F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.159711.1 Research Article Articles On t-derivations of PMS-algebras

[version 1; peer review: 2 approved]

 Melese Kassahun Nibret Conceptualization Data Curation Formal Analysis Funding Acquisition Investigation Methodology Project Administration Resources Software Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0009-0004-5010-6943 a 1 2 Assaye Alaba Berhanu Conceptualization Supervision Validation Writing – Review & Editing 1 Gedamu Wondifraw Yohannes Resources Supervision Validation Visualization 1 Teshome Wale Zelalem Investigation Methodology Supervision Validation Visualization Writing – Review & Editing 3 Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, 6000, Ethiopia Mathematics, Debre Markos University College of Natural and Computational Science, Debre Markos, Amhara, None, Ethiopia Mathematics, Addis Ababa University College of Natural Sciences, Addis Ababa, Addis Ababa, 1000, Ethiopia nibret.melese2010@gmail.com

No competing interests were disclosed.

 13 1 2025 2025 14 70 19 12 2024 Copyright: © 2025 Melese Kassahun N et al. 2025 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

PMS algebras are a type of algebraic structure that has been studied extensively in recent years. They are a generalization of several other algebraic structures, such as Boolean algebras and MV-algebras.

 Methods

In this paper, we introduce the concept of t-derivations on PMS algebras. T-derivations are a type of mapping between PMS algebras that satisfies certain properties. We then study the properties of t-derivations and regular t-derivations on PMS algebras.

 Results

We characterize further properties of t-derivations in the context of PMS algebras. We also investigate a novel result of t-derivations on the G-part of a PMS-algebra. Finally, we prove that the set of all t-derivations on a PMS-algebra forms a semigroup.

 Conclusions

This paper provides a comprehensive study of t-derivations on PMS algebras. We have established several new results and characterized the properties of t-derivations in detail. Our results contribute to the further understanding of PMS algebras and their associated structures.

 PMS-algebras derivations t-derivations semigroup The author(s) declared that no grants were involved in supporting this work. 1. Introduction

Iseki 3 introduced BCK-algebras, which are a type of abstract algebra, in his 1978 work. Additionally, Iseki 4 established another class of abstract algebras called BCI- algebras, which are an extension of BCK-algebras, in his research work. Furthermore, Iseki 4 studied and explored the algebraic properties of BCI-algebras. Also, Tamalibrasi and Megalai 8 introduced and studied TM-algebras based on propositional calculus. Moreover, Mostafa et al. 9 studied the concepts of KUS-algebras, KUS-ideals, and KUS-subalgebras, and their relationships were investigated. Sithar and Nagalakshmi 11 continued studying. PMS-algebras, which encompass the notions of BCK/BCI/TM/KUS algebras. They also discussed the ideals of PMS-algebras and studied their properties.

Derivations are essential areas of study in algebraic structure theory, originating from invariant theory and Galois theory. Derivations were introduced in rings and near rings by many researchers. Based on derivations in rings and near-rings theory, Jun and Xin 7 expanded the notion of derivations to BCI-algebras. After derivations were intro- duced in various algebraic structures, derivations were extended to new concepts called t-derivations. As a result, t-derivations are studied and characterized in various algebraic structures. Muhiuddin and Al-Roqi 10 introduced the notion of t-derivations on BCI- algebras and characterized their interesting results based on the context of BCI-algebras in their 2012 work. In 2013, Ganeshkumar and Chandramouleeswaran 2 introduced the idea of t-derivations on TM-algebras, and they explored their basic properties. Jana et al. 5 established the notion of t-derivations on subtraction algebras with their essential properties in their 2017 work. Furthermore, in 2020, t-derivations were introduced on lattices and explored their properties based on lattice structures by Javed et al. 6 Re- cently, in 2021, Siswanti et al. 12 t-derivations were studied on BP-algebras, and their detailed properties were examined. Additionally, Ganesan and Kandaraj 1 t-derivations on BH-algebras were introduced in their 2021 research work. In this paper, we introduce t-derivations on PMS-algebras and investigate their nice properties based on the structure of PMS-algebras. We also define regular derivations on PMS-algebras and characterize their properties. Finally, we show that the set of all t-derivations on PMS-algebras is a semigroup.

 2. Preliminaries

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. The following definitions and properties are taken from Sithar and Nagalakshmi. 11

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

0 ⋆ a = a;

(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.

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

a ⋆ a = 0.

(b ⋆ a) ⋆ a = b.

a ⋆ (b ⋆ a) = b ⋆ 0.

(b ⋆ a) ⋆ c = (c ⋆ a) ⋆ b.

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

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.

Let X be a PMS-algebra. The set G (X) = {a ∈ X: a⋆ 0 = a} is called the G-part of X.

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

3. t-derivations

Let (X; ⋆, 0) be a PMS-algebra.

If b ⋆ a = c ⋆ a, then b = c;

If a ⋆ b = a ⋆ c, then b = c for all a, b, c ∈ X.

Let a, b, c ∈ X.

Assume that b ⋆ a = c ⋆ a

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

Suppose that a ⋆ b = a ⋆ c

Now b ⋆ a = b ⋆ ((a ⋆ b) ⋆ b) = b ⋆ ((a ⋆ c) ⋆ b) = (a ⋆ c) ⋆ 0 = (0 ⋆ c) ⋆ a = c ⋆ a. Hence by (1), we get b = c. ■

A PMS-algebra X is said to be associative if (x ⋆ y) ⋆ z = x ⋆ (y ⋆ z) for all x, y, z ∈ X.

Let X be the PMS-algebra with the following Cayley table.
⋆ 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 (X; ⋆, 0) is an associative PMS-algebra.

Let X be a PMS-algebra. Then, for t ∈ X, define a self-map d t : X→ X by d t (x) = t ⋆ x for all x ∈ X.

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

Let d t be a (l, r)-t-derivation on a PMS-algebra X. Then d t (x ⋆ y) = d t (x) ⋆ y for all x, y ∈ X.

Let X be an associative PMS-algebra with the following Cayley Table.
⋆ 0 1 2
0 0 1 2
1 1 0 1
2 2 1 0

Define, when t = 0, d t (x) = x, ∀x ∈ X. when t = 1, d t (0) = 1, d t (1) = 0, d t (2) = 1. when t = 2, d t (0) = 2, d t (1) = 1, d t (2) = 0. Therefore, for each t ∈ X, d t is a (l, r)-t-derivation on X.

Let X be a PMS-algebra. Then, for t ∈ X, a self-map d t : X→ X is said to be a (l, r)-t-derivation of X, denoted by, (r, l)-t-derivation if d t (x ⋆ y) = (x ⋆ d t (y)) ∧ (d t (x) ⋆ y) for all x, y ∈ X.

Let d t be a (r, l)-t-derivation on a PMS-algebra X. Then d t (x ⋆ y) = x ⋆ d t (y) for all x, y ∈ X.

Let X be a non-associative PMS-algebra with the following Cayley Table.
⋆ 0 1 2
0 0 1 2
1 2 0 1
2 1 2 0

Define, when t = 0, d t (x) = x, ∀x ∈ X. when t = 1, d t (0) = 2, d t (1) = 0, d t (2) = 1.

when t = 2, d t (0) = 1, d t (1) = 2, d t (2) = 0.

Therefore, for each t ∈ X, d t is a (r, l)-t-derivation on X.

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

Let d t : X→ X defined by d t (x) = t ⋆ x. Now for x, y ∈ X, d t ( x ⋆ y ) = t ⋆ ( x ⋆ y ) = ( ( t ⋆ y ) ⋆ y ) ⋆ ( x ⋆ y ) = x ⋆ ( t ⋆ y ) = x ⋆ d t ( t ⋆ y )

Hence d t is a (r, l)-t-derivation on X. ■

Let X be a PMS-algebra. Then, for t ∈ X, a self-map d t : X→ X is called the t-derivation on X if d t is both a (l, r)-t-derivation and a (r, l)-t-derivation on X.

Consider the PMS-algebra X in Example 3.3. Define the mapping d t as follows:

When t = 0, d t (x) = x, ∀x ∈ X.

When t = 1, d t (0) = 1, d t (1) = 0, d t (2) = 3, d t (3) = 2.

When t = 2, d t (0) = 2, d t (1) = 3, d t (2) = 0, d t (3) = 1.

When t = 3, d t (0) = 3, d t (1) = 2, d t (2) = 1, d t (3) = 0. Therefore, for each t ∈ X, d t is a t-derivation of X.

Let d t be a self-map on an associative PMS -algebra X . Then d t is a (l, r) -t -derivation on X.

Let X be an associative PMS-algebra and d t be a self-map on X. d t ( x ⋆ y ) = t ⋆ ( x ⋆ y ) = ( t ⋆ x ) ⋆ y = d t ( x ) ⋆ y

Hence d t is (r, l)-t-derivation on X.

By combining Proposition 3.11 and Proposition 3.14, we get the following theorem. ■

Let X be an associative PMS -algebra. For t ∈ X , a self-map d t is a t -derivation on X.

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

In Example 3.13, d t is a regular t-derivation on X where t = 0. However, for t = 1, t = 2, or t = 3, d t is not a regular t-derivation on X.

Let X be the p -radical of a PMS -algebra and d t be a self-map on X . Then d t is t -regular.

Let X be a p-radical PMS-algebra.

Let x ∈ X. Then x ⋆ 0 = 0. Now d t (0) = t ⋆ 0 = 0. Hence d t is t-regular. ■

Let d t be a self-map on a PMS -algebra X . Then

If d t is a (l, r) -t -derivation on X , then d t (x) = x ∧ d t (x) ∀x ∈ X if and only if d t is t -regular.

If d t is a (r, l) -t -derivation on X . If d t is t -regular on X , then d t (x) = d t (x) ∧ x ∀x ∈ X.

(1) Let d t is a (l, r)-t-derivation on X. Suppose that d t (x) = x ∧ d t (x).

d t (0) = 0 ∧ d t (0) = (0 ⋆ d t (0)) ⋆ d t (0) = (d t (0) ⋆ d t (0)) ⋆ 0 = 0 ⋆ 0 = 0.

d t is regualr on X.

Conversely, assume that d t is regular.

Now d t (x) = d t (0 ⋆ x) = (d t (0) ⋆ x) ∧ (0 ⋆ d t (x)) = (0 ⋆ x) ∧ d t (x) = x ∧ d t (x). Hence d t (x) = x ∧ d t (x)

(2) Let d t is a (r, l)-t-derivation on X.

Suppose d t (0) = 0.

Now d t (x) = d t (0 ⋆ x) = (0 ⋆ d t (x) ∧ (d t (0) ⋆ x) = d t (x) ∧ (0 ⋆ x) = d t (x) ∧ x ■

Let d t be a (r, l) -t -derivation on a PMS -algebra. Then the following holds true.

d t (0) = x ⋆ d t (x), ∀x ∈ X.

d t is one-to-one.

If d t is t -regular, then it is the identity map.

If there is an element x ∈ X such that d t (x) = x , then d t is the identity map.

If x ≤ y , then d t (x) ≤ d t (y) for all x, y ∈ X.

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

Let x ∈ X. Then d t (0) = t ⋆ 0 = t ⋆ (x ⋆ x) = x ⋆ (t ⋆ x) = x ⋆ d t (x).

Let x, y ∈ X such that d t (x) = d t (y). Then t ⋆ x = t ⋆ y. Now x ⋆ t = x ⋆ ( ( t ⋆ x ) ⋆ x ) = x ⋆ ( ( t ⋆ y ) ⋆ x ) = ( 0 ⋆ x ) ⋆ ( ( t ⋆ y ) ⋆ x ) = ( t ⋆ y ) ⋆ 0 = ( 0 ⋆ y ) ⋆ t = y ⋆ t .

Hence by right cancellation, x = y. Therefore d t is an identity map.

Let d t be a t-regualr derivation on X.

By (1), we have 0 = d t (0) = x ⋆ d t (x) implies that x ⋆ x = x ⋆ d t (x) Hence by left cancellation, d t (x) = x.

Let d t (x) = x for some x ∈ X.

Now 0 = x ⋆ x = x ⋆ d t (x) = d t (x ⋆ x) = d t (0) implies that d t is regular. Hence by (3) d t is an identity map.

Let x ≤ y. Then x ⋆ y = 0.

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

Hence d t (x) ≤ d t (y). ■

Let X be a PMS -algebra and d t be a t -derivation on X . If y ≤ x and d t (x ⋆ y) = d t (x) ⋆ d t (y) for all x, y ∈ X ,t hen d t (x) = d t (y).

Let x ≤ y and d t (x ⋆ y) = d t (x) ⋆ d t (y).

Now for x ∈ X, d t (x) = d t (0⋆x) = d t ((y⋆x)⋆x) = d t (y⋆x)⋆d t (x) = (d t (y)⋆d t (x))⋆d t (x) = (d t (x) ⋆ d t (x)) ⋆ d t (y) = 0 ⋆ d t (y) = d t (y).

Hence d t (x) = d t (y).■

Let d t be a t -regular (l, r) -t -derivation on a PMS -algebra X . Then the following hold.

d t (x) = x.

d t (x) ⋆ y = x ⋆ d t (y) for all x, y ∈ X.

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

Ker(d t ) = {x ∈ X: d t (x) = 0} is a subalgebra of X .

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

Let x ∈ X. Then d t (x) = d t (0 ⋆ x) = d t (0) ⋆ x = 0 ⋆ x = x.

Let x, y ∈ X. Then d t (x) ⋆ y = x ⋆ y = x ⋆ d t (y).

Let x, y ∈ X. Then d t (x ⋆ y) = d t (x) ⋆ y = d t (x) ⋆ d t (y) = x ⋆ y.

Since d t is regular, d t (0) = 0. Hence 0 ∈ Ker(d t ) Thus ker(d t ) is nonempty.

Let x, y ∈ Ker(d t ). Then d t (x) = 0, d t (Y) = 0. Now d t (x ⋆ y) = x ⋆ y = d t (x) ⋆ d t (y) = 0 ⋆ 0 = 0. Hence x ⋆ y ∈ Ker(d t ).

Therefore Ker(d t ) is a subalgebra of X. ■

Let X be a PMS -algebra. Then Ker(d t ) = {0} if and only if d t is t -regular.

Suppose Ker(d t ) = {0}. Then d t (x) = 0 ∀x ∈ X. In particular, d t (0) = 0. Hence d t is regular.

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

Thus x = 0 implies that Ker(d t ) = {0} ■

Let X be a PMS-algebra and d t d t ′ be self-maps on X. Then define a mapping d t ◦ d t ′ : X → X by (d t ◦ d t ′ )(x) = d t ( d t ′ (x)) for all x ∈ X.

Let X be a PMS -algebra and d t d t ′ be a (l, r) -t -derivation on X . Then (d t ◦ d t ′ ) is a (l, r) -t -derivation on X.

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

For x, y ∈ X, we have ( d t ◦ d t ′ ) ( x ⋆ y ) = d t ( d t ′ ( x ⋆ y ) ) = d t ( d t ′ ( x ) ⋆ y ) = d t ( d t ′ ( x ) ) ⋆ y = ( d t   ◦   d t ′ ) ( x ) ⋆ y

Hence (d t ◦ d t ′ ) is a (l, r)- t-derivation of X. ■

Let X be a PMS algebra and d t d ' t be a (r, l) - t -derivation of X . Then (d t ◦ d t ′ ) is a (r, l) -t -derivation on X.

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

For x, y ∈ X, we have ( d t   ◦   d t ′ ) ( x ⋆ y ) = d t ( d t ′ ( x ⋆ y ) ) = d t ( x ⋆ d t ′ ( y ) ) = x ⋆ d t ( d t ′ ( y ) ) = x ⋆ ( d t   ◦   d t ′ ) ( y )

Hence (d t ◦ d t ′ ) is a (r, l)-t-derivation of X.

Combining Theorem 3.25 and Theorem 3.26, we get the following theorem. ■

Let X be a PMS -algebra and d t d t ′ be t -derivation of X . Then (d t ◦ d t ′ ) is a t -derivation on X.

Let X be a PMS -algebra. Let d t be a (r, l) -t -derivation on X and d t ′ be a (l, r)-t- derivation on X. Then d t ◦ d t ′ = d t ′ ◦ d t.

Let d t be a (r, l)-t-derivation on X and d t ′ be a (l, r)-t-derivation on X. Now ( d t   ◦   d t ′ ) ( x ) = ( d t   ◦   d t ′ ) ( 0 ⋆ x ) = d t ( d t ′ ( 0 ⋆ x ) ) = d t ( d t ′ ( 0 ) ⋆ x ) = d t ′ ( 0 ) ⋆ d t ( x ) ( d t ′   ◦   d t ) ( x ) = ( d t ′   ◦   d t ) ( 0 ⋆ x ) = d t ′ ( d t ( 0 ⋆ x ) ) = d t ′ ( 0 ⋆ d t ( x ) ) = d t ′ ( 0 ) ⋆ d t ( x )

From (3.1) and (3.2) we have, (d t ◦ d t ′ )(x) = ( d t ′ ◦ d t )(x), ∀x ∈ X. Thus d t ◦ d t ′ = d t ′ ◦ d t. ■

Let X be a PMS -algebra and d t d t ′ be two t -derivations on X , then d t ◦ d t ′ = d t ′ ◦ d t.

Let X be a PMS-algebra and d t d t ′ be self-maps on X. Then define d t ⋆ d t ′ : X→ X by (d t ⋆ d t ′ )(x) = d t (x) ⋆ d t ′ (x) for all x ∈ X.

Let X be a PMS -algebra and d t d t ′ be t -derivations on X . Then d t ⋆ d t ′ = d t ′ ⋆ d t.

Let LDer t (X) be the set of all (l, r)-t-derivations on a PMS-algebra X. Then define a binary operation “∧” on LDer t (X) by (d t ∧ d t ′ )(x) = d t (x) ∧ d t ′ (x), ∀x ∈ X, where d t d t ′ ∈ LDer t (X).

If d t and d ' t are (l, r) -t -derivations on X . Then (d t ∧ d t ′ ) is a (l, r) -t -derivation on X.

( d t ∧ d t ′ ) ( x ⋆ y ) = d t ( x ⋆ y ) ∧ d t ′ ( x ⋆ Y ) = ( d t ( x ) ⋆ y ) ∧ ( d t ′ ( x ) ⋆ y ) = ( ( d t ( x ) ⋆ y ) ⋆ ( d t ′ ( x ) ⋆ y ) ) ⋆ ( d t ′ ( x ) ⋆ y ) = ( ( d t ′ ( x ) ⋆ y ) ⋆ ( d t ′ ( x ) ⋆ y ) ) ⋆ ( d t ( x ) ⋆ y ) = 0 ⋆ ( d t ( x ) ⋆ y ) = d t ( x ) ⋆ y ( d t ∧ d t ′ ) ( x ) ⋆ y = ( d t ( x ) ∧ d t ′ ( x ) ) ⋆ y = ( ( d t ( x ) ⋆ d t ′ ( x ) ) ⋆ d t ′ ( x ) ) ⋆ y = ( ( d t ′ ( x ) ⋆ d t ′ ( x ) ) ⋆ d t ( x ) ) ⋆ y = ( 0 ⋆ d t ( x ) ) ⋆ y = d t ( x ) ⋆ y

From (3.3) and (3.4), we get (d t ∧ d t ′ )(x ⋆ y) = (d t ∧ d t ′ )(x) ⋆ y. ■

The binary operation ∧ defined on L t Der(X) is associative.

Let d t d t ′ d t ″ ∈ LDer t (X). Now for x ∈ X, ( ( d t ∧ d t ′ ) ∧ d t ′ ′ ) ( x ) = ( d t ( x ) ∧ d t ′ ) ( x ) ∧ d t ″ ( x ) ( ( d t ( x ) ⋆ d t ′ ( x ) ) ⋆ d t ′ ( x ) ) ∧ d t ″ ( x ) ( ( d t ′ ( x ) ⋆ d t ′ ( x ) ) ⋆ d t ( x ) ) ∧ d t ″ ( x ) = ( 0 ⋆ d t ( x ) ) ∧ d t ″ ( x ) = d t ( x ) ∧ d t ″ ( x ) = ( d t ( x ) ⋆ d t ″ ( x ) ) ⋆ d t ″ ( x ) = ( d t ″ ( x ) ⋆ d t ″ ( x ) ) ⋆ d t ( x ) = 0 ⋆ d t ( x ) = d t ( x ) ( d t ∧ ( d t ′ ∧ d t ″ ) ) ( x ) = d t ( x ) ∧ ( d t ′ ∧ d t ″ ) ( x ) = d t ( x ) ∧ ( ( d t ′ ( x ) ⋆ d ′ ′ t ( x ) ) ⋆ d ′ ′ t ( x ) ) = d t ( x ) ∧ ( ( d ″ t ( x ) ⋆ d ″ t ( x ) ) ⋆ d t ′ ( x ) ) = d t ( x ) ∧ ( 0 ⋆ d t ′ ( x ) ) = d t ( x ) ∧ d t ′ ( x ) = ( d t ( x ) ⋆ d t ′ ( x ) ) ⋆ d t ′ ( x ) = ( d t ′ ( x ) ⋆ d t ′ ( x ) ) ⋆ d t ( x ) = 0 ⋆ d t ( x ) = d t ( x )

From (3.5) and (3.6), we have ((d t ∧ d t ′ ) ∧ d t ″ )(x) = (d t ∧ ( d t ′ ∧ d t ″ ))(x), ∀x ∈ X.

Thus “∧” is associative. ■

L t Der(X) is a semigroup under the binary operation ∧ defined by (d t ∧ d t ′ )(x) = d t (x) ∧ d t ′ (x) for all x ∈ X , and d t d t ′ ∈ L t Der(X).

Let R t Der(X) be the set of all (r, l)-t-derivations on a PMS-algebra X. Then define a binary operation ∧ on R t Der(X) by (d t ∧ d t ′ )(x) = d t (x) ∧ d t ′ (x), ∀x ∈ X, where d t d t ′ ∈ R t Der(X).

If d t and d ' t are (r, l) -t -derivations on X . Then (d t ∧ d t ′ ) is also a (r, l) -t -derivation on X.

Let d t d t ′ ∈ RDer t (X). For x ∈ X ,   ( d t ∧ d t ′ ) ( x ⋆ y ) = d t ( x ⋆ y ) ∧ d t ′ ( x ⋆ y ) = ( x ⋆ d t ( y ) ) ∧ ( x ⋆ d t ′ ( y ) ) = ( ( x ⋆ d t ( y ) ) ⋆ ( x ⋆ d t ′ ( y ) ) ) ⋆ ( x ⋆ d t ′ ( y ) ) = ( ( x ⋆ d t ′ ( y ) ) ⋆ ( x ⋆ d t ′ ( y ) ) ) ⋆ ( x ⋆ d t ( y ) ) = x ⋆ d t ( y ) x ⋆ ( d t ∧ d t ′ ) ( y ) = x ⋆ ( d t ( y ) ∧ d t ′ ( Y ) ) = x ⋆ ( ( d t ( y ) ⋆ d t ′ ( y ) ) ⋆ d t ′ ( y ) ) = x   ⋆ ( ( d t ′ ( y )   ⋆ d t ′ ( y ) )   ⋆ d t ( y ) ) = x   ⋆ ( 0   ⋆ d t ( y ) ) = x ⋆ d t ( y )

From (3.7) and (3.8), we have, (d t ∧ d t ′ )(x ⋆ y) = x ⋆ (d t ∧ d t ′ )(y). ■

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

Let d t d t ′ d t ″ ∈ RDer t (X). Now for, x ∈ X. ( ( d t ∧ d t ′ ) ∧ d t ″ ) ( x ) = ( ( d t ∧ d t ′ ) ∧ d t ″ ) ( 0 ⋆ x ) = ( d t ∧ d t ′ ) ( 0 ⋆ x ) ∧ d t ″ ( 0 ⋆ x ) = ( d t ( 0 ⋆ x ) ∧ d t ′ ( 0 ⋆ x ) ) ∧ d t ″ ( 0 ⋆ x ) = ( ( 0 ⋆ d t ( x ) ) ∧ ( 0 ⋆ d t ′ ( x ) ) ) ∧ ( 0 ⋆ d ″ t ( x ) ) = ( d t ( x ) ∧ d t ′ ( x ) ) ∧ d t ″ ( X ) = ( d t ( x ) ⋆ d t ′ ( x ) )   ⋆ d t ″ ( x ) = ( d t ″ ( x )   ⋆ d t ′ ( x ) )   ⋆ d t ( x ) = d t ( x ) And ( d t ∧ ( d t ′ ∧ d t ″ ) ) ( x ) = ( d t ∧ ( d t ′ ∧ d t ″ ) ) ( 0   ⋆ x ) = d t ( 0 ⋆ x ) ∧ ( d t ′ ∧ d t ″ ) ( 0 ⋆ x ) = ( 0   ⋆ d t ( x ) )   ∧ ( ( d t ′ ( 0 ⋆ x )   ∧   d t ″ ( 0 ⋆ x ) ) = d t ( x ) ∧ ( ( 0   ⋆ d t ′ ( x ) )   ∧ ( 0 ⋆ d t ″ ( x ) ) ) = d t ( x )   ∧ ( d t ′ ( x ) ∧ d t ″ ( x ) ) = d t ( x )   ∧ d t ′ ( x ) = d t ( x )

From (3.9) and (3.10), we have ((d t ∧ d t ′ ) ∧ d t ″ )(x) = (d t ∧ ( d t ′ ∧ d t ″ ))(x), ∀x ∈ X. Thus the binary operation “∧” is associative on RDer t (X). ■

RDer t (X) is a semigroup under the binary operation “∧ ” defined by (d t ∧ d t ′ )(x) = d t (x) ∧ d t ′ (x) for all x ∈ X , where d t d t ′ ∈ R t Der(X).

Combining Theorem (3.35) and Theorem (3.39), we get the following theorem.

If Der t (X) denotes the set of all t -derivations on X . Then it is a semigroup under the binary operation “∧” defined by (d t ∧ d t ′ )(x) = d t (x) ∧ d t ′ (x) for all x ∈ X , where d t d t ′ ∈ Der t (X).

Conclusions

The study of propositional logic-based algebraic structures, such as PMS-algebras, is crucial for understanding derivation concepts. Derivations are applied to coding theory. The concept of t-derivations in PMS-algebras is explored, and it is proven that the set of all t-derivations forms a semigroup. This research can be applied to other algebras like BF- algebras, UP-algebras, and KUS-algebras, providing a foundation for further research on derivations.

 Ethics and consent

Ethical approval and consent were not required.

 Data availability statement

No data are associated with this article.

 Acknowledgements

We would like to thank the editor in chief, associate editor, and anonymous reviewers for their valuable comments and suggestions for improving this manuscript.

 References Ganesan P Kandaraj N : A note on t-derivations of BH-algebras. Adv. Appl. Math. Sci. 2021;20(11):2959–2969. Ganeshkumar T Chandramouleeswaran M : t-derivations on TM-algebras. Int. J.Pure Appl. Math. 2013;85(1):95–107. 10.12732/ijpam.v85i1.8 Iseki K : An introduction to the theory of BCK-algebras. Math. Japon. 1978;23(1):1–26. Iseki K : On bci-algebras. Math. Seminar Notes. 1980;8:235–236. Jana C Senapati T Pal M : t-derivations on com- plicated subtraction algebras. J. Discret. Math. Sci. Cryptogr. 2017;20(8):1583–1595. 10.1080/09720529.2017.1308663 Javed MA Malik SN : On t-derivations of lattices. Int. J. Contemp. Math. Sci. 2020;15(4):207–214. 10.12988/ijcms.2020.91444 Jun YB Xin XL : On derivations of bci-algebras. Inf. Sci. 2004;159(3-4):167–176. 10.1016/j.ins.2003.03.001 Megalai K Tamilarasi A : Classification of TM-algebra, IJCA Special Issue on “Computer Aided Soft Computing Techniques for Imaging and Biomedical Applications” CASCT. 2010. Mostafa SM Abdel Naby MA Abdel Halim F : On KUS-algebras. Int. J. Algebra. 2013;7(3):131–144. 10.12988/ija.2013.13013 Muhiuddin G Abdullah Al-roqi M : On t-Derivations of BCI-Algebras. Abstr. Appl. Anal. 2012;2012:Article ID 872784. 10.1155/2012/872784 Sithar Selvam PM Nagalakshmi KT : On PMS-algebras, Transylvanian Review. 2016;24(10):1622–1628. Siswanti FT : Sri Gemawati, and Syamsudhuha Syamsudhuha, t-derivations in BP-algebras. Sci. Technol. Commun. J. 2021;1(3):97–103. 10.59190/stc.v1i3.199 10.5256/f1000research.175478.r375060 Reviewer response for version 1 P Hemavathi 2 Referee https://orcid.org/0000-0003-0607-2817 R Dr. Vinod Kumar 1 Co-referee Mathematics, Rajalakshmi Engineering College, Thandalam/Chennai, Tamil Nadu, India Saveetha Institute of Medical and Technical Sciences (SIMATS), Thandalam, India

Competing interests: No competing interests were disclosed.

 28 6 2025 Copyright: © 2025 R DVK and P H 2025 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

The manuscript presents a rigorous and comprehensive study of t-derivations in PMS-algebras, an emerging algebraic framework that generalizes several known systems such as BCK-, BCI-, TM-, and KUS-algebras. The paper introduces and explores both left-right and right-left t-derivations, regular t-derivations, and their algebraic behavior within PMS-algebras, culminating in the significant result that the set of all t-derivations forms a semigroup.

Strengths

The topic is mathematically relevant and original, expanding derivation concepts to a broader class of algebraic systems.

The definitions are precise, and the theorems are stated clearly with logically consistent proofs.

The paper is well-structured, gradually developing the theory with examples and counterexamples to support understanding.

The establishment of semigroup properties for derivation mappings is a valuable algebraic contribution.

 Concerns and Suggestions

While the paper is mathematically sound, some phrases are awkwardly constructed, which may affect reader comprehension. A language refinement or minor copy-editing is recommended to improve clarity (e.g., consistent use of “self-map”, “identity map”, etc.).

veral notations such as dt, d’t, ∧, and various operation compositions (◦, ⋆) are heavily reused. A notation table at the beginning or end would be helpful for reference.

The introduction could be strengthened by briefly discussing potential applications or motivations for studying derivations in PMS-algebras, especially in logic, coding theory, or information systems, as hinted in the conclusion.

 Minor Comments

Fix typographical issues (e.g., “regualr” → “regular”; inconsistent capitalizations in definitions and remarks).

The references are appropriate but could benefit from standardized formatting for better consistency (some include first names; others don’t).

 Consider revising the abstract to be more informative currently, it repeats “we study” and “we prove” without summarizing the key findings in a concise, technical tone.

Recommendation:

Minor Revision

The manuscript provides a well-structured and significant contribution to algebraic logic and derivation theory. A minor revision focusing on language, presentation, and notational clarity will enhance the paper’s accessibility and impact.

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

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

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

Yes

Are the conclusions drawn adequately supported by the results?

Yes

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

Yes

Reviewer Expertise:

Fuzzy algebra, Fuzzy Decision Making, Neutrosophic Sets

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.

Melese Kassahun Nibret Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, Ethiopia

Competing interests: No competing interests were disclosed.

 28 6 2025

We have accepted the reviewers' comments and thank you for the valuable feedback.

Thank you,

Dr. Nibret Melese Kassahun

Corresponding author of the article.

 10.5256/f1000research.175478.r375059 Reviewer response for version 1 Yedlapalli Phani 1 Referee Shri Vishnu Engineering College for Women, Vishnupur, India

Competing interests: No competing interests were disclosed.

 26 5 2025 Copyright: © 2025 Yedlapalli P 2025 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

The paper presents a novel study on t-derivations and regular derivations in PMS-algebras, introducing  concepts with mathematical rigor. The authors successfully explore key properties of these derivations, grounded in the structural framework of PMS-algebras. A notable result is that the set of all t-derivations forms a semigroup, which adds depth to the algebraic structure. The work is original, clearly written, and logically organized. The results have potential for further exploration and applications. Overall, this is a valuable and publishable contribution to algebraic research.

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

Yes

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

Yes

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

Yes

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

Yes

Are the conclusions drawn adequately supported by the results?

Yes

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

Yes

Reviewer Expertise:

Algebras

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.

Melese Kassahun Nibret Mathematics, Bahir Dar University College of Science, Bahir Dar, Amhara, Ethiopia

Competing interests: No competing interests were disclosed.

 26 5 2025

We have read the comments made by the reviewer and we have been accept all of the valuable comments which are constructive.

Thank you for your valid comments  

Nibret Melese kassahun