EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5783
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Sheffer Stroke BN-algebras and Connected Topics

Sri Gemawati1,∗, Mashadi1, Kartini1, Musraini1, Elsi Fitria1

1 Department of Mathematics, Faculty of Mathematics and Natural Science,
Universitas Riau, Pekanbaru, Indonesia

Abstract. This article introduces the concept of a Sheffer stroke BN-algebra by applying the
Sheffer stroke operator | to the BN-algebra axioms and aligning it with the axioms of the Sheffer
stroke groupoid. From this definition, properties of Sheffer stroke BN-algebras are derived, focusing
on the relationship between the axioms and the properties of the special element 0. Furthermore,
the notions of Sheffer stroke BN-subalgebras, BN-ideals, and BN-homomorphisms are defined,
along with normal subsets of Sheffer stroke BN-algebras, and the relationships between these
concepts are explored. It is shown that every normal subset in a Sheffer stroke BN-algebra is a
Sheffer stroke BN-subalgebra, but the converse is not necessarily true. This implies that every
normal BN-ideal in a Sheffer stroke BN-algebra is also a Sheffer stroke BN-subalgebra. Finally,
the properties of the kernel of the Sheffer stroke BN-homomorphism are investigated.

2020 Mathematics Subject Classifications: 03G25, 03G10

Key Words and Phrases: BN-algebra, Sheffer stroke, Sheffer stroke BN-algebra, BN-ideal,
BN-subalgebra

1. Introduction

The Sheffer stroke operator is equivalent to the NAND logic gate, a fundamental
component in digital electronics. A NAND gate is a type of integrated logic gate with
two inputs and one output, essentially functioning as a combination of a NOT gate and
an AND gate. NAND gates work closely with other logic gates to regulate the flow of
information and instructions in computers, the behavior of devices such as electric motors
and water pumps in industrial control systems, and access to buildings or spaces in security
systems. In addition to having a wide range of practical applications, the Sheffer stroke
operator has inspired many theoretical developments in mathematics, including in the
context of algebra.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5783

Email addresses: sri.gemawati@lecturer.unri.ac.id (S. Gemawati),
mashadi@lecturer.unri.ac.id (Mashadi), kartini@lecturer.unri.ac.id (Kartini),
Musraini@lecturer.unri.ac.id (Musraini), elsifitria823@gmail.com (E. Fitria)

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



S. Gemawati et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5783 2 of 12

An operator | is called a Sheffer stroke operator in a groupoid T if it satisfies
the following four conditions: (S1) a|b = b|a, (S2) (a|a)|(a|b) = a, (S3) a|[(b|c)|(b|c)] =
[(a|b)|(a|b)]|c, (S4) [a|((a|a)|(b|b))]|[a|((a|a)|(b|b))] = a, for all a, b, c ∈ T [2].

The functions and axioms of Boolean algebra can be expressed using the Sheffer stroke
operator, as can those of Hilbert algebras [15]. The Sheffer stroke operator has also been
defined for R0-algebras [10]. An R0-algebra contains a unary operation and three binary
operations. However, after defining the Sheffer stroke operator in the R0-algebra, the
resulting set contains only two binary operations with the special properties of the Sheffer
stroke operator. Further research has explored the Sheffer stroke and its properties in
additional algebraic systems [1, 12–14, 16, 17].

The extensive study of BCI-algebra and BCK-algebra influenced the development
of BN-algebras. A non-empty set A equipped with a constant 0 and a binary operation ∗
is called a BN-algebra if it meets the following axioms: (BN1) a ∗ a = 0, (BN2) a ∗ 0 = a,
(BN3) (a∗ b)∗ c = (0∗ c)∗ (b∗a) for all a, b, c ∈ A [11]. Building on a work that researched
ideals in BN-algebra [3], various further studies have developed the ring-theoretic concepts
of ideals and normality in BN-algebra and related algebras, as seen in [4, 5, 9].

Motivated by the close relationship between ideals and normal subsets in BN-algebra,
as discussed in previous research, and the author’s experience in abstract algebra, espe-
cially work concerning the concept of algebraic structures, such as in [6–8], in this study,
the concept of a Sheffer stroke BN-algebra is defined and its properties are determined.
Then, the notions of Sheffer stroke BN-subalgebras, BN-ideals, and BN-homomorphisms
and their kernels, as well as normal subsets of Sheffer stroke BN-algebras, are introduced,
and the relationships between these concepts are investigated.

2. Preliminaries

This section presents several definitions and properties necessary for the construction
of Sheffer stroke BN-algebras and the other concepts explored in this study.

Definition 1. [11] A BN-algebra is a non-empty set A equipped with a constant 0 and a
binary operation ∗ that meets the following axioms:

(BN1) a ∗ a = 0 for all a ∈ A,

(BN2) a ∗ 0 = 0 for all a ∈ A,

(BN3) (a ∗ b) ∗ c = (0 ∗ c) ∗ (b ∗ a) for all a, b, c ∈ A.

Theorem 1. [11] Suppose (A; ∗, 0) be a BN-algebra, then for all a, b, c ∈ A:

(i) 0 ∗ (0 ∗ a) = a,

(ii) b ∗ a = (0 ∗ a) ∗ (0 ∗ b),

(iii) (0 ∗ a) ∗ b = (0 ∗ b) ∗ a,

(iv) if a ∗ b = 0, then b ∗ a = 0,



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(v) if 0 ∗ a = 0 ∗ b, then a = b,

(vi) (a ∗ c) ∗ (b ∗ c) = (c ∗ b) ∗ (c ∗ a).

Definition 2. [11] A BN-algebra (A; ∗, 0) that satisfies (D) (a ∗ b) ∗ c = a ∗ (c ∗ b) for all
a, b, c ∈ A is called a BN-algebra with condition (D).

Theorem 2. [11] Suppose (A; ∗, 0) is a BN-algebra with condition (D). Then, the follow-
ing properties are satisfied for all a, b, c ∈ A:

(i) 0 ∗ a = a,

(ii) a ∗ b = b ∗ a.

Definition 3. [3] Suppose (A; ∗, 0) is a BN-algebra. A non-empty subset I of A is defined
as an ideal in A if it meets the following conditions:

(i) 0 ∈ I, and

(ii) for all a, b ∈ A, if b ∈ I and a ∗ b ∈ I, implies a ∈ I.

Suppose (A; ∗, 0) is a BN-algebra. A non-empty subset S is considered a subalgebra of
A if it satisfies a ∗ b ∈ S for all a, b ∈ S. A non-empty subset N of A is termed a normal if
it satisfies (x ∗ a) ∗ (y ∗ b) ∈ N for all x ∗ y, a ∗ b ∈ N , where a, b, x, y ∈ A. In a BN-algebra
(A; ∗, 0), the operation ∧ is defined as a ∧ b = b ∗ (b ∗ a) for all a, b ∈ A.

Definition 4. [2] The operator denoted by | is called a Sheffer stroke on a groupoid T if
the following four conditions are met: for all a, b, c ∈ T

(S1) a|b = b|a,

(S2) (a|a)|(a|b) = a,

(S3) a|((b|c)|(b|c)) = ((a|b)|(a|b))|c,

(S4) (a|((a|a)|(b|b)))|(a|((a|a)|(b|b))) = a.

3. Sheffer stroke BN-algebras

In this section, the Sheffer stroke BN-algebra is defined as an algebra of type (2,0) with
the Sheffer stroke operation | and the constant element 0. In this algebra, we integrate the
Sheffer stroke operation with the BN-algebra axioms by examining the interrelationships
among these axioms. Two main axioms are defined for Sheffer stroke BN-algebra, (SBN1)
and (SBN2). Then, its properties are determined.

Definition 5. A Sheffer stroke BN-algebra is an algebra (PSBN ; |, 0) of type (2, 0) con-
taining the constant 0, where | is the Sheffer stroke operation in PSBN and the following
axioms are satisfied for all a, b, c ∈ PSBN :



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(SBN1) (a|(0|0))|(a|(0|0)) = a,

(SBN2) (a|(b|b))|((0|c)|(0|c)) = [(0|(0|(c|c))|(0|(0|(c|c))]|(b|(a|a)).

Lemma 1. If (PSBN ; |, 0) is a Sheffer stroke BN-algebra, then the axioms (SBN1) and
(SBN2) are independent.

Proof. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. We give examples of groupoids
(P ; |, 0) and (Q; |, 0) to show that each axiom can be satisfied when the other is not.

(i) Independence of (SBN1):
We provide an example in which (SBN1) does not hold but (SBN2) does. Consider
the groupoid (P ; |, 0), where P = {0, 1}, defined as in Table 1.

Table 1: Composition table for (P ; |, 0)

| 0 1

0 0 1
1 0 0

It can be demonstrated that (P ; |, 0) fulfills (SBN2), but it fails to satisfy (SBN1)
when a = 1, since (1|(0|0))|(1|(0|0)) = (1|0)|(1|0) = 0|0 = 0 ̸= 1.

(ii) Independence of (SBN2):
We present an example in which (SBN2) does not apply while (SBN1) remains valid.
Take the groupoid (Q; |, 0), where Q = {0, 1}, defined as in Table 2.

Table 2: Composition table for (Q; |, 0)

| 0 1

0 0 0
1 1 1

It can be shown that (P ; |, 0) satisfies (SBN2), but it does not fulfill (SBN1) when
a = 1, b = 1, and c = 0 since the left-hand side (1|(1|1))|((0|0)|(0|0)) = (1|1)|(0|0) =
1|0 = 1, whereas the right-hand side
[(0|(0|(0|0))|(0|(0|(0|0))]|(1|(1|1)) = (0|(0|0))|(0|(0|0))])|(1|1) = 0|1 = 0.

Next, an example of an algebra that satisfies the Sheffer stroke BN-algebra property is
presented, showing the structure of the algebra with elements {0, p, q, 1} that satisfy the
specified axioms.

Example 1. Consider (PSBN ; |, 0), an algebra with the Sheffer stroke operation and
the constant 0, defined as in Table 3.



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Table 3: Composition table for (PSBN ; |, 0)

| 0 p q 1

0 1 1 1 1
p 1 q q q
q 1 q p p
1 1 q p 0

It can be shown that (PSBN ; |, 0) is a Sheffer stroke BN-algebra.

By applying the Sheffer stroke operation to the elements in the algebra (PSBN ; |, 0),
we can discover relationships that apply to all elements in PSBN . These relationships
show how the elements interact through the operation | and constant 0, for example, how
this operation relates to the identity element or changes when applied to the same or
different elements. In the following theorem, some basic properties are established to help
understand the structure of the Sheffer stroke BN-algebra and provide a solid basis for
further analysis of the relationships between elements in this algebra.

Theorem 3. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. Then, the following proper-
ties are satisfied for all a, b ∈ PSBN :

(i) a|a = a|(0|0),

(ii) [(a|b)|(a|b)]|b = a|b,

(iii) (a|(a|a))|(a|a) = a,

(iv) 0|(0|(a|a)) = 0|(a|a),

(v) 0|(a|(b|b)) = 0|(b|(a|a)),

(vi) 0|(a|a) = 0|a.

Proof. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra.

(i) By substituting b := a and c := 0|0 into (S3), for all a ∈ PSBN , we have

a|[(a|(0|0))|(a|(0|0))] = ((a|a)|(a|a))|(0|0).

Then, by applying axioms (SBN1) on the left-hand side and (S2) on the right-hand
side, we obtain a|a = a|(0|0).

(ii) By substituting c := b into (S3), we have [(a|b)|(a|b)]|b = a|[(b|b)|(b|b)]. Then, by
using (S2) we obtain [(a|b)|(a|b)]|b = a|b for all a, b ∈ PSBN .

(iii) By using axioms (S1) and (S2), we find that

(a|(a|a))|(a|a) = (a|a)|(a|(a|a)) = a,

for all a ∈ PSBN .



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(iv) By substituting c := a and b := 0 into (SBN2), and by using (i), (ii), (S1), and (S2),
for all a ∈ PSBN , we obtain

(a|(0|0))|((0|a)|(0|a)) = [(0|(0|(a|a))|(0|(0|(a|a))]|(0|(a|a)),
(a|(0|0))|((0|a)|(0|a)) = 0|(0|(a|a)),

(a|a)|((0|a)|(0|0)) = 0|(0|(a|a)),
(a|a)|((0|0)|(0|a)) = 0|(0|(a|a)),

(a|a)|0 = 0|(0|(a|a)),
0|(a|a) = 0|(0|(a|a)).

(v) By substituting c := 0 into (SBN2), and by using (S2), (iv), (SBN1), and (S1), for
all a, b ∈ PSBN , we have

(a|(b|b))|((0|0)|(0|0)) = [(0|(0|(0|0))|(0|(0|(0|0))]|(b|(a|a)),
(a|(b|b))|0 = [(0|(0|0))|(0|(0|0))]|(b|(a|a)),
(a|(b|b))|0 = 0|(b|(a|a)),
0|(a|(b|b)) = 0|(b|(a|a)).

(vi) By substituting b := a|a into (v), we have

0|(a|((a|a)|(a|a))) = 0|((a|a)|(a|a)),

for all a ∈ PSBN . Then, by using (S2) we obtain 0|(a|a) = 0|a for all a ∈ PSBN .

In a Sheffer stroke BN-algebra, we will observe how certain conditions involving the
Sheffer stroke operation and constant 0 can yield important conclusions about the rela-
tionships between elements. Theorem 4 describes a special case in which, if element b has
a certain association with element a through this operation, then the result of the Sheffer
stroke operation between 0 and b will be the same as the result of the operation between
0 and a.

Theorem 4. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. If a = b|(a|a) for all a, b ∈
PSBN , then 0|b = 0|a.

Proof. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. Since a := b|(a|a) for all
a, b ∈ PSBN , by using (v), (S1), and (S2), we have

0|(b|(a|a)) = 0|(a|(b|b)),
0|a = 0|((b|(a|a))|(b|b)),
0|a = 0|((b|b)|(b|(a|a))),
0|a = 0|b.



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We next define some additional concepts, but first let us look at the role of subsets
in the study of Sheffer stroke BN-algebra, namely, Sheffer stroke BN-subalgebras, normal
subsets, and Sheffer stroke BN-ideals, have an important function in maintaining the
regularity of this algebra. A Sheffer stroke BN-subalgebra is a part of the algebra that is
closed under the Sheffer stroke operation, meaning that the result of each such operation
will be in the subset itself. A normal subset is a special subset that makes an algebraic
structure more symmetrical, preserving order among certain elements. A Sheffer stroke
BN-ideal is a subset that contains the constant 0 and meets certain rules so that it remains
consistent under Sheffer stroke operations. These three types of subsets help understand
patterns and stability in Sheffer stroke BN-algebra.

Definition 6. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra and S a non-empty sub-
set of PSBN . The set S is called a Sheffer stroke BN-subalgebra of PSBN if it satisfies
(a|(b|b))|(a|(b|b)) ∈ S for all a, b ∈ S.

Definition 7. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra and N a non-empty subset
of PSBN . The set N is called a normal subset of the Sheffer stroke BN-algebra PSBN if
it satisfies

(((a|(p|p))|(a|(p|p)))|(b|(q|q)))|(((a|(p|p))|(a|(p|p)))|(b|(q|q))) ∈ N ,

for any (a|(b|b))|(a|(b|b)), (p|(q|q))|(p|(q|q)) ∈ N .

Definition 8. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra and I a non-empty subset
of PSBN . The set I is called a Sheffer stroke BN-ideal of PSBN if it satisfies the following
conditions:

(i) 0 ∈ I,

(ii) b ∈ I and (a|(b|b))|(a|(b|b)) ∈ I imply a ∈ I for all a, b ∈ PSBN .

Before deriving results about Sheffer stroke BN-ideals, Sheffer stroke BN-subalgebras,
and normal subsets in Sheffer stroke BN-algebras, we will look at a concrete example.
This example aims to show how the elements in the algebra fit into these categories of
subset categories according to the properties of the Sheffer stroke operation.

Example 2. Consider the same Sheffer stroke BN-algebra (PSBN ; |, 0) defined in Example
1. The set of Sheffer stroke BN-ideals of PSBN consists of {0}, {0, p}, {0, q}, and PSBN .
The subsets {0}, {0, p}, {0, 1}, {0, p, q}, and PSBN are Sheffer stroke BN-subalgebras of
PSBN . The normal subsets in PSBN are {0}, {0, p}, and PSBN .

In the following theorem, we will see how the existence of a Sheffer stroke BN-ideal
can affect the membership of other elements. The theorem shows that if an element is in
the Sheffer stroke BN-ideal and fulfills certain relationships, then there are other elements
that are also included in the Sheffer stroke BN-ideal.

Theorem 5. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra and let I be a Sheffer stroke
BN-ideal of PSBN . If b ∈ I and (a|(b|b))|(a|(b|b)) = 0, then a ∈ I for all a, b ∈ PSBN .



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Proof. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. Since I is a Sheffer stroke
BN-ideal of PSBN , we have 0 ∈ I. Given that

b ∈ I, (a|(b|b))|(a|(b|b)) ∈ I,

it follows that a ∈ I. Moreover, since (a|(b|b))|(a|(b|b)) = 0 ∈ I, we conclude that a ∈ I
for all a, b ∈ PSBN .

The following theorem addresses the relationship between Sheffer stroke BN-subalgebras,
normal subsets, and Sheffer stroke BN-ideals.

Theorem 6. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. If N is a normal subset of
PSBN , then N is a Sheffer stroke BN-subalgebra of PSBN .

Proof. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra and a, b ∈ N . By using axiom
(SBN1), we obtain (a|(0|0))|(a|(0|0)) ∈ N and (b|(0|0))|(b|(0|0)) ∈ N . Then, by using
axiom (SBN1), Theorem 3 (iv), and the fact that N is a normal subset of PSBN , we
obtain

(a|(b|b))|(a|(b|b)) = (((a|(b|b))|(a|(b|b)))|(0|0))|(((a|(b|b))|(a|(b|b)))|(0|0)),

such that,

(((a|(b|b))|(a|(b|b)))|(0|(0|0)))|(((a|(b|b))|(a|(b|b)))|(0|(0|0))) ∈ N .

Thus, it has been proven that N is a BN-subalgebra of PSBN .

The converse of Theorem 6 does not hold in general.
In Sheffer stroke BN-algebras, it is important to understand that normal sets always

form BN-subalgebras. Theorem 6 shows that if a subset is normal, then its elements will
remain connected via the Sheffer stroke operation, satisfying the condition for it to be
a BN-subalgebra. However, not all BN-subalgebras are normal, as shown in Example 2.
The subset N = {0, 1} is a Sheffer stroke BN-subalgebra of PSBN , but it is not a normal
subset of PSBN , since

(1|(1|1))|1|(1|1)) = (1|0)|(1|0) = 1|1 = 0 ∈ I,

and
(0|(p|p))|(0|(p|p)) = (0|q)|(0|q) = 1|1 = 0 ∈ I.

However,

((1|(0|0))|(1|(0|0)))|(1|(p|p)) = ((1|1)|(1|1))|(1|q) = (0|0)|p = 1|p = q,

and hence

((1|(0|0))|(1|(0|0)))|(1|(p|p))|((1|(0|0))|(1|(0|0)))|(1|(p|p)) = q|q = p ∈ N .



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Corollary 1. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. If I is a Sheffer stroke
normal BN-ideal of PSBN , then I is a Sheffer stroke BN-subalgebra of PSBN .

Proof. Let (PSBN ; |, 0) be a Sheffer stroke BN-algebra. Since I is both a BN-ideal and
normal in PSBN , it follows from Theorem 6 that I is a Sheffer stroke BN-subalgebra of
PSBN .

The converse of Corollary 1 does not hold in general.
Corollary 1 reinforces the idea that in a Sheffer stroke BN-algebra, sets that are both

BN-ideal and normal necessarily form BN-subalgebras. This underlines that the normality
and ideality properties in this algebra have a direct effect on the structure of its BN-
subalgebra. However, not all BN-subalgebras are both BN-ideal and normal. Example 2
shows that although the sets {0, 1} and {0, p, q} are BN-subalgebras, they do not qualify as
BN-ideals or normal subsets in the Sheffer stroke BN-algebra. This emphasizes that normal
subsets and BN-ideals have additional structure compared with ordinary BN-subalgebras.

For a Sheffer stroke BN-algebra, we will define a mapping called a Sheffer stroke BN-
homomorphism. This mapping connects two Sheffer stroke BN-algebras by ensuring that
the Sheffer stroke operations in one set are applied consistently in the other set. The
following definition explains the conditions that must be met by this mapping.

Definition 9. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras.
A map φ : PSBN → QSBN is called a Sheffer stroke BN-homomorphism if it satisfies
φ(a|P b) = φ(a)|Qφ(b) for all a, b ∈ PSBN . We assume φ(0) = 0.

In Sheffer stroke BN algebras, homomorphisms are mappings that keep algebraic op-
erations consistent. For example, if we have two Sheffer stroke BN-algebras, (PSBN ; |P , 0)
and (QSBN ; |Q, 0), and a mapping φ : PSBN → QSBN that satisfies the properties of a
Sheffer stroke BN-homomorphism, we can examine the kernel of this mapping. The ker-
nel is the set of elements that map to the zero elements in the goal algebra. We define
Kerφ = {a ∈ PSBN : φ(a) = 0}.

The following theorem will show that the kernel of a Sheffer stroke BN-homomorphism
always forms a BN-subalgebra in the source algebra. In other words, the elements that
map to zero form the corresponding algebraic structure in the source algebra.

Theorem 7. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras. If
φ : PSBN → QSBN is a Sheffer stroke BN-homomorphism, then Kerφ is a Sheffer stroke
BN-subalgebra of PSBN .

Proof. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras and
φ : PSBN → QSBN a Sheffer stroke BN-homomorphism. Let a, b ∈ Kerφ, we have
φ(a) = 0 and φ(b) = 0. By using (SBN1), we obtain

φ((a|P (b|P b))|P (a|P (b|P b))) = φ(a|P (b|P b))|Qφ(a|P (b|P b)),
= (φ(a)|Q(φ(b)|Q(b)))|Q(φ(a)|Q(φ(b)|Qφ(b))),
= (0|Q(0|Q0))|Q(0|Q(0|Q0)),



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φ((a|P (b|P b))|P (a|P (b|P b))) = 0.

Hence, we have ((a|P (b|P b))|P (a|P (b|P b))) ∈ Kerφ. This shows that Kerφ is a Sheffer
stroke BN-subalgebra of PSBN .

In Theorem 8, the kernel of the Sheffer stroke BN-homomorphism φ is the set of ele-
ments in PSBN that map to zero elements in QSBN . The property proven by Theorem 8
is that the kernel of a Sheffer stroke BN-homomorphism not only forms a Sheffer stroke
BN-subalgebra of PSBN but also forms a BN-ideal. It provides an understanding of the
algebraic structures preserved and changed by homomorphisms, as well as the impor-
tance of kernels in preserving ideal properties and algebraic structures in the context of
homomorphisms.

Theorem 8. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras. If
φ : PSBN → QSBN is a Sheffer stroke BN-homomorphism, then Kerφ is a Sheffer stroke
BN-ideal of PSBN .

Proof. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras and φ :
PSBN → QSBN a Sheffer stroke BN-homomorphism. Since φ(0) = 0, we have 0 ∈ Kerφ.
Let b ∈ Kerφ, we have φ(b) = 0. Let (a|(b|b))|(a|(b|b)) ∈ Kerφ, then by using (SBN1),
we obtain

φ((a|P (b|P b))|P (a|P (b|P b))) = 0,

= (φ(a)|Q(φ(b)|Qφ(b)))|Q(φ(a)|Q(φ(b)|Qφ(b))),
= (φ(a)|Q(0|Q0))|Q(φ(a)|Q(0|Q0)),

φ(a) = 0.

We have a ∈ Kerφ. Hence, Kerφ is a Sheffer stroke BN-ideal of PSBN .

The following two theorems further characterize the properties of the kernel of a Sheffer
stroke BN-homomorphism Kerφ. Theorem 9 shows that if an element a is in Kerφ, then
the result of a particular operation involving a and other elements in PSBN remains within
Kerφ. The same applies to pairs of elements a and b that are both in Kerφ, as shown
by Theorem 10. These two theorems provide further insight into the relationship between
kernels and operations in a Sheffer stroke BN-algebra, as well as how elements in the kernel
behave under certain operations in the algebraic structure.

Theorem 9. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras and
φ : PSBN → QSBN a Sheffer stroke BN-homomorphism. If a ∈ Kerφ, then (0|P (a|Pa))|P (a|P b) ∈
Kerφ.

Proof. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras and
φ : PSBN → QSBN a Sheffer stroke BN-homomorphism. Let a ∈ Kerφ, we have φ(a) = 0.
By using Theorem 3 (vi) and (S2), we obtain

φ((0|P (a|Pa))|P (a|P b)) = φ(0|P (a|Pa))|Q(φ(a|P b),



S. Gemawati et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5783 11 of 12

= (φ(0)|Q(φ(a)|Qφ(a)))|Q((φ(a)|Q(φ(b)),
= (0|Q(0|Q0))|Q((0|Q(φ(b)),
= (0|Q0)|Q((0|Q(φ(b)),

φ((0|P (a|Pa))|P (a|P b)) = 0.

Therefore, we have ((0|P (a|Pa))|P (a|P b)) ∈ Kerφ.

Theorem 10. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras
and φ : PSBN → QSBN a Sheffer stroke BN-homomorphism. If a, b ∈ Kerφ, then
(b|P (b|P b))|P (a|Pa) ∈ Kerφ.

Proof. Let (PSBN ; |P , 0) and (QSBN ; |Q, 0) be two Sheffer stroke BN-algebras and
φ : PSBN → QSBN a Sheffer stroke BN-homomorphism. Let a, b ∈ Kerφ, we have
φ(a) = 0 and φ(b) = 0. By using Theorem 3 (vi) and (S2), we obtain

φ((b|P (b|P b))|P (a|Pa)) = φ(b|P (b|P b))|Qφ(a|Pa),
= (φ(b)|Q(φ(b)|Qφ(b)))|Q((φ(a)|Q(φ(a)),
= (0|Q(0|Q0))|Q((0|Q(0),

φ((b|P (b|P b))|P (a|Pa)) = 0.

Hence, it is proven that ((b|P (b|P b))|P (a|Pa)) ∈ Kerφ.

4. Conclusions

In this paper, the concept of a Sheffer stroke BN-algebra is introduced, and its key
properties are thoroughly examined. These include the independence of the axioms and
the special properties related to the element 0. Furthermore, the paper defines the notions
of Sheffer stroke BN-subalgebras, BN-ideals, and BN-homomorphisms, along with normal
subsets of Sheffer stroke BN-algebras. The interrelationships between these concepts are
explored, providing a clear framework for understanding their connections. Finally, the pa-
per outlines potential avenues for future research, particularly focusing on the exploration
of the filter and derivation concepts within the context of Sheffer stroke BN-algebras.

Acknowledgements

We would like to thank you to The Directorate General of Higher Education, Research,
and Technology (DGHERT) of the Ministry of Education, Culture, Research, and Tech-
nology (MOECRT) of the Republic of Indonesia for the funding support provided through
the 2024 research scheme with contract number 083/E5/PG.02.00.PL/2024, so that this
research can be carried out well.



S. Gemawati et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5783 12 of 12

References

[1] S. S. Ahn, H. S. Kim, S. Song, and Y. B. Jun. Dokdo filters and deductive systems of
sheffer stroke hilbert algebras. European Journal of Pure and Applied Mathematics,
16:1862–1877, 2023.

[2] I. Chajda. Sheffer operation in ortholattices. Acta Univ. Palacki. Olomuc., Fac. rer.
nat., Mathematica, 44:19–23, 2005.

[3] G. Dymek and A. Walendziak. (fuzzy) ideals of bn-algebras. Scientific World Journal,
pages 1–9, 2015.

[4] E. Fitria, S. Gemawati, and Kartini. Prime ideals in b-algebras. International Journal
of Algebra, 11:301–309, 2017.

[5] S. Gemawati, E. Fitria, A. Hadi, and M. Musraini. Complete ideal and n-ideal of
bn-algebras. International Journal of Mathematics Trends and Technology, 66:52–59,
2020.

[6] S. Gemawati, Mashadi, M. M, and E. Fitria. fq-derivation of bp-algebras. Journal of
the Indonesian Mathematical Society, pages 235–244, 2023.

[7] S. Gemawati, M. Musraini, A. Hadi, L. Zakaria, and E. Fitria. On r-ideals and
m-k-ideals in bn-algebras. Axioms, 11, 2022.

[8] S. Gemawati, A. Sirait, M. M, and E. Fitria. fq-derivations of bn1-algebras. Interna-
tional Journal of Mathematics Trends and Technology, 67:1–13, 2021.

[9] T. Gerima, Y. Endris, and G. Fasil. Ideals and filters on implication algebras. Ad-
vances in Mathematics: Scientific Journal, 10:1167–1174, 2020.

[10] T. Katican, T. Oner, and A. B. Saeid. Sheffer stroke r0-algebras. Algebraic Structures
and their Applications, 10:65–85, 2023.

[11] C. B. Kim and H. S. Kim. On bn-algebras. Kyungpook Mathematical Journal, 53:175–
184, 2013.

[12] A. Molkhasi. Representations of sheffer stroke algebras and visser algebras. Soft
Computing, 25:8533–8538, 2021.

[13] T. Oner, T. Kalkan, and A. Borumand Saeid. Class of sheffer stroke bck-algebras.
Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 30:247–269,
2022.

[14] T. Oner, T. Katican, A. B. Saeid, and M. Terziler. Filters of strong sheffer stroke non-
associative mv-algebras. Analele Universitatii Ovidius Constanta-Seria Matematica,
29:143–164, 2021.

[15] T. Oner, T. Katican, and A. Borumand Saeid. Relation between sheffer stroke
and hilbert algebras. Categories and General Algebraic Structures with Application,
14:245–268, 2021.

[16] A. Borumand Saeid, T. Katican, and T. Oner. On sheffer stroke up-algebras. Dis-
cussiones Mathematicae - General Algebra and Applications, 41:381–394, 2021.

[17] I. Senturk. A view on state operators in sheffer stroke basic algebras. Soft Computing,
25:11471–11484, 2021.