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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XV, Issue III, March 2026
Cartesian Product and Direct Product of Finite Prime Fuzzy BD-
Ideals of BD-Algebras
Esraa Kareem Kadhim
1
, Huda Qusay Hashim
2
, Hanan Hayder Mohammed
3
1,3
Department of Mathematic, Faculty of Basic Education, University of Kufa, Najaf, Iraq
2
Department of Mathematic, Faculty of Management Technical, University of AL-Furat AL-Awsat
Technical, Kufa, Iraq
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150300129
Received: 01 April 2026; 06 April 2026; Published: 25 April 2026
ABSTRACT
Fuzzy Bd-ideals provide a useful framework for studying Bd-algebras. This paper focuses on the Cartesian and
direct products of finite prime fuzzy Bd-ideals. We extend the idea of primeness to these constructions and
analyze how it behaves under product operations. Key properties of the Cartesian product are obtained,
including conditions that keep primeness intact. We also study the direct product case and identify when the
prime structure is preserved. These findings give a clearer view of how fuzzy ideals interact within product
structures in Bd-algebras.
Keywords: Bd-algebra, Cartesian Product prime fuzzy Bd-ideal, Cartesian Product semiprime fuzzy Bd-ideal,
Direct Product of Finite Prime.
INTRODUCTION
In 2022, Bantaojai and colleagues [6] introduced a novel algebraic framework known as Bd-algebras by
merging certain features of B-algebras and d-algebras. More recently, in 2024, Nakkhasen et al. [4] extended
fuzzy set theory to Bd-algebras by defining fuzzy Bd-ideals and examining a range of their structural
properties.In this work, we present the notions of prime and semiprime subsets, together with their fuzzy
counterparts, within the setting of Bd-algebras. We further explore the connections between classical prime
(and semiprime) Bd-ideals and their fuzzy analogues. In addition, we introduce the concept of the Cartesian
product of prime and semiprime fuzzy Bd-ideals and investigate several related properties. Finally, we analyze
the behavior of the direct product of finite prime and semiprime fuzzy Bd-ideals in Bd-algebras.
METHODOLOGY
This study uses a theoretical approach to investigate prime and semiprime fuzzy ideals in Bd-algebras. First,
the main definitions of Bd-algebras and fuzzy sets are reviewed. Then, prime and semiprime fuzzy ideals are
introduced and their basic properties are derived using algebraic methods.
After that, the Cartesian and direct products of finite fuzzy ideals are constructed and analyzed to study their
behavior. Several results are proved to show how these properties are preserved under these operations. Finally,
simple examples are given to support the theoretical results.
Preliminaries:This section presents several foundational results that support the demonstration of the principal
theorem.
Definition [6]. An algebraic structure (ℵ,⁕,0) is called a Bd-algebra if ℵ is a non-empty set, ⁕ is a binary
operation defined on ℵ, and 0 is a special element in, such that for all ɕ, ȥ ℵ the following conditions hold: ₺
(i) ɕ ⁕ 0 = ɕ;
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(ii) if ɕ ⁕ ȥ = 0 and ȥ ⁕ ɕ = 0, then ȥ = ɕ.
Definition [4]. Let I be a nonempty subset of a Bd-algebra (ℵ,⁕,0). Then I is called a Bd-ideal of ℵ if it
satisfies the following properties:
(i) 0 I;
(ii) if for all
ꭗ
, ℒ ℵ,
ꭗ
⁕ ℒ I and ℒ I, then
ꭗ
I;
(iii) for every
ꭗ
I and
ꭗ
ℵ,
ꭗ
⁕ℒ I.
Definition [4]. Let (ℵ,⁕,0) be a Bd-algebra. A fuzzy set ᶆ of ℵ is called a fuzzy Bd-ideal of ℵ if, for each
ꭗ
,ℒ
ℵ it satisfies the following conditions:
(i) ᶆ (0) ≥ ᶆ (ꭗ );
(ii) ᶆ(ꭗ ) ≥ min{ᶆ(ꭗ ⁕ ℒ),ᶆ(ℒ)};
(iii) ᶆ(ꭗ ⁕ ℒ) ≥ ᶆ(ꭗ ).
Definition [2]. Let
be a Bd-algebra and let
be a nonempty subset of
Then.
i.
is called prime, if for any ꭗ ,ℒ ℵ, (ꭗ ⁕ ℒ)
, then ꭗ
or ℒ
ii.
is called semiprime, if for any
ꭙ
,
ꭗ ⁕ ꭗ
, then
ꭙ
Definition [2]. Let ℵ be a Bd-algebra and let ᶆ be a fuzzy set of ℵ. Then ᶆ is said to be:
(i) prime if ᶆ(ꭗ ⁕ℒ)≤ max{ᶆ(ꭗ ),ᶆ(ℒ)}, for all ꭗ ,ℒ ℵ;
(ii) semiprime if ᶆ(ꭗ ⁕ ꭗ ) ≤ ᶆ(ꭗ ), for all ꭗ ℵ. It is known that every prime fuzzy set of a Bd-algebra
ℵ is also a semiprime fuzzy set of ℵ.
In general, the converse of this statement is not true as the following example shows:
Example [3]. let
with the binary operation (⁕) on ℵ as follows:
⁕
0
ỿ
e
0
0
y
y
y
ỿ
ỿ
y
ỿ
ỿ
e
e
y
y
e
Then (ℵ,⁕,0). is a Bd-algebra. ᶆ: ℵ→[0,1] , ₺:ℵ→[0,1]
The fuzzy sets ᶆ and ₺ of
defined by
ᶆ (0) =
, ᶆ (ỿ) =
, ᶆ() =ᶆ(e) =
,
and ₺ (0) =
, ₺ (ỿ) =
, ₺() =
,
(e) =
By routine calculate that ᶆ is prime fuzzy set of ℵ. ₺ is semiprime but not prime is prime fuzzy set of
,
because
RESULTS
The outcomes of this study show that prime and semiprime fuzzy ideals in Bd-algebras largely retain their
structural behavior when combined through Cartesian and direct product constructions, under appropriate
conditions. The derived statements describe when these characteristics remain stable within product formations.
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Additionally, the illustrative cases demonstrate the validity of the theoretical results and help clarify how the
concepts operate in a clearer and more intuitive manner.
Definition. Let ꭗ
ꭗ
ꭗ and ꭗ
ꭗ ꭗ
are two prime fuzzy Bd-ideals of ℵ, Then the Cartesian product of and is defined by
such that
is defined by
(ꭗ ,ℒ)
min{
ꭗ
}, for all
ꭗ ,ℒ ℵ
.
Theorem. Let ꭗ
ꭗ
ꭗ and ꭗ
ꭗ ꭗ be is prime fuzzy Bd-ideals of
, then
is prime fuzzy Bd-ideals of
×
.
Proof. For all (ꭗ ⁕ ℒ)
, Now, let
ꭗ
ꭗ
, then
ꭗ
ꭗ
⁕
ꭗ
⁕
ꭗ
⁕
r
,
ꭗ
ꭗ
⁕ ꭗ
,
⁕
ꭗ
ꭗ
Hence
is prime fuzzy Bd-ideals of
.
Definition. Let
ꭗ
ꭗ
ꭗ
and
ꭗ
ꭗ
ꭗ
are prime fuzzy Bd-ideals of
, For s [0,1], the set
ꭗ
ꭗ
is called an upper s-level
Theorem. Let
ꭗ
ꭗ ꭗ
and
ꭗ
ꭗ ꭗ
be is semiprime fuzzy Bd-
ideals of
, then
×
is semiprime fuzzy Bd-ideals of
×
.
Theorem. Let ꭗ
ꭗ ꭗ and ꭗ
ꭗ ꭗ be prime fuzzy Bd-ideals of ,
then the nonempty set upper s-level cut U
is prime fuzzy Bd-ideals of
×
.
Proof. Let A and B be prime fuzzy Bd-ideals of X, therefore for any (ꭗ , ) × ,
Let
ꭗ
ꭗ
,
×
, such that
ꭗ
ꭗ
⁕
, then
ꭗ
ꭗ
⁕
ꭗ
⁕
ꭗ
⁕
ꭗ
ꭗ
max{s, s}= s. Hence U
is prime fuzzy Bd-ideals of ×
In a similar way, we can prove that
is semiprime fuzzy Bd-ideals of
×
Theorem. Let
ꭗ
ꭗ ꭗ
and
ꭗ
ꭗ ꭗ
be semiprime fuzzy Bd-ideals of
, then the nonempty set upper s-level cut U
is semiprime fuzzy.
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Bd-ideals of
×
direct product of finite prime and semiprime fuzzy Bd-ideals of Bd-algebra
Definition. Let
prime fuzzy set of a Bd-algebra
, respectively i=1,2,…,c Then
is called
direct product of finite prime fuzzy set of a Bd-algebra of
ꭗ
ꭗ
ꭗ
⁕
ꭗ
ꭗ
ꭗ
,
for all ꭗ
ꭗ
ꭗ
. And is called direct product of finite semiprime fuzzy set of a Bd-algebra of
ꭗ
ꭗ
ꭗ
⁕ ꭗ
ꭗ
ꭗ
ꭗ
ꭗ
ꭗ
, for all
ꭗ
ꭗ
ꭗ
.
Example. Let
{0,y, d} and
{0,y, d, e} are Bd-algebras by the following tables:
Then
{(0,0),(0,y),(0,d),(0,e),(y,0),(y,y),(y,d),(y,e),(d,0),(d,y),(d,d),(d,e)}
is an Bd-algebra. We defined prime fuzzy Bd-ideals
on
as
by and
(0) =
,
(ỿ) =
,
() =
We also introduced the concept of prime fuzzy Bd-ideals.
on
as
by and
(0) = ,
(y)= ,
() =
(e) = ,
By a standard calculation
is prime fuzzy Bd-ideals of
We also introduced the notion of semiprime fuzzy Bd-ideals.
on
as
(0) =
,
(ỿ) =
,
() =
Defined semiprime fuzzy Bd-ideals
on
as
by
(0) =
,
(ỿ) =
,
() =
,
(e) =
,
By routine calculation
is semiprime fuzzy Bd-ideals of
Theorem. Let
be prime fuzzy Bd-ideals of Bd-algebra
, respectively i=1,2,3,…,c,
is
prime fuzzy Bd-ideals of
Proof. Let
ꭗ
ꭗ
Then
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ꭗ
ꭗ
⁕
ꭗ
⁕
ꭗ
⁕
ꭗ
ꭗ
=
ꭗ
ꭗ
.
Proposition. Let
be semiprime fuzzy set of a Bd-algebra
, respectively i=1,2,3,…,c.
Is
semiprime fuzzy Bd-ideals of
.
Proof. Let ꭗ
ꭗ
Then
ꭗ
ꭗ
⁕ ꭗ
ꭗ
ꭗ
⁕
ꭗ
ꭗ
⁕
ꭗ
ꭗ
ꭗ
DISCUSSION
The results of this study suggest that prime and semiprime fuzzy ideals in Bd-algebras do not lose their main
features when they are combined using Cartesian or direct product, as long as the needed conditions are met. In
most cases, the product structures still reflect the behavior of the original ideals in a clear way.
This shows that there is a close connection between the starting ideals and the structures formed from them,
where the main properties are passed on in a natural manner.
CONCLUSION
This section studies the Cartesian and direct products of finite prime fuzzy Bd-ideals in Bd-algebras and
examines the conditions under which primeness is preserved.
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3. Kadhim E., I.V. of Prime Fuzzy Bd-ideal in Bd-algebra
4. Nakkhasen W., Phimkota S, Phoemkhuen K, Iampan A. , Characteriza tions of fuzzy Bd-ideals in Bd-
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