EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6189 ISSN 1307-5543 – ejpam.com Published by New York Business Global Applications of Strongest Fuzzy Dot Bd-Subalgebras in Bd-Algebras Warud Nakkhasen1,∗, Narinthon Jaroenwan1, Panida Huekkhunthod1, Atthchai Chada2 1 Department of Mathematics, Faculty of Science, Mahasarakham University, Maha Sarakham 44150, Thailand 2 Department of Mathematics, Faculty of Science and Technology, Rajabhat Mahasarakham University, Maha Sarakham 44000, Thailand Abstract. In 2024, Nakkhasen et al. introduced the concept of fuzzy Bd-subalgebras of Bd- algebras. This paper will present the concept of fuzzy dot Bd-subalgebras in Bd-algebras as a generalization of fuzzy Bd-subalgebras. Subsequently, we evaluate the connections of fuzzy dot Bd-subalgebras in the framework of a homomorphism of Bd-algebras. Finally, we propose the concept of strongest fuzzy dot Bd-subalgebras and explain their characteristics in relation to Bd- subalgebras. 2020 Mathematics Subject Classifications: 08A30, 08A72 Key Words and Phrases: Bd-algebra, Bd-subalgebra, fuzzy Bd-subalgebra, fuzzy dot Bd- subalgebra 1. Introduction The subalgebras, such as BCK/BCI-subalgebras [1], BE-subalgebras [2], and BG- subalgebras [3], are the most frequently examined concepts when examining the charac- teristics of different ideas in each algebraic structure. Exploring the characteristics of non-empty subsets of an algebra and applying the same operations as that algebra while preserving the structure of the original algebra are key components of the subalgebra notion. In B-algebras, Walendziak [4] investigated the concept of normal subalgebras by showing that the notion of a normal subalgebra is equivalent to the normal subgroup of the derived group.Jun et al. [5] researched d-algebras using the theory of a falling shadow. To accomplish this, they developed the concept of falling d-subalgebras and examined their various characteristics, including how to classify the properties of falling d-subalgebras in d-algebras.In BCK/BCI-algebras, Balami et al. [6] presented the concepts of soft ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6189 Email addresses: warud.n@msu.ac.th (W. Nakkhasen), 64010213033@msu.ac.th (N. Jaroenwan), 64010213041@msu.ac.th (P. Huekkhunthod), atthchaichada@gmail.com (A. Chada) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 2 of 13 BCK/BCI-algebras and soft BCK/BCI-subalgebras and discussed some of their char- acteristics. Mathematicians also study the features of subalgebras in additional intriguing algebraic structures, such as [7], [8], [9], and [10]. Those who are interested can learn more about these structures. The concept of the fuzzy set of δ in a non-empty set X is a function δ from X to the closed interval [0, 1] in the real numbers. This concept was introduced by Zadeh [11], and it has since become a fundamental tool in various fields, such as artificial intelligence, control systems, and decision-making processes. Fuzzy sets allow for the representation of uncertain and imprecise information, enabling more flexible and realistic modelling than traditional binary sets. Rosenfeld [12] applied fuzzy sets to establish the concepts of fuzzy subgroups and fuzzy ideals in groups. Subsequently, Kuroki [13] examined the classifi- cations of fuzzy subsemigroups, investigating the features and uses in semigroups. Next, Rezaei and Saeid [14] developed the concept of fuzzy subalgebras into BE-algebras and studied various characterizations of these fuzzy subalgebras. Afterwards, Muhiuddin [15] defined the concept of (∈,∈ ∨qδ0)-fuzzy subalgebras of BCK/BCI-algebras as a more gen- eral type of (∈,∈ ∨q)-fuzzy subalgebras. Following that, Tacha et al. [16] introduced the concepts of length fuzzy UP -subalgebras and mean fuzzy UP -subalgebras of UP -algebras. The researchers also investigated the relationships between length fuzzy UP -subalgebras (mean fuzzy UP -subalgebras) and hyper fuzzy UP -subalgebras of UP -subalgebras. For research related to fuzzy subalgebras, further studies can be found in [17], [18], [19], and [20]. For the general concept of fuzzy subalgebras, another concept that has been con- tinuously studied is the fuzzy dot. Saeid [21] presented the notion of fuzzy dot BCK- subalgebras and fuzzy dot topological BCK-algebras within the framework of BCK- algebras. Following this, Senapati et al. [22] provided definitions and examined related features of fuzzy dot subalgebras, fuzzy normal dot subalgebras, and fuzzy dot ideals of BG-algebras. In the same year, Senapati et al. [23] introduced fuzzy dot subalgebras, fuzzy normal dot subalgebras, and fuzzy dot ideals to the investigation in B-algebras. Later, Dejen [24] gave the idea of fuzzy dot subalgebras in the structure of fuzzy dot d-subalgebras and studied several of its characteristics. In addition, Jiang [25] established the notions of hesitant fuzzy dot subalgebras, hesitant fuzzy normal dot subalgebras, and hesitant fuzzy dot ideals of B-algebras and explored properties related to these concepts of B-algebras. In 2022, the concept of Bd-algebras is derived from certain properties of d-algebras and B-algebras, introduced by Bantaojai et al. [26], who defined various concepts, one of which is Bd-subalgebras. Thereafter, the notion of fuzzy Bd-subalgebras of Bd-algebras was recently defined by Nakkhasen et al. [27]. In their presentation, they discussed an op- portunity of fuzzy multiplications, fuzzy magnified translations, and fuzzy translations to characterize fuzzy Bd-subalgebras in Bd-algebras. To further investigate the general idea of fuzzy Bd-subalgebras, this article will introduce the notion of fuzzy dot Bd-subalgebras, which serves as a generalization of fuzzy Bd-subalgebras in Bd-algebras. In Section 3, we explore certain features of fuzzy dot Bd-subalgebras in Bd-algebras. Moreover, we further examine the relationships of fuzzy dot Bd-subalgebras under a homomorphism of Bd- W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 3 of 13 algebras. Finally, Section 4 presents the concept of strongest fuzzy dot Bd-subalgebras in Bd-algebras and studies the connections between strongest fuzzy dot Bd-subalgebras and fuzzy dot Bd-subalgebras in Bd-algebras, while characterizing the strongest fuzzy dot Bd-subalgebras via Bd-subalgebras of Bd-algebra. 2. Preliminaries In this section, we will review the fundamental concepts necessary for use in the fol- lowing sections. Let X be a nonempty set. A fuzzy set [11] ζ of X is a function from X into the interval [0, 1]. Let ζ and ξ be any two fuzzy sets of a nonempty set X. Then we denote: (i) (ζ ∩ ξ)(x) := min{ζ(x), ξ(x)} for all x ∈ X; (ii) (ζ ∪ ξ)(x) := max{ζ(x), ξ(x)} for all x ∈ X. Let {ri | i ∈ Λ} be a family of real numbers. Then we denote ⋂ i∈Λ ri = min i∈Λ ri if Λ is finite, inf i∈Λ ri otherwise. Let X be a nonempty set and ζ be a fuzzy set of X. The set ζt := {x ∈ X | ζ(x) ≥ t}, where t ∈ [0, 1] is celled a level subset of ζ (see, [27]). Let A be a subset of a nonempty set X. The characteristic function (see, [27]) CA of A is a fuzzy set of X, defined by for every x ∈ X, CA(x) = { 1 if x ∈ A, 0 otherwise. Lemma 1. If a, b, c, d are elements in real numbers, then min{ab, cd} ≥ min{a, c}min{b, d}. Proof. Without loss of generality, assume that a ≤ c. Then min{a, c} = a. If b ≤ d, then min{b, d} = b. Thus, ab ≤ cb ≤ cd. It turns out that min{ab, cd} = ab = min{a, c}min{b, d}. On the other case, if d < b, then min{b, d} = d. Since a ≤ c and d < b, we have ad ≤ cd and ad < ab. This implies that min{ab, cd} ≥ ad = min{a, c}min{b, d}. Therefore, min{ab, cd} ≥ min{a, c}min{b, d}. Definition 1. [26] Let X be a nonempty set and ∗ be a binary operation on X. An algebraic structure (X, ∗, 0) is called a Bd-algebra if it satisfies the following conditions: for each x, y ∈ X, (i) x ∗ 0 = x; (ii) if x ∗ y = 0 and y ∗ x = 0, then x = y. Throughout this study, we denote a Bd-algebra (X, ∗, 0) by X the bold letter of its universe set. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 4 of 13 Definition 2. [26] Let X be a Bd-algebra. A nonempty subset A of X is said to be a Bd-subalgebra of X if 0 ∈ A and x ∗ y ∈ A for all x, y ∈ A. Definition 3. [27] Let X be a Bd-algebra. A fuzzy set ζ of X is called a fuzzy Bd- subalgebra of X if it satisfies the following inequality: for any x, y ∈ X, (i) ζ(0) ≥ ζ(x); (ii) ζ(x ∗ y) ≥ min{ζ(x), ζ(y)}. 3. Fuzzy dot Bd-subalgebras of Bd-algebras In this section, we apply the usual multiplication in real numbers to the fuzzy sets by introducing the notion of fuzzy dot Bd-subalgebras, which is useful as a generalization of fuzzy Bd-subalgebras in the Bd-algebras. Then we examine certain characteristics of fuzzy dot Bd-subalgebras within the Bd-algebras. Subsequently, we investigate the connections of fuzzy dot Bd-subalgebras under a homomorphism of Bd-algebras. Definition 4. Let X be a Bd-algebra, and let ζ be a fuzzy sets of X. Then ζ is called a fuzzy dot Bd-subalgebra of X if for every x, y ∈ X: (i) ζ(0) ≥ ζ(x); (ii) ζ(x ∗ y) ≥ ζ(x) · ζ(y), where “·” denotes ordinary multiplication in real numbers. Example 1. Let X = {0, a, b, c} and ∗ be a binary operation on X as defined in the following table: ∗ 0 a b c 0 0 0 a 0 a a b a a b b b b c c c a a c Table 1: The binary operation ∗ on X. Then X := (X, ∗, 0) is a Bd-algebra. Define a fuzzy set ζ of X by ζ(0) = 0.80, ζ(a) = 0.60, ζ(b) = 0.70, and ζ(c) = 0.70. By calculate routine, we have ζ is a fuzzy dot Bd-subalgebra of X. Proposition 1. Every fuzzy Bd-subalgebra of a Bd-algebra X is also a fuzzy dot Bd- subalgebra. Proof. Let ζ is a fuzzy Bd-subalgebra of a Bd-algebra X. Then ζ(0) ≥ ζ(a) for all a ∈ X. Now, let x, y ∈ X. If ζ(x) ≤ ζ(y), then min{ζ(x), ζ(y)} = ζ(x). So, ζ(x ∗ y) ≥ min{ζ(x), ζ(y)} = ζ(x) ≥ ζ(x) · ζ(y). On the other hand, if ζ(x) > ζ(y), then W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 5 of 13 min{ζ(x), ζ(y)} = ζ(y). Thus, ζ(x ∗ y) ≥ min{ζ(x), ζ(y)} = ζ(y) ≥ ζ(x) · ζ(y). Hence, ζ is a fuzzy dot Bd-subalgebra of X. Generally, the fuzzy dot Bd-subalgebras need not be a fuzzy Bd-subalgebras in Bd- algebras, as shown in the following example. Example 2. In Example 1, we have the fuzzy set ζ is a fuzzy dot Bd-subalgebra of a Bd-algebra X. However, ζ is not a fuzzy Bd-subalgebra of X, because ζ(0 ∗ b) = 0.60 ≱ 0.70 = min{ζ(0), ζ(b)}. Proposition 2. Let ζ be a fuzzy dot Bd-subalgebra of a Bd-algebra X. If there exists a nonempty subset A of X such that supa∈A ζ(a) = 1, then ζ(0) = 1. Proof. Assume that X contains a nonempty subset A of X such that supa∈A ζ(a) = 1. Since ζ is a fuzzy dot Bd-subalgebra of X, we have ζ(0) ≥ ζ(x) for all x ∈ X. Then 1 ≥ ζ(0) ≥ supa∈A ζ(a) = 1. This implies that ζ(0) = 1. Let ζ be any fuzzy set of a nonempty set X, and m be a positive integer. Define a fuzzy set ζm of X by ζm(x) = (ζ(x))m for all x ∈ X. Proposition 3. Let X be a Bd-algebra. If ζ is a fuzzy dot Bd-subalgebra of X, then ζm is also a fuzzy dot Bd-subalgebra of X whenever m is a positive integer. Proof. Let ζ be a fuzzy dot Bd-subalgebra of X and m be a positive integer. For every x, y ∈ X, we have ζm(0) = (ζ(0))m ≥ (ζ(x))m = ζm(x) and ζm(x ∗ y) = (ζ(x ∗ y))m ≥ (ζ(x) · ζ(y))m = (ζ(x))m · (ζ(y))m = ζm(x) · ζm(y). Consequently, ζm is a fuzzy dot Bd-subalgebra of X. Theorem 1. Let X be a Bd-algebra. If ζ and ξ are fuzzy dot Bd-subalgebras of X, then ζ ∩ ξ is a fuzzy dot Bd-subalgebra of X. Proof. Assume that ζ and ξ are fuzzy dot Bd-subalgebras of X. Let x, y ∈ X. By Lemma 1 and assumption, we have (ζ ∩ ξ)(0) = min{ζ(0), ξ(0)} ≥ min{ζ(x), ξ(x)} = (ζ ∩ ξ)(x) and (ζ ∩ ξ)(x ∗ y) = min{ζ(x ∗ y), ξ(x ∗ y)} ≥ min{ζ(x) · ζ(y), ξ(x) · ξ(y)} ≥ min{ζ(x), ξ(x)} ·min{ζ(y), ξ(y)} = (ζ ∩ ξ)(x) · (ζ ∩ ξ)(y). Therefore, ζ ∩ ξ is a fuzzy dot Bd-subalgebra of X. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 6 of 13 Corollary 1. Let {ζi | i ∈ Λ} be a family of fuzzy dot bd-subalgebras of a Bd-algebra X. Then ⋂ i∈Λ ζi is also a fuzzy dot Bd-subalgebra of X, where Λ is any index set. Proof. Let ζ := ⋂ i∈Λ ζi. We recall that ζ(x) = ⋂ i∈Λ ζi(x) = infi∈Λ ζi(x) for all x ∈ X. For every x, y ∈ X, we have ζ(0) = infi∈Λ ζi(0) ≥ infi∈Λ ζi(x) = ζ(x) and ζ(x ∗ y) = infi∈Λ ζi(x ∗ y) ≥ infi∈Λ ζi(x) · ζi(y) = infi∈Λ ζi(x) · infi∈Λ ζi(y) = ζ(x) · ζ(y). Hence, ⋂ i∈Λ ζi is a fuzzy dot Bd-subalgebra of X. Example 3. By Example 1, we have the ζ is a fuzzy dot Bd-subalgebra of a Bd-algebra X where ζ(0) = 0.80, ζ(a) = 0.60, ζ(b) = 0.70, and ζ(c) = 0.70. Additionally, we define a fuzzy dot Bd-subalgebra ξ of X by ξ(0) = 0.90, ξ(a) = 0.60, ξ(b) = 0.60, and ξ(c) = 0.50. We obtain that (ζ ∪ ξ)(0 ∗ b) = 0.60 ≱ 0.63 = (ζ ∪ ξ)(0) · (ζ ∪ ξ)(b). This show that ζ ∪ ξ is not a fuzzy dot Bd-subalgebra of X. From Example 3, we conclude that the union of fuzzy dot Bd-subalgebras of Bd- algebras doesn’t necessarily have to be a fuzzy dotBd-subalgebra ofBd-algebras in general. Theorem 2. Let X be a Bd-algebra, and A be a nonempty subset of X. Then A is a Bd-subalgebra of X if and only if CA is a fuzzy dot Bd-subalgebra of X. Proof. Assume that A is a Bd-subalgebra of X. Then 0 ∈ A. So, CA(0) = 1 ≥ CA(x) for all x ∈ X. Suppose that there exist a, b ∈ X such that CA(a ∗ b) < CA(a) · CA(b). Thus, CA(a ∗ b) = 0 and CA(a) · CA(b) = 1; that is, CA(a) = 1 and CA(b) = 1. It follows that a∗ b ̸∈ A and a, b ∈ A. By assumption, we have a∗ b ∈ A, which is a contradiction. Hence, CA(x ∗ y) ≥ CA(x) · CA(y) for all x, y ∈ X. Therefore, CA is a fuzzy dot Bd-subalgebra of X. Conversely, assume that CA is a fuzzy dot Bd-subalgebra of X. If 0 ̸∈ A, then 0 = CA(0) ≥ CA(x) for all x ∈ X. Also, CA(x) = 0 for all x ∈ X, implies that A = ∅. This is a contradiction. So, 0 ∈ A. Next, let x, y ∈ A. Then, CA(x ∗ y) ≥ CA(x) · CA(y) = 1. We obtain that CA(x ∗ y) = 1; that is, x ∗ y ∈ A. Consequently, A is a Bd-subalgebra of X. Theorem 3. Let X be a Bd-algebra, and ζ be a fuzzy set of X. If a nonempty level subset ζt is a Bd-subalgebra of X for all t ∈ [0, 1], then ζ is a fuzzy dot Bd-subalgebra of X. Proof. Let x, y ∈ X. Take ζ(x) = t′ for some t′ ∈ [0, 1]. Then x ∈ ζt′ , and so ζt′ ̸= ∅. By assumption, we have ζt′ is a Bd-algebra of X. That is, 0 ∈ ζt′ . It follows that ζ(0) ≥ t′ = ζ(x). Next, letting ζ(x) · ζ(y) = s′ for some s′ ∈ [0, 1]. Since ζ(x), ζ(y) ∈ [0, 1], we get ζ(x) ≥ ζ(x) · ζ(y) = s′ and ζ(y) ≥ ζ(x) · ζ(y) = s′. Thus, x, y ∈ ζs′ . By the given assumption, we have x ∗ y ∈ ζs′ . This implies that ζ(x ∗ y) ≥ s′ = ζ(x) · ζ(y). Therefore, ζ is a fuzzy dot Bd-subalgebra of X. The converse of Theorem 3 is not always true, as shown in the following example. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 7 of 13 Example 4. In Example 1, we have the fuzzy set ζ is a fuzzy dot Bd-subalgebra of X, but the level subset ζ0.70 = {0, b, c} of ζ is not a Bd-subalgebra of X, since 0 ∗ b = a ̸∈ ζ0.70. Let X := (X, ∗, 0X) and Y := (Y, ◦, 0Y ) be Bd-algebras. Let ω : X → Y be a mapping of Bd-algebras X and Y, and let ζ be a fuzzy set of Y. The fuzzy set ζω of X is defined by ζω(x) = ζ(ω(x)) for all x ∈ X. A function ω : X → Y is called a homomorphism if ω(0X) = 0Y and ω(x ∗ y) = ω(x) ◦ ω(y) for all x, y ∈ X, and a homomorphism ω is called an epimorphism if ω is onto. Theorem 4. Let ω : X → Y be a homomorphism of Bd-algebras X := (X, ∗, 0X) and Y := (Y, ◦, 0Y ). If ζ is a fuzzy dot Bd-subalgebra of Y, then ζω is a fuzzy dot Bd-subalgebra of X. Proof. Assume that ζ is a fuzzy dot Bd-subalgebra of Y. Let x, y ∈ X. Then we have ζω(0X) = ζ(ω(0X)) = ζ(0Y ) ≥ ζ(ω(x)) = ζω(x) and ζω(x ∗ y) = ζ(ω(x ∗ y)) = ζ(ω(x) ◦ ω(y)) ≥ ζ(ω(x)) · ζ(ω(y)) = ζω(x) · ζω(y). Thus, ζω is a fuzzy dot Bd-subalgebra of X. By adding specific properties into Theorem 4, the converse of this Theorem will ulti- mately hold true as delineated below. Theorem 5. Let ω : X → Y be an epimorphism of Bd-algebras X := (X, ∗, 0X) and Y := (Y, ◦, 0Y ). If ζω is a fuzzy dot Bd-subalgebra of X, then ζ is a fuzzy dot Bd- subalgebra of Y. Proof. Assume that ζω is a fuzzy dot Bd-subalgebra of X. Let a, b ∈ Y . Then there exist x, y ∈ X such that ω(x) = a and ω(y) = b. Thus, we have ζ(0Y ) = ζ(ω(0X)) = ζω(0X) ≥ ζω(x) = ζ(ω(x)) = ζ(a) and ζ(a◦b) = ζ(ω(x)◦ω(y))) = ζ(ω(x∗y)) = ζω(x∗y) ≥ ζω(x)·ζω(y) = ζ(ω(x))·ζ(ω(y)) = ζ(a)·ζ(b). Therefore, ζ is a fuzzy dot Bd-subalgebra of Y. 4. Strongest fuzzy dot Bd-subalgebras on Bd-algebras In this section, we present some properties of the Cartesian product of fuzzy dot Bd- subalgebras of Bd-algebras. After that, we introduce the concept of strongest fuzzy dot Bd-subalgebras on Bd-algebras and investigate some of its properties and the relationships between strongest fuzzy dot Bd-subalgebras and fuzzy dot Bd-subalgebras in Bd-algebras. W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 8 of 13 Finally, we characterize the strongest fuzzy dot Bd-subalgebras by Bd-subalgebras of Bd- algebras. Let X := (X, ∗, 0X) and Y := (Y, ◦, 0Y ) be Bd-algebras. The mapping ⊛ : (X × Y )× (X × Y ) → X × Y is defined by (x1, y1)⊛ (x2, y2) = (x1 ∗ x2, y1 ◦ y2) for all (x1, y1), (x2, y2) ∈ X × Y . We have that X × Y := (X × Y,⊛, (0X , 0Y )) is a Bd- algebra. In particular, if Y = X, we have X×X := (X ×X,⊛, (0X , 0X)) is a Bd-algebra where the binary operation ⊛ onX×X is defined by (x1, y1)⊛(x2, y2) = (x1∗x2, y1∗y2) for all (x1, y1), (x2, y2) ∈ X×X. Throughout this section, the Bd-algebra (X×X,⊛, (0X , 0X)) will be replaced by the symbol X×X. Let ζ and ξ be a fuzzy sets of a nonempty set X. The Cartesian product [28] ζ × ξ : X ×X → [0, 1] is defined by (ζ × ξ)(x, y) = ζ(x) · ξ(y) for all x, y ∈ X. Theorem 6. Let X be a Bd-algebra. If ζ and ξ are fuzzy dot Bd-subalgebras of X, then ζ × ξ is a fuzzy dot Bd-subalgebra of X×X. Proof. Assume that ζ and ξ are fuzzy dot Bd-subalgebras of X. Let (x1, y1), (x2, y2) ∈ X ×X. Then we have (ζ × ξ)(0, 0) = ζ(0) · ξ(0) ≥ ζ(x1) · ξ(y1) = (ζ × ξ)(x1, y1) and (ζ × ξ)((x1, y1)⊛ (x2, y2)) = (ζ × ξ)(x1 ∗ x2, y1 ∗ y2) = ζ(x1 ∗ x2) · ξ(y1 ∗ y2) ≥ [ζ(x1) · ζ(x2)] · [ξ(y1) · ξ(y2)] = [ζ(x1) · ξ(y1)] · [ζ(x2) · ξ(y2)] = (ζ × ξ)(x1, y1) · (ζ × ξ)(x2, y2). Consequently, ζ × ξ is a fuzzy dot Bd-subalgebra of X×X. The converse of Theorem 6 is not true, as proved by the following example. Example 5. Let X = {0, 1, 2} be a set with the binary operation ∗ on X define in the following table: ∗ 0 1 2 0 0 2 2 1 1 0 2 2 2 1 1 Table 2: The binary operation ∗ on X. Then X := (X, ∗, 0) is a Bd-algebra. Define two fuzzy sets ζ and ξ of X by W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 9 of 13 ζ(0) = 0.70, ζ(1) = 0.70, ζ(2) = 0.40 and ξ(0) = 0.60, ξ(1) = 0.50, ξ(2) = 0.60. It is not difficult to verify that ξ is a fuzzy dot Bd-subalgebra of X, but ζ is not a fuzzy dot Bd-subalgebra of X. At the same time, ζ × ξ is also a fuzzy dot Bd-subalgebra of X×X as shown below. Now, we consider the results of the Cartesian product ζ × ξ as follows. (ζ × ξ)(0, 0) = 0.42,(ζ × ξ)(0, 1) = 0.35, (ζ × ξ)(0, 2) = 0.42, (ζ × ξ)(1, 0) = 0.42,(ζ × ξ)(1, 1) = 0.35, (ζ × ξ)(1, 2) = 0.42, (ζ × ξ)(2, 0) = 0.24,(ζ × ξ)(2, 1) = 0.20, (ζ × ξ)(2, 2) = 0.24. We see that (ζ × ξ)(0, 0) ≥ (ζ × ξ)(x, y) for all (x, y) ∈ X ×X. Subsequently, below are some computed results. (ζ × ξ)((0, 2)⊛ (2, 1)) = (ζ × ξ)(2, 1) = 0.20 > 0.08 = (ζ × ξ)(0, 2) · (ζ × ξ)(2, 1), (ζ × ξ)((1, 1)⊛ (0, 1)) = (ζ × ξ)(1, 0) = 0.42 > 0.12 = (ζ × ξ)(1, 1) · (ζ × ξ)(0, 1), (ζ × ξ)((2, 0)⊛ (1, 2)) = (ζ × ξ)(1, 2) = 0.42 > 0.10 = (ζ × ξ)(2, 0) · (ζ × ξ)(1, 2), (ζ × ξ)((2, 2)⊛ (2, 1)) = (ζ × ξ)(1, 0) = 0.42 > 0.08 = (ζ × ξ)(2, 2) · (ζ × ξ)(1, 1). By meticulous computations, we have (ζ × ξ)((x1, y1) ⊛ (x2, y2)) ≥ (ζ × ξ)(x1, y1) · (ζ × ξ)(x2, y2) for all (x1, y1), (x2, y2) ∈ X×X. Consequently, ζ×ξ is a fuzzy dot Bd-subalgebra of X×X. Let X be a nonempty set and ζ be any fuzzy set of X. A fuzzy relation S on X [28] is a fuzzy set of X ×X. Then the fuzzy relation Sζ on X is called a fuzzy ζ-product relation on X [22] if Sζ(x, y) ≥ ζ(x) · ζ(y) for all x, y ∈ X. Moreover, the strongest fuzzy ζ-relation Sζ on X [22] given by Sζ(x, y) = ζ(x) · ζ(y) for all x, y ∈ X. For any t ∈ [0, 1], the set (Sζ)t := {(x, y) | Sζ(x, y) ≥ t} is called a level subset of Sζ , see [28]. Let A be any subset of a nonempty set X and ζ be a fuzzy set of X. The characteristic function CA ζ of A×A is defined by for every x, y ∈ X, CA ζ (x, y) := { 1 if (x, y) ∈ A×A, 0 otherwise. Next, the notion of strongest fuzzy dot Bd-subalgebras on Bd-algebras is further in- troduced as follows. Definition 5. Let X be a Bd-algebra, ζ be a fuzzy set of X, and Sζ be a strongest fuzzy ζ-relation on X. Then Sζ is called a strongest fuzzy dot Bd-subalgebra on X if for every x1, x2, y1, y2 ∈ X: (i) Sζ(0, 0) ≥ Sζ(x1, y1); (ii) Sζ(x1 ∗ x2, y1 ∗ y2) ≥ Sζ(x1, y1) · Sζ(x2, y2). W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 10 of 13 Example 6. Consider the Bd-algebra X := (X, ∗, 0) as defined in Example 1. Afterward, we define a fuzzy set ζ of X by ζ(0) = 0.90, ζ(a) = 0.70, ζ(b) = 0.80, ζ(c) = 0.80. Then the strongest fuzzy ζ-relation Sζ on X is as follows. Sζ(0, 0) = 0.81,Sζ(0, a) = 0.63,Sζ(0, b) = 0.72,Sζ(0, c) = 0.72, Sζ(a, 0) = 0.63,Sζ(a, a) = 0.49,Sζ(a, b) = 0.56,Sζ(a, c) = 0.56, Sζ(b, 0) = 0.72,Sζ(b, a) = 0.56,Sζ(b, b) = 0.64,Sζ(b, c) = 0.64, Sζ(c, 0) = 0.72,Sζ(c, a) = 0.56,Sζ(c, b) = 0.56,Sζ(c, c) = 0.64. It turns out that Sζ(0, 0) ≥ Sζ(x, y) for all x, y ∈ X. A select few of results are calculated below. Sζ(a ∗ 0, b ∗ a) = Sζ(a, b) = 0.56 > 0.35 = Sζ(a, b) · Sζ(0, a), Sζ(c ∗ a, 0 ∗ b) = Sζ(a, a) = 0.49 > 0.40 = Sζ(c, 0) · Sζ(a, b), Sζ(b ∗ c, 0 ∗ c) = Sζ(c, 0) = 0.72 > 0.46 = Sζ(b, 0) · Sζ(c, c). By routine calculations, we obtain Sζ(x1 ∗ x2, y1 ∗ y2) ≥ Sζ(x1, y1) · Sζ(x2, y2) for all x1, x2, y1, y2 ∈ X. Therefore, Sζ is a strongest fuzzy dot Bd-subalgebra on X. Theorem 7. Let X be a Bd-algebra, ζ be a fuzzy set of X, and Sζ be a strongest fuzzy ζ-relation on X. Then ζ is a fuzzy dot Bd-subalgebra of X if and only if Sζ is a strongest fuzzy dot Bd-subalgebra on X. Proof. Assume that ζ is a fuzzy dot Bd-subalgebra of X. Let x1, x2, y1, y2 ∈ X. Then we have Sζ(0, 0) = ζ(0) · ζ(0) ≥ ζ(x1) · ζ(y1) = Sζ(x1, y1) and Sζ(x1 ∗ x2, y1 ∗ y2) = ζ(x1 ∗ x2) · ζ(y1 ∗ y2) ≥ [ζ(x1) · ζ(x2)] · [ζ(y1) · ζ(y2)] = [ζ(x1) · ζ(y1)] · [ζ(x2) · ζ(y2)] = Sζ(x1, y1) · Sζ(y1, y2). Therefore, Sζ is a strongest fuzzy dot Bd-subalgebra on X. Conversely, assume that Sζ is a strongest fuzzy dot Bd-subalgebra on X. Let x, y ∈ X. We consider (ζ(0))2 = ζ(0) · ζ(0) = Sζ(0, 0) ≥ Sζ(x, x) = ζ(x) · ζ(x) = (ζ(x))2 and (ζ(x ∗ y))2 = ζ(x ∗ y) · ζ(x ∗ y) W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 11 of 13 = Sζ(x ∗ y, x ∗ y) ≥ Sζ(x, x) · Sζ(y, y) = [ζ(x) · ζ(x)] · [ζ(y) · ζ(y)] = [ζ(x) · ζ(y)] · [ζ(x) · ζ(y)] = (ζ(x) · ζ(y))2. Since ζ(0), ζ(x), ζ(x ∗ y), ζ(x) · ζ(y) ≥ 0, we have ζ(0) ≥ ζ(x) and ζ(x ∗ y) ≥ ζ(x) · ζ(y). Consequently, ζ is a fuzzy dot Bd-subalgebra of X. Theorem 8. Let X be a Bd-algebra and Sζ be a strongest fuzzy ζ-relation on X, where ζ is a fuzzy set of X. If a nonempty level subset (Sζ)t is a Bd-subalgebra of X ×X for all t ∈ [0, 1], then Sζ is a strongest fuzzy dot Bd-subalgebra on X. Proof. Let x1, x2, y1, y2 ∈ X. Choose t′ = Sζ(x1, y1) for some t′ ∈ [0, 1]. Then (x1, y1) ∈ (Sζ)t′ ; that is, (Sζ)t′ ̸= ∅. By assumption, we have (Sζ)t′ is a Bd-subalgebra of X×X. This implies that (0, 0) ∈ (Sζ)t′ , and then Sζ(0, 0) ≥ t′ = Sζ(x1, y1). Next, take s ′ = Sζ(x1, y1)·Sζ(x2, y2) for some s′ ∈ [0, 1]. So, we have Sζ(x1, y1) ≥ Sζ(x1, y1)·Sζ(x2, y2) = s′ and Sζ(x2, y2) ≥ Sζ(x1, y1) · Sζ(x2, y2) = s′. It turns out that (x1, y1), (x2, y2) ∈ (Sζ)s′ . By the hypothesis, we get (x1 ∗x2, y1 ∗ y2) = (x1, y1)⊛ (x2, y2) ∈ (Sζ)s′ . Thus, Sζ(x1 ∗x2, y1 ∗ y2) ≥ s′ = Sζ(x1, y1) · Sζ(x2, y2). Therefore, Sζ is a strongest fuzzy dot Bd-subalgebra on X. The converse of Theorem 8 is generally not valid, as seen by the following example. Example 7. By Example 6, we have Sζ is a strongest fuzzy dot Bd-subalgebra on X := (X, ∗, 0). Then the level subset (Sζ)0.72 = {(0, b), (0, c), (b, 0), (c, 0)}. We observe that (Sζ)0.72 is not a Bd-subalgebra of X×X, since (0, b)⊛ (b, 0) = (a, b) ̸∈ (Sζ)0.72. Theorem 9. Let X be a Bd-algebra, A be a nonempty subset of X, and ζ be a fuzzy set of X. Then A×A is a Bd-subalgebra of X×X if and only if CA ζ is a strongest fuzzy dot Bd-subalgebra on X. Proof. Assume that A×A is a Bd-subalgebra of X×X. Then (0, 0) ∈ A×A, implies that CA ζ (0, 0) = 1 ≥ CA ζ (x, y) for all (x, y) ∈ X × X. Now, suppose that there exist (a1, b1), (a2, b2) ∈ X ×X such that CA ζ (a1 ∗ a2, b1 ∗ b2) < CA ζ (a1, b1) · CA ζ (a2, b2). We obtain that CA ζ (a1 ∗a2, b1 ∗b2) = 0 and CA ζ (a1, b1) ·CA ζ (a2, b2) = 1. Since CA ζ (a1, b1) ·CA ζ (a2, b2) = 1, we have CA ζ (a1, b1) = 1 and CA ζ (a2, b2) = 1. It follows that (a1 ∗ a2, b1 ∗ b2) ̸∈ A × A and (a1, b1), (a2, b2) ∈ A×A. By the hypothesis, we have (a1 ∗a2, b1 ∗ b2) = (a1, b1)⊛ (a2, b2) ∈ A×A. This is a contradiction. Hence, CA ζ (x1 ∗ x2, y1 ∗ y2) ≥ CA ζ (x1, y1) · CA ζ (x2, y2) for all (x1, y1), (x2, y2) ∈ X ×X. Therefore, CA ζ is a strongest fuzzy dot Bd-subalgebra on X. Conversely, assume that CA ζ is a strongest fuzzy dot Bd-subalgebra on X. If (0, 0) ̸∈ A × A, then 0 = CA ζ (0, 0) ≥ CA ζ (x, y) for all (x, y) ∈ X × X. Thus, CA ζ (x, y) = 0 for all (x, y) ∈ X × X. This means that A × A = ∅. This is a contradiction, because A is a nonempty subset of X. Hence, (0, 0) ∈ A× A. Now, let (x1, y1), (x2, y2) ∈ A× A. Then, W. Nakkhasen et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6189 12 of 13 CA ζ (x1 ∗ x2, y1 ∗ y2) ≥ CA ζ (x1, y1) · CA ζ (x2, y2) = 1, and so CA ζ (x1 ∗ x2, y1 ∗ y2) = 1. This implies that (x1, y1) ⊛ (x2, y2) = (x1 ∗ x2, y1 ∗ y2) ∈ A × A. Consequently, A × A is a Bd-subalgebra of X×X. 5. Conclusions In 2024, Nakkhasen et al. [27] applied the concept of fuzzy sets to Bd-algebras, defining the concept of fuzzy Bd-subalgebras. This article presents the notion of fuzzy dot Bd- subalgebras, which provide as a generalization of fuzzy Bd-subalgebras. That means that some of the results obtained from this work will generalize those from [27]. For example, Theorem 1 will be a general implication of Proposition 3.1 in [27]. In Section 3, we studied certain properties of fuzzy dot Bd-subalgebras of the Bd-algebras. Also, the relationships between fuzzy dot Bd-subalgebras under a homomorphism of Bd-algebras were then considered. 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