EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 116-123
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Filter of Cyclic B-Algebras

Maliwan Phattarachaleekul

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

Abstract. This paper introduces the notion of a B-filter in a B-algebra (X, ∗, 0) and presents
characteristics of its properties : for any a ∈ X, the set ⟨a⟩B = {ak : k ∈ Z} forms a B-ideal and
B-filter of X. Moreover, this paper showns some properties of exponents on B-algebra.

2020 Mathematics Subject Classifications: 06F35, 03G25

Key Words and Phrases: B-algebras; cyclic B-algebras; B-ideal ; B-filter

1. Introduction

J. Neggers and H. S. Kim introduced in [1] the notion of B-algebras and some properties
of exponents on its. Furthermore, they investigated the relationship between B-algebras
and groups and asked whether a group determines a B-algebra, and conversely. In [5],
D. Al-Kadi introduced the notion of B-ideal and then K. E. Belleza and J. P. Vilela in
[7] presents the characterizations and properties of B-ideals in a topological B-algebra
and introduces the uniform topology on a B-algebra in terms of its B-ideals. Moreover,
they have shown that a uniform B-topological space is a topological B-algebra. K. E.
Belleza and J. R. Albaracin introduces and characterized the notion of a dual B-algebra,
in [6]. Moreover in the year 2022 , K. E. Belleza introduces the dual B-topological space,
dual B-ideals and dual B-subalgebras. Also, some properties of a filterbase on a dual
B-topological space are provided. In [2], N. C. Gonzaga, Jr and J. P. Vilela introduced
the notion of cyclic B-algebras and some of its properties. Moreover the authors had
investigated the relationship between the class of cyclic B-algebras and the class of cyclic
groups coincide. In [8], K. E. Belleza and J. R. Albaracin introduced the notion of tdB-
algebra, presents characteristics and properties of dual B-filters and dual B-subalgebras
in a tdB-algebra, and introduces the uniform topology on a dual B-algebra in terms of its
dual B-subalgebras.

Specifically, this paper introduces the notion of the B-filter in a B-algebra and we have
show that for any a ∈ X, the set ⟨a⟩B = {ak : k ∈ Z} form a B-ideal and B-filter of X.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4977

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

https://www.ejpam.com 116 © 2024 EJPAM All rights reserved.



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 17 (1) (2024), 116-123 117

2. Preliminaries

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

Definition 1. [4] A B-algebra is a non-empty set X with a constant 0 and a binary
operation ∗ satisfying the following axioms :

(B1) x ∗ x = 0,

(B2) x ∗ 0 = x,

(B3) (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)) for all x, y, z ∈ X.

A B-algebra (X, ∗, 0) is said to be commutative if x ∗ (0 ∗ y) = y ∗ (0 ∗ x) for any
x, y ∈ X and a nonempty subset S of a X is called a sub-algebra of X if x ∗ y ∈ S for any
x, y ∈ S. Also, the authors of [1] have proved that a B-algebra X is commutative if and
only if the equality x ∗ (x ∗ y) = y holds for all x, y ∈ X.

Example 1. ([4],[5]) : Let X = {0, 1, 2, 3} and Y = {0, 1, 2, 3, 4, 5}. Define binary opera-
tions ∗ on X and ⊙ on Y defined by the following two tables respectively :

∗ 0 1 2 3

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

⊙ 0 1 2 3 4 5

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

Then (X, ∗, 0) is a commutative B-algebra, but (Y,⊙, 0) is a non commutative B-algebra,
since 2 ∗ (0 ∗ 5) = 2 ∗ 5 = 4 ̸= 3 = 5 ∗ 1 = 5 ∗ (0 ∗ 2).

We recall the following axioms for the laws of Exponents for B-algebras.

Theorem 1. [1] Let (X, ∗, 0) be a B-algebra. Then the following conditions hold for any
x, y, z ∈ X:

(i) x = (x ∗ y) ∗ (0 ∗ y),

(ii) y ∗ x = 0 ∗ (x ∗ y),

(iii) 0 ∗ (0 ∗ x) = x,

(iv) x ∗ (y ∗ z) = (x ∗ (0 ∗ z)) ∗ y,



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 17 (1) (2024), 116-123 118

(v) x ∗ y = x ∗ (0 ∗ (0 ∗ y),

(vi) x ∗ y = 0 implies x = y,

(vii) x ∗ z = y ∗ z implies x = y and

(viii) 0 ∗ x = 0 ∗ y, implies x = y.

Definition 2. [5] Let (X, ∗, 0) be a B-algebra. A nonempty subset I of X is called a
B-ideal of X if it satisfies the following conditions for any x, y, z ∈ X:

(i) 0 ∈ I,

(ii) If x ∗ y ∈ I and y ∈ I, then x ∈ I.

Definition 3. [6] Let X be a non-empty set with a binary operation ∗ and a constant
0. Then the triple (X, ∗, 0) is a dual B-algebra if its satisfies the following axioms for all
x, y, z ∈ X:

(i) x ∗ x = 0,

(ii) 0 ∗ x = x,

(iii) x ∗ (y ∗ z) = ((y ∗ 0) ∗ x) ∗ z.

Definition 4. [8] Let (X, ∗, 0) be a dual B-algebra. A nonempty subset F of X is called
a dual B-filter of X if it satisfies the following axioms for all x, y, z ∈ X:

(i) 0 ∈ F ,

(ii) If x ∗ y ∈ F and x ∈ F , then y ∈ F .

There is a B-algebra that is also a dual B-algebra in the following example.

Example 2. [6] Let X = {0, a, b, c} and a binary operations ∗ on X satisfying the following
table :

∗ 0 a b c

0 0 a b c
a a 0 c b
b b c 0 a
c c b a 0



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 17 (1) (2024), 116-123 119

Example 3. [8] Let X = {0, a, b, c, d, e} and a binary operations ∗ on X satisfies the
following table :

∗ 0 a b c d e

0 0 a b c d e
a b 0 a d e c
b d b 0 e c d
c c d e 0 a b
d d e c b 0 a
e e c d a b 0

Then (X, ∗, 0) is a dual B-algebra. The sets F0 = {0}, F2 = {0, c}, F3 = {0, d}, F4 = {0, e}
and F5 = {0, a, b} are dual B-filters of X while A = {0, a, e} is not a dual B-filter since
e ∗ c = a ∈ A where e ∈ A but c /∈ A.

3. Some Axioms of Exponents for B-algebras

In this section, we recall the axioms for a B-algebra (X, ∗, 0). The paper [1] and [2]
introduced the notions of exponents of B-algebra and some of its properties. For any
x, y ∈ X and n ∈ Z+, defined the relation:

xn = xn−1 ∗ (0 ∗ x) and −x = 0 ∗ x

where x0 = 0 and x1 = x0 ∗ (0 ∗ x) = 0 ∗ (0 ∗ x) = x and denote that expression

x ∗
n∏
y = (...((x ∗ y) ∗ y) ∗ ...) ∗ y,

where y occurs n times. By convention, x∗
0∏
y means x∗0 = x, so that xn = x∗(0∗

n−1∏
x)

and x−n = (−x)n = −(x)n = 0 ∗ xn implies that (x−1)−n = (x−n)−1 = xn.

Theorem 2. [1] Let (X, ∗, 0) be a B-algebra , g ∈ X and m,n ∈ Z+. Then

gm ∗ gn =

{
gm−n if m ≥ n

0 ∗ gn−m if m < n

Corollary 1. [1] Let (X, ∗, 0) be a B-algebra. Then the following equalities hold for all
g ∈ X and m,n ∈ Z+:

(i) gm ∗ gn = gm−n ,

(ii) g ∗ g−n = gn+1 and g−n ∗ g = g−(n+1),



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 17 (1) (2024), 116-123 120

(iii) −g ∗ gn = g−(n+1),

(iv) gm ∗ (−g) = g(n+1),

(v) gm ∗ g−n = g(m+n) and g−m ∗ gn = g−(m+n).

Corollary 2. [2] Let (X, ∗, 0) be a B-algebra, g ∈ X and m,n ∈ Z. Then gm∗gn = gm−n.

4. On Exponents and cyclic B-Algebras

We shall give some elementary properties of cyclic B-algebras. Recall that for a
B-algebra (X, ∗, 0) (see [2]) if there is an a ∈ X such that

⟨a⟩B = {ak : k ∈ Z} = X,

then X is called a cyclic B-algebra generated by a. Also, the authors of [2] have proved
that every cyclic B-algebra is commutative.

Theorem 3. Let (X, ∗, 0) be a B-algebra and x, y ∈ X with n ∈ Z+, then

0 ∗ (x ∗ y)n = (y ∗ x)n

Proof. Cleary 0 ∗ (x ∗ y) = (0 ∗ (0 ∗ y) ∗ x) = y ∗ x. Thus, the equality holds for n = 1
Next, suppose that 0 ∗ (x ∗ y)n = (y ∗ x)n for any n > 1.

So 0 ∗ (x ∗ y)n+1 = 0 ∗ {(x ∗ y) ∗ (0 ∗
n∏
(x ∗ y))}

= 0 ∗ {(x ∗ y) ∗ [[...[[0 ∗ (x ∗ y)] ∗ (x ∗ y)] ∗ ...] ∗ (x ∗ y)]︸ ︷︷ ︸
(x∗y) occurs n times

}

= 0 ∗ {[(x ∗ y) ∗ [0 ∗ (x ∗ y)]] ∗ [[...[[0 ∗ {(x ∗ y)] ∗ (x ∗ y)] ∗ ...] ∗ (x ∗ y)]︸ ︷︷ ︸
(x∗y) occurs n−1 times

}

= 0 ∗ {[(x ∗ y) ∗ (y ∗ x) ∗ [[...[[0 ∗ (x ∗ y)] ∗ (x ∗ y)] ∗ ...] ∗ (x ∗ y)]︸ ︷︷ ︸
(x∗y) occurs n−1 times

}

= 0 ∗ {[[...[(x ∗ y) ∗ (y ∗ x)] ∗ ...] ∗ (y ∗ x)] ∗ (y ∗ x)︸ ︷︷ ︸
(y∗x) occurs n−1 times

] ∗ [0 ∗ (x ∗ y)]}

= 0 ∗ {[[[...[[0 ∗ (y ∗ x)] ∗ ...] ∗ (y ∗ x)] ∗ (y ∗ x)︸ ︷︷ ︸
(y∗x) occurs n−1 times

∗[0 ∗ (x ∗ y)] ∗ 0]}

= (y ∗ x) ∗ [[...[[0 ∗ (y ∗ x)] ∗ (y ∗ x)]] ∗ ...] ∗ (y ∗ x)] ∗ (y ∗ x)︸ ︷︷ ︸
(y∗x) occurs n times

= (y ∗ x) ∗ (0 ∗
n∏
(y ∗ x))



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 17 (1) (2024), 116-123 121

= (y ∗ x)n+1

Therefore, 0 ∗ (x ∗ y)n = (y ∗ x)n for any n ∈ Z+.

Theorem 4. Let (X, ∗, 0) be a B-algebra and x, y ∈ X with n ∈ Z+, then

0 ∗ (0 ∗ x)n = 0 ∗ (0 ∗ xn)

Proof. Since 0 ∗ xn = x−n in [2],

0 ∗ (0 ∗ x)n = (0 ∗ x)−n

= (x−1)−n

= (x−n)−1

= 0 ∗ (x−n)
= 0 ∗ (0 ∗ xn).

Theorem 5. Let (X, ∗, 0) be a B-algebra and a ∈ X, then ⟨a⟩B is a B-ideal of X.

Proof. Clearly, 0 = a0 ∈ ⟨a⟩B. Next, let x ∗ y ∈ ⟨a⟩B and y ∈ ⟨a⟩B, then x ∗ y = ak

and y = ar for some k, r ∈ Z. So, y∗x = 0∗(x∗y) = 0∗ak = a0∗ak = a0−k = a−k ∈< a >
and hance by [1], x = y ∗ (y ∗ x) = ar ∗ a−k = ar−(−k) = ar+k ∈ ⟨a⟩B. Therefore, ⟨a⟩B is
an ideal of X.

In 2023, K. E. Belleza, J. R. Albaracin [8] introduced the concept of dual B-filters of
dual B-algebra. Thus, we can have the following definition:

Definition 5. Let (X, ∗, 0) be a B-algebra. A nonempty subset F of X is called a B-filter
of X if it satisfies the following axioms for all x, y ∈ X:

(i) 0 ∈ F ,

(ii) If x ∗ y ∈ F and x ∈ F , then y ∈ F .

Example 4. Consider the B-algebra X = {0, 1, 2, 3} from Example 1. The sets F1 = {0},
F2 = {0, 2} are B-filters of X while F3 = {0, 3} is not a B-filter of X, since 0 ∗ 1 = 3 ∈ F3

where 0 ∈ F3 but 1 /∈ F3.
Consider (Y,⊙, 0), the sets F4 = {0}, F5 = {0, 3}, F6 = {0, 4}, F7 = {0, 5}, are

B-filters of Y .

Theorem 6. Let (X, ∗, 0) be a B-algebra and a ∈ X, then ⟨a⟩B is a B-filter of X.

Proof. Clearly 0 = a0 ∈ ⟨a⟩B. Next, let x ∗ y ∈ ⟨a⟩B and x ∈ ⟨a⟩B , then x ∗ y = ak

and x = ar for some k, r ∈ Z and hence by theory 3.2 in [1],

y = x ∗ (x ∗ y) = ar ∗ ak = ar−k ∈ ⟨a⟩B

Therefore, ⟨a⟩B is a filter of X.



M. Phattarachaleekul / Eur. J. Pure Appl. Math, 17 (1) (2024), 116-123 122

Theorem 7. Let (X, ∗, 0) be a B-algebra and a ∈ X with 0 ∗ a = a, then ⟨a⟩B = {0, a}
form a B-filter of X.

Proof. Let a ∈ X with 0 ∗ a = a. Consider
a0 = 0,
a1 = a,
a2 = a ∗ a = 0,
a3 = a2 ∗ a = 0 ∗ a = a,
a4 = a3 ∗ a = a ∗ 0 = 0,
.
.
.

This implies that ⟨a⟩B = {0, a}.

Next, Let x, y ∈ X, F = {0, a} with 0 ∗ a = a , x ∗ y ∈ F and x ∈ F.

Case 1 : If x ∗ y = 0 and x = 0, then x ∗ y = 0 ∗ y = 0 = y ∗ y by theorem 1 (vii), y = 0 ∈ F.

Case 2 : If x ∗ y = 0 and x = a, then x ∗ y = a ∗ y = 0 = y ∗ y by theorem 1 (vi), y = a ∈ F.

Case 3 : If x ∗ y = a and x = 0, then x ∗ y = 0 ∗ y = a by assumption a = 0 ∗ a and theorem
1 (viii), y = a ∈ F.

Case 4 : If x ∗ y = a and x = a, then x ∗ y = a ∗ y = a by theorem 1, a = a ∗ y = 0 ∗ (y ∗ a) =
0 ∗ (0 ∗ a) = a implies that (y ∗ a) = (0 ∗ a) and hence y = 0 ∈ F .

Therefore, ⟨a⟩B = F = {0, a} is a B-filter of X.

5. Conclusion

In this paper shown some properties of exponents on B-algebra and we introduces the
notion of B-filter on a B-algebra and presented together with some of its properties on a
cyclic B-algebra, that is for any an element a in a B-algebra (X, ∗, 0), we show that the
set ⟨a⟩B = {ak : k ∈ Z} form a B-ideal and B-filter of X. Moreover, if 0 ∗ a = a, we
obtain ⟨a⟩B = {0, a} form a B-filter of X.

Acknowledgements

This research project was financially supported by Thailand Science Research and
Innovation (TSRI).



REFERENCES 123

References

[1] J. Neggers, and H. S. Kim. On B-algebras. Matematički Vesnik, 54:21–29, 2002.

[2] N. C. Gonzaga and J P. Vilela. On Cyclic B-algebras. Applied Mathematical Sciences,
9:5507–5522, 2015.

[3] Y. B. Jun, E. H. Roh, Chinju and H. S. Kim. On Fuzzy B-algebras. Czechoslovak
Mathematical Journal, 52(127):375–384, 2002.

[4] J. Neggers and H. S. Kim. A fundamental theorem of B-homomorphism for B-
algebras. International Mathematics Journal, 2:207–214, 2002.

[5] D. Al-Kadi. Anti Fuzzy Ideals of B-algebra. International Journal of Pure and Applied
Mathematics, 117(3):437–445, 2017.

[6] K. E. Belleza and J. P. Vilela. The Dual B-Algebra. European Journal of Pure and
Applied Mathematics, 12(4):1497–1507, 2019.

[7] K. E. Belleza and J. P. Vilela. On B-ideals in a Topological B-algebra and the Uniform
B-topological Space. European Journal of Pure and Applied Mathematics, 13(4):830–
839, 2020.

[8] K. E. Belleza and J. R. Albaracin. On dual B-filters and dual B-subalgebras in a
topological dual B-algebra. Journal of Mathematics and Computer Science, 28:1–10,
2023.