EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1957-1965
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fuzzification of the Dual B-algebra

Ma. Rizal V. Dicen1,∗, Katrina E. Belleza-Fuentes1

1 Department of Computer, Information Sciences and Mathematics, School of Arts and
Sciences, University of San Carlos, 6000 Cebu City, Philippines

Abstract. This paper introduces the fuzzification of the dual B-algebra and presented some of
its related properties. A characterization of a fuzzy (normal) dual B-algebra is also investigated.

2020 Mathematics Subject Classifications: 47L45, 08C05, 08A72

Key Words and Phrases: Dual algebras, categories of algebras, fuzzy algebraic structures

1. Introduction

In 1965, after the publication of Lotfi Zadeh’s article entitled ”Fuzzy Sets”, the era of
fuzzy mathematics had started. Fuzzy mathematics is the branch of mathematics which
includes fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in
a set on a spectrum, as opposed to simple binary ”yes” or ”no” (0 or 1) inclusion [8].
A fuzzy set is a class of objects with a continuum of grades of membership [8]. The
term fuzzification refers to the process of mapping the numerical input of a system to
fuzzy sets with some degree of membership. This degree of membership can be anywhere
within the closed interval from 0 to 1 [4]. Fuzzy technology is very productive in many
applications especially in the field of computer science and control engineering, expert
system, management science, robotics, to name a few [7].

In 1999, Y.B. Jun,et. al introduced the fuzzification of B-algebras and at the same
time investigated the characterization of fuzzy normal B-algebras [5]. By 2020, M. Ahmed
and E. Ahmed investigated the properties of fuzzy BCK-algebras. In their research,they
applied the concept of fuzzy sets and gave properties to some algebraic structures such as
an ideal, semilattice, lower semilattice, lattice and sub-algebra [1].

In 2019, K. Belleza and J. Vilela, introduced and gave the characterization of the
notion of dual B-algebra and its initial properties [2]. In their research, they investigated
the relationships between the dual B-algebra and the BCK-algebra.

This paper considers the fuzzification of the dual B-algebra and its related properties.
This also aims to give a characterization of a fuzzy dual B-algebra.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4561

Email addresses: marizverdadicen@gmail.com (M. Dicen), kebelleza@usc.edu.ph (K. Belleza-Fuentes)

https://www.ejpam.com 1957 © 2022 EJPAM All rights reserved.



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1958

2. Preliminaries

An algebra of type (2, 0) is an algebra with a binary operation and a constant element.

Definition 1. [3] A binary operation “∗” on a set S is a function mapping S×S→S.
For each element (a, b) ∈ S × S, we will denote ∗((a, b)) ∈ S by a∗b. That is, ∗ is a binary
operation on S if and only if for all a, b ∈ S, a ∗ b ∈ S.

Definition 2. [2] A Dual B-algebra X is a triple (X, ◦, 1) where X is a non-empty
set with a binary operation “◦” and a constant “1” satisfying the following axioms for all
x, y, z in X:

(DB1) x ◦ x = 1; (DB2) 1 ◦ x = x; (DB3) x ◦ (y ◦ z) = ((y ◦ 1) ◦ x) ◦ z.

A non-empty subset N of a dual B-algebra X is called a dual B-subalgebra of X if
x ◦ y ∈ N for any x, y ∈ N . A non-empty subset N of a dual B-algebra X is said to be
normal if (x ◦ f) ◦ (y ◦ g) ∈ N whenever x ◦ y ∈ N and f ◦ g ∈ N . Note that any normal
subset N of a dual B algebra X is a B-subalgebra of X, the converse need not be true [6].
A non-empty subset N of a dual B-algebra X is called a normal B-subalgebra of X if it
is both a dual B-subalgebra and normal.

Lemma 1. [2] Let X be a dual B-algebra. Then for any x, y ∈ X, we have:

(i) x ◦ y = 1; (ii) (x ◦ 1) ◦ 1 = x.

3. Fuzzy Dual B-Algebra

In what follows, let X denotes a dual B-algebra unless otherwise specified.

Definition 3. A fuzzy set µ inX is called a fuzzy dual B-algebra if it satisfies the following
inequality for all x, y ∈ X:

µ(x ◦ y) ≥ min{µ(x), µ(y)}.

Example 1. Let X = {1, A,B,C,D,E} be the set with the following table:

◦ 1 A B C D E

1 1 A B C D E
A B 1 A D E C
B A B 1 E C D
C C D E 1 A B
D D E C B 1 A
E E C D A B 1

Then (X, ◦, 1) is a dual B-algebra by routine calculations.
Define a fuzzy set µ : X → [0, 1] by µ(1)=µ(C)= 0.8 >0.3 = µ(x) for all x ∈ X\{1, C}.
Then, µ is a fuzzy dual B-algebra.



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1959

Proposition 1. Every fuzzy dual B- algebra µ satisfies the inequality µ(1) ≥ µ(x) for all
x ∈ X.

Proof. Note that by (DB1) x ◦ x = 1 for all x ∈ X , which implies that,

µ(1) = µ(x ◦ x) ≥ min {µ(x), µ(x)} = µ(x).

This proves that µ(1) ≥ µ(x) for all x ∈ X.

The following corollary follows immediately from Proposition 1.

Corollary 1. Let X be a fuzzy dual B-algebra. Then for all x ∈ X, µ(x) = µ(x ◦ 1).

Proof. Note that x = (x ◦ 1) ◦ 1 [by Lemma 1(ii)]. If µ is a fuzzy dual B-algebra, then,

µ(x) = µ
(
(x ◦ 1) ◦ 1

)
[by Lemma 1(ii)]

≥ min{µ(x ◦ 1), µ(1)} [by Definition 3]

≥ min{min{µ(x), µ(1)}, µ(1)}
≥ min{µ(x), µ(1)} [by Proposition 1]

= µ(x ◦ 1).

That is, µ(x) = µ(x ◦ 1) for all x ∈ X.

For any elements x and y of X, let us write
n∏
x ◦ y for ((((x ◦ y) ◦ y) ◦ y)...) where y

occurs n times.

Proposition 2. Let a fuzzy set µ in X be a fuzzy dual B-algebra and let n ∈ N. Then
for all x ∈ X,

(i) µ
( n∏

x ◦ x
)
≥ µ(x) whenever n is odd, and

(ii) µ
( n∏

x ◦ x
)
= µ(x) whenever n is even.



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1960

Proof. Let x ∈ X.
(i) Suppose that n is odd. Then n = 2k−1 for some positive integer k. Note that by DB1
and Proposition 1, µ(x ◦ x) = µ(1) ≥ µ(x) when k = 1.

Assume that µ

(
2k−1∏

x ◦ x
)

≥ µ(x) for a positive integer k. Then,

µ

( 2(k+1)−1∏
x ◦ x

)
= µ

( 2k+1∏
x ◦ x

)

= µ

( 2k−1∏
((x ◦ x) ◦ x) ◦ x

)

= µ

( 2k−1∏
(1 ◦ x) ◦ x

)
[by (DB1)]

= µ

( 2k−1∏
x ◦ x

)
[by (DB2)]

= µ(1)

≥ µ(x). [by Proposition 1]

Hence, the above implies that (i) holds.

(ii) Suppose that n is even. Then n = 2k for some positive integer k. For k = 1, n = 2,
we have, µ((x ◦ x) ◦ x) = µ(1 ◦ x) = µ(x) by (DB1) and (DB2).

Suppose that µ

(
2k∏

x ◦ x
)

= µ(x) for a positive integer k. Then,

µ

( 2(k+1)∏
x ◦ x

)
= µ

( 2k+2∏
x ◦ x

)

= µ

( 2k∏
((x ◦ x) ◦ x) ◦ x

)

= µ

( 2k∏
(1 ◦ x) ◦ x

)
[by (DB1)]

= µ

( 2k∏
(x ◦ x)

)
[by (DB2)]

= µ(x).

Hence, the above implies that (ii) holds.



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1961

Proposition 3. If a fuzzy set µ in X is a fuzzy dual B-algebra, then for all x, y ∈ X,

(fDB1) µ(x ◦ 1) ≥ µ(x); (fDB2) µ((y ◦ 1) ◦ x) ≥ min{µ(x), µ(y)}.

Proof. Suppose X is a fuzzy dual B-algebra and µ is a fuzzy set for all x, y ∈ X.

(fDB1): By Definition 3, µ(x ◦ 1) ≥ min{µ(x), µ(1)} ≥ µ(x), by Proposition 1. Hence,
(fDB1) holds.

(fDB2): By Definition 3,

µ((y ◦ 1) ◦ x) ≥ min{µ(y ◦ 1), µ(x)}
≥ min{min{µ(y), µ(1)}, µ(x)}
≥ min{µ(y), µ(x)} [by Proposition 1]

≥ min{µ(x), µ(y)}.

Hence, (fDB2) holds.

The next theorem is a characterization of a fuzzy dual B-algebra which is the converse
of Proposition 3.

Theorem 1. If a fuzzy set µ in X satisfies (fDB1) and (fDB2), then µ is a fuzzy dual
B-algebra.

Proof. Suppose that a fuzzy set µ in X satisfies the conditions in (fDB1) and (fDB2).
Let x, y ∈ X. Then we have,

µ(x ◦ y) = µ(((x ◦ 1) ◦ 1) ◦ y) [by Lemma 1(i)]

≥ min{µ((x ◦ 1) ◦ 1), µ(y)}
≥ min{min{µ(x ◦ 1), µ(1)}, µ(y)}
≥ min{µ(x ◦ 1), µ(y)} [by (fDB2)]

= min{µ(x), µ(y)}. [by (fDB1)]

Hence, µ is a fuzzy dual B-algebra.

In what follows is an example of the above theorem.

Example 2. The fuzzy set µ discussed in Example 1 satisfies (fDB1) and (fDB2). Con-
sidering the example below:

Let x = A and y = B.
Note that µ(A ◦B) = µ(((A ◦ 1) ◦ 1) ◦B) = µ(A) = 0.3. Then we have

µ(A ◦B) = µ(((A ◦ 1) ◦ 1) ◦B)



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1962

≥ min{µ((A ◦ 1) ◦ 1), µ(B)}
≥ min{µ(B), µ(B)}
= 0.3.

Hence, µ is indeed a fuzzy dual B-algebra.

Definition 4. A fuzzy set µ in X is said to be fuzzy normal if it satisfies the following
inequality for all f, g, x, y ∈ X:

µ((f ◦ x) ◦ (g ◦ y)) ≥ min{µ(f ◦ g), µ(x ◦ y)}.

Example 3. Refer to the dual B-algebra from Example 1. Define a fuzzy set δ : X → [0, 1]
by δ(1) = δ(A) = δ(B) = 0.6 and δ(C) = δ(D) = δ(E) = 0.1, then δ is a fuzzy normal set
in X by routine calculations.

Example 4. Let X = {1, a, b, c} be a set with the following table:

◦ 1 a b c

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

Then (X, ◦, 1) is a dual B-algebra by routine calculations. Suppose we define a map
µ : X → [0, 1] by µ(1) > µ(b) > µ(a) = µ(c), then µ is a fuzzy normal set in X. And also,
if we define a map φ : X → [0, 1] by φ(1) = φ(b) > φ(a) = φ(c), then φ is also a fuzzy
normal set in X.

Theorem 2. Every fuzzy normal set µ in X is a fuzzy dual B-algebra.

Proof. Since µ is fuzzy normal, recall Definition 4. For any x, y ∈ X, we have,

µ((x ◦ y) = µ((1 ◦ 1) ◦ (x ◦ y))
≥ min{µ(1 ◦ x), µ(1 ◦ y)}
= min{µ(x), µ(y)}. [by (DB2)]

Hence, µ is a fuzzy dual B-algebra.

Remark 1. The converse of Theorem 2 is not always true.
Example 1 above is the correct representation of this remark.
Now, note that µ((C ◦D) ◦ (1 ◦B)) = µ(B ◦A) = µ(A) = 0.3. Then we have,

µ((C ◦D) ◦ (1 ◦B)) = min{µ(C ◦ 1), µ(D ◦B)}
= min{µ(C), µ(C)}
< 0.8.



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1963

Manipulation above shows that the fuzzy dual B-algebra is not fuzzy normal. Thus,
the remark holds true.

Definition 5. A fuzzy set µ in X is called fuzzy normal dual B-algebra if it is a fuzzy
dual B-algebra which is fuzzy normal.

Example 5. The fuzzy sets discussed in Example 3 and Example 4 are indeed fuzzy
normal dual B-algebras by routine calculations.

Proposition 4. If a fuzzy set µ in X is a fuzzy normal dual B-algebra, then
µ(x ◦ y) = µ(y ◦ x) for all x, y ∈ X.

Proof. Let x, y ∈ X. Then

µ(x ◦ y) = µ(1 ◦ (x ◦ y)) [by DB2]

= µ((y ◦ y) ◦ (x ◦ y)) [by DB1]

≥ min{µ(y ◦ x), µ(y ◦ y)} [since µ is fuzzy normal]

= min{µ(y ◦ x), µ(1)} [by DB1]

≥ µ(y ◦ x). [by Proposition 1]

Interchanging x with y, we obtain

µ(y ◦ x) = µ(1 ◦ (y ◦ x)) [by DB2]

= µ((x ◦ x) ◦ (y ◦ x)) [by DB1]

≥ min{µ(x ◦ y), µ(x ◦ x)} [since µ is fuzzy normal]

= min{µ(x ◦ y), µ(1)} [DB1]

≥ µ(x ◦ y). [by Proposition 1]

By the manipulation above, we conclude that µ(x ◦ y) = µ(y ◦ x) for all x, y ∈ X.

The following result will be used in the characterization of fuzzy normal dual B-algebra.

Theorem 3. Let µ be a fuzzy normal dual B-algebra. Then the following set is a normal
dual B-subalgebra of X,

Xµ = {x ∈ X | µ(x) = µ(1)}.

Proof. It is sufficient to show that Xµ is normal since µ is a fuzzy normal dual B-
algebra. Let f, g, x, y ∈ X be such that x ◦ y ∈ Xµ and f ◦ g ∈ Xµ. Then µ(x ◦ y) = µ(1) =
µ(f ◦ g). Considering µ is fuzzy normal, it follows that

µ((x ◦ f) ◦ (y ◦ g)) ≥ min{µ(x ◦ y), µ(f ◦ g)} ≥ min{µ(1), µ(1)} = µ(1).

Now, by Proposition 1,

µ(1) ≥ min{µ(x ◦ y), µ(f ◦ g)} ≥ µ((x ◦ f) ◦ (y ◦ g)).



K. Belleza, M. R. V. Dicen / Eur. J. Pure Appl. Math, 15 (4) (2022), 1957-1965 1964

Therefore, by the manipulation above, µ((x ◦ f) ◦ (y ◦ g)) = µ(1), which implies that
(x ◦ f) ◦ (y ◦ g) ∈ Xµ.

Theorem 4. The intersection of any set of a fuzzy normal dual B-algebras is also fuzzy
normal dual B-algebra.

Proof. Let {µω | ω ∈ I} be a family of fuzzy normal dual B-algebras and let f, g, x, y ∈
X. Then by the greatest lower bound of fuzzy set, definition of fuzzy normal, and by the
minimum or infimum property, we have:( ⋂

ω∈I
µω

)
((x ◦ f) ◦ (y ◦ g)) = inf

ω∈I
µω((x ◦ f) ◦ (y ◦ g))

≥ inf
ω∈I

{
min{µω(x ◦ y), µω(f ◦ g)}

}
= min{ inf

ω∈I
(x ◦ y), inf

ω∈I
(f ◦ g)}

= min
{( ⋂

ω∈I
µω

)
(x ◦ y),

( ⋂
ω∈I

µω

)
(f ◦ g)

}
,

which implies that
⋂
ω∈I

µω is a fuzzy normal set in X. By Theorem 2, therefore
⋂
ω∈I

µω is

a fuzzy normal dual B-algebra.

Remark 2. The union of any set of fuzzy dual B-algebras may not be a fuzzy dual
B-algebra as shown in the next example.

Example 6. Referring to Example 1. Define a fuzzy set δ : X → [0, 1] by δ(1) = δ(D) =
0.6 > 0.1 = δ(A) = δ(B) = δ(C) = δ(E), then it is also a fuzzy dual B-algebra. Now,

(µ ∪ δ)(C ◦D) = (µ ∪ δ)(A) = 0.3 [by union property of fuzzy set]

and min{(µ ∪ σ)(C), (µ ∪ σ)(D)} = min{0.8, 0.6} = 0.6.

Notice that (µ ∪ δ)(C ◦D) < min{(µ ∪ σ)(C), (µ ∪ σ)(D)} which does not satisfy the
definition. Thus, µ ∪ δ is not a fuzzy dual B-algebra. Recall from Theorem 2 that every
fuzzy normal dual B-algebra is a fuzzy dual B-algebra. However, in view of Remark 2,
the union of fuzzy normal dual B-algebras may not be a fuzzy normal B-algebra.

4. Conclusion

In this paper, the notion of a fuzzy dual B-algebra is presented together with some of
its properties and characterizations. Here it is considered that every fuzzy normal set is a
fuzzy dual B algebra but the converse is not always true. While undergoing an operation
in fuzzy sets, the union of any set of fuzzy dual B-algebra may not be a fuzzy dual B-
algebra. However, the intersection of any set of a fuzzy normal dual B-algebra is also a
fuzzy normal dual B-algebra. The difference between a fuzzy dual B-algebra and fuzzy
normal dual B-algebra is also presented.



REFERENCES 1965

Acknowledgements

The authors are grateful to the Department of Science and Technology (DOST) through
the Accelerated Science and Technology Human Resource Development Program (ASTHRDP)
and its partner university, University of San Carlos who funded and made this research
possible.

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