EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6501
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Quotient INK-Algebras

Maliwan Phattarachaleekul

Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham
44150, Thailand

Abstract. This paper introduces the notion of quotient INK-algebra (X, ∗, 0). We formalize the
notion of congruence relations in INK-algebras, proving that the set of equivalence classes forms
a partition. Additionally, we define a normal subset N of X and show that the quotient set
X/N = {[x]N |x ∈ X} where [x]N = {y ∈ X|x ∼

N
y} and a binary operation ⊙ on X/N such that

[x]N ⊙ [y]N = [x ∗ y]N for all [x]N , [y]N ∈ X/N forms a quotient INK-algebra.

2020 Mathematics Subject Classifications: 06F35, 03G25

Key Words and Phrases: INK-algebras, ideal, normal subset, partitions, quotient INK-algebras

1. Introduction

Algebraic structures play a crucial role in various branches of mathematics, including
logic, topology, and theoretical computer science. Among these, BE-algebras and INK-
algebras have been extensively studied due to their applicability in non-classical logics and
algebraic systems. BE-algebras, introduced as a generalization of BCK/BCI-algebras,
provide a framework for handling abstract algebraic operations with specific axiomatic
structures. Meanwhile, INK-algebras have been recently developed as an extension of
BE-algebras with additional constraints and properties.

The study of INK-algebras has garnered interest due to their unique structural prop-
erties and potential applications in algebraic logic. Understanding congruence relations,
ideals, and quotient structures within these algebras is fundamental in advancing the the-
oretical framework of algebraic systems. The notion of congruences provides insight into
the partitioning of algebraic structures, while ideals and quotient algebras help in forming
new algebraic systems from existing ones.

In 2002, J. Neggers and H. S. Kim ([1]) introduced the notion of B-algebras and some
properties of exponents on them. One such structure is the INK-algebras, introduced in
2017 ([2]) by Kaviyarasu and Indhira, who introduced a new notion called INK-algebras
to generalize and extend the properties of B-algebras. INK-algebras have been widely
studied in relation to their ideal structures, congruences, and algebraic operations. In

DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6501

Email address: maliwan.t@msu.ac.th (M. Phattarachaleekul)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 2 of 8

particular, the concept of quotient INK-algebras arises naturally when considering the
role of congruences and normal subsets in these algebras. A quotient INK-algebra is
constructed by partitioning an INK-algebra using a congruence relation induced by an
ideal. This process mirrors the formation of quotient groups in group theory and quotient
rings in ring theory. The primary objective of forming a quotient INK-algebra is to
simplify the structure while preserving essential algebraic properties. Given an INK-
algebra (X, ∗, 0), an ideal I ofX induces a congruence relation∼

I
onX as x ∼

I
y iff x∗y ∈ I

and y ∗ x ∈ I for any x, y ∈ X. Define an equivalence classes [x]I = {y ∈ X|x ∼
I
y}

and hence the set of all equivalence classes forms a partition of X. Moreover, this paper
aims to explore the construction and properties of quotient INK-algebras (X/N,⊙, [0]I),
including the role of normal subsets defining these structures. We establish key theorems
that demonstrate how the quotient operation preserves the axioms of an INK-algebra.
Furthermore, we provide illustrative examples to highlight the behavior of quotient INK-
algebras and their applications within algebraic systems.

This paper aims to explore fundamental properties of INK-algebras, particularly fo-
cusing on congruence relations and their impact on the structure of INK-algebras. We
establish necessary definitions, investigate the conditions for congruences, and present
results related to the formation of quotient INK-algebras. The organization of the pa-
per is as follows: Section 2 introduces preliminaries, including essential definitions and
fundamental results on INK-algebras. Section 3 discusses congruence relations and their
characterization in INK-algebras. Finally, Section 4 explores the construction of quotient
INK-algebras and their properties.

2. Preliminaries

First, we will review some essential notations and definitions of INK-algebras and
ordinary senses that are needed for this study in this section. Throughout this paper, X
will denote the INK-algebra (X, ∗, 0) unless otherwise specified.

Definition 1. [2] A BE-algebras (X, ∗, 0) is a non-empty set X with a constant 0 and a
binary operation ∗ satisfying the following axioms:

(BE1) x ∗ x = 0,
(BE2) x ∗ 0 = 0,
(BE3) 0 ∗ x = x,
(BE4) x ∗ (y ∗ z) = y ∗ (x ∗ z) for all x, y, z ∈ X.

Definition 2. [3] A INK-algebras (X, ∗, 0) is a non-empty set X with a constant 0 and a
binary operation ∗ satisfies the following axioms for all x, y, z ∈ X.

(INK1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0,

(INK2) ((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y) = 0,

(INK3) x ∗ 0 = x,



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 3 of 8

(INK4) x ∗ y = 0 and y ∗ x = 0 implies x = y.

A non-empty subset S of X is said to be a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S.

Example 1. [4]: Let X = {0, 1, 2, 3} and a binary operations ∗ on X defined by the
following table:

∗ 0 1 2 3

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

Then (X, ∗, 0) is a INK-algebra but not a BE-algebra, since 0 ∗ 1 = 0 ̸= 1 and
0 ∗ 3 = 2 ̸= 3.

Example 2. [5] Let X = {1, a, b, c, d} with a binary operations ∗ on X defined by the
following table:

∗ 1 a b c d

1 1 a b c d
a 1 1 b c d
b 1 a 1 c c
c 1 1 b 1 b
d 1 1 1 1 1

Then (X, ∗, 0) is a BE-algebra but not an INK-algebra, since a ∗ 1 = 1 ̸= a.

Definition 3. [3] Let (X, ∗, 0) be a INK-algebra. A nonempty subset I of X is called an
ideal of X if it satisfies the following conditions:

(i) 0 ∈ I,

(ii) x ∗ y ∈ I and y ∈ I imply that x ∈ I for all x, y ∈ X.

Theorem 1. [2] Let (X, ∗, 0) be an INK-algebra. Then the following conditions hold for
all x, y, z ∈ X;

(i) x ∗ 0 = 0 implies x = 0,

(ii) (x ∗ (x ∗ y) ∗ y) = 0,

(iii) 0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y),



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 4 of 8

(iv) x ∗ y = 0 and y ∗ x = 0 implies x = y,

(v) (x ∗ y) ∗ z = (x ∗ z) ∗ y,

3. Congruence relations and Partitions on INK-algebras

In the study of algebraic structures, congruence relations play a fundamental role in
constructing quotient structures and analyzing their properties. For INK-algebras, the
characterization of congruence relations provides a pathway to understanding the internal
symmetry and decomposability of such algebras via partitions induced by these relations.

Definition 4. Let ∼ be a binary relation on a set X. Then ∼ is called:

(i) reflexive if x ∼ x for all x ∈ X;

(ii) symmetric if x ∼ y implies y ∼ x for all x, y ∈ X;

(iii) transitive if x ∼ y and y ∼ z implies x ∼ z for all x, y, z ∈ X;

(iv) compatible if a ∼ b and x ∼ y implies a ∗ x ∼ b ∗ y for all a, b, x, y ∈ X.

A relation ∼ is said to be an equivalence relation if ∼ is reflexive, symmetric and
transitive. An equivalence relation on X that is compatible with the operation ∗ is called
a congruence on X.

Example 3. [3] Let X = {0, 1, a, b} with a binary operations ∗ on X defined by the
following table:

∗ 0 1 a b

0 0 0 a a
1 1 0 b a
a a a 0 0
b b a 1 0

Then (X, ∗, 0) is a INK-algebra. The set I = {0, a} is an ideal of X and the relation
∼
I
= {(0,0),(1,1),(a,a),(b,b),(0,a),(a,0),(1,b),(b,1)} is a congruence on X.

Theorem 2. Let (X, ∗, 0) be a INK-algebra and I is an ideal of X. Then the relation ∼
I

= {(x, y) ∈ X ×X| x ∗ y ∈ I and y ∗ x ∈ I} is a congruence on X.

Proof. Since x ∗ x = 0 ∈ I for all x ∈ X, that is x ∼
I
x. If x, y ∈ X and x ∼

I
y, clearly

y ∼
I
x. Next if x ∼

I
y and y ∼

I
z, then x ∗ y ∈ I, y ∗ x ∈ I, y ∗ z ∈ I and z ∗ y ∈ I. By



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 5 of 8

(INK2), ((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y) = 0 ∈ I and I is ideal, implies that (x ∗ z) ∗ (y ∗ z) ∈ I,
and hence x ∗ z ∈ I. Similarly, we can prove that ((z ∗ x) ∗ (y ∗ x)) ∗ (z ∗ y) = 0 ∈ I, then
(z ∗ x) ∗ (y ∗ x) ∈ I, thus z ∗ x ∈ I and hence x ∼

I
z. Next to show ∼

I
is compatible, let

w, x, y, z ∈ X such that w ∼
I
y and x ∼

I
z, then w∗y ∈ I, y∗w ∈ I, x∗z ∈ I and z∗x ∈ I.

Since (INK1) satisfying the identity ((w∗x)∗(w∗z))∗(z∗x) = 0 ∈ I, so (w∗x)∗(w∗z) ∈ I
and consider ((w ∗ z) ∗ (w ∗ x)) ∗ (x ∗ z) = 0 ∈ I, we have (w ∗ z) ∗ (w ∗ x) ∈ I, implies that
(w ∗ x) ∼

I
(w ∗ z). Because ((w ∗ z) ∗ (w ∗ y)) ∗ (y ∗ z) = ((w ∗ z) ∗ (y ∗ z)) ∗ (w ∗ y) = 0 ∈ I

by theorem 1(iv) and (INK1), implies that (y ∗ z) ∗ (w ∗ z) ∈ I and hence (w ∗ z) ∼
I
(y ∗ z).

Since ∼
I
is transitive ,so (w ∗ x) ∼

I
(y ∗ z). Therefore ∼

I
is a congruence relation on X.

Lemma 1. Let I be an ideal of a INK-algebra (X, ∗, 0) and ∼
I

is a congruence relation

on X. Then the following conditions hold;

(i) x ∈ [x]I for all x ∈ X,

(ii) x ∼
I
y if and only if [x]I = [y]I for all x, y ∈ X.

Proof. (i) Clearly, for any x ∈ X, x ∗ x = 0 ∈ I that is x ∼
I
x, and hence x ∈ [x]I .

(ii) Assume that x ∼
I
y , then y ∼

I
x. Let a ∈ [x]I , then x ∼

I
a and hence a ∼

I
x, thus

a ∼
I
y, so a ∈ [y]I . Therefore [x]I ⊆ [y]I . Simmilary, to prove [y]I ⊆ [x]I . Conversely,

suppose that [x]I = [y]I , so x ∈ [x]I = [y]I by (i), and hence y ∼
I
x, thus x ∼

I
y.

Theorem 3. Let (X, ∗, 0) be a INK-algebra and I is an ideal of X. Then the set P =
{[x]I |x ∈ X} forms a partition of X.

Proof. Let [x]I , [y]I ∈ P such that [x]I ̸= [y]I . Suppose that [x]I ∩ [y]I ̸= ∅, there
is a b ∈ [x]I ∩ [y]I , then x ∼

I
b, y ∼

I
b and hence b ∼

I
x, so we have y ∼

I
x, implies

that [x]I = [y]I that’s a contradiction, thus [x]I ∩ [y]I = ∅. Next to show
⋃
x∈X

[x]I = X.

Clearly for any x ∈ X, [x]I ⊆ X and hence
⋃
x∈X

[x]I ⊆ X. And for any x ∈ X,

x ∈ [x]I ⊆
⋃
x∈X

[x]I . Thus P = {[x]I |x ∈ X} is a partition of X.

Example 4. From example 3, we see that I = {0, a} is an ideal of X and relation
∼
I
= {(0, 0), (1, 1), (a, a), (b, b), (0, a), (a, 0), (1, b), (b, 1)}, is a congruence.

Consider [0]I = {0, a} = [a]I and [1]I = {1, b} = [b]I , and hence the set
P = {[0]I , [1]I} is a partition of X.



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 6 of 8

4. Quotient INK-algebras

The notion of quotient structures is fundamental in universal algebra, serving as a tool
for classifying and analyzing algebraic systems via congruence relations. In this section,
we introduce the concept of quotient INK-algebras, establish their basic properties, and
explore structural implications arising from the induced congruences.

Definition 5. Let (X, ∗, 0) be a INK-algebra. A nonempty subset N of X is said to be
normal of X if (x ∗ a) ∗ (y ∗ b) ∈ N for any x ∗ y, a ∗ b ∈ N.

Theorem 4. Every normal subset of an INK-algebra is a subalgebra.

Proof. Assume that N is a normal subset of a INK-algebra (X, ∗, 0) and x, y ∈ N . By
(INK3), x ∗ 0 = x ∈ N and y ∗ 0 = y ∈ N , hence x ∗ y = (x ∗ y) ∗ 0 = (x ∗ y) ∗ (0 ∗ 0) ∈ N .

The next example is shown that the converse of theorem 4 is not true.

Example 5. Consider the example 3,
I = {0, a} is a subalgebra of X and so is ideal but not normal, because 1 ∗ b = a ∈ I

and a ∗ b = 0 ∈ I, while (1 ∗ a) ∗ (b ∗ b) = b ∗ 0 = b /∈ I.
N = {0, 1} is a normal subset of X and so is a subalgebra.

Lemma 2. If S is a subalgebra of INK-algebra (X, ∗, 0), then 0 ∈ S.

Proof. Let x ∈ S, By INK-3, x ∗ 0 = x ∈ S and hence by INK-2,
0 = ((0 ∗ x) ∗ (0 ∗ x) ∗ (x ∗ x)) ∈ S.

Lemma 3. Let (X, ∗, 0) be a INK-algebra and I is an ideal of X such that 0 ∗ a ∈ I for
all a ∈ I, then I is a subalgebra of X.

Proof. Let x, y ∈ I, then 0 ∗ y = (x ∗ x) ∗ y = (x ∗ y) ∗ x ∈ I implies that x ∗ y ∈ I.

Definition 6. Let (X, ∗, 0) be an INK-algebra and let I be a subset of X with an associated
congruence relation ∼

I
. Define the set X/I = {[x]I | x ∈ X}, where [x]I = {y ∈ X|x ∼

I
y},

and a binary operation ⊙ on X/I defined by [x]I ⊙ [y]I = [x ∗ y]I for all [x]I , [y]I ∈ X/I.
We say that (X/N,⊙, [0]N ) is a quotient INK-algebra if it satisfies Definition 2.

Theorem 5. Let (X, ∗, 0) be a INK-algebra and N is a normal subset of X, then (X/N,⊙, [0]I)
is a quotient INK-algebra.

Proof. Clearly the operation ⊙ is well-defined. That is, for any [a]N , [b]N , [x]N , [y]N ∈
X/N , if ([a]N , [b]N ) = ([x]N , [y]N ), then [a]N = [x]N and [b]N = [y]N , which implies that



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 7 of 8

a ∼
N

x and b ∼
N

y, thus (a ∗ b) ∼
N

(x ∗ y), and hence [a ∗ b]N = [x ∗ y]N , implies that

[a]N ⊙ [b]N = [x]N ⊙ [y]N , and satisfies the following;

1) [([x]N ⊙ [y]N )⊙ ([x]N ⊙ [z]N )]⊙ ([z]N ⊙ [y]N ) = [([x ∗ y]N )⊙ ([x ∗ z]N )]⊙ ([z ∗ y]N )
= [(x ∗ y) ∗ (x ∗ z)]N ⊙ [z ∗ y]N
= [((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)]N
= [0]N ,

2) [([x]N ⊙ [z]N )⊙ ([y]N ⊙ [z]N )]⊙ ([z]N ⊙ [y]N ) = [([x ∗ y]N )⊙ ([x ∗ z]N )]⊙ ([x ∗ y]N )
= [(x ∗ z) ∗ (y ∗ z)]N ⊙ [x ∗ y]N
= [((x ∗ z) ∗ (y ∗ z))N ∗ (x ∗ y)]N
= [0]N ,

3) [x]N ⊙ [0]N = [x ∗ 0]N = [x]N
4) Assume that [x]N ⊙ [y]N = [0]N , then [x∗y]N = [0]N , implies that x∗y = 0, so x = y.

Therefore (X/N,⊙, [0]N ) is a quotient INK-algebra.

5. Conclusion

In this paper, we investigated the structure of congruence relations induced by ideals
in INK-algebras. We began by reviewing the foundational definitions and properties of
INK-algebras, a class of non-classical algebras characterized by specific axioms on binary
operation axioms. Notably, we established the necessary and sufficient conditions under
which a binary relation derived from an ideal constitutes a congruence relation. The main
results demonstrated that for any ideal I of an INK-algebra (X, ∗, 0), the relation ∼

I
, de-

fined by x ∼
I
y if and only if x ∗ y ∈ I and y ∗ x ∈ I, is a congruence relation on X.

Furthermore, it was shown that the equivalence classes under this congruence relation
form a partition of the underlying set X, enabling a quotient-like decomposition of the
algebraic structure. In addition, we introduced the notion of normal subsets and demon-
strated how they play a central role in defining well-structured quotient INK-algebras.
Our construction ensures that the quotient algebra X/N inherits the fundamental axioms
of INK-algebras. The provided theorems and examples confirm that the operations on the
quotient set are well-defined and consistent with the original algebra. These results not
only extend the theory of INK-algebras but also pave the way for further exploration of
categorical, topological, or fuzzy extensions of quotient structures in algebraic logic and
non-classical algebraic systems.

Acknowledgements

This research project was financially supported by Thailand Science Research and In-
novation (TSRI). The author would also like to express sincere gratitude to Mahasarakham
University for its continuous academic and institutional support throughout the develop-
ment of this work.



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 18 (4) (2025), 6501 8 of 8

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