Group Structures and Derivations on&nbsp;<i>PMS</i>-algebras

Nibret Melese Kassahun; Berhanu Assaye Alaba; Yohannes Gedamu Wondifraw; Zelalem Teshome Wale

doi:10.12688/f1000research.159712.1

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Research Article

Group Structures and Derivations on PMS-algebras

[version 1; peer review: 2 approved]

Nibret Melese Kassahun ^1,2, Berhanu Assaye Alaba¹, Yohannes Gedamu Wondifraw¹, Zelalem Teshome Wale³

PUBLISHED 13 Jan 2025

Author details Author details

¹ Mathematics, Bahir Dar University, Bahir Dar, Amhara, 6000, Ethiopia
² Mathematics, Debre Markos University, Debre Markos, Amhara, None, Ethiopia
³ Mathematics, Addis Ababa University, Addis Ababa, Addis Ababa, 1000, Ethiopia

Nibret Melese Kassahun
Roles: Conceptualization, Data Curation, Formal Analysis, Funding Acquisition, Investigation, Methodology, Project Administration, Resources, Software, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Berhanu Assaye Alaba
Roles: Conceptualization, Supervision, Validation, Visualization

Yohannes Gedamu Wondifraw
Roles: Methodology, Supervision, Validation, Visualization

Zelalem Teshome Wale
Roles: Conceptualization, Investigation, Methodology, Supervision, Validation, Visualization, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS

Abstract

Background

PMS-algebras are a specific algebraic structure that generalizes a propositional algebra called BCK-algebra. This paper delves into the intricate group structure of these algebras and the concept of derivations within this framework.

Methods

We employ rigorous mathematical techniques to analyze the properties of derivations in P MS-algebras. This involves examining various characteristics of derivations and investigating their behavior in specific subcategories, such as torsion-free P MS-algebras.

Results

Our research reveals several key findings. Firstly, we establish that the set of all derivations associated with the binary operation defined on a P MS-algebra constitutes a semigroup. Secondly, we provide a comprehensive analysis of generalized derivations, d-invariant ideals, fixed sets, and torsion-free P MS-algebras within the context of P MS-algebras.

Conclusions

This study contributes to a deeper understanding of the algebraic structure of P MS-algebras and the role of derivations within this context. The findings presented here have implications for further research in abstract algebra and related fields.

Keywords

P MS-algebras, derivations, generalized derivation, torsion free P MS-algebra, semigroup

Corresponding author: Nibret Melese Kassahun

Competing interests: No competing interests were disclosed.

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

Copyright: © 2025 Melese Kassahun N 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: Melese Kassahun N, Assaye Alaba B, Gedamu Wondifraw Y and Teshome Wale Z. Group Structures and Derivations on PMS-algebras [version 1; peer review: 2 approved]. F1000Research 2025, 14:75 (https://doi.org/10.12688/f1000research.159712.1) First published: 13 Jan 2025, 14:75 (https://doi.org/10.12688/f1000research.159712.1) Latest published: 13 Jan 2025, 14:75 (https://doi.org/10.12688/f1000research.159712.1)

1. Introduction

Iseki¹ introduced a type of abstract algebra known as BCK-algebras. In a subse- quent publication, Iseki² introduced another class of algebra called BCI-algebras, which can be seen as a generalization of BCK-algebras. Iseki studied some properties of BCI-algebras. Tamalibrasi and Megalai³ introduced T M-algebras based on propositional calculus. Mostafa et al.⁴ introduced the concepts of KUS-algebras, KUS-ideals, and KUS-subalgebras, and investigated their relationships. Selvam and Nagalakshmi⁵ introduced an algebraic structure called PMS-algebra, which generalizes the notions of BCK/BCI/TM/KUS-algebras. They also explored additional properties of PMS-algebras and their ideals.

Derivations are an important research area in algebraic structure theory, developed from the concepts of Galois theory and invariant theory. Derivations are a crucial research area in the theory of algebraic structures in mathematics. The ring with derivation is an ancient concept that significantly integrates analysis, algebraic geometry, and algebra. Jun and Xin⁶ extended the concept of derivations in ring and near-ring theory to BCI algebras and related properties. Ganeshkumar and Chandramouleeswaran⁷ derivations and generalized derivations on TM algebras were studied. Also, Kim and Lee⁸ introduced the notion of derivations on BE algebras and studied their properties. Furthermore, Jana et al.⁹ introduced derivations, generalized, and f-derivations and characterized their properties on KUS algebras. Sawika et al.¹⁰ studied the notions of (l, r)-derivations and derivations of UP-algebras, and their properties were investigated. Recently, by Rong et al.,¹¹ the properties of derivations on FI-algebras and the relationship between derivations and ideals were investigated. Sugianti et al.¹² introduced the notion of a (l, r)-derivation, a (r, l)-derivation, and derivation in BM-algebras and investigated related properties. Ganesan and Kandaraj¹³ studied the idea of generalized derivations of BH-algebras and characterized new properties based on the context of BH-algebras in their work. Additionally, Muangkarn et al.¹⁴ studied the derivations of d-algebras based on endomorphisms.

This paper focuses on introducing the fundamental concepts of derivations in PMS-algebras, exploring the properties of generalized derivations, and providing a definition of torsion-free PMS-algebras. The study also investigates the properties of a d-invariant ideal and establishes conditions for its existence.

2. Methods

In this section, we will address the concepts and basic properties of PMS-algebras that we need for the main results in the next sections. These concepts are taken from Selvam and Nagalakshmi.⁵

Definition 1.

⁵ A PMS-algebra is an algebra (X; ⋆, 0) of type (2, 0) with a constant “0” and a binary operation “⋆” satisfying the following axioms:

1. 0 ⋆ a = a;
2. (b ⋆ a) ⋆ (c ⋆ a) = c ⋆ b for all a, b, c ∈ X.

In X, we define a binary relation “≤” by a ≤ b if and only if a ⋆ b = 0.

Proposition 1.

⁵ In a PMS-algebra (X; ⋆, 0) the following properties hold for all a, b, c ∈ X.

1. a ⋆ a = 0.
2. (b ⋆ a) ⋆ c = (c ⋆ a) ⋆ b.
3. (b ⋆ a) ⋆ a = b.
4. a ⋆ (b ⋆ a) = b ⋆ 0.
5. If a ⋆ b = 0 and b ⋆ a = 0, then a = b.

Definition 2.

⁵ Let S be a nonempty subset of a PMS-algebra (X; ⋆, 0). Then S is called a PMS-subalgebra of X if a*b ∈ S for all a, b ∈ S.

Example 1.

⁵ Consider (Z; ⋆, 0) where a ⋆ b = b - a for all a, b $\in$ X. Then (Z; ⋆, 0) is a PMS-algebra, and the set E of even integers is a PMS-subalgebra of Z.

Definition 3.

⁵ Let X be a PMS-algebra and I be a subset of X. Then I is called a PMS-ideal of X if

1. 0 ∈ I
2. c ⋆ b ∈ I and c ⋆ a ∈ I ⇒ b ⋆ a ∈ I for all a, b, c ∈ X.

Definition 4.

⁵ Let X be a PMS-algebra. The set $G (X) = {a \in X | a ⋆ 0 = a}$ is called the G-part of X.

Definition 5.

Let X be a PMS-algebra. The set $B (X) = {a \in X | a ⋆ 0 = 0}$ is called the p-radical of X.

Definition 6.

⁵ A PMS-algebra X is said to be Medial if (a ⋆ b) ⋆ (c ⋆ d) = (a ⋆ c) ⋆ (b ⋆ d) for all a, b, c, d ∈ X.

3. Group structure and Derivations on P MS-algebras

In this section, we will obtain an abelian group from a PMS-algebra by defining binary operations on it. Moreover, we introduce derivations and similarly the basic properties of derivations on PMS-algebras.

Remark 1.

Let (X; ⋆, 0) be a PMS-algebra. If b ⋆ a = c ⋆ a, then b = c for all a, b, c ∈ X.

Proof.

Let b ⋆ a = c ⋆ a.

Now b = 0 ⋆ b = (a ⋆ a) ⋆ b = (b ⋆ a) ⋆ a = (c ⋆ a) ⋆ a = (a ⋆ a) ⋆ c = 0 ⋆ c = c.

Theorem 2.

Let (X, ⋆, 0) be a PMS-algebra and define ′^′+^′′ as a + b = (b ⋆ 0) ⋆ a, ∀a, b ∈ X. Then the following holds:

1. a + 0 = a = 0 + a.
2. Addition is associative.
3. If a + b = 0, then a = b ⋆ 0.
4. Addition is commutative.
5. The additive inverse of a is a ⋆ 0.

Proof.

1. a + 0 = (0 ⋆ 0) ⋆ a = 0 ⋆ a = a and 0 + a = (a ⋆ 0) ⋆ 0 = (0 ⋆ 0) ⋆ a = 0 ⋆ a = a.
Hence a + 0 = a = 0 + a.
2. By applying the definition of “+” and by simplifying, we get the result.
3. a + b = 0 $\Rightarrow$ (b ⋆ 0) ⋆ a = a ⋆ a, by applying right cancellation we get a = b ⋆ 0.
4. a + b = (b ⋆ 0) ⋆ a = (a ⋆ 0) ⋆ b = b + a.
5. a + (a ⋆ 0) = ((a ⋆ 0) ⋆ 0) ⋆ a = (a ⋆ 0) ⋆ (a ⋆ 0) = a ⋆ a = 0.

Hence a ⋆ 0 is the inverse of a.

Definition 7.

Let (X; ⋆, 0) be a PMS-algebra. Define a + b = (b⋆0) ⋆ a and -a = (a⋆0). Then (X, +) is an abelian group.

Remark 2.

If we have a PMS-algebra (X, ⋆, 0) it follows from the above definition that (X, +) is an abelian group with -b = b ⋆ 0, $\forall$ b $\in$ X. Then we have b - a = a ⋆ b, ∀ a, b ∈ X. On the other hand if we have an abelian group (X, +) with an identity “0” and define a ⋆ b = b - a, we get a PMS-algebra (X, ⋆, 0), where a + b = (b ⋆ 0) ⋆ a, $\forall$ a, b $\in$ X.

Example 2.

Let (X, ⋆, 0) be a PMS-algebra as shown in Cayley Table 1.

Table 1. PMS-algebra.

⋆	0	1	2	3	4
0	0	1	2	3	4
1	4	0	1	2	3
2	3	4	0	1	2
3	2	3	4	0	1
4	1	2	3	4	0

Define a + b = (b ⋆ 0) ⋆ a

Then (X, +) is an abelian group as shown in Table 2.

Table 2. Abelian group.

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

Definition 8.

Let (X; ⋆, 0) be a PMS-algebra. Define a binary operation“∧” on X by x ∧ y = (x ⋆ y) ⋆ y for all x, y ∈ X.

Definition 9.

Let (X; ⋆, 0) be a PMS-algebra. A self-map d: X $\to$ X is said to be a (left, right)-derivation of X, denoted by, (l, r)-derivation if d(x ⋆ y) = (d(x) ⋆ y) ∧ (x ⋆ d(y)) for all x, y ∈ X.

Example 3.

Consider the PMS-algebra X = {0, 1, 2, 3} as shown in Table 3:

Table 3. (l, r)-derivation on X.

⋆	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

Then the self-map d: X $\to$ X defined by d(0) = 3, d(1) = 2, d(2) = 1 and d(3) = 0 is a (l, r)-derivation on X.

Remark 3.

Let d: X $\to$ X be a (l, r)-derivation on X. Then d(x ⋆ y) = d(x) ⋆ y for all x, y $\in$ X.

Definition 10.

Let (X; ⋆, 0) be a PMS-algebra. A self-map d: X $\to$ X is said to be a (right, left)-derivation of X, denoted by, (r, l)-derivation if d(x ⋆ y) = (x ⋆ d(y)) $\land$ (d(x) ⋆ y) for all x, y $\in$ X.

Example 4.

Let X be the PMS-algebra defined in the above example 3, the self-map d: X $\to$ X defined by d(0) = 1, d(1) = 0, d(2) = 3 and d(3) = 2 is a (r, l)-derivation on X.

Remark 4.

Let d: X $\to$ X be a (r, l)-derivation on X. Then d(x ⋆ y) = x ⋆ d(y) for all x, y ∈ X.

Definition 11.

Let d: X $\to$ X be a self-map on a PMS-algebra (X; ⋆, 0). Then the self-map d is said to be a derivation on X if d is both a (l, r)-derivation and a (r, l)-derivation on X.

Example 5.

Let (Z; ⋆, 0) be a PMS-algebra with x ⋆ y = y − x. Then the map d: Z → Z defined

1. by d(x) = x is a derivation on Z.
2. by d(x) = x-1 is a (r, l) derivation on Z but not a (l, r)-derivation. Hence, d is not a derivation.

Remark 5.

Let d: X→ X be a derivation on X. Then d(x ⋆ y) = d(x) ⋆y = x ⋆ d(y) for all x, y ∈ X.

Theorem 3.

Let (X; ⋆, 0) be a PMS-algebra, and d: X→ X be a (l, r) -derivation. If y ≤ x, then d(x) = d(y) for all x, y ∈ X.

Proof.

Let y ≤ x. Then y ⋆ x = 0.

Now

\begin{array}{l} d (x) = d (0 ⋆ x) \\ = d ((y ⋆ x) ⋆ x) \\ = d ((y ⋆ x) ⋆ (0 ⋆ x)) \\ = d (0 ⋆ y) \\ = d (y) \end{array}

Hence, d(x) = d(y).

Definition 12.

Let (X; ⋆, 0) be a PMS-algebra. A self-map d on X is said to be regular if d(0) = 0.

Example 6.

Let X be the PMS-algebra defined as in Example 3. The self-map d: X→ X defined by d(0) = 0, d(1) = 1, d(2) = 2, d(3) = 3 is a regular derivation.

Theorem 4.

Let (X; ⋆, 0) be a PMS-algebra. If d: X→ X is a regular (l, r)-derivation, then d(x) ≤ x for all x ∈ X.

Proof.

Suppose d is a regular (l, r)-derivation on X. Then d(0) = d(x ⋆ x) implies d(x) ⋆ x = 0. Thus d(x) ≤ x.

Theorem 5.

Let (X, ⋆, 0) be a PMS-algebra. If d: X→ X be a regular (r, l)-derivation, then x ≤ d(x) for all x ∈ X.

Proof.

Suppose d is a regular (r, l)-derivation on X. Then d(0) = d(x ⋆ x) implies that x ⋆ d(x) = 0. Thus x ≤ d(x).

Theorem 6.

Let (X; ⋆, 0) a PMS-algebra and d: X $\to$ X be a (l, r)-derivation. Then d is regular if and only if d(x) = x.

Proof.

Let d be a derivation on X. Suppose that d(0) = 0.

Now

\begin{array}{l} d (x) = d (0 ⋆ x) \\ = (d (0) ⋆ x) \land (0 ⋆ d (x)) \\ = (0 ⋆ x) \land (d (x)) \\ = x \land d (x) \\ = (x ⋆ d (x)) ⋆ d (x) \\ = (d (x) ⋆ d (x)) ⋆ x \\ = 0 ⋆ x \\ = x . \end{array}

Therefore, d(x) = x.

Conversely, assume that d(x) = x, $\forall$ x $\in$ X.

In particular, d(0) = 0. Hence d is regular.

Theorem 7.

Let G(X) be the G-part of a PMS-algebra X and d be a (l, r)-derivation on X. Then d(x) ∈ G(X), ∀x ∈ G(X).

Proof.

Let x∈G(X). Then x ⋆ 0 = x. Now

\begin{array}{l} d (x) = d (x ⋆ 0) \\ = d (x) ⋆ 0 . \end{array}

Hence d(x) ∈ G(X).

Theorem 8.

Let B(X) be the P-radical of a PMS-algebra X and d be a (l, r)-derivation on B(X). Then d is regular.

Proof.

Let x ∈ B(X). Then x ⋆ 0 = 0.

Now

\begin{array}{l} d (0) = d (x ⋆ 0) \\ = d (x) ⋆ 0 \\ = 0 \end{array}

Hence d(0) = 0:

Therefore, d is a regular derivation on B(X).

Theorem 9.

Let (X; ⋆, 0) be a PMS-algebra and d: X $\to$ X be a (l, r)-derivation. Then

1. d(x) ⋆ x = 0 ∀x ∈ X if and only if d is regular.
2. d(x ⋆ d(x)) = 0 ∀x ∈ X.

Proof.

1. Suppose that d(x) ⋆ x = 0. Now
$\begin{array}{l} d (0) = d (x ⋆ x) \\ = d (x) ⋆ x \\ = 0 \end{array}$
Hence, d is regular.
Conversely, assume that d(0) = 0
Now d(x ⋆ x) = 0 implies that d(x) ⋆ x = 0 Hence d(x) ⋆ x = 0.
2. d(x ⋆ d(x)) = d(x) ⋆ d(x) = 0.

Theorem 10.

Let (X; ⋆, 0) be a PMS-algebra and d: X $\to$ X be a (r, l)-derivation. Then

1. x ⋆ d(x) = 0, ∀x ∈ X if and only if d is regular.
2. d(d(x) ⋆ x) = 0, ∀x ∈ X.

Proof.

1. Suppose that d(x) ⋆ x = 0 Now
$\begin{array}{l} d (0) = d (x ⋆ x) \\ = d (x) ⋆ x \\ = 0 \end{array}$
Hence, d is regular.
Conversely, assume that d(0) = 0
Now d(x ⋆ x) = d(0) implies that d(x) ⋆ x = 0
Hence d(x) ⋆ x = 0.
2. d(d(x) ⋆ x) = d(x) ⋆ d(x) = 0.

Theorem 11.

Let d be a self-map of a PMS-algebra X.

1. Let d be a (l, r) -derivation on X. Then d regular if and only if d(x) = x ∧ d(x).
2. Let d be a (r, l) -derivation on X. If d is regular, then d(x) = d(x) ∧ x.

Proof.

1. Let d be a self-map on a PMS-algebra. Suppose that d is regular (l, r)-derivation on X.
$\begin{array}{l} d (x) = d (0 ⋆ x) \\ = (d (0) ⋆ x) \land (0 ⋆ d (x)) \\ = (0 ⋆ x) \land (0 ⋆ d (x)) \\ = x \land d (x) . \end{array}$
Conversely, assume that d(x) = x∧d(x). Then d(0) = 0∧d(0) = (0 ⋆ d(0)) ⋆ d(0) = (d(0) ⋆ d(0)) ⋆ 0 = 0 ⋆ 0 = 0.
2. Let d be regular (l, r)-derivation.
$\begin{array}{l} d (x) = d (0 ⋆ x) \\ = (0 ⋆ d (x)) \land (d (0) ⋆ x) \\ = (0 ⋆ d (x)) \land (0 ⋆ x) \\ = d (x) \land x . \end{array}$

Theorem 12.

Let d be a (r, l)-derivation on a PMS-algebra X. Then

1. d(0) = x ⋆ d(x).
2. d is 1-1.
3. If there is an element x∈X such that d(x) = x, then d is the identity map.
4. If there is an element x∈X such that d(y) ⋆ x = 0 for all y∈X, then d(y) = x, that is, d is a constant map.

Proof.

1. d(0) = d(x ⋆ x) = x ⋆ d(x) Because d is a (r, l)-derivation on X.
2. Let x, y ∈ X and d(x) = d(y).
(1)
$d (0) = d (x ⋆ x) = x ⋆ d (x)$

(2)
$d (0) = d (y ⋆ y) = y ⋆ d (y)$
From (1) and (2) we have by x ⋆ d(x) = y ⋆ d(y) implies that x ⋆ d(x) = y ⋆ d(x). Hence by right cancellation we have x = y.
Therefore d is a one to one map.
3. Let x ∈ X such that d(x) = x.
Let y ∈ X be any element of X. Then y = 0 ⋆ y = (x ⋆ x) ⋆ y = (y ⋆ x) ⋆ x. Then
$\begin{array}{l} d (y) = d ((y ⋆ x) ⋆ x) \\ = (y ⋆ x) ⋆ d (x) \\ = (y ⋆ x) ⋆ x \\ = y \end{array}$
Hence, d(x) = x, ∀x∈X.
This proves that d is the identity map on X.
4. Given d(y) ⋆ x = 0 implies that d(y) ⋆ x = x ⋆ x, and hence by right cancellation we have d(y) = x.

Corollary 1.

Let (X; ⋆, 0) be a PMS-algebra. If d is a (r, l)-derivation on X, then x ⋆ d(x) = y ⋆ d(y) ∀x, y ∈ X.

Proof.

Let x, y ∈ X. Then

(3)

d (0) = d (x ⋆ x) = x ⋆ d (x)

(4)

d (0) = d (y ⋆ y) = y ⋆ d (y)

From (3) and (4) we have x ⋆ d(x) = y ⋆ d(y)

Theorem 13.

Let X be a PMS-algebra and d be a regular (l, r)-derivation on X. Then Ker d = {x ∈ X: d(x) = 0} is a PMS-subalgebra of X.

Proof.

Since d is a regular, d(0) = 0.

Hence 0 ∈ Ker d. It follows that Ker d $\neq \emptyset$ .

Let x, y ∈ Ker d. Then d(x) = 0,

d(y) = 0. Now we have

\begin{array}{l} d (x ⋆ y) = d (x) ⋆ y \\ = x ⋆ y \\ = d (x) ⋆ d (y) \\ = 0 ⋆ 0 \\ = 0 . \end{array}

Hence x ⋆ y ∈ Ker d.

Therefore Ker d is a PMS-subalgebra of X.

Theorem 14.

Let d be a (r, l)-derivation on a PMS-algebra X. If x ≤ y and x ∈ Ker d, then y ∈ Ker d.

Proof.

Let x≤y and x∈Ker d. Then x ⋆ y = 0 and d(x) = 0.

Now d(y) = d(0 ⋆ y) = d((x ⋆ y) ⋆ y) = d((y ⋆ y) ⋆ x) = d(0 ⋆ x) = d(x) = 0. Hence y ∈ Ker d.

Theorem 15.

Let X be medial of a PMS-algebra and d be a regular (l, r)-derivation on X. Then Ker d is a PMS-ideal of X.

Proof.

Let z ⋆ x $\in$ Ker d and z ⋆ y $\in$ Ker d. Then d(z ⋆ x) = 0 and d(z ⋆ y) = 0. Now

\begin{array}{l} d (x ⋆ y) = d (0 ⋆ (x ⋆ y)) \\ = d ((z ⋆ z) ⋆ (x ⋆ y)) \\ = d ((z ⋆ x) ⋆ (z ⋆ y)) \\ = d (z ⋆ x) ⋆ (z ⋆ y) \\ = d (z ⋆ x) ⋆ d (z ⋆ y) \\ = 0 ⋆ 0 \\ = 0 . \end{array}

Hence x ⋆ y ∈ Ker d

Therefore Ker d is a PMS-ideal of X.

Theorem 16.

Let X be a PMS-algebra and d be a (l, r)-derivation on X. Then Ker d = {0} if and only if d is regular.

Proof.

Suppose that Ker d = {0}.

Let x ∈ Ker d. Then d(x) = 0, ∀x ∈ X in particular d(0) = 0. Hence d is regular.

Conversely, assume that d is regular Let x ∈ Ker d. Then d(x) = 0.

Now 0 = d(0) = d(x ⋆ x) = d(x) ⋆ x = 0 ⋆ x = x. Hence, x = 0

Thus Ker d = {0}.

Definition 13.

Let X be a PMS-algebra and d: X→ X be a derivation of X. Define ${Fix}_{d} (X) = {x \in X | d (x) = x}$ is called the fixed set of X.

Theorem 17.

Let (X; ⋆, 0) be a PMS-algebra and d be a (l, r)-derivation on X. Then Fix_d(X) is a PMS-subalgebra of X.

Proof.

Since 0 ∈ X, and d(0) = 0. Hence 0 ∈ Fix_d(X). It follows that Fix_d(X) $\neq \emptyset$ .

Let x, y ∈ Fix_d(X). Then d(x) = x and d(y) = y.

Now

\begin{array}{l} d (x ⋆ y) = d (x) ⋆ y \\ = x ⋆ d (y) \\ = x ⋆ y \end{array}

Hence d(x ⋆ y) = x ⋆ y implies that x ⋆ y ∈ Fix_d(X). Therefore, Fix_d(X) is a PMS-subalgebra of X.

Theorem 18.

Let d be a (r, l)-derivation on a PMS-algebra X. If x ≤ y and y ∈ Fix_d(X), then x ∈ Fix_d(X).

Proof.

Let x $\leq$ y and y $\in$ Fix_d(X). Then x ⋆ y = 0 and d(y) = y. Now

\begin{array}{l} d (x) = d (0 ⋆ x) \\ = d ((y ⋆ y) ⋆ x) \\ = d ((x ⋆ y) ⋆ y) \\ = (x ⋆ y) ⋆ d (y) \\ = (x ⋆ y) ⋆ y \\ = (y ⋆ y) ⋆ x \\ = 0 ⋆ x \\ = x \end{array}

Hence, x ∈ Fix_d(X).

Theorem 19.

Let X be a medial PMS-algebra and d be a (r, l)-derivation on X. Then Fix_d(X) is a PMS-ideal of X.

Proof.

1. Since d(x) = x for all x ∈ Fix_d(X), d(0) = 0.
Hence, 0 ∈ Fix_d(X).
2. Let z ⋆ x∈Fix_d(X) and z ⋆ y∈Fix_d(X). Then d(z ⋆ x) = z ⋆ x and d(z ⋆ y) = z ⋆ y.
Now
$\begin{array}{l} d (x ⋆ y) = d (0 ⋆ (x ⋆ y)) \\ = d ((z ⋆ z) ⋆ (x ⋆ y)) \\ = d ((z ⋆ x) ⋆ (z ⋆ y)) \\ = (z ⋆ x) ⋆ d (z ⋆ y) \\ = (z ⋆ x) ⋆ (z ⋆ y) \\ = (z ⋆ z) ⋆ (x ⋆ y) \\ = 0 ⋆ (x ⋆ y) \\ = x ⋆ y \end{array}$

Hence, x ⋆ y ∈ Fix_d(X).

Therefore, Fix_d(X) is an ideal of X.

Theorem 20.

Let X be a PMS-algebra and d be a (r, l)-derivation on X. If y≤x and d(x) = x, then d(y) = y.

Proof.

d(y) = d(0 ⋆ y) = d((x ⋆ x) ⋆ y) = d((y ⋆ x) ⋆ x) = (y ⋆ x) ⋆ d(x) = (y ⋆ x) ⋆ x = (x ⋆ x) ⋆ y = 0 ⋆ y = y.

Definition 14.

Let X be a PMS-algebra. A self-map d on X is said to be an isotone if x≤y implies that d(x) ≤d(y) for all x, y∈X.

Theorem 21.

Let X be a PMS-algebra and d be a regular (l, r)-derivation on X. Then the following properties are equivalent:

1. d is an isotone derivation on X.
2. x ≤ y implies d(x ⋆ y) = d(x) ⋆ y.

Proof.

Suppose that d is an isotone (l, r)-derivation on X. Let x, y∈X such that x≤y. Then we have d(x ⋆ y) = d(0) = 0. Since d is isotone, we get d(x) ⋆ d(y) = 0. Now

\begin{array}{l} d (x ⋆ y) = 0 \\ = d (x) ⋆ d (y) \\ = d (x) ⋆ d (0 ⋆ y) \\ = d (x) ⋆ (d (0) ⋆ y) \\ = d (x) ⋆ (0 ⋆ y) \\ = d (x) ⋆ y . \end{array}

Therefore, d(x ⋆ y) = d(x) ⋆ y.

Assume that x ≤ y implies d(x ⋆ y) = d(x) ⋆ y. Let x, y∈X such that x ≤ y. Now

\begin{array}{l} d (x) ⋆ d (y) = d (x) ⋆ d (0 ⋆ y) \\ = d (x) ⋆ (d (0) ⋆ y) \\ = d (x) ⋆ (0 ⋆ y) \\ = d (x) ⋆ y \\ = d (x ⋆ y) \\ = d (0) \\ = 0 \end{array}

Hence, d(x) ⋆ d(y) = 0 implies that d(x) ≤ d(y).

Therefore, d is an isotone derivation on X.

Definition 15.

Let LDer(X) be the set of all (l, r)-derivations on X. Define a binary operation “∧” on LDer(X) by (d₁ ∧ d₂)(x) = d₁(x) ∧ d₂(x) for all x ∈ X, where d₁, d₂ ∈ LDer(X).

Lemma 1.

If d₁ and d₂ are (l, r)-derivations on X, then (d₁ $\land$ d₂) is also a (l, r) derivation on X. Proof.

(5)

\begin{array}{l} (d_{1} \land d_{2}) (x ⋆ y) = d_{1} (x ⋆ y) \land d_{2} (x ⋆ y) \\ = (d_{1} (x) ⋆ y) \land (d_{2} (x) ⋆ y) \\ = ((d_{1} (x) ⋆ y) ⋆ (d_{2} (x) ⋆ y)) ⋆ (d_{2} (y) ⋆ y) \\ = d_{1} (x) ⋆ y \end{array}

(6)

\begin{array}{l} (d_{1} \land d_{2}) (x) ⋆ y = (d_{1} (x) \land d_{2} (x)) ⋆ y \\ = ((d_{1} (x) ⋆ d_{2} (x)) ⋆ d_{2} (x)) ⋆ y \\ = d_{1} (x) ⋆ y \end{array}

From (5) and (6) we have (d₁ ∧ d₂)(x ⋆ y) = (d₁ ∧ d₂)(x) ⋆ y. Hence, (d₁ ∧ d₂) (l, r)-derivation.

Lemma 2.

The binary operation “∧” defined on LDer(X) is associative. Proof. Let X be a PMS-algebra.

Let d₁, d₂, d₃ be (l, r)-derivations on X. Now

(7)

\begin{array}{l} ((d_{1} \land d_{2}) \land d_{3}) (x) = ((d_{1} \land d_{2}) \land d_{3}) (0 ⋆ x) \\ = (d_{1} \land d_{2}) (0 ⋆ x) \land d_{3} (0 ⋆ x) \\ = (d_{1} (0) ⋆ x) \land (d_{3} (0) ⋆ x) \\ = ((d_{1} (0) ⋆ x) ⋆ (d_{3} (0) ⋆ x)) ⋆ (d_{3} (0) ⋆ x) \\ = d_{1} (0) ⋆ x \end{array}

(8)

\begin{array}{l} (d_{1} \land (d_{2} \land d_{3})) (x) = (d_{1} \land (d_{2} \land d_{3})) (0 ⋆ x) \\ = d_{1} (0 ⋆ x) \land (d_{2} \land d_{3}) (0 ⋆ x) \\ = (d_{1} (0) ⋆ x) \land (d_{2} (0) ⋆ x) \\ = ((d_{1} (0) ⋆ x) ⋆ (d_{2} (0) ⋆ x)) ⋆ ((d_{2} (0) ⋆ x) \\ = d_{1} (0) ⋆ x \end{array}

From (7) and (8) we have (d₁ ∧ d₂) ∧ d₃)(x) = d₁ ∧ (d₂ ∧ d₃)(x), ∀x ∈ X. Hence “∧” is associative.

Combining the above two lemmas we get following theorem.

Theorem 22.

LDer(X) is a semigroup under the binary operation defined by (d₁ ∧ d₂)(x) = d₁(x)∧d₂(x) ∀x∈X, where d₁,d₂∈LDer(X).

Definition 16.

Let RDer(X) be the set of all (l, r)-derivations on X. Define a binary operation “∧” on RDer(X) by (d₁ ∧ d₂)(x) = d₁(x) ∧ d₂(x) for all x ∈ X, where d₁, d₂ ∈ RDer(X).

Lemma 3.

If d₁ and d₂ are (r, l)-derivations on X, then (d₁∧d₂) is also a (r, l)-derivation on X.

Proof.

Let d₁ and d₂ are (r, l)-derivations on X. Then

(9)

\begin{array}{l} (d_{1} \land d_{2}) (x ⋆ y) = d_{1} (x ⋆ y) \land d_{2} (x ⋆ y) \\ = (x ⋆ d_{1} (y)) \land (x ⋆ d_{2} (y)) \\ = (((x ⋆ d_{1} (y)) ⋆ (x ⋆ d_{2} (y))) ⋆ (x ⋆ d_{2} (y))) \\ = x ⋆ d_{1} (y) \end{array}

(10)

\begin{array}{l} x ⋆ (d_{1} \land d_{2}) (y) = x ⋆ ((d_{1} (y) \land d_{2} (y)) \\ = x ⋆ ((d_{1} (y) ⋆ d_{2} (y)) ⋆ d_{2} (y)) \\ = x ⋆ d_{1} (y) \end{array}

From (9) and (10) we have (d₁ ∧ d₂)(x ⋆ y) = x ⋆ (d₁ ∧ d₂)(y). Hence, (d₁ ∧ d₂) (l, r)-derivation on X.

Lemma 4.

The binary operation “∧” defined on RDer(X) is associative.

Proof.

Let X be a PMS-algebra.

Let d₁, d₂, d₃ be (r, l)-derivations on X.

Now

(11)

\begin{array}{l} ((d_{1} \land d_{2}) \land d_{3}) (x) = ((d_{1} \land d_{2}) \land d_{3}) (0 ⋆ x) \\ = (d_{1} \land d_{2}) (0 ⋆ x) \land d_{3} (0 ⋆ x) \\ = (d_{1} (0) \land d_{2} (x)) \land (0 ⋆ d_{3} (x)) \\ = (0 ⋆ d_{1} (x)) ⋆ (d_{3} (x) ⋆ y)) ⋆ (d_{3} (x) ⋆ y) \\ = d_{1} (x) ⋆ y \end{array}

(12)

\begin{array}{l} (d_{1} \land (d_{2} \land d_{3})) (x ⋆ y) = d_{1} (x ⋆ y) \land (d_{2} \land d_{3}) (x ⋆ y) \\ = (d_{1} (x) ⋆ y) \land (d_{2} (x) ⋆ y) \\ = ((d_{1} (x) ⋆ y) ⋆ (d_{2} (x) ⋆ y)) ⋆ ((d_{2} (x) ⋆ y) \\ = d_{1} (x) ⋆ y \end{array}

From (11) and (12) we have (d₁ ∧ d₂) ∧ d₃) = d₁ ∧ (d₂ ∧ d₃). Combining the above two lemmas we get following theorem.

Theorem 23.

RDer(X) is a semigroup under the binary composition defined by (d₁ ∧ d₂)(x) = d₁(x) ∧ d₂(x) ∀x ∈ X and d₁, d₂ ∈ RDer(X).

Lemma 5.

Let (X; ⋆, 0) be a PMS-algebra and d₁, d₂ be (l, r)-derivations on X. Then the composition (d₁ ◦ d₂) is a (l, r)-derivation on X.

Proof.

(d₁ ◦ d₂)(x⋆ y) = d₁(d₂(x⋆ y)) = d₁(d₂(x) ⋆ y) = d₁(d₂(x)) ⋆ y = (d₁ ◦ d₂)(x) ⋆ y. Hence, (d1 ◦ d2) is a (l, r)-derivation on X.

Lemma 6.

Let (X; ⋆, 0) be a PMS-algebra and d₁, d₂ be (r, l)-derivations on X. Then the composition (d₁ ◦ d₂) is a (r, l)-derivation on X.

Proof.

(d₁ ◦ d₂)(x⋆ y) = d₁(d₂(x⋆ y)) = d₁(x⋆ d₂(y)) = x⋆ d₁(d₂(y)) = x⋆ (d₁ ◦ d₂)(y). Hence, (d₁ ◦ d₂) is a (r, l)-derivation on X.

Theorem 24.

Let (X; ⋆, 0) be a PMS-algebra and d₁, d₂ be derivations on X. Then the composition (d₁ ◦ d₂) is also a derivation on X.

Theorem 25.

Let (X; ⋆, 0) be a PMS-algebra and d₁, d₂ be derivations on X. Then d₁ ◦ d₂ = d₂ ◦ d₁.

Proof.

Given d₁, d₂ be derivation on X. Then d₁, d₂ are both (l, r) and (r, l)-derivations on X. Now

(13)

\begin{array}{l} (d_{1} ◦ d_{2}) (x) = (d_{1} ◦ d_{2}) (0 ⋆ x)) \\ = d_{1} (d_{2} (0 ⋆ x)) \\ = d_{1} (d_{2} (0) ⋆ x) \\ = d_{2} (0) ⋆ d_{1} (x) \end{array}

(14)

\begin{array}{l} (d_{2} ◦ d_{1}) (x) = (d_{2} ◦ d_{1}) (0 ⋆ x) \\ = d_{2} (d_{1} (0 ⋆ x)) \\ = d_{2} (0 ⋆ d_{1} (x)) \\ = d_{2} (0) ⋆ d_{1} (x) \end{array}

From (13) and (14) we have (d₁ ◦ d₂)(x) = (d₂ ◦ d₁)(x). Therefore, d₁ ◦ d₂ = d₂ ◦ d₁.

Theorem 26.

Let X be a PMS-algebra and d be a regular (l, r)-derivation on X. Define d²(x) = d(d(x)) for all x ∈ X. Then d² = d.

Proof.

Now

\begin{array}{l} d^{2} (x) = d (d (x)) \\ = d (d (0 ⋆ x)) \\ = d ((d (0) ⋆ x) ⋆ (0 ⋆ d (x)) \\ = d ((0 ⋆ x) \land d (x)) \\ = d (x \land d (x)) \\ = d ((x ⋆ d (x)) ⋆ d (x)) \\ = d ((d (x) ⋆ d (x)) ⋆ x) \\ = d (0 ⋆ x) \\ = d (x), \forall x \in X . \end{array}

Hence d² = d.

Definition 17.

Let (X; ⋆, 0) be a PMS-algebra. Let d₁, d₂ be self-maps on X. We define (d₁ ⋆ d₂): X→ X as (d₁ ⋆ d₂)(x) = d₁(x) ⋆ d₂(x) ∀x ∈ X.

Theorem 27.

Let (X; ⋆, 0) be a PMS-algebra and d₁, d₂ be derivations on X. Then d₁ ⋆ d₂ = d₂ ⋆ d₁.

Proof.

Let x ∈ X.

(15)

\begin{array}{l} (d_{1} ◦ d_{2}) (x) = (d_{1} ◦ d_{2}) (0 ⋆ x) \\ = d_{1} (d_{2} (0 ⋆ x)) \\ = d_{1} (d_{2} (0) ⋆ x) \\ = d_{2} (0) ⋆ d_{1} (x) \\ = d_{2} (0) ⋆ d_{1} (x) . \end{array}

(16)

\begin{array}{l} (d_{1} ◦ d_{2}) (x) = (d_{1} ◦ d_{2}) (0 ⋆ x) \\ = d_{1} (d_{2} (0 ⋆ x)) \\ = d_{1} (0 ⋆ d_{2} (x)) \\ = d_{1} (0) ⋆ d_{2} (x) \\ = d_{1} (0) ⋆ d_{2} (x) \end{array}

From (15) and (16) we have d₁(0) ⋆ d₂(x) = d₂(0) ⋆ d₁(x) implies that (d₁ ⋆ d₂)(x) = (d₂ ⋆ d₁)(x), ∀x ∈ X.

Hence d₁ ⋆ d₂ = d₂ ⋆ d₁.

Definition 18.

Let d a self-map on PMS-algebra of X. A PMS-ideal I of X is said to be d-invariant if d(I) ⊆ I, where d(I) = {d(x): x ∈ I}.

Theorem 28.

Let d be a derivation on a PMS-algebra X. Then d is regular if and only if every PMS-ideal of X is d-invariant.

Proof.

Suppose that d be a regular derivation.

Let y ∈ d(I). Then y = d(x) for some x ∈ I. Now y ⋆ x = d(x) ⋆ x = x ⋆ x = 0, since d is regular d(x) = x and hence y ⋆ x ∈ I as 0 ∈ I.

y = 0 ⋆ y = (x ⋆ x) ⋆ y = (y ⋆ x) ⋆ x ∈ I, since x ∈ I and y ⋆ x ∈ I. Thus, y ∈ I implies that d(I) ⊆ I.

Conversely, assume that every ideal of X is d-invariant, then d(0) ⊆ 0, and hence d(0) = 0.

Therefore, d is regular.

4. Generalized derivations on P MS-algebras

Definition 19.

Let X be a PMS-algebra. A mapping D: X $\to$ X is called a generalized (l, r)-derivation on X if there exists a (l, r)-derivation d: X $\to$ X such that D(x ⋆ y) = (D(x) ⋆ y) $\land$ (x ⋆ d(y), $\forall$ x, y $\in$ X.

Definition 20.

Let X be a PMS-algebra. A mapping D: X $\to$ X is called a generalized (r, l)-derivation on X if there exists a (r, l)-derivation d: X $\to$ X such that D(x ⋆ y) = (x ⋆ D(y)) $\land$ (d(x) ⋆ y) for all x, y $\in$ X.

Definition 21.

Let X be a PMS-algebra. A mapping D: X $\to$ X is called a generalized derivation on X, if there exists a derivation d: X $\to$ X such that D is both a generalized (l, r)-derivation and a generalized (r, l)-derivation on X.

Example 7.

Consider the PMS-algebra X = {0, 1, 2, 3, 4} as shown in Table 4:

Table 4. Generalized (r, l)-derivation on X.

⋆	0	1	2	3	4
0	0	1	2	3	4
1	4	0	1	2	3
2	3	4	0	1	2
3	2	3	4	0	1
4	1	2	3	4	0

Define a self-map d: X $\to$ X by d(0) = 3, d(1) = 4, d(2) = 0, d(3) = 1, d(4) = 2.

Then d is a (r, l)-derivation on X. But d is not an (l, r)-derivation on X.

Define a map D: X $\to$ X by D(0) = 1, D(1) = 2, D(2) = 3, D(3) = 4, D(4) = 0.

Then D satisfies the equality, D(x ⋆ y) = (x ⋆ D(y)) ∧ (d(x) ⋆ y) for all x, y∈X. Thus D is a generalized (r, l)-derivation on X. But D is not a (l, r)-derivation on X.

Example 8.

Let (X; ⋆, 0) be the PMS-algebra as shown in Cayley Table 5.

Table 5. Generalized derivation on X.

⋆	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

A self-map d: X $\to$ X be defined by d(0) = 3, d(1) = 2, d(2) = 1, d(3) = 0.

Then d is a derivation on X.

Define a map D: X $\to$ X by D(0) = 2, D(1) = 3, D(2) = 0, D(3) = 1. D satisfies the equality, D(x ⋆ y) = (D(x) ⋆ y) $\land$ (x ⋆ d(y)) for all x, y $\in$ X and D(x ⋆ y) = (x ⋆ D(y)) $\land$ (d(x) ⋆ y) for all x, y $\in$ X. Thus D is a generalized derivation on X.

Theorem 29.

Let D be a self-map on a PMS-algebra X. Then the following holds true.

1. If D is a generalized (r, l)-derivation on X and d be a regular (r, l)-derivation on X, then D(x) = D(x) $\land$ x for all x $\in$ X.
2. If D is a generalized (l, r)-derivation on X, then D(0) = 0 if and only if D(x) = x ∧ d(x) for all x ∈ X and for some regular (l, r)-derivation d of X.

Proof.

1. Let d be a regular derivation on X. Now
$\begin{array}{l} D (x) = D (0 ⋆ x) \\ = (0 ⋆ D (x)) \land (d (0) ⋆ x) \\ = D (x) \land (0 ⋆ x) \\ = D (x) \land x \end{array}$
2. Suppose that D(0) = 0. Now
$\begin{array}{l} D (x) = D (0 ⋆ x) \\ = (D (0) ⋆ x) \land (0 ⋆ d (x)) \\ = (0 ⋆ x) \land d (x) \\ = x \land d (x) \end{array}$
Hence, D(x) = x∧d(x)
Conversely, assume D(x) = x∧d(x) Now
$\begin{array}{l} D (0) = 0 \land d (0) \\ = (0 ⋆ d (0)) ⋆ d (0) \\ = (d (0) ⋆ d (0)) ⋆ 0 \\ = 0 ⋆ 0 \\ = 0 \end{array}$
Hence D(0) = 0.

Theorem 30.

Let D be a generalized (r, l)-derivation on a PMS-algebra X. Then

1. D(a) = a + D(0), ∀a ∈ X.
2. D(x + a) = x + D(a), ∀a, x ∈ X.
3. D(a + b) = D(a) + b = a + D(b), ∀a, b ∈ X.

Proof.

1.
$\begin{array}{l} D (a) = D ((a ⋆ 0) ⋆ 0) \\ = (a ⋆ 0) ⋆ D (0) \\ = (D (0) ⋆ 0) ⋆ a \\ = a + D (0) . \end{array}$
Hence, D(a) = a + D(0).
2.
$\begin{array}{l} D (x + a) = D ((a ⋆ 0) ⋆ x) \\ = D ((x ⋆ 0) ⋆ a) \\ = (x ⋆ 0) ⋆ D (a) \\ = ((D (a) ⋆ 0) ⋆ x) \\ = x + D (a) \end{array}$
Hence, D(x + a) = x + D(a).
3. Since (X, +) is an abelian group, a + b = b + a, ∀a, b ∈ X. Now
- i.
  $\begin{array}{l} D (a) + b = D (a + b) \\ = D (b + a) \\ = D (b) + a \end{array}$
  Hence D(a + b) = D(a) + b = a + D(b), ∀a, b ∈ X.

Theorem 31.

Let D be a generalized (l, r)-derivation on a PMS algebra X. Then

1. D(a)∈G(X) ∀a∈G(X).
2. D(a) = a ⋆ D(0) = a + D(0) ∀a∈X.
3. D(a + b) = D(a) + D(b) - D(0), $\forall$ a, b $\in$ X.
4. D is the identity map on X if and only if D(0) = 0.

Proof.

1. Let a ∈ G(X), then a ⋆ 0 = a.
$\begin{array}{l} D (a) = D (a ⋆ 0) \\ = (D (a) ⋆ 0) \land (a ⋆ d (0)) \\ = ((D (a) ⋆ 0) ⋆ (a ⋆ d (0))) ⋆ (a ⋆ d (0)) \\ = ((a ⋆ d (0)) ⋆ (a ⋆ d (0))) ⋆ ((D (a) ⋆ 0) \\ = 0 ⋆ ((D (a) ⋆ 0) \\ = D (a) ⋆ 0 \end{array}$
Therefore, D(a)∈G(X).
2.
$\begin{array}{l} D (a) = D (0 ⋆ a) \\ = (D (0) ⋆ a) \land (0 ⋆ d (a)) \\ = ((D (0) ⋆ a) ⋆ d (a)) \\ = (d (a) ⋆ d (a)) ⋆ (D (0) ⋆ a) \\ = D (0) ⋆ a \end{array}$
Hence, D(a) = D(0) ⋆ a Again
$\begin{array}{l} D (a) = D (0 ⋆ a) \\ = D (0) ⋆ a \\ = D (0 ⋆ 0) ⋆ a \\ = (D (0) ⋆ 0) ⋆ a \\ = D (0) + a \end{array}$
Therefore, D(a) = D(0) ⋆ a = D(0) + a.
3.
$\begin{array}{l} D (a + b) = a + b + D (0) \\ = a + D (0) + b + D (0) - D (0) \\ = D (a) + D (b) - D (0) \end{array}$
Hence, D(a + b) = D(a) + D(b)-D(0).
4. Suppose D be an identity map on X. Then D(a) = a for all a∈X. Thus, D(0) = 0.
Conversely, assumed (0) = 0. Now
$\begin{array}{l} D (a) = D (0 ⋆ a) \\ = D (0) ⋆ a \\ = 0 ⋆ a \\ = a \end{array}$
Thus, D is an identity map on X.

5. Torsion-Free P MS-algebras

Definition 22.

A PMS-algebra X is said to be torsion-free if x = 0 whenever x + x = 0, $\forall$ x $\in$ X.

Example 9.

Let (X, ⋆, 0) be the PMS-algebra as shown in Cayley Table 6.

Table 6. Torsion free PMS-algebra.

⋆	0	1	2	3	4
0	0	1	2	3	4
1	4	0	1	2	3
2	3	4	0	1	2
3	2	3	4	0	1
4	1	2	3	4	0

0 + 0 = 0

1 + 1 = (1 ⋆ 0) ⋆ 1 = 4 ⋆ 1 = 2

2 + 2 = (2 ⋆ 0) ⋆ 2 = 3 ⋆ 2 = 4

3 + 3 = (3 ⋆ 0) ⋆ 3 = 2 ⋆ 3 = 1

4 + 4 = (4 ⋆ 0) ⋆ 4 = 1 ⋆ 4 = 3

Thus X is a Torsion-free PMS-algebra.

Theorem 32.

Let X be a Torsion-free PMS-algebra and let D₁ and D₂ be two generalized derivations on X. If D₁ ◦ D₂ = 0 on X, then D₂ = 0 on X.

Proof.

Let x ∈ X, then x + x ∈ X.

\begin{array}{l} 0 = (D_{1} ◦ D_{2}) (x + x) \\ = D_{1} (D_{2} (x + x)) \\ = D_{1} (0) + D_{2} (x + x) \\ = D_{1} (0) + D_{2} (x) + D_{2} (x) - D_{2} (0) \\ = D_{1} (0) - D_{2} (0) + D_{2} (x) + D_{2} (x) \\ = (D_{2} (0) ⋆ D_{1} (0)) + D_{2} (x) + D_{2} (x) \\ = (D_{1} (0) ⋆ 0) ⋆ D_{2} (0) + D_{2} (x) + D_{2} (x) \\ = D_{1} (0) + D_{2} (0) + D_{2} (x) + D_{2} (x) \\ = D_{1} (D_{2} (0)) + D_{2} (x) + D_{2} (x) \\ = 0 + D_{2} (x) + D_{2} (x) \end{array}

Hence D₂(x) + D₂(x) = 0.

Since X is a Torsion-free, D₂(x) = 0, ∀x ∈ X. Thus D₂ = 0 on X.

Corollary 2.

Let X be a Torsion-free PMS-algebra and D be a generalized derivation on X. If D² = 0 on X, then D = 0 on X.

Proof.

Let x ∈ X, then x + x ∈ X.

\begin{array}{l} 0 = D^{2} (x ⋆ x) \\ = D (D (x + x)) \\ = D (0) + D (x + x) \\ = D (0) + D (x) + D (x) - D (0) \end{array}

Hence D(x) + D(x) = 0.

Since X is torsion free D₂(x) = 0, $\forall$ x $\in$ X. Thus D₂ = 0 on X

6. Conclusions

Derivation is a crucial research area in the field of algebraic structure in mathematics. The derivations on PMS-algebras were applied, and further properties of the derivations on PMS-algebras were discussed. The study focused on generalized derivations in PMS-algebras, including regular derivations, fixed set derivations, and composition of derivations. We demonstrated that the set of all derivations on a PMS-algebra X forms a semigroup under a suitable binary composition. The definitions and main results can be applied to other algebraic systems and to computational physics and computer sciences. Further research is needed to explore the application of these concepts in computer science for information processing, and derivations have also been employed in coding theory. This idea may extend to other algebraic structures.

Ethics and consent

Ethical approval and consent were not required.

Data availability

No data are associated with this article.

References

1. Iseki K: An introduction to the theory of BCK-algebras. Math. Japon. 1978; 23(1): 1–26.
2. Iseki K: On BCI-algebras. Math. Seminar Notes. 1980; 8: 235–236.
3. Megalai K, Tamilarasi A: Classification of TM-algebra. IJCA. 2010; CASCT: 11–16. Special Issue on “Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications” CASCT. Publisher Full Text
4. Mostafa SM, Abdel Naby M, Abdel Halim F, et al.: On KUS-algebras. Int. J. Algebra. 2013; 7(3): 131–144. Publisher Full Text
5. Sithar Selvam P, Nagalakshmi K: On PMS-algebras. Transylv. Rev. 2016; 24(10): 1622–1628.
6. Jun YB, Xin XL: On derivations of BCI-algebras. Inf. Sci. 2004; 159(3-4): 167–176. Publisher Full Text
7. Ganeshkumar T, Chandramouleeswaran M: Generalized derivation on TM-algebras. Int. J. Algebra. 2013; 7: 251–258. Publisher Full Text
8. Kim KH, Lee SM: On derivations of BE-algebras. Honam Math. J. 2014; 36(1): 167–178. Publisher Full Text
9. Jana C, Senapati T, Pal M: Derivation, f-derivation and generalized derivation of KUS-algebras. Cogent Math. 2015; 2(1): 1064602. Publisher Full Text
10. Sawika K, Intasan R, Kaewwasri A, et al.: Derivations of UP-algebras. Korean J. Math. 2016; 24(3): 345–367. Publisher Full Text
11. Rong J-X, Liao Z-H, Xi Y, et al.: On derivations of FI- algebras. Fuzzy Sets and Operations Research. Springer; 2019; 9. : pp. 68–80.
12. Sugianti K, Gemawati S: Generalized derivations of BM-algebras. Int. J. Contemp. Math. Sci. 2020; 15(4): 225–233. Publisher Full Text
13. Ganesan P, Kandaraj N: Some results on generalized derivations of BH-algebras. Commun. Math. Appl. 2021; 12(4): 1069–1077. Publisher Full Text
14. Muangkarn P, Suanoom C, Iampan A: New derivations of d-algebras based on endomorphisms. Int. J. Math. Comput. Sci. 2022; 17: 1025–1032.

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VERSION 1 PUBLISHED 13 Jan 2025

Author details Author details

Berhanu Assaye Alaba
Roles: Conceptualization, Supervision, Validation, Visualization

Yohannes Gedamu Wondifraw
Roles: Methodology, Supervision, Validation, Visualization

Zelalem Teshome Wale
Roles: Conceptualization, Investigation, Methodology, Supervision, Validation, Visualization, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

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

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Published: 13 Jan 2025, 14:75

https://doi.org/10.12688/f1000research.159712.1

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Melese Kassahun N, Assaye Alaba B, Gedamu Wondifraw Y and Teshome Wale Z. Group Structures and Derivations on PMS-algebras [version 1; peer review: 2 approved]. F1000Research 2025, 14:75 (https://doi.org/10.12688/f1000research.159712.1)

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PUBLISHED 13 Jan 2025

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Reviewer Report 02 Jul 2025

Dr. Vinod Kumar R, Mathematics, Rajalakshmi Engineering College, Thandalam/Chennai, Tamil Nadu, India

Hemavathi P, Saveetha Institute of Medical and Technical Sciences (SIMATS), Chennai, Tamil Nadu, India

Approved

https://doi.org/10.5256/f1000research.175479.r393129

This paper investigates the concept of ideals within the framework of PMS-algebras—a class of algebraic structures that generalizes several well-known systems such as BCK-, BCI-, and TM-algebras. The authors define and explore various types of ideals (left, right, two-sided, bi-ideals, and interior ideals), providing structural results and demonstrating the interplay among them through propositions and examples

Positive Aspects

The manuscript addresses an important and underexplored topic in algebraic systems, especially concerning the ideal structure of PMS-algebras.
The mathematical definitions are precise, and the results are stated and proved rigorously.
A good balance of theory and examples is maintained. The examples are well-chosen and help in clarifying the abstract notions.
The discussion of absorbing elements and identity-related properties within the context of ideals is a welcome addition and contributes to a more complete understanding of PMS-algebra structures.

Areas for Improvement

Abstract and Introduction:
- The abstract could better reflect the core results and contributions of the paper. It is currently too general and lacks specificity regarding the key findings.
- A clearer motivation for studying ideals in PMS-algebras would enhance the introduction. For instance, how do these structures relate to or extend previous work in logic or algebra?
Clarity and Readability:
- Some sentences are awkwardly constructed or overly formal. Simplifying and clarifying the language would improve accessibility, particularly for readers new to PMS-algebras.
- Consider breaking down longer paragraphs, especially in the proof sections, for better flow and readability.
Notational Consistency:
- Ensure all notations (especially for operations and mappings) are defined consistently and used uniformly across the manuscript.
- A glossary of notations at the start or end of the paper would be beneficial.
Structure and Formatting:
- Some minor typographical issues (e.g., inconsistent punctuation, misaligned equations) should be corrected for professional presentation.

Minor Suggestions

Capitalization in section titles and definitions should be made consistent.
Ensure that all cited references are correctly formatted and updated.
It may be worth summarizing key results at the end of each section to improve digestibility.

Recommendation:
Minor Revision
This is a promising and well-developed theoretical manuscript. With some refinements to the abstract, presentation, and organization, it will make a valuable contribution to the literature on PMS-algebras and generalized algebraic structures.

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?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Fuzzy algebra, Neutrosophic Sets, Fuzzy Decision Making, Topology

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

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Author Response 09 Aug 2025

Nibret Melese Kassahun, Mathematics, Bahir Dar University, Bahir Dar, 6000, Ethiopia

09 Aug 2025

Author Response

We have accepted all the strength comments and suggestion including the minor comments to improve the paper, we will revise to correct some typographical errors and mathematical notations.

Nibret ... Continue reading We have accepted all the strength comments and suggestion including the minor comments to improve the paper, we will revise to correct some typographical errors and mathematical notations.

Nibret Melese
Corresponding author of the article

We have accepted all the strength comments and suggestion including the minor comments to improve the paper, we will revise to correct some typographical errors and mathematical notations.

Nibret Melese
Corresponding author of the article
Competing Interests: No competing interests were disclosed. Close
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Author Response 09 Aug 2025

Nibret Melese Kassahun, Mathematics, Bahir Dar University, Bahir Dar, 6000, Ethiopia

09 Aug 2025

Author Response

We have accepted all the strength comments and suggestion including the minor comments to improve the paper, we will revise to correct some typographical errors and mathematical notations.

Nibret ... Continue reading We have accepted all the strength comments and suggestion including the minor comments to improve the paper, we will revise to correct some typographical errors and mathematical notations.

Nibret Melese
Corresponding author of the article

We have accepted all the strength comments and suggestion including the minor comments to improve the paper, we will revise to correct some typographical errors and mathematical notations.

Nibret Melese
Corresponding author of the article
Competing Interests: No competing interests were disclosed. Close
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Reviewer Report 21 Mar 2025

Mezgebu Manmekto Gereme, Department of Mathematics, Debark University, Debark, Amhara, Ethiopia

Approved

https://doi.org/10.5256/f1000research.175479.r369263

The article, “Group Structure and Derivation on PMS-algebras,” presents a compelling investigation. Using mathematical techniques, the researchers analyzed the properties and behavior of derivations within PMS-algebras. Specifically, they demonstrated that the set of all derivations associated with the defined binary operations forms a semi-group. This analysis offers a comprehensive understanding of generalized derivations, d-invariant ideals, fixed sets, and torsion-free PMS-algebras within this algebraic framework.
Overall, the article effectively explores various aspects of torsion-free and generalized PMS-algebras, as well as the group structure and derivations on PMS-algebras. Consequently, this manuscript makes a significant contribution to the study of derivations on PMS-algebras. While the article is rich in proven results, the introduction could be strengthened by explicitly stating the motivation for this research. However, the authors adequately highlighted the study's importance and main contributions in the conclusion part of the manuscript. Therefore, I approved the acceptance for indexing in the journal.

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?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Fuzzy algebras, algebraic coding theory, Abstract algebras, Fuzzy coding theory, cryptography and fuzzy modeling.

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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Author Response 16 Jun 2025

Nibret Melese Kassahun, Mathematics, Bahir Dar University, Bahir Dar, 6000, Ethiopia

16 Jun 2025

Author Response

We have read the comments made by the reviewer and we have been accepted all of the comments. The comments and suggestions are very constructive which motivates our next works.
... Continue reading We have read the comments made by the reviewer and we have been accepted all of the comments. The comments and suggestions are very constructive which motivates our next works.
Thank yu
Nibret Melese Kassahun
nibret.melese2010@gmail.com
We have read the comments made by the reviewer and we have been accepted all of the comments. The comments and suggestions are very constructive which motivates our next works.
Thank yu
Nibret Melese Kassahun
nibret.melese2010@gmail.com
Competing Interests: No competing interests were disclosed. Close
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Author Response 16 Jun 2025

Nibret Melese Kassahun, Mathematics, Bahir Dar University, Bahir Dar, 6000, Ethiopia

16 Jun 2025

Author Response

We have read the comments made by the reviewer and we have been accepted all of the comments. The comments and suggestions are very constructive which motivates our next works.
... Continue reading We have read the comments made by the reviewer and we have been accepted all of the comments. The comments and suggestions are very constructive which motivates our next works.
Thank yu
Nibret Melese Kassahun
nibret.melese2010@gmail.com
We have read the comments made by the reviewer and we have been accepted all of the comments. The comments and suggestions are very constructive which motivates our next works.
Thank yu
Nibret Melese Kassahun
nibret.melese2010@gmail.com
Competing Interests: No competing interests were disclosed. Close
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Version 1 13 Jan 25	read	read

Mezgebu Manmekto Gereme, Debark University, Debark, Ethiopia
Dr. Vinod Kumar R, Rajalakshmi Engineering College, Thandalam/Chennai, India

Hemavathi P, Saveetha Institute of Medical and Technical Sciences (SIMATS), Chennai, India

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02 Jul 2025 | for Version 1

Dr. Vinod Kumar R, Mathematics, Rajalakshmi Engineering College, Thandalam/Chennai, Tamil Nadu, India

Hemavathi P, Saveetha Institute of Medical and Technical Sciences (SIMATS), Chennai, Tamil Nadu, India

8 Views Cite this report Responses(1)

Approved

The manuscript addresses an important and underexplored topic in algebraic systems, especially concerning the ideal structure of PMS-algebras.
The mathematical definitions are precise, and the results are stated and proved rigorously.
A good balance of theory and examples is maintained. The examples are well-chosen and help in clarifying the abstract notions.
The discussion of absorbing elements and identity-related properties within the context of ideals is a welcome addition and contributes to a more complete understanding of PMS-algebra structures.

Areas for Improvement

Abstract and Introduction:
- The abstract could better reflect the core results and contributions of the paper. It is currently too general and lacks specificity regarding the key findings.
- A clearer motivation for studying ideals in PMS-algebras would enhance the introduction. For instance, how do these structures relate to or extend previous work in logic or algebra?
Clarity and Readability:
- Some sentences are awkwardly constructed or overly formal. Simplifying and clarifying the language would improve accessibility, particularly for readers new to PMS-algebras.
- Consider breaking down longer paragraphs, especially in the proof sections, for better flow and readability.
Notational Consistency:
- Ensure all notations (especially for operations and mappings) are defined consistently and used uniformly across the manuscript.
- A glossary of notations at the start or end of the paper would be beneficial.
Structure and Formatting:
- Some minor typographical issues (e.g., inconsistent punctuation, misaligned equations) should be corrected for professional presentation.

Minor Suggestions

Capitalization in section titles and definitions should be made consistent.
Ensure that all cited references are correctly formatted and updated.
It may be worth summarizing key results at the end of each section to improve digestibility.

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?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Fuzzy algebra, Neutrosophic Sets, Fuzzy Decision Making, Topology

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

Respond to this report

Responses (1)

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Reviewer Report

8 Views

21 Mar 2025 | for Version 1

Mezgebu Manmekto Gereme, Department of Mathematics, Debark University, Debark, Amhara, Ethiopia

8 Views Cite this report Responses(1)

Approved

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?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Fuzzy algebras, algebraic coding theory, Abstract algebras, Fuzzy coding theory, cryptography and fuzzy modeling.

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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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. Iseki K: An introduction to the theory of BCK-algebras. Math. Japon. 1978; 23(1): 1–26.

[2] 2. Iseki K: On BCI-algebras. Math. Seminar Notes. 1980; 8: 235–236.

[3] 3. Megalai K, Tamilarasi A: Classification of TM-algebra. IJCA. 2010; CASCT: 11–16. Special Issue on “Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications” CASCT. Publisher Full Text

[4] 4. Mostafa SM, Abdel Naby M, Abdel Halim F, et al.: On KUS-algebras. Int. J. Algebra. 2013; 7(3): 131–144. Publisher Full Text

[5] 5. Sithar Selvam P, Nagalakshmi K: On PMS-algebras. Transylv. Rev. 2016; 24(10): 1622–1628.

[6] 6. Jun YB, Xin XL: On derivations of BCI-algebras. Inf. Sci. 2004; 159(3-4): 167–176. Publisher Full Text

[7] 7. Ganeshkumar T, Chandramouleeswaran M: Generalized derivation on TM-algebras. Int. J. Algebra. 2013; 7: 251–258. Publisher Full Text

[8] 8. Kim KH, Lee SM: On derivations of BE-algebras. Honam Math. J. 2014; 36(1): 167–178. Publisher Full Text

[9] 9. Jana C, Senapati T, Pal M: Derivation, f-derivation and generalized derivation of KUS-algebras. Cogent Math. 2015; 2(1): 1064602. Publisher Full Text

[10] 10. Sawika K, Intasan R, Kaewwasri A, et al.: Derivations of UP-algebras. Korean J. Math. 2016; 24(3): 345–367. Publisher Full Text

[11] 11. Rong J-X, Liao Z-H, Xi Y, et al.: On derivations of FI- algebras. Fuzzy Sets and Operations Research. Springer; 2019; 9. : pp. 68–80.

[12] 12. Sugianti K, Gemawati S: Generalized derivations of BM-algebras. Int. J. Contemp. Math. Sci. 2020; 15(4): 225–233. Publisher Full Text

[13] 13. Ganesan P, Kandaraj N: Some results on generalized derivations of BH-algebras. Commun. Math. Appl. 2021; 12(4): 1069–1077. Publisher Full Text

[14] 14. Muangkarn P, Suanoom C, Iampan A: New derivations of d-algebras based on endomorphisms. Int. J. Math. Comput. Sci. 2022; 17: 1025–1032.