EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 999-1014
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Direct product of infinite family of B-Algebras

Chatsuda Chanmanee1, Ronnason Chinram2, Rukchart Prasertpong3,
Pongpun Julatha4, Aiyared Iampan1,∗

1 Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics,
School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand
2 Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat
Yai, Songkhla 90110, Thailand
3 Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan
Rajabhat University, Nakhon Sawan 60000, Thailand
4 Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat
University, Phitsanulok 65000, Thailand

Abstract. The concept of the direct product of finite family of B -algebras is introduced by Ling-
cong and Endam [J. A. V. Lingcong and J. C. Endam, Direct product of B -algebras, Int. J. Algebra,
10(1):33-40, 2016.]. In this paper, we introduce the concept of the direct product of infinite family
of B -algebras, we call the external direct product, which is a generalization of the direct product
in the sense of Lingcong and Endam. Also, we introduce the concept of the weak direct product
of B -algebras. Finally, we provide several fundamental theorems of (anti-)B -homomorphisms in
view of the external direct product B -algebras.

2020 Mathematics Subject Classifications: 03G25, 20K25

Key Words and Phrases: B -algebra, external direct product, weak direct product, B -homomorphism,
anti-B -homomorphism

1. Introduction and Preliminaries

Imai and Iséki introduced two classes of abstract algebras called BCK -algebras and
BCI -algebras. It is known that the class of BCK -algebras is a proper subclass of the
class of BCI -algebras [8, 9]. In 2002, Neggers and Kim [17] constructed a new algebraic
structure. They took some properties from BCI and BCK -algebras be called a B -algebra.
A B -algebra X = (X; ∗, 0) is an algebra of type (2, 0), that is, a nonempty set X together
with a binary operation ∗ and a constant 0 satisfying some axioms.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4383

Email addresses: chatsuda.chanmanee@gmail.com (C. Chanmanee),
ronnason.c@psu.ac.th (R. Chinram), rukchart.p@nsru.ac.th (R. Prasertpong),
pongpun.j@psru.ac.th (P. Julatha), aiyared.ia@up.ac.th (A. Iampan)

https://www.ejpam.com 999 © 2022 EJPAM All rights reserved.



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1000

B -algebras and some of their properties have been discussed, e.g., some axiomatizations
of B -algebras by Walendziak in 2006 [22], medial B -algebras by Kim in 2014 [11], fuzzy
order relative to fuzzy B -algebras by Gonzaga, Jr. and Vilela in 2019 [7], B -ideals in a
topological B -algebra and the uniform B -topological space by Belleza and Vilela in 2020
[3]. In 2021, Gan et al. [6] guaranteed the existences of both direct limits and inverse
limits in the categories of quantum B -algebras with morphisms of exact ones or spectral
ones, etc.

The concept of the direct product [19] was first defined in the group and obtained some
properties. For example, a direct product of the group is also a group, and a direct product
of the abelian group is also an abelian group. Then, direct product groups are applied to
other algebraic structures. In 2016, Lingcong and Endam [12] discussed the notion of the
direct product of B -algebras, 0-commutative B -algebras, and B -homomorphisms and ob-
tained related properties, one of which is a direct product of two B -algebras, which is also
a B -algebra. Then, they extended the concept of the direct product of B -algebra to finite
family B -algebra, and some of the related properties were investigated. Also, they intro-
duced two canonical mappings of the direct product of B -algebras and we obtained some of
their properties [13]. In the same year, Endam and Teves [5] defined the direct product of
BF -algebras, 0-commutative BF -algebras, and BF -homomorphism and obtained related
properties. In 2018, Abebe [1] introduced the concept of the finite direct product of BRK -
algebras and proved that the finite direct product of BRK -algebras is a BRK -algebra.
In 2019, Widianto et al. [23] defined the direct product of BG-algebras, 0-commutative
BG-algebras, and BG-homomorphism, including related properties of BG-algebras. In
2020, Setiani et al. [19] defined the direct product of BP -algebras, which is equivalent to
B -algebras. They obtained the relevant property of the direct product of BP -algebras and
then defined the direct product of BP -algebras as applied to finite sets of BP -algebras,
finite family 0-commutative BP -algebras, and finite family BP -homomorphisms. In 2021,
Kavitha and Gowri [10] defined the direct product of GK algebra. They derived the finite
form of the direct product of GK algebra and function as well. They investigated and
applied the concept of the direct product of GK algebra in GK function and GK kernel
and obtained interesting results.

In this paper, we introduce the concept of the direct product of infinite family of
B -algebras, we call the external direct product, which is a generalization of the direct
product in the sense of Lingcong and Endam [12]. Moreover, we introduce the concept of
the weak direct product of B -algebras. Finally, we discuss several (anti-)B -homomorphism
theorems in view of the external direct product B -algebras.

First of all, we start with the definitions and examples of B -algebras as well as other
relevant definitions for the study in this paper as follows:

Definition 1. [17] A B-algebra P = (P ; ∗, 0) is an algebra of type (2, 0), that is, a
nonempty set P together with a binary operation ∗ and a constant 0 satisfying the fol-
lowing axioms:

(∀x ∈ P )(x ∗ x = 0), (B-1)

(∀x ∈ P )(x ∗ 0 = x), (B-2)



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1001

(∀x, y, z ∈ P )((x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y))). (B-3)

Example 1. Let P = {0, 1, 2, 3, 4, 5} be a set with the Cayley table as follows:

∗ 0 1 2 3 4 5

0 0 1 4 5 2 3
1 1 0 5 4 3 2
2 2 3 0 1 4 5
3 3 2 1 0 5 4
4 4 5 2 3 0 1
5 5 4 3 2 1 0

Then P = (P ; ∗, 0) is a B-algebra.

Definition 2. [17] A B-algebra (P ; ∗, 0) is said to be commutative if

(∀x, y ∈ P )(x ∗ (0 ∗ y) = y ∗ (0 ∗ x)).

Example 2. From Example 1, we have P = (P ; ∗, 0) is commutative.

Definition 3. [16] A nonempty subset N of a B-algebra P = (P ; ∗, 0) is said to be a
B-subalgebra of P if

(∀x, y ∈ N)(x ∗ y ∈ N).

Definition 4. [2] A nonempty subset I of a B-algebra P = (P ; ∗, 0) is called a B-ideal of
P if it satisfies following conditions:

0 ∈ I, (BI-1)

(∀x, y ∈ P )((x ∗ y ∈ I, y ∈ I) ⇒ x ∈ I). (BI-2)

By (B-1), we have every B -subalgebra of a B -algebra satisfies (BI-1).

Definition 5. [16] A nonempty subset N of a B-algebra P = (P ; ∗, 0) is said to be normal
of P if

(∀x, y, a, b ∈ P )(x ∗ y, a ∗ b ∈ N ⇒ (x ∗ a) ∗ (y ∗ b) ∈ N).

Theorem 1. [16] Every normal subset of a B-algebra is a B-subalgebra and hence, it
satisfies (BI-1).

The concept of B -homomorphisms was also introduced by Neggers and Kim [16]. Let
A = (A; ∗A, 0A) and B = (B; ∗B, 0B) be B -algebras. A map φ : A → B is called a
B-homomorphism if

(∀x, y ∈ A)(φ(x ∗A y) = φ(x) ∗B φ(y)),

an anti-B-homomorphism if

(∀x, y ∈ A)(φ(x ∗A y) = φ(y) ∗B φ(x)).



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1002

The kernel of φ, denoted by kerφ, is defined to be the {x ∈ A | φ(x) = 0B}. The
kerφ is a normal B -subalgebra of A, and kerφ = {0A} if and only if φ is injective. A
(anti-)B -homomorphism φ is called a (anti-)B -monomorphism, (anti-)B -epimorphism, or
(anti-)B -isomorphism if φ is injective, surjective, or bijective, respectively.

Theorem 2. [16] Let N be a nonempty subset of a B-algebra P = (P ; ∗, 0). Then the
following statements are equivalent:

(i) N is a B-subalgebra of P .

(ii) x ∗ (0 ∗ y), 0 ∗ y ∈ N for all x, y ∈ N .

2. External Direct Product of B-algebras

Lingcong and Endam [12] discussed the notion of the direct product of B -algebras,
0-commutative B -algebras, and B -homomorphisms and obtained related properties, one
of which is a direct product of two B -algebras, which is also a B -algebra. Then, they
extended the concept of the direct product of B -algebra to finite family B -algebra, and
some of the related properties were investigated as follows:

Definition 6. [12] Let (Pi; ∗i) be an algebra for each i ∈ {1, 2, ..., k}. Define the direct
product of algebras P1, P2, ..., Pk to be the structure (

∏k
i=1 Pi;⊗), where

k∏
i=1

Pi = P1 × P2 × ...× Pk = {(p1, p2, ..., pk) | pi ∈ Pi ∀i = 1, 2, ..., k}

and whose operation ⊗ is given by

(p1, p2, ..., pk)⊗ (q1, q2, ..., qk) = (p1 ∗1 q1, p2 ∗2 q2, ..., pk ∗k qk)

for all (p1, p2, ..., pk), (q1, q2, ..., qk) ∈
∏k
i=1 Pi.

Theorem 3. [12] (Pi; ∗i, 0i) is a B-algebra for all i = 1, 2, ..., k if and only if

(

k∏
i=1

Pi;⊗, (01, 02, ..., 0k))

is a B-algebra, where the binary operation ⊗ is defined in Definition 6.

Now, we extend the concept of the direct product to infinite family of B -algebras and
provide some of its properties.

Definition 7. Let Pi be a nonempty set for each i ∈ I. Define the external direct product
of sets Pi for all i ∈ I to be the set

∏
i∈I Pi, where∏

i∈I
Pi = {f : I →

⋃
i∈I

Pi | f(i) ∈ Pi ∀i ∈ I}.

For convenience, we define an element of
∏
i∈I Pi with a function (pi)i∈I : I →

⋃
i∈I Pi,

where i 7→ pi ∈ Pi for all i ∈ I.



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1003

Definition 8. Let Pi = (Pi; ∗i) be an algebra for all i ∈ I. Define the binary operation ⊗
on the external direct product

∏
i∈I Pi = (

∏
i∈I Pi;⊗) as follows:

(∀(pi)i∈I , (qi)i∈I ∈
∏
i∈I

Pi)((pi)i∈I ⊗ (qi)i∈I = (pi ∗i qi)i∈I). (2.1)

We shall show that ⊗ is a binary operation on
∏
i∈I Pi. Let (pi)i∈I , (qi)i∈I ∈

∏
i∈I Pi.

Since ∗i is a binary operation on Pi for all i ∈ I, we have pi ∗i qi ∈ Pi for all i ∈ I. Then
(pi ∗i qi)i∈I ∈

∏
i∈I Pi such that

(pi)i∈I ⊗ (qi)i∈I = (pi ∗i qi)i∈I .

Let (pi)i∈I , (qi)i∈I , (p
′
i)i∈I , (q

′
i)i∈I ∈

∏
i∈I Pi be such that (pi)i∈I = (qi)i∈I and (p′i)i∈I =

(q′i)i∈I . We shall show that (pi)i∈I ⊗ (p′i)i∈I = (qi)i∈I ⊗ (q′i)i∈I . Then

pi = qi for all i ∈ I and p′i = q′i for all i ∈ I.

Since ∗i is a binary operation on Pi for all i ∈ I, we have pi ∗i p′i = qi ∗i q′i for all i ∈ I.
Thus

(pi)i∈I ⊗ (p′i)i∈I = (pi ∗i p′i)i∈I
= (qi ∗i q′i)i∈I
= (qi)i∈I ⊗ (q′i)i∈I .

Hence, ⊗ is a binary operation on
∏
i∈I Pi.

Let Pi = (Pi; ∗i, 0i) be a B -algebra for all i ∈ I. For i ∈ I, let pi ∈ Pi. We define the
function fpi : I →

⋃
i∈I Pi as follows:

(∀j ∈ I)

(
fpi(j) =

{
pi if j = i

0j otherwise

)
. (2.2)

Then fpi ∈
∏
i∈I Pi.

Lemma 1. Let Pi = (Pi; ∗i, 0i) be a B-algebra for all i ∈ I. For i ∈ I, let pi, qi ∈ Pi.
Then fpi ⊗ fqi = fpi∗iqi.

Proof. Now,

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j ∗j 0j otherwise

)
.

By (B-1), we have

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j otherwise

)
.

By (2.2), we have fpi ⊗ fqi = fpi∗iqi .

The following theorem shows that the direct product of B -algebras in term of infinite
family of B -algebras is also a B -algebra which is more generalized than Theorem 3.



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1004

Theorem 4. Pi = (Pi; ∗i, 0i) is a B-algebra for all i ∈ I if and only if
∏
i∈I Pi =

(
∏
i∈I Pi;⊗, (0i)i∈I) is a B-algebra, where the binary operation ⊗ is defined in Definition

8.

Proof. Assume that Pi = (Pi; ∗i, 0i) is a B -algebra for all i ∈ I.
(B-1) Let (pi)i∈I ∈

∏
i∈I Pi. Since Pi satisfies (B-1), we have pi ∗i pi = 0i for all i ∈ I.

Thus

(pi)i∈I ⊗ (pi)i∈I = (pi ∗i pi)i∈I = (0i)i∈I .

(B-2) Let (pi)i∈I ∈
∏
i∈I Pi. Since Pi satisfies (B-2), we have pi ∗i 0i = pi for all i ∈ I.

Thus

(pi)i∈I ⊗ (0i)i∈I = (pi ∗i 0i)i∈I = (pi)i∈I .

(B-3) Let (pi)i∈I , (qi)i∈I , (ri)i∈I ∈
∏
i∈I Pi. Since Pi satisfies (B-3), we have (pi ∗i qi) ∗i

ri = pi ∗i (ri ∗i (0i ∗i qi)) for all i ∈ I. Thus

((pi)i∈I ⊗ (qi)i∈I)⊗ (ri)i∈I = (pi ∗i qi)i∈I ⊗ (ri)i∈I

= ((pi ∗i qi) ∗i ri)i∈I
= (pi ∗i (ri ∗i (0i ∗i qi)))i∈I
= (pi)i∈I ⊗ (ri ∗i (0i ∗i qi))i∈I
= (pi)i∈I ⊗ ((ri)i∈I ⊗ (0i ∗i qi)i∈I)
= (pi)i∈I ⊗ ((ri)i∈I ⊗ ((0i)i∈I ⊗ (qi)i∈I)).

Hence,
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I) is a B -algebra.

Conversely, assume that
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I) is a B -algebra, where the bi-

nary operation ⊗ is defined in Definition 8. Let i ∈ I.
(B-1) Let pi ∈ Pi. Then fpi ∈

∏
i∈I Pi, which is defined by (2.2). Since

∏
i∈I Pi satisfies

(B-1), we have fpi ⊗ fpi = (0i)i∈I . Now,

(∀j ∈ I)

(
(fpi ⊗ fpi)(j) =

{
pi ∗i pi if j = i

0j ∗j 0j otherwise

)
,

this implies that pi ∗i pi = 0i.
(B-2) Let pi ∈ Pi. Then fpi ∈

∏
i∈I Pi, which is defined by (2.2). Since

∏
i∈I Pi satisfies

(B-2), we have fpi ⊗ (0i)i∈I = fpi . Now,

(∀j ∈ I)

(
(fpi ⊗ (0i)i∈I)(j) =

{
pi ∗i 0i if j = i

0j ∗j 0j otherwise

)
,

this implies that pi ∗i 0i = pi.



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1005

(B-3) Let pi, qi, ri ∈ Pi. Then fpi , fqi , fri ∈
∏
i∈I Pi, which is defined by (2.2). Since∏

i∈I Pi satisfies (B-3), we have (fpi ⊗ fqi)⊗ fri = fpi ⊗ (fri ⊗ ((0i)i∈I ⊗ fqi)). Now,

(∀j ∈ I)

(
(fpi ⊗ fqi)⊗ fri)(j) =

{
(pi ∗i qi) ∗i ri if j = i

(0j ∗j 0j) ∗j 0j otherwise

)

and

(∀j ∈ I)

(
fpi ⊗ (fri ⊗ ((0i)i∈I ⊗ fqi)))(j) =

{
pi ∗i (ri ∗i (0i ∗i qi)) if j = i

0j ∗j (0j ∗j (0j ∗j 0j)) otherwise

)
,

this implies that

(pi ∗i qi) ∗i ri = pi ∗i (ri ∗i (0i ∗i qi)).

Hence, Pi = (Pi; ∗i, 0i) is a B -algebra for all i ∈ I.

We call the B -algebra
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I) in Theorem 4 the external direct

product B -algebra induced by a B -algebra Pi = (Pi; ∗i, 0i) for all i ∈ I.

Theorem 5. Let Pi = (Pi; ∗i, 0i) be a B-algebra for all i ∈ I. Then Pi is commutative for
all i ∈ I if and only if

∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I) is commutative, where the binary

operation ⊗ is defined in Definition 8.

Proof. By Theorem 4, we have Pi = (Pi; ∗i, 0i) is a B -algebra for all i ∈ I if and only
if
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I) is a B -algebra, where the binary operation ⊗ is defined

in Definition 8. We are left to prove that Pi is commutative for all i ∈ I if and only if∏
i∈I Pi is commutative.
Assume that Pi is commutative for all i ∈ I. Let (pi)i∈I , (qi)i∈I ∈

∏
i∈I Pi. Since Pi is

commutative for all i ∈ I, we have pi ∗i (0i ∗i qi) = qi ∗i (0i ∗i pi) for all i ∈ I. Thus

(pi)i∈I ⊗ ((0i)i∈I ⊗ (qi)i∈I) = (pi)i∈I ⊗ (0i ∗i qi)i∈I
= (pi ∗i (0i ∗i qi))i∈I
= (qi ∗i (0i ∗i pi))i∈I
= (qi)i∈I ⊗ (0i ∗i pi)i∈I
= (qi)i∈I ⊗ ((0i)i∈I ⊗ (pi)i∈I).

Hence,
∏
i∈I Pi is commutative.

Conversely, assume that
∏
i∈I Pi is commutative. Let i ∈ I. Let pi, qi ∈ Pi. Then

fpi , fqi ∈
∏
i∈I Pi, which is defined by (2.2). Since

∏
i∈I Pi is commutative, we have

fpi ⊗ ((0i)i∈I ⊗ fqi) = fqi ⊗ ((0i)i∈I ⊗ fpi). Now,

(∀j ∈ I)

(
(fpi ⊗ ((0i)i∈I ⊗ fqi))(j) =

{
pi ∗i (0i ∗i qi) if j = i

0j ∗j (0j ∗j 0j) otherwise

)



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1006

and

(∀j ∈ I)

(
(fqi ⊗ ((0i)i∈I ⊗ fpi))(j) =

{
qi ∗i (0i ∗i pi) if j = i

0j ∗j (0j ∗j 0j) otherwise

)
,

this implies that

pi ∗i (0i ∗i qi) = qi ∗i (0i ∗i pi).

Hence, Pi is commutative for all i ∈ I.

Next, we introduce the concept of the weak direct product of infinite family of B -
algebras and obtain some of its properties as follows:

Definition 9. Let Pi = (Pi; ∗i, 0i) be a B-algebra for all i ∈ I. Define the weak direct
product of a B-algebra Pi for all i ∈ I to be the structure

∏w
i∈I Pi = (

∏w
i∈I Pi;⊗), where

w∏
i∈I

Pi = {(pi)i∈I ∈
∏
i∈I

Pi | pi ̸= 0i, where the number of such i is finite}.

Then (0i)i∈I ∈
∏w
i∈I Pi ⊆

∏
i∈I Pi.

Theorem 6. Let Pi = (Pi; ∗i, 0i) be a B-algebra for all i ∈ I. Then
∏w
i∈I Pi is a B-

subalgebra of the external direct product B-algebra
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I).

Proof. We see that (0i)i∈I ∈
∏w
i∈I Pi ̸= ∅. Let (pi)i∈I , (qi)i∈I ∈

∏w
i∈I Pi, where

I1 = {i ∈ I | pi ̸= 0i} and I2 = {i ∈ I | qi ̸= 0i} are finite. Then |I1 ∪ I2| is finite. Thus

(∀j ∈ I)

((pi)i∈I ⊗ (qi)i∈I)(j) =


pj ∗j 0j if j ∈ I1 − I2

pj ∗j qj if j ∈ I1 ∩ I2
0j ∗j qj if j ∈ I2 − I1

0j ∗j 0j otherwise

 .

By (B-1) and (B-2), we have

(∀j ∈ I)

((pi)i∈I ⊗ (qi)i∈I)(j) =


pj if j ∈ I1 − I2

pj ∗j qj if j ∈ I1 ∩ I2
0j ∗j qj if j ∈ I2 − I1

0j otherwise

 .

This implies that the number of such ((pi)i∈I⊗(qi)i∈I)(j) is not more than |I1∪I2|, that is,
it is finite. Thus (pi)i∈I ⊗ (qi)i∈I ∈

∏w
i∈I Pi. Hence,

∏w
i∈I Pi is a B -subalgebra of

∏
i∈I Pi.

Theorem 7. Let Pi = (Pi; ∗i, 0i) be a B-algebra and Qi a subset of Pi for all i ∈ I. Then
Qi is a B-subalgebra of Pi for all i ∈ I if and only if

∏
i∈I Qi is a B-subalgebra of the

external direct product B-algebra
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I).



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1007

Proof. Assume that Qi is a B -subalgebra of Pi for all i ∈ I. Then 0i ∈ Qi for all i ∈ I,
so (0i)i∈I ∈

∏
i∈I Qi ̸= ∅. Let (pi)i∈I , (qi)i∈I ∈

∏
i∈I Qi. Then pi, qi ∈ Qi for all i ∈ I.

Thus pi ∗i qi ∈ Qi for all i ∈ I, so (pi)i∈I ⊗ (qi)i∈I = (pi ∗i qi)i∈I ∈
∏
i∈I Qi.

Hence,
∏
i∈I Qi is a B -subalgebra of

∏
i∈I Pi.

Conversely, assume that
∏
i∈I Qi is a B -subalgebra of

∏
i∈I Pi. Then (0i)i∈I ∈

∏
i∈I Qi,

so 0i ∈ Qi ̸= ∅ for all i ∈ I. Let i ∈ I and let pi, qi ∈ Qi. Then fpi , fqi ∈
∏
i∈I Qi, which is

defined by (2.2). Since
∏
i∈I Qi is a B -subalgebra of

∏
i∈I Pi, we have fpi ⊗ fqi ∈

∏
i∈I Qi.

Now,

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j ∗j 0j otherwise

)
,

this implies that pi ∗i qi ∈ Qi.
Hence, Qi is a B -subalgebra of Pi for all i ∈ I.

Theorem 8. Let Pi = (Pi; ∗i, 0i) be a B-algebra and Qi a subset of Pi for all i ∈ I. Then
Qi is normal of Pi for all i ∈ I if and only if

∏
i∈I Qi is normal of the external direct

product B-algebra
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I).

Proof. Assume that Qi is normal of Pi for all i ∈ I. Then Qi ̸= ∅ for all i ∈
I, so

∏
i∈I Qi ̸= ∅. Let (pi)i∈I , (p

′
i)i∈I , (qi)i∈I , (q

′
i)i∈I ∈

∏
i∈I Pi be such that (pi)i∈I ⊗

(qi)i∈I , (p
′
i)i∈I ⊗ (q′i)i∈I ∈

∏
i∈I Qi. Then pi ∗i qi, p′i ∗i q′i ∈ Qi for all i ∈ I. Since Qi is

normal of Pi, we have (pi ∗i p′i) ∗i (qi ∗i q′i) ∈ Qi for all i ∈ I. Thus

((pi)i∈I⊗(p′i)i∈I)⊗((qi)i∈I⊗(q′i)i∈I) = (pi∗ip′i)i∈I⊗(qi∗iq′i)i∈I = ((pi∗ip′i)∗i(qi∗iq′i))i∈I ∈
∏
i∈I

Qi.

Hence,
∏
i∈I Qi is normal of

∏
i∈I Pi.

Conversely, assume that
∏
i∈I Qi is normal of

∏
i∈I Pi. Then

∏
i∈I Qi ̸= ∅, so Qi ̸= ∅ for

all i ∈ I. By Theorem 1, we have (0i)i∈I ∈
∏
i∈I Qi. Thus 0i ∈ Qi for all i ∈ I. Let i ∈ I

and let pi, qi, p
′
i, q

′
i ∈ Pi be such that pi ∗i qi, p′i ∗i q′i ∈ Qi. Then fpi , fqi , fp′i , fq′i ∈

∏
i∈I Pi,

which is defined by (2.2). Now,

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j ∗j 0j otherwise

)
.

By (B-1), we have

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j otherwise

)
,

this implies that fpi ⊗ fqi ∈
∏
i∈I Qi. Similarly, fp′i ⊗ fq′i ∈

∏
i∈I Qi. Since

∏
i∈I Qi is

normal of
∏
i∈I Pi, we have (fpi ⊗ fp′i)⊗ (fqi ⊗ fq′i) ∈

∏
i∈I Qi. Now,

(∀j ∈ I)

(
((fpi ⊗ fp′i)⊗ (fqi ⊗ fq′i))(j) =

{
(pi ∗i p′i) ∗i (qi ∗i q′i) if j = i

(0j ∗j 0j) ∗j (0j ∗j 0j) otherwise

)
,



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1008

this implies that (pi ∗i p′i) ∗i (qi ∗i q′i) ∈ Qi.
Hence, Qi is normal of Pi for all i ∈ I.

Theorem 9. Let Pi = (Pi; ∗i, 0i) be a B-algebra and Qi a subset of Pi for all i ∈ I. Then
Qi is a B-ideal of Pi for all i ∈ I if and only if

∏
i∈I Qi is a B-ideal of the external direct

product B-algebra
∏
i∈I Pi = (

∏
i∈I Pi;⊗, (0i)i∈I).

Proof. Assume that Qi is a B -ideal of Pi for all i ∈ I.
(BI-1) By (BI-1), we have 0i ∈ Qi for all i ∈ I. Then (0i)i∈I ∈

∏
i∈I Qi.

(BI-2) Let (pi)i∈I , (qi)i∈I ∈
∏
i∈I Pi be such that (pi)i∈I ⊗ (qi)i∈I ∈

∏
i∈I Qi and

(qi)i∈I ∈
∏
i∈I Qi. Then (pi ∗i qi)i∈I ∈

∏
i∈I Qi. Thus pi ∗i qi ∈ Qi and qi ∈ Qi, it

follows from (BI-2) that pi ∈ Qi for all i ∈ I. Thus (pi)i∈I ∈
∏
i∈I Qi.

Hence,
∏
i∈I Qi is a B -ideal of

∏
i∈I Pi.

Conversely, assume that
∏
i∈I Qi is a B -ideal of

∏
i∈I Pi. Then

∏
i∈I Qi ̸= ∅, so Qi ̸= ∅

for all i ∈ I. Let i ∈ I.
(BI-1) By (BI-1), we have (0i)i∈I ∈

∏
i∈I Qi. Then 0i ∈ Qi.

(BI-2) Let pi, qi ∈ Pi be such that pi ∗i qi ∈ Qi and qi ∈ Qi. By (BI-1), we have 0i ∈ Qi
for all i ∈ I. Then fpi ∈

∏
i∈I Pi and fpi∗iqi , fqi ∈

∏
i∈I Qi, which are defined by (2.2). By

Lemma 1, we have fpi ⊗ fqi = fpi∗iqi ∈
∏
i∈I Qi. By (BI-2), we have fpi ∈

∏
i∈I Qi. By

(2.2), we have pi ∈ Qi.
Hence, Qi is a B -ideal of Pi for all i ∈ I.

Moreover, we discuss several homomorphism theorems in view of the external direct
product of B -algebras.

Definition 10. Let Pi = (Pi; ∗i) and Qi = (Qi; ◦i) be algebras and ψi : Pi → Qi be a
function for all i ∈ I. Define the function ψ :

∏
i∈I Pi →

∏
i∈I Qi given by

(∀(pi)i∈I ∈
∏
i∈I

Pi)(ψ(pi)i∈I = (ψi(pi))i∈I). (2.3)

We shall show that ψ :
∏
i∈I Pi →

∏
i∈I Qi is a function. Let (pi)i∈I ∈

∏
i∈I Pi. Since

ψi : Pi → Qi is a function and pi ∈ Pi for all i ∈ I, we have ψi(pi) ∈ Qi for all i ∈ I. Thus
(ψi(pi))i∈I ∈

∏
i∈I Qi. such that ψ(pi)i∈I = (ψi(pi))i∈I . Let (pi)i∈I , (p

′
i)i∈I ∈

∏
i∈I Pi be

such that (pi)i∈I = (p′i)i∈I . Then pi = p′i for all i ∈ I, so ψi(pi) = ψi(p
′
i). Thus

ψ(pi)i∈I = (ψi(pi))i∈I = (ψi(p
′
i))i∈I = ψ(p′i)i∈I .

Therefore, ψ :
∏
i∈I Pi →

∏
i∈I Qi is a function.

Theorem 10. Let Pi = (Pi; ∗i) and Qi = (Qi; ◦i) be algebras and ψi : Pi → Qi be a
function for all i ∈ I.

(i) ψi is injective for all i ∈ I if and only if ψ is injective which is defined in Definition
10,



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1009

(ii) ψi is surjective for all i ∈ I if and only if ψ is surjective,

(iii) ψi is bijective for all i ∈ I if and only if ψ is bijective.

Proof. (i) Assume that ψi is injective for all i ∈ I. Let (pi)i∈I , (qi)i∈I ∈
∏
i∈I Pi be

such that ψ(pi)i∈I = ψ(qi)i∈I . Then (ψi(pi))i∈I = (ψi(qi))i∈I . Thus ψi(pi) = ψi(qi) for all
i ∈ I. Since ψi is injective for all i ∈ I, we have pi = qi for all i ∈ I. Thus (pi)i∈I = (qi)i∈I .
Hence, ψ is injective.

Conversely, assume that ψ is injective. Let i ∈ I. Let pi, p
′
i ∈ Pi be such that

ψi(pi) = ψi(p
′
i).

Let pj = p′j ∈ Pj for all j ∈ I and j ̸= i. Then ψj(pj) = ψj(p
′
j) ∈ Qj . Let hψi(pi) : I →⋃

i∈I Qi and hψi(p′i)
: I →

⋃
i∈I Qi are functions defined by

(∀j ∈ I)

(
hψi(pi)(j) =

{
ψi(pi) if j = i

ψj(pj) otherwise

)
(2.4)

and

(∀j ∈ I)

(
hψi(p′i)

(j) =

{
ψi(p

′
i) if j = i

ψj(p
′
j) otherwise

)
. (2.5)

Then hψi(pi), hψi(p′i)
∈
∏
i∈I Qi such that ψ(pi)i∈I = hψi(pi) = hψi(p′i)

= ψ(p′i)i∈I . Since ψ
is injective, we have (pi)i∈I = (p′i)i∈I . Thus pi = p′i. Hence, ψi is injective for all i ∈ I.

(ii) Assume that ψi is surjective for all i ∈ I. Let (qi)i∈I ∈
∏
i∈I Qi. Then qi ∈ Qi for

all i ∈ I. Since ψi is surjective, there exists pi ∈ Pi such that ψi(pi) = qi for all i ∈ I.
Thus (pi)i∈I ∈

∏
i∈I Pi such that

ψ(pi)i∈I = (ψi(pi))i∈I = (qi)i∈I .

Hence, ψ is surjective.

Conversely, assume that ψ is surjective. Let i ∈ I. Let ki ∈ Qi.
Let kj ∈ Qj for all j ∈ I and j ̸= i. Then (ki)i∈I ∈

∏
i∈I Qi. Since ψ is surjective,

there exists (pi)i∈I ∈
∏
i∈I Pi such that

(ki)i∈I = ψ(pi)i∈I = (ψi(pi))i∈I .

Thus ki = ψi(pi). Hence, ψi is surjective for all i ∈ I.
(iii) It is straightforward from (i) and (ii).

Theorem 11. Let Pi = (Pi; ∗i, 0i) and Qi = (Qi; ◦i, 1i) be B-algebras and ψi : Pi → Qi be
a function for all i ∈ I. Then

(i) ψi is a B-homomorphism for all i ∈ I if and only if ψ is a B-homomorphism which
is defined in Definition 10,

(ii) ψi is a B-monomorphism for all i ∈ I if and only if ψ is a B-monomorphism,



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1010

(iii) ψi is a B-epimorphism for all i ∈ I if and only if ψ is a B-epimorphism,

(iv) ψi is a B-isomorphism for all i ∈ I if and only if ψ is a B-isomorphism,

(v) kerψ =
∏
i∈I kerψi and ψ(

∏
i∈I Pi) =

∏
i∈I ψi(Pi).

Proof. (i) Assume that ψi is a B -homomorphism for all i ∈ I. Let (pi)i∈I , (p
′
i)i∈I ∈∏

i∈I Pi. Then

ψ((pi)i∈I ⊗ (p′i)i∈I) = ψ(pi ∗i p′i)i∈I
= (ψi(pi ∗i p′i))i∈I
= (ψi(pi) ∗i ψi(p′i))i∈I
= (ψi(pi))i∈I ⊗ (ψi(p

′
i))i∈I

= ψ(pi)i∈I ⊗ ψ(p′i)i∈I .

Hence, ψ is a B -homomorphism.

Conversely, assume that ψ is a B -homomorphism. Let i ∈ I. Let pi, qi ∈ Pi. Then
fpi , fqi ∈

∏
i∈I Pi, which is defined by (2.2). Since ψ is a B -homomorphism, we have

ψ(fpi ⊗ fqi) = ψ(fpi)⊗ ψ(fqi). Since

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j ∗j 0j otherwise

)
,

we have

(∀j ∈ I)

(
ψ(fpi ⊗ fqi)(j) =

{
ψi(pi ∗i qi) if j = i

ψj(0j ∗j 0j) otherwise

)
. (2.6)

Since

(∀j ∈ I)

(
ψ(fpi)(j) =

{
ψi(pi) if j = i

ψj(0j) otherwise

)

and

(∀j ∈ I)

(
ψ(fqi)(j) =

{
ψi(qi) if j = i

ψj(0j) otherwise

)
,

we have

(∀j ∈ I)

(
(ψ(fpi)⊗ ψ(fqi))(j) =

{
ψi(pi) ◦i ψi(qi) if j = i

ψj(0j) ◦j ψj(0j) otherwise

)
. (2.7)

By (2.6) and (2.7), we have ψi(pi ∗i qi) = ψi(pi) ◦i ψi(qi). Hence, ψi is a B -homomorphism
for all i ∈ I.



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1011

(ii) It is straightforward from (i) and Theorem 10 (i).
(iii) It is straightforward from (i) and Theorem 10 (ii).
(iv) It is straightforward from (i) and Theorem 10 (iii).
(v) Let (pi)i∈I ∈

∏
i∈I Pi. Then

(pi)i∈I ∈ kerψ ⇔ ψ(pi)i∈I = (1i)i∈I

⇔ (ψi(pi))i∈I = (1i)i∈I

⇔ ψi(pi) = 1i ∀i ∈ I

⇔ pi ∈ kerψi ∀i ∈ I

⇔ (pi)i∈I ∈
∏
i∈I

kerψi.

Hence, kerψ =
∏
i∈I kerψi. Now,

(qi)i∈I ∈ ψ(
∏
i∈I

Pi) ⇔ ∃(pi)i∈I ∈
∏
i∈I

Pi s.t. (qi)i∈I = ψ(pi)i∈I

⇔ ∃(pi)i∈I ∈
∏
i∈I

Pi s.t. (qi)i∈I = (ψi(pi))i∈I

⇔ ∃pi ∈ Pi s.t. qi = ψi(pi) ∈ ψ(Pi) ∀i ∈ I

⇔ (qi)i∈I ∈
∏
i∈I

ψi(Pi).

Hence, ψ(
∏
i∈I Pi) =

∏
i∈I ψi(Pi).

Finally, we discuss several anti-B -homomorphism theorems in view of the external
direct product of B -algebras.

Theorem 12. Let Pi = (Pi; ∗i, 0i) and Qi = (Qi; ◦i, 1i) be B-algebras and ψi : Pi → Qi be
a function for all i ∈ I. Then

(i) ψi is an anti-B-homomorphism for all i ∈ I if and only if ψ is an anti-B-homomorphism
which is defined in Definition 10,

(ii) ψi is an anti-B-monomorphism for all i ∈ I if and only if ψ is an anti-B-monomorphism,

(iii) ψi is an anti-B-epimorphism for all i ∈ I if and only if ψ is an anti-B-epimorphism,

(iv) ψi is an anti-B-isomorphism for all i ∈ I if and only if ψ is an anti-B-isomorphism.

Proof. (i) Assume that ψi is an anti-B -homomorphism for all i ∈ I. Let (pi)i∈I , (p
′
i)i∈I ∈∏

i∈I Pi. Then

ψ((pi)i∈I ⊗ (p′i)i∈I) = ψ(pi ∗i p′i)i∈I
= (ψi(pi ∗i p′i))i∈I
= (ψi(p

′
i) ∗i ψi(pi))i∈I



A. Iampan et al. / Eur. J. Pure Appl. Math, 15 (3) (2022), 999-1014 1012

= (ψi(p
′
i))i∈I ⊗ (ψi(pi))i∈I

= ψ(p′i)i∈I ⊗ ψ(pi)i∈I .

Hence, ψ is an anti-B -homomorphism.

Conversely, assume that ψ is an anti-B -homomorphism. Let i ∈ I. Let pi, qi ∈ Pi.
Then fpi , fqi ∈

∏
i∈I Pi, which is defined by (2.2). Since ψ is an anti-B -homomorphism,

we have ψ(fpi ⊗ fqi) = ψ(fqi)⊗ ψ(fpi). Since

(∀j ∈ I)

(
(fpi ⊗ fqi)(j) =

{
pi ∗i qi if j = i

0j ∗j 0j otherwise

)
,

we have

(∀j ∈ I)

(
ψ(fpi ⊗ fqi)(j) =

{
ψi(pi ∗i qi) if j = i

ψj(0j ∗j 0j) otherwise

)
. (2.8)

Since

(∀j ∈ I)

(
ψ(fqi)(j) =

{
ψi(qi) if j = i

ψj(0j) otherwise

)

and

(∀j ∈ I)

(
ψ(fpi)(j) =

{
ψi(pi) if j = i

ψj(0j) otherwise

)
,

we have

(∀j ∈ I)

(
(ψ(fqi)⊗ ψ(fpi))(j) =

{
ψi(qi) ◦i ψi(pi) if j = i

ψj(0j) ◦j ψj(0j) otherwise

)
. (2.9)

By (2.8) and (2.9), we have ψi(pi ∗i qi) = ψi(qi) ◦i ψi(pi). Hence, ψi is an anti-B -
homomorphism for all i ∈ I.

(ii) It is straightforward from (i) and Theorem 10 (i).
(iii) It is straightforward from (i) and Theorem 10 (ii).
(iv) It is straightforward from (i) and Theorem 10 (iii).

3. Conclusions and Future Work

In this paper, we have introduced the concept of the direct product of infinite family
of B -algebras, we call the external direct product, which is a generalization of the direct
product in the sense of Lingcong and Endam [12]. We proved that the external direct
product of B -algebras is also a B -algebra. Also, we have introduced the concept of the
weak direct product of B -algebras. We proved that the weak direct product of B -algebras



REFERENCES 1013

is a B -subalgebra of the external direct product B -algebras. Finally, we have provided
several fundamental theorems of (anti-)B -homomorphisms in view of the external direct
product B -algebras.

Based on the concept of the external direct product of B -algebras in this article, we can
apply it to the study of the external direct product in other algebraic systems. Researching
the external and weak direct products that we will study in the future will be UP-algebras.

The research topics of interest by our research team being studied in the external direct
product of B -algebras are as follows:

(1) to study fuzzy set theory (with respect to a triangular norm) based on the concept
of Somjanta et al. [20] and Thongarsa et al. [4, 21],

(2) to study bipolar fuzzy set theory based on the concept of Muhiuddin [14],

(3) to study interval-valued fuzzy set theory based on the concept of Muhiuddin et al.
[15],

(4) to study interval-valued intuitionistic fuzzy set theory based on the concept of Sena-
pati et al. [18].

Acknowledgements

This work was supported by the revenue budget in 2022, School of Science, University
of Phayao, Thailand.

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