EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7055 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exploring Categorical Perspectives on Soft BCK/BCI-Algebras G. Muhiuddin1,∗, Mohamed E. Elnair1,2, Ahmed A. Khidir1, Mohammed Hassan1 1 Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia 2 Department of Mathematics and Physics, Gezira University, P. O. Box 20, Sudan Abstract. In this manuscript, we present new ideas concerning the domain of soft BCK/BCI- algebras and outline specific categorical frameworks, including equalizers and finite products. Ad- ditionally, we demonstrate that the category of soft BCK/BCI-algebras conforms to a topological construct. Moreover, we establish that the category of soft BCK/BCI-algebras features distinctive elements such as terminal objects, initial objects, and zero objects. 2020 Mathematics Subject Classifications: 18A20, 18D05, 06F35 Key Words and Phrases: BCK/BCI-algebra, soft BCK/BCI-algebra, category of soft BCK/BCI- algebra, soft BCK/BCI-homomorphism 1. Introduction Imai and Iséki introduced two classes of abstract algebras: BCK-algebras and BCI- algebras [1, 2]. It is established that the category of BCK-algebras is a proper subset of the category of BCI-algebras. In the realm of fuzzy set theory [3, 4], Molodtsov [5] proposed the concept of soft sets as a novel mathematical approach to address uncertainties without the presence of errors found in existing theories. Subsequently, Maji et al. [6, 7] introduced fuzzy soft sets. Ali et al. [8] explored new operations on soft sets, while ongoing research continues to advance soft set theory. In [9], the application of soft set theory is extended to various concepts including (i) filters in R0-algebras; (ii) positive implicative ideals of BCK-algebras [10]; (iii) decision-making problems using fuzzy soft sets [11]; (iv) fuzzy soft groups [12]; (v) fuzzy soft sets in BCK/BCI-algebras [13]; (vi) Normal Unisoft Filters in R0-algebras [14]. Additionally, Muhiuddin et al. studied the application of soft set theory in areas such as filter theory in MTL-algebras [15], Unisoft Filters in R0-algebras [16], Cubic Soft ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7055 Email addresses: chishtygm@gmail.com, gmuhiuddin@ut.edu.sa (G. Muhiuddin), abomunzir124@gmail.com (M. E. Elnair), akhidir@ut.edu.sa (A. A. Khidir), m.salih@ut.edu.sa (M. Hassan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 2 of 13 BCK/BCI-Algebras [17, 18], and soft ordered semigroups [19]. Moreover, category theory is broadly defined as a general mathematical theory of struc- tures and systems of structures. For detailed insights into categorical concepts, readers are directed to [20–24]. In recent years, researchers have merged category theory with soft set theory to estab- lish the category of soft sets across various domains. Notable contributions include: • In 2013, Zahiri [25] introduced the concept of the “category of soft sets.” • In the same year, Sardar and Gupta [26] constructed a soft category and investigated several intriguing properties. • In 2014, Zhou et al. [27] delved into the categorical properties of soft sets. • Borzooei et al. [28] explored key concepts related to the category of soft sets in 2015. • In 2016, Öztunç [29] examined specific properties of soft categories. • Shirmohammadi and Rasouli [30] presented a categorical approach to soft S-acts in 2017. • In 2022, Sharma et al. [31] introduced the notion of the category of intuitionistic fuzzy modules. 2. Purpose for conducting this research The study of soft BCK/BCI-algebras represents a fascinating area of research that offers intriguing insights into algebraic systems. By investigating and elucidating the categorical structures and properties within this domain, we aim to expand the theoretical foundations of soft BCK/BCI-algebras. Our exploration of concepts such as equalizers and finite products sheds light on the organizational principles governing these algebraic systems. Furthermore, our discovery of terminal, initial, and zero objects in the category of soft BCK/BCI-algebras reveals unique elements that contribute to a deeper understanding of their structural nuances. Through this work, we not only enhance our knowledge of categorical properties but also pave the way for future research and applications in the realm of algebraic structures and categorical theory. 3. Targets of the planned technique • To develop a systematic framework for analyzing soft BCK/BCI-algebras and ex- ploring their categorical structures. • To investigate the presence of equalizers and finite products within the category of soft BCK/BCI-algebras. G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 3 of 13 • To demonstrate the adherence of the category of soft BCK/BCI-algebras to a topo- logical construct. • To identify and characterize special objects such as terminal objects, initial objects, and zero objects in the category of soft BCK/BCI-algebras. • To contribute to a deeper understanding of the categorical properties and structural nuances inherent in soft BCK/BCI-algebras. • To lay the groundwork for further exploration and applications of the proposed method in the broader context of algebraic structures and categorical theory. This paper is structured as follows: Section 2 presents fundamental notions of BCK/BCI- algebras and soft BCK/BCI-algebras. Section 3 entails the construction of the soft BCK/BCI-algebras category and various category-related concepts. Finally, in Section 4, we explore the special objects within the category of soft BCK-algebras. 4. Preliminaries K. Iséki introduced the significant class of logical algebras known as BCK/BCI- algebras, described as the most important class of logical algebras [1, 2]. A nonempty subset T is referred to as a BCK/BCI-subalgebra of X̃ if ϖ∗ϱ ∈ T for all ϖ, ϱ ∈ T where X̃ is a BCK/BCI-algebra. please consult [32], for further details regarding BCK/BCI- algebras. A “fuzzy set” µ in a “BCK/BCI-algebra” X̃ is termed a ”fuzzy BCK/BCI-algebra” if it satisfies the condition: (∀ϖ, ϱ ∈ X̃ ) “(µ(ϖ ∗ ϱ) ≥ min{µ(ϖ), µ(ϱ)})”. Molodtsov defined a soft set as follows: Consider an initial universe set U and a set of parameters E, where P(U) denotes the power set of U and Ω ⊂ E. Definition 1 ([5]). A pair (ζ,Ω) is termed a soft set over U , where ζ is a function defined by ζ : Ω → P(U). For further insights, Molodtsov presented “many examples” in [5]. Definition 2. [33] Let X̃ be a BCK/BCI-algebra. Let (ζ,Ω) be called a Soft BCK/BCI- algebra over X̃ if ζ(ϖ) is a BCK/BCI sub-algebra over X̃ ,∀ϖ ∈ Ω. ζ : Ω → P (ϖ) and ϖ 7→ ζ(ϖ), therefore ζ(ϖ) is BCK/BCI sub-algebra. Example 1. [33] Let “X̃ = {0, α, β, γ, δ} be a BCK-algebra with the following Cayley table”: Let (ζ,Ω) be soft set over X̃ where Ω = X̃ and ζ : Ω → P (X̃ ) is a set-valued function defined by, “ζ(ϖ) = {ϱ ∈ X̃ : ϖRϱ ⇔ ϱ ∈ ϖ−1I}” For all ϖ ∈ Ω “where I = {0, α} and ϖ−1 = {ϱ ∈ X̃ : ϖ ∧ ϱ ∈ I}” G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 4 of 13 ∗ 0 α β γ δ 0 0 0 0 0 0 α α 0 α α α β β β 0 β β γ γ γ γ 0 γ δ δ δ δ δ 0 we have ζ(0) = ζ(α) = X̃ , ζ(β) = {0, α, β, δ}, ζ(δ) = {0, α, β, δ} and ζ(δ) = {0, α, β, δ} are BCK-subalgebras of X̃ . “Therefore (ζ,Ω) is a soft BCK-algebra over X̃ ”. Definition 3. [33] Let “(ζ,Ω) and (ϕ,B) be two soft BCK/BCI-algebras over X̃”. Then the BCK/BCI-homomorphism µ : Ω → B is “called” a soft BCK/BCI-homomorphism” if ζ(α) ⊆ (ϕ ◦ µ)(α), ∀α ∈ Ω. Figure 1: i.e. ζ(α) ⊆ (ϕ ◦ µ)(α) = ϕ(µ(α), ∀α ∈ Ω. 5. Category of Soft BCI/BCK-algebra Definition 4. To form a category, we need a class of objects: “Soft BCK/BCI-algebras and a class of morphisms: Soft BCK- homomorphisms”. (I) Composition of maps. (II) Identity. Proposition 1. (Composition of soft-BCK/BCI-homomorphisms is again a soft BCK/BCI- homomorphism). Let (ζ,Ω),(ϕ,B) and (φ,ℵ) be any three BCK/BCI-algebras. Let µ : Ω → B and ℏ : B → ℵ be two soft BCK/BCI-homomorphisms. Then ℏ ◦ µ : Ω → ℵ is again a “soft BCK/BCI-homomorphism”. Proof. Let µ : Ω → B and ℏ : B → ℵ be two soft BCK/BCI-homomorphisms. Then by definition, we have, ζ(α) ⊆ (ϕ ◦ µ)(α), ∀α ∈ Ω and ϕ(β) ⊆ (φ ◦ ℏ)(β), ∀β ∈ B. G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 5 of 13 Figure 2: To show that ℏ ◦ µ is a soft BCK/BCI-homomorphism, we have to prove “ζ(α) ⊆ (φ ◦ (ℏ ◦ µ)(α), ∀α ∈ Ω.” Now, ζ(α) ⊆ (ϕ ◦ µ)(α) = (ϕ ◦ µ(α)) = (ϕ(β))forsomeβ = µ(α) ∈ B. ⊆ ϕ(φ ◦ ℏ)(β) = φ(ℏ(β)) = φ(ℏ ◦ (µ(α))) = φ((ℏ ◦ µ)(α) = φ ◦ (ℏ ◦ µ))(α) Consequently, “ℏ ◦ µ is a soft BCK/BCI-homomorphism”. Definition 5. Let “(ζ,Ω) be a soft-BCK-algebra over X̃”. Then the soft identity BCK/BCI- homomorphism is defined as ζ(α) ⊆ (ζ ◦ IdΩ)(α). This implies that the following diagram commutes. Figure 3: G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 6 of 13 Then IdΩ is a soft identity BCK/BCI-homomorphism. Moreover, IdΩ : (ζ,Ω) → (ζ,Ω) is a soft BCK/BCI-homomorphism if ζ(α) ⊆ (ζ ◦ IdΩ)(a), ∀α ∈ Ω. Now, in view of the above discussion, we have the following. Definition 6. The “class of all soft BCK/BCI-algebras together with the class of all soft BCK/BCI-homomorphisms from a category”. It is called the “category of soft BCK/BCI- algebra and is denoted by” SBCKI . Next, we prove the following results: Theorem 1. The category SBCKI has equilizers. Proof. We begin this proof with the following diagram Figure 4: Let (ζ,Ω) and (ϕ,B) be two soft-BCK/BCI-algebras in SBCKI -objects over X̃ . Again, let µ and λ be two SBCKI−morphisms from (ζ,Ω) to (ϕ,B). Define ℵ = {α ∈ Ω : µ(α) = λ(α)}, u : ℵ → Ω an embedding, and φ = ζ ◦ u. From assumption, we have (φ,ℵ) is a SBCKI − object, µ ◦ u = λ ◦ u and φ(c) = (ζ ◦ u)(c), for all c ∈ ℵ. Thus by definition, u is a SBCKI −morphism. We will demonstrate that ((φ,ℵ), u) functions as the equilizer between µ and λ. Let (K, X̃ ) be a SBCKI -object and suppose v is a SBCKI -morphism from (K, X̃ ) to (ζ,Ω) such that ℵ = {α ∈ Ω : µ(α) = λ(α)},ℵ 6= ∅. Since µ(0) = λ(0) =⇒ 0 ∈ ℵ. Let ϖ1, ϖ2 ∈ ℵ. Then µ(ϖ1) = λ(ϖ1) and µ(ϖ2) = λ(ϖ2). We have µ(ϖ1 ∗ϖ2) = µ(ϖ1) ∗ µ(ϖ2) = λ(ϖ1) ∗ λ(ϖ2) = λ(ϖ1 ∗ϖ2). G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 7 of 13 This implies that ϖ1 ∗ϖ2 ∈ ℵ. This show that ℵ is a subalgebra of Ω. Now, define a map η : X̃ → ℵ and η = v. In what follows, Our focus is to demonstrate that η is a SBCKI -morphism from (K, X̃ ) to (ϕ,ℵ), and that v = u ◦ η. Firstly by µ ◦ v = λ ◦ v, we get µ(v(ϖ)) = λ(v(ϖ)), ∀ϖ ∈ X̃ ⇒ v(ϖ) ∈ ℵ. Hence η = v is well defined. Again, if φ = ζ ◦ u, η = v and v being a SBCKI −morphism, then form Figure 4, we have K(ϖ) ⊆ ζ(v(ϖ)) = ζ(u ◦ η(ϖ)) = ζ(u(η(ϖ))) = (ζ ◦ u)(η(ϖ)) = φ(η(ϖ)) for all ϖ ∈ X̃ =⇒ η is a SBCKI −morphism (by definition). Finally from assumption, we know that v = u ◦ η and η is unique. Consequently (φ,ℵ) is the equilizer of µ and λ is SBCKI . Proposition 2. The category SBCKI has a finite product. Proof. Let (ζ,Ω) and (ϕ,B) are two SBCI − objects. Define three mappings Figure 5: φ : Ω×B → P (U) (α, β) 7→ ζ(α) ∩ ϕ(β) p1 : Ω×B → Ω (α, β) 7→ α ⇒ p1(α, β) = α p2 : Ω×B → B (α, β) 7→ β Foreach (α, β) ∈ Ω×B. φ(α, β) = ζ(α) ∩ ϕ(β) ⊆ ζ(α) = ζ(p1((α, β)). This implies that p1 is an SBCKI −morphism. By the some argument, p2 is a SBCKI − morphism. Again for each SBCKI − objects (I,D), suppose that µ and λ are SBCKI − morphism from (I,D) → (ζ,Ω) and (ϕ,B). G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 8 of 13 Figure 6: Then I(δ) ⊆ ζ(µ(δ)) and I(δ) ⊆ ϕ(λ(δ)), ∀δ ∈ D. Further, define ℏ : D → Ω×B δ 7→ (µ(δ), λ(δ)) ℏ(δ) = (µ(δ), λ(δ)) for each δ ∈ D, then we get I(δ) ⊆ ζ(µ(δ) ∩ ϕ(λ(δ))) = φ(µ(δ), λ(δ)) = φ(ℏ(δ) = (φ ◦ ℏ)(δ) This implies that ℏ is a SBCI −morphism. Finallyforevery δ ∈ D, we get (p1 ◦ ℏ)(δ) = p1(ℏ(δ)) = p1(µ(δ), λ(δ)) = µ(δ) Therefore, p1 ◦ ℏ = µ. Similarity, p2 ◦ ℏ = λ. Clearly, ℏ is unique. Consequently, {(φ,ℵ), p1, p2} is a finite product of (ζ,Ω) and (ϕ,B). Proposition 3. The category SBCKI is a “topological construct”. Proof. Let {(ζi,Ωi)}i∈I be a family of SBCKI -objects indexed by a class I, and let {(ζi : Ω → Ωi)}i∈I be a family of mappings. We define asoftset.over.U as follows: ζ : Ω → P (U) α → ∩i∈I(ζi(µi(α))). Then, (ζ,Ω) ∈ Ob(SBCKI). It suffices to show that {µi : (ζ,Ω) → (ζi,Ωi)}i∈I is the unique SBCKI initial lift of {µi : Ω → Ωi}i∈I . Now, to “complete the proof, we take the following two cases”. Case 1: We show that {µi : (ζ,Ω) → (ζi,Ωi)}i∈I is a SBCKI initial.lift.of {µi : Ω → Ωi}i∈I . Firstly, we claim that {µi : (ζ,Ω) → (ζi,Ωi)} is a family of SBCKI -morphisms for every i ∈ I. By the assumption, for each α ∈ Ω and i ∈ I, one yields ζ(α) = ∩i∈I(ζi(µi(α))) ⊆ ζi(µi(α)), G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 9 of 13 Where {µi}i∈I is a family of SBCKI -morphisms, Furthermore, suppose that ϕ,B ∈ Ob(SBCKI), λ : B → Ω is a mapping such that λi = µi ◦ λ for every i ∈ I, and λi : (ϕ,B)) → (ζi,Ωi) is is a family of SBCKI -morphisms. Then, we can infer that ϕ(β) ⊆ ∩i∈I(ζi(λi)β)) = ∩i∈I(ζi(µi ◦ g)(β)) = ∩i∈I(ζi(µi(g(β)))) = ζ(g(β)) Therefore, λ represents a SBCKI -morphism from ϕ,B to ζ,Ω. As per the definition, it is evident that the family {µi : (ζ,Ω) → (ζi,Ωi)}i∈I constitutes a SBCKI initial lift of {µi : Ω → Ωi}i∈I . In Case 2, we address the ”uniqueness of the initial lift”. Assuming that {µi : (ζ̃,Ω) → (ζi,Ωi)}i∈I also stands as a SBCKI initial lift of {µi : Ω → Ωi}i∈I , distinct from {µi : (ζ,Ω) → (ζi,Ωi)}i∈I , then {µi : (ζ̃,Ω) → (ζi,Ωi)}i∈I forms a set of SBCKI -morphisms. It is evident that ζ̃(α) ⊆ ζi(µi(α)) forall- i ∈ I and α ∈ Ω. Thus, ζ̃(α) ⊆ ζi(µi(α)) = ζ(α), implying ζ̃ ⊆ ζ. Conversely, for the SBCKI -entities (ζ,Ω) and identity mapping IδΩ : Ω → Ω, given that {µi : (ζ̃,Ω) → (ζi,Ωi)}i∈I serves as a SBCKI initial lift of {µi : Ω → Ωi}i∈I , we find that µi◦iδΩ = µi, where µi represents a SBCKI -morphism for all i ∈ I, and iδΩ : (ζ,Ω) → (ζ̃,Ω) is also a SBCKI -morphism. Consequently, ζ(α) ⊆ ζ̃(iδΩ(α)) = ζ̃(α) for each α ∈ Ω, indicating ζ ⊆ ζ̃. In summary, it is established that ζ ⊆ ζ̃. Combining the outcomes of steps (1) and (2), it can be concluded that SBCKI forms a topological construct. 6. Special objects in SBCKI It is evident that the set {0} constitutes a BCI/BCK-algebra. Consequently, it follows trivially that the pair (ζ, {0}) represents a ”soft BCI/BCK-algebra over X̃”. Now, for any other object (ϕ,Ω) in SBCKI ,there is only one morphism from 0 → Ω i.e, 0 → 0 and also there is only morphism Ω → {0} i.e, α → 0, ∀α ∈ Ω.Trivially, the 0-morphism is a soft morphism. Thus SBCKI has zero objects. Proposition 4. The empty set (accompanied by an empty function mapping into P (X̃ )) serves as a zero object within SBCKI . Proof. For each object (ζ,Ω) ∈ SBCKI , there exist unique morphism The inequalities are satisfied by default. therefore soft BCI/BCK-algebras ((ζ,Ω), (Φ,Φ)) and ((Φ,Φ), (ζ,Ω)) each contain only one -morphism. Proposition 5. The category SBCKI has a terminal object. Proof. Define a mapping T{Φ} : Φ → P (U) Φ 7→ U G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 10 of 13 Figure 7: Figure 8: Trivially (T{Φ}, {Φ} in a soft BCK-algebra for any object (TM ,M), define a map µ : M → {Φ} by m → Φ. Then, for each m ∈ M , the relationship TM (m) ⊆ T{Φ}(µ(m)) = T{Φ}(Φ) = U holds, indicating that µ functions as a soft BCK-morphism from (TM ,M) to (T{Φ},Φ). It is evident that µ is uniquely defined. Hence, (T{Φ},Φ) stands as a terminal object in SBCKI . Proposition 6. The category SBCKI has an initial object. Proof. Similar proof for a terminal object. Proposition 7. The category SBCKI has zero objects. Proof. Trivially, the empty set Φ forms a BCK-algebra. Then (Φ,Φ) is a soft BCK- algebra (Φ : Φ → P (U)). Again, for each object (ζ,Ω) in SBCKI , there exists a unique morphism. Figure 9: G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7055 11 of 13 The inequalities are satisfied by default. Thus there is only one morphism from (ζ,Ω) to (Φ,Φ) and also only one morphism from (Φ,Φ) to (ζ,Ω). Thus, SBCK has no objects. 7. Conclusion In conclusion, this study delves into the realm of soft BCK/BCI-algebras, introduc- ing novel concepts and elucidating categorical structures such as finite products, equal- izers, and co-equalizers within this domain. By demonstrating that the category of soft BCK/BCI-algebras adheres to a topological construct, we provide insights into the under- lying organizational principles of these algebraic systems. 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