hence, ρ ⋉ ς, ρ ⋊ ς ∈ n.

If at most one of $ρ, ς \in n$ , then $n (ρ)$ or $n (ς)$ is equal to 0.

\begin{array}{l} n (ρ ⋉ ς) = 0 \geq \land {n (ρ), n (ς)} and \\ n (ρ ⋊ ς) = 0 \geq \land {n (ρ), n (ς)} . \end{array}

Hence, ρ ⋉ ς, ρ ⋊ ς ∈ n.

This indicates that $W$ is the PTMSA level of $Σ$ , corresponding to an FPTMSA of $Σ$ .

Corollary 3.16.

Let $W$ be a subset of a PTMA $Σ$ . Characteristic mapping is defined as

χ_{W} (ρ) = {\begin{cases} 1 & if ρ \in W \\ 0 & if ρ \notin W \end{cases}

is a FPTMSA of

Σ

iff

W

is a PTMSA of

Σ

Homomorphism on fuzzy pseudo TM-subalgebras

In this section, we explore FPTMSAs within a PTMA under homomorphic conditions. We examined the HI and II of FPTMSAs in a PTMA. In addition, we discuss the composition of two epimorphic images and inverse images of FPTMSAs, along with other related results.

Definition 4.1.

Let $(Σ; ⋉_{ρ}, ⋊_{ρ}, 0_{ρ})$ and $(ψ; ⋉_{ς}, ⋊_{ς}, 0_{ς})$ be two pseudo-TMAs. A mapping $h : Σ \to ψ$ is called PTMH if $h (ρ ⋉ ς) = h (ρ) ⋉ h (ς)$ and $h (ρ ⋊ ς) = h (ρ) ⋊ h (ς)$ for every $ρ, ς \in Σ$ .

Note that if $h : Σ \to ψ$ is a pseudo-TM homomorphism, then $h (0_{Σ}) = 0_{r}$ where $0_{Σ}$ and $0_{r}$ are zero constant elements of $Σ$ and $ψ$ respectively.

Theorem 4.2.

Let $(Σ; ⋉_{ρ}, ⋊_{ρ}, 0_{ρ})$ and $(ψ; ⋉_{ς}, ⋊_{ς}, 0_{ς})$ be two PTMAs. Let $h : Σ \to ψ$ be the epimorphism of the PTMAs. If n is an FPTMSA of $Σ$ with the sup-property, then $h (n)$ is an FPTMSA of $r$ .

Theorem 4.3.

$Let (Σ; ⋉_{ρ}, ⋊_{ρ}, 0_{ρ})$ and $(r; ⋉_{ς}, ⋊_{ς}, 0_{ς})$ be two PTMAs. Let $h : Σ \to ψ$ be an epimorphism of a PTMA. If $r$ is an FPTMSA of $ψ$ , then $h^{- 1} (r)$ is an FPTMSA of $Σ$ .

Proof.

We assumed that $r$ is an FPTMSA of $ψ$ . Claim: $h^{- 1} (r)$ is the FPTMSA of $Σ$ . For any $ρ, ς \in Σ$ , we have: $h^{- 1} (r (ρ ⋉_{ρ} ς)) = r (h (ρ ⋉_{ρ} ς)) = r (h (ρ) ⋉_{ς} h (ς)) \geq \land {r (h (ρ)), r (h (ς))} =$ $\land {h^{- 1} (r) (ρ), h^{- 1} (r) (ς)}$ . Similarly, $f^{- 1} (r (ρ ⋊_{ρ} ς)) = r (h (ρ ⋊_{ρ} ς)) = r (h (ρ) ⋊_{ς} h (ς)) \geq \land {r (h (ρ)), r (h (ς))} =$ $\land {h^{- 1} (r) (ρ), h^{- 1} (r) (ς)}$

Therefore, $h^{- 1} (r)$ is a FPTMSA of $Σ$ .

Lemma 4.4.

Let $h : Σ \to ψ$ be an epimorphism and $r$ be an FS in $ψ$ . If $h^{- 1} (r)$ is an FPTMSA of $Σ$ , then $r$ must be an FPTMSA of $r$ .

Definition 4.5.

Let $h : Σ \to ψ$ be the homomorphism of the PTMAs. Then, for any FS $n$ in $ψ, n^{h}$ in $Σ$ by $n^{h} (ρ) = n (h (ρ))$ for all $ρ \in Σ$ is called the epimorphism of a PTMA $Σ$ .

Theorem 4.6.

Let $h : Σ \to ψ$ be an epimorphism of a PTMA.Then the FS $n$ is an FPTMSA of $ψ$ iff $n^{h}$ is an FPTMSA of $Σ$ .

Proposition 4.7.

Let $h : Σ \to ψ$ and $g : ψ \to Z$ be the epimorphism of a PTMAs, and $r$ be an FPTMSA. Then, ${(goh)}^{- 1} (r)$ is an FPTMSA of $Σ$ iff $(goh) (r)$ is an FPTMSA of $Z$ , assuming that the sup-property holds.

Cartesian product of fuzzy pseudo TM-subalgebra

In this section, we explore the concept of CP in relation to the level subsets of FPTMSA. We demonstrate that the CP of two FPTMSA yields another FPTMSA and investigate additional related results.

Definition 5.1.

Suppose $Σ$ and $ψ$ are PTMAs, with $n_{1}$ as an FPTMSA of $Σ$ and $n_{2}$ as an FPTMSA of $ψ$ . The CP of $n_{1}$ and $n_{2}$ , denoted as $n_{1} \times n_{2}$ , is defined by

(n_{1} \times n_{2}) ((ρ_{1}, ρ_{2}) ⋉ (ς_{1}, ς_{2})) = \land {n_{1} (ρ_{1} ⋉ ς_{1}), n_{2} (ρ_{2} ⋉ ς_{2})}

and

(n_{1} \times n_{2}) ((ρ_{1}, ρ_{2}) ⋊ (ς_{1}, ς_{2})) = \land {n_{1} (ρ_{1} ⋊ ς_{1}), n_{2} (ρ_{2} ⋊ ς_{2})}

for all

(ρ_{1}, ρ_{2})

and

(ς_{1}, ς_{2}) \in Σ \times ψ

Lemma 5.2.

Let $n_{1}$ and $n_{2}$ any two FPTMSA of $Σ$ and $ψ$ respectively. Then $(n_{1} \times n_{2}) (0, 0) \geq$ $(n_{1} \times n_{2}) (ρ, ς)$ , for all $(ρ, ς) \in Σ \times ψ$ .

Proof.

For every $(ρ, ς) \in Σ \times ψ$ .

\begin{array}{l} (n_{1} \times n_{2}) (0, 0) = (n_{1} \times n_{2}) (ρ ⋉ ρ, ς ⋉ ς) \\ = \land {n_{1} (ρ ⋉ ρ), n_{2} (ς ⋉ ς)} \\ \geq \land {\land {n_{1} (ρ), n_{1} (ρ)}, \land {n_{2} (ς), n_{2} (ς)}} \\ = \land {n_{1} (ρ), n_{2} (ς)} \\ = (n_{1} \times n_{2}) (ρ, ς) and \\ (n_{1} \times n_{2}) (0, 0) = (n_{1} \times n_{2}) (ρ ⋊ ρ, ς ⋊ ς) \\ = \land {n_{1} (ρ ⋊ ρ), n_{2} (ς ⋊ ς)} \\ \geq \land {\land {n_{1} (ρ), n_{1} (ρ)}, \land {n_{2} (ς), n_{2} (ς)}} \\ = \land {n_{1} (ρ), n_{2} (ς)} \\ = (n_{1} \times n_{2}) (ρ, ς) \end{array}

Hence, $(n_{1} \times n_{2}) (0, 0) \geq (n_{1} \times n_{2}) (ρ, ς)$ , for all $(ρ, ς) \in Σ \times ψ$ .

Theorem 5.3.

If $n_{1}$ and $n_{2}$ are FPTMSA of $Σ$ and $ψ$ respectively, then the $FS (n) = n_{1} \times n_{2}$ defined on $Σ \times ψ$ is a FPTMSA of $Σ \times ψ$ .

Proof.

Assume that $n_{1}$ and $n_{2}$ be a FPTMSA of $Σ$ and $ψ$ respectively. For any $(ρ_{1}, ρ_{2})$ and $(ς_{1}, ς_{2}) \in Σ \times r$ .

\begin{array}{l} n ((ρ_{1}, ρ_{2}) ⋉ (ς_{1}, ς_{2})) = n (ρ_{1} ⋉ ς_{1}, ρ_{2} ⋉ ς_{2}) \\ = (n_{1} \times n_{2}) (ρ_{1} ⋉ ς_{1}, ρ_{2} ⋉ ς_{2}) \\ = \land {n_{1} (x_{1} ⋉ ς_{1}), n_{2} (ρ_{2} ⋉ ς_{2}))} \\ \geq \land {\land {n_{1} (ρ_{1}), n_{1} (ς_{1})}, \land {n_{2} (ρ_{2}), n_{2} (ς_{2})}} \\ = \land {\land {n_{1} (ρ_{1}), n_{2} (ρ_{2})}, \land {n_{1} (ς_{1}), n_{2} (ς_{2})}} \\ = \land {(n_{1} \times n_{2}) (ρ_{1}, ρ_{2}), (n_{1} \times n_{2}) (ς_{1}, ς_{2})} \\ = \land {n (ρ_{1}, ρ_{2}), n (ς_{1}, ς_{2})} and \\ n ((ρ_{1}, ρ_{2}) ⋊ (ς_{1}, ς_{2})) = n (ρ_{1} ⋊ ς_{1}, ρ_{2} ⋊ ς_{2}) \\ = (n_{1} \times n_{2}) (ρ_{1} ⋊ ς_{1}, ρ_{2} ⋊ ς_{2}) \\ = \land {n_{1} (ρ_{1} ⋊ ς_{1}), n_{2} (ρ_{2} ⋊ ς_{2})} \\ \geq \land {\land {n_{1} (ρ_{1}), n_{1} (ς_{1})}, \land {n_{2} (ρ_{2}), n_{2} (ς_{2})}} \\ = \land {\land {n_{1} (ρ_{1}), n_{2} (ρ_{2})}, \land {n_{1} (ς_{1}), n_{2} (ς_{2})}} \\ = \land {(n_{1} \times n_{2}) (x_{1}, x_{2}), (n_{1} \times n_{2}) (ς_{1}, ς_{2})} \\ = \land {n (ρ_{1}, ρ_{2}), n (ς_{1}, ς_{2})} \end{array}

Definition 5.4.

Let $n_{1}$ and $n_{2}$ be the two FS of $Σ$ and $ψ$ respectively. For any $e \in [0, 1]$ the set $v (n_{1} \times n_{2}, t) = {(ρ, ς) \in Σ \times r / (n_{1} \times n_{2}) (ρ, ς) \geq e}$ is called the upper-level subset of $n_{1} \times n_{2}$ .

Theorem 5.5.

Let $n_{1}$ and $n_{2}$ be the FSs of $Σ$ and $r$ , respectively. The $CP n_{1} \times n_{2}$ forms an FPTMSA of $Σ \times ψ$ iff the nonvoid upper e-level set $v (n_{1} \times n_{2}, e)$ is an FPTMSA of $Σ \times ψ$ .

Conclusion

In this article, we introduce the concept of FSs applied to PTMSAs within PTMAs. We define the idea of an FPTMSA of a PTMA and explore the various results. The characterization of an FPTMSA is provided through its level subsets in PTMA. We investigated several properties of FPTMSAs with homomorphisms and CPs. The HIs and preimages of FPTMSA of a PTMA were examined, and the CP of two FPTMSA was studied, yielding related results. Additionally, we characterized CPs of FPTMSA in their level sets.

In future work, we plan to extend these concepts to the fuzzy pseudo-ideals of a PTMA, aiming to achieve new results.

Ethics and consent

Ethical approval and consent were not required.

Data availability statement

No data are associated with this article.

References

1. Dudek WA, Jun YB: Pseudo-BCI algebras. East Asian Math. J. 2008; 24(2): 187–190.
2. Gedamu WY, Alaba BA, Derseh BL: Intuitionistic Fuzzy PMS-ideals in a PMS-algebra. Thai J. Math. 2022; 20(4): 1535–1548.
3. Georgescu G, Iorgulescu A: Pseudo-BCK algebras: an extension of BCK algebras. Combinatorics, Computability and Logic: Proceedings of the Third International Conference on Combinatorics, Computability and Logic,(DMTCS01). London: Springer London; 2001; pp. 97–114.
4. Iseki K: On BCI-algebras. Math. Seminar Notes. 1980; 8: 125–130.
5. Iseki K, Shotaro T: An introduction to the theory of the BCK-algebras.1978; pp. 1–26.
6. Jawad HK: Some Types of Fuzzy Pseudo Ideals of Pseudo Q-algebra. University of Kufa; 2019. Thesis.
7. Jun YB, Kim HS, Ahn SSH: Structuctures of pseudo ideal and pseudo Atom in a pseudo Q-algebra. Kyungpook Math. J. 2016; 56: 95–106. Publisher Full Text
8. Jun Y: On BH-algebras. Sci. Math. 1998; 1(3): 347–354.
9. Jun Y: Pseudo-ideals of pseudo-BCK- algebras. Sci. Math. Jpn. 2003; 8: 87–91.
10. Kim CB, Kim HS: On BG-algebras. Demonstr. Math. 2008; 41(3): 497–506.
11. Megalai K, Tamilarasi A: TM-algebra-an introduction, Special Issue on Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications, CASCT.2010.
12. Mostafa SM, Naby M, Elgendy OR: Fuzzy TM-ideals of TM- algebras. J. Am. Sci. 2011; 79): 17–21.
13. Neggers J, Ahn SS, Kim HS, et al.: On Q-algebras. Int. J. Math. Math. Sci. 2001; 27: 749–757. Publisher Full Text
14. Nouri AH, Sabbar MG: Pseudo TM-algebra. Int. J. Algebra. 2019; 13(8): 341–348. Publisher Full Text
15. Prabpayak C, Leerawat U: On ideals and congruences in KU-algebras. Sci. Magna. 2009; 5(1): 54–57.
16. Prabpayak C: Some homomorphism properties of TM-algebras. Appl. Mech. Mater. 2017; 866: 406–409. Publisher Full Text
17. Sharma P: Homomorphism of intuitionistic fuzzy groups. Int. Math. Forum. 2011; 6: 3169–3178.
18. Somjanta J, Thuekaew N, Kumpeangkeaw P, et al.: Fuzzy sets in UP-algebras. Ann. Fuzzy Math. Inform. 2016; 12(6): 739–756.
19. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8: 338–353. Publisher Full Text

Comments on this article Comments (0)

Version 1

VERSION 1 PUBLISHED 06 Jan 2025

Author details Author details

¹ Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia
² Debre Markos University College of Natural and Computational Science, Debre Markos, Amhara, Ethiopia

Competing interests

No competing interests were disclosed.

Grant information

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

Article Versions (1)

version 1

Published: 06 Jan 2025, 14:32

https://doi.org/10.12688/f1000research.160239.1

© 2025 Melaku AG et al. 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.

Download

Export To

metrics

	Views	Downloads
F1000Research	-	-
PubMed Central Data from PMC are received and updated monthly.	-	-

Citations

SEE MORE DETAILS

CITE

how to cite this article

Melaku AG, Alaba BA, Bitew BT and Derseh BL. Fuzzy PTM subalgebra of PTM - algebra [version 1; peer review: 1 approved with reservations]. F1000Research 2025, 14:32 (https://doi.org/10.12688/f1000research.160239.1)

NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article.

track

receive updates on this article

Track an article to receive email alerts on any updates to this article.

Open Peer Review

Version 1

VERSION 1

PUBLISHED 06 Jan 2025

Views

Reviewer Report 27 Feb 2025

Munusamy Premkumar, Sathyabama Institute of Science and Technology, Chennai, Tamil Nadu, India

Approved with Reservations

https://doi.org/10.5256/f1000research.176100.r366638

Reviewer’s Recommendation: Accept with Minor Revisions
1. General Comments
The author's work is highly valuable and contributes significantly to the field. I recommend acceptance of the paper after addressing the following minor revisions to improve clarity, readability, and presentation.
2. Specific Comments and Suggested Revisions
Abstract:
The abstract should explicitly mention the importance of the work to highlight its contribution.
The author should clearly state the key properties studied in the paper to provide a concise overview of the research.

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?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
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: my research are is 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, however I have significant reservations, as outlined above.

CITE

Report a concern

Author Response 28 Feb 2025

Alemayehu Girum Melaku, Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia

28 Feb 2025

Author Response

This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work ... Continue reading This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work lies in its exploration of the intricate relationships between FPTMSAs and their level sets, which are crucial for understanding the characterization of these subalgebras.

The study meticulously investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby laying a foundational framework for future research into fuzzy pseudo-ideals within PTMAs.

The Key properties studied in this research include the definition and examination of level sets for FPTMSAs, the behavior of FPTMSAs under homomorphisms, and the implications of epimorphic and inverse images. Also, the study establishes that the Cartesian product of two FPTMSAs are remains a FPTMSA. Moreover, the relationships between homomorphic images and inverse images of FPTMSAs are analyzed, providing further algebraic insights.

Generally, this article introduced the concepts of FPTMSAs in the structural of PTMAs, particularly regarding their interactions with homomorphisms, epimorphisms, and Cartesian products.
This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work lies in its exploration of the intricate relationships between FPTMSAs and their level sets, which are crucial for understanding the characterization of these subalgebras.

The study meticulously investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby laying a foundational framework for future research into fuzzy pseudo-ideals within PTMAs.

The Key properties studied in this research include the definition and examination of level sets for FPTMSAs, the behavior of FPTMSAs under homomorphisms, and the implications of epimorphic and inverse images. Also, the study establishes that the Cartesian product of two FPTMSAs are remains a FPTMSA. Moreover, the relationships between homomorphic images and inverse images of FPTMSAs are analyzed, providing further algebraic insights.

Generally, this article introduced the concepts of FPTMSAs in the structural of PTMAs, particularly regarding their interactions with homomorphisms, epimorphisms, and Cartesian products.
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 28 Feb 2025

Alemayehu Girum Melaku, Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia

28 Feb 2025

Author Response

This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work ... Continue reading This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work lies in its exploration of the intricate relationships between FPTMSAs and their level sets, which are crucial for understanding the characterization of these subalgebras.

The study meticulously investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby laying a foundational framework for future research into fuzzy pseudo-ideals within PTMAs.

The Key properties studied in this research include the definition and examination of level sets for FPTMSAs, the behavior of FPTMSAs under homomorphisms, and the implications of epimorphic and inverse images. Also, the study establishes that the Cartesian product of two FPTMSAs are remains a FPTMSA. Moreover, the relationships between homomorphic images and inverse images of FPTMSAs are analyzed, providing further algebraic insights.

Generally, this article introduced the concepts of FPTMSAs in the structural of PTMAs, particularly regarding their interactions with homomorphisms, epimorphisms, and Cartesian products.
This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work lies in its exploration of the intricate relationships between FPTMSAs and their level sets, which are crucial for understanding the characterization of these subalgebras.

The study meticulously investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby laying a foundational framework for future research into fuzzy pseudo-ideals within PTMAs.

The Key properties studied in this research include the definition and examination of level sets for FPTMSAs, the behavior of FPTMSAs under homomorphisms, and the implications of epimorphic and inverse images. Also, the study establishes that the Cartesian product of two FPTMSAs are remains a FPTMSA. Moreover, the relationships between homomorphic images and inverse images of FPTMSAs are analyzed, providing further algebraic insights.

Generally, this article introduced the concepts of FPTMSAs in the structural of PTMAs, particularly regarding their interactions with homomorphisms, epimorphisms, and Cartesian products.
Competing Interests: No competing interests were disclosed. Close
Report a concern

Comments on this article Comments (0)

Version 1

VERSION 1 PUBLISHED 06 Jan 2025

Open Peer Review

Reviewer Status

Reviewer Reports

	Invited Reviewers
	1
Version 1 06 Jan 25	read

Munusamy Premkumar, Sathyabama Institute of Science and Technology, Chennai, India

Comments on this article

All Comments(0)

Add a comment

Browse by related subjects

Back to all reports

Reviewer Report

12 Views

27 Feb 2025 | for Version 1

Munusamy Premkumar, Sathyabama Institute of Science and Technology, Chennai, Tamil Nadu, India

12 Views Cite this report Responses(1)

Approved With Reservations

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?

No
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
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

my research are is Algebra and Fuzzy Algebra

Respond to this report

Responses (1)

Author Response

28 Feb 2025

Alemayehu Girum Melaku, Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia

This article explains a significant advancement in the field of algebraic structures by introducing the concept of fuzzy pseudo-TM subalgebra (FPTMSA) within pseudo-TM algebra (PTMA). The importance of this work lies in its exploration of the intricate relationships between FPTMSAs and their level sets, which are crucial for understanding the characterization of these subalgebras.

The study meticulously investigates the homomorphic behavior of FPTMSAs under various algebraic operations, thereby laying a foundational framework for future research into fuzzy pseudo-ideals within PTMAs.

The Key properties studied in this research include the definition and examination of level sets for FPTMSAs, the behavior of FPTMSAs under homomorphisms, and the implications of epimorphic and inverse images. Also, the study establishes that the Cartesian product of two FPTMSAs are remains a FPTMSA. Moreover, the relationships between homomorphic images and inverse images of FPTMSAs are analyzed, providing further algebraic insights.

Generally, this article introduced the concepts of FPTMSAs in the structural of PTMAs, particularly regarding their interactions with homomorphisms, epimorphisms, and Cartesian products.

View more View less

Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

[1] 1. Dudek WA, Jun YB: Pseudo-BCI algebras. East Asian Math. J. 2008; 24(2): 187–190.

[2] 2. Gedamu WY, Alaba BA, Derseh BL: Intuitionistic Fuzzy PMS-ideals in a PMS-algebra. Thai J. Math. 2022; 20(4): 1535–1548.

[3] 3. Georgescu G, Iorgulescu A: Pseudo-BCK algebras: an extension of BCK algebras. Combinatorics, Computability and Logic: Proceedings of the Third International Conference on Combinatorics, Computability and Logic,(DMTCS01). London: Springer London; 2001; pp. 97–114.

[4] 4. Iseki K: On BCI-algebras. Math. Seminar Notes. 1980; 8: 125–130.

[5] 5. Iseki K, Shotaro T: An introduction to the theory of the BCK-algebras.1978; pp. 1–26.

[6] 6. Jawad HK: Some Types of Fuzzy Pseudo Ideals of Pseudo Q-algebra. University of Kufa; 2019. Thesis.

[7] 7. Jun YB, Kim HS, Ahn SSH: Structuctures of pseudo ideal and pseudo Atom in a pseudo Q-algebra. Kyungpook Math. J. 2016; 56: 95–106. Publisher Full Text

[8] 8. Jun Y: On BH-algebras. Sci. Math. 1998; 1(3): 347–354.

[9] 9. Jun Y: Pseudo-ideals of pseudo-BCK- algebras. Sci. Math. Jpn. 2003; 8: 87–91.

[10] 10. Kim CB, Kim HS: On BG-algebras. Demonstr. Math. 2008; 41(3): 497–506.

[11] 11. Megalai K, Tamilarasi A: TM-algebra-an introduction, Special Issue on Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications, CASCT.2010.

[12] 12. Mostafa SM, Naby M, Elgendy OR: Fuzzy TM-ideals of TM- algebras. J. Am. Sci. 2011; 79): 17–21.

[13] 13. Neggers J, Ahn SS, Kim HS, et al.: On Q-algebras. Int. J. Math. Math. Sci. 2001; 27: 749–757. Publisher Full Text

[14] 14. Nouri AH, Sabbar MG: Pseudo TM-algebra. Int. J. Algebra. 2019; 13(8): 341–348. Publisher Full Text

[15] 15. Prabpayak C, Leerawat U: On ideals and congruences in KU-algebras. Sci. Magna. 2009; 5(1): 54–57.

[16] 16. Prabpayak C: Some homomorphism properties of TM-algebras. Appl. Mech. Mater. 2017; 866: 406–409. Publisher Full Text

[17] 17. Sharma P: Homomorphism of intuitionistic fuzzy groups. Int. Math. Forum. 2011; 6: 3169–3178.

[18] 18. Somjanta J, Thuekaew N, Kumpeangkeaw P, et al.: Fuzzy sets in UP-algebras. Ann. Fuzzy Math. Inform. 2016; 12(6): 739–756.

[19] 19. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8: 338–353. Publisher Full Text

Fuzzy PTM subalgebra of PTM - algebra

Abstract

Background

Methods

Results

Conclusions

Keywords

Introduction

Preliminaries

Definition 2.1.

Lemma 2.2.

Definition 2.3.

Definition 2.4.

Remark 2.5.

Definition 2.6.

Definition 2.7.

Definition 2.8.

Lemma 2.9.

Fuzzy pseudo TM-Subalgebra of pseudo TM-algebra

Definition 3.1.

Example 3.2.

Lemma 3.3.

Proof.

Theorem 3.4.

Proof.

Theorem 3.5.

Proof.

Corollary 3.6.

Remark 3.7.

Example 3.8.

Proposition 3.9.

Proof.

Definition 3.10.

Theorem 3.11.

Proof.

Theorem 3.12.

Proof.

Corollary 3.13.

Theorem 3.14.

Proof.

Lemma 3.15.

Proof.

Corollary 3.16.

Homomorphism on fuzzy pseudo TM-subalgebras

Definition 4.1.

Theorem 4.2.

Theorem 4.3.

Proof.

Lemma 4.4.

Definition 4.5.

Theorem 4.6.

Proposition 4.7.

Cartesian product of fuzzy pseudo TM-subalgebra

Definition 5.1.

Lemma 5.2.

Proof.

Theorem 5.3.

Proof.

Definition 5.4.

Theorem 5.5.

Conclusion

Ethics and consent

Data availability statement

References

Comments on this article Comments (0)

Open Peer Review

Comments on this article Comments (0)

Open Peer Review

Reviewer Status

Reviewer Reports

Comments on this article

Browse by related subjects

Competing Interests Policy

Stay Updated