EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 423-430 ISSN 1307-5543 – ejpam.com Published by New York Business Global Inverse Limit of an Inverse System of BE-algebras Jimboy R. Albaracin1,∗, Jocelyn P. Vilela2 1 Mathematics Program, College of Science, University of the Philippines Cebu, 6000 Cebu City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. This paper covers the notion of the inverse limit of an inverse system of BE-algebras and investigates some of its properties. Moreover, this study deals with the completion of a BE- algebra. 2020 Mathematics Subject Classifications: 06F35, 03G25 Key Words and Phrases: BE-algebra, inverse limit, inverse system, completion 1. Introduction The notion of BCK-algebras was initiated by Y. Imai and K. Iséki [1] in 1966 as a generalization of the concept of set theoretic difference and propositional calculi. In [3], K. H. Kim and Y. H. Yon introduced the dual BCK-algebra and study its relation to MV-algebra. As a generalization of dual BCK-algebra, H. S. Kim and Y. H. Kim [2] introduced the BE-algebra. Today, BE-algebras have been studied by many authors and many branches of mathematics have been applied to BE-algebras, such as probability theory, topology, fuzzy set theory and so on. In this paper we will apply the concept of inverse limit in the sense of category theory to some collections of BE-algebras. This notion in category theory has been studied in different kinds of categories. This study introduces the inverse limit of an inverse system of BE-algebras and investigates some of its properties. Through this concept, we present the idea of the completion of any BE-algebra. An algebra (X; ∗, 1X) is called a BE-algebra if the following hold: for all x, y, z ∈ X, (BE1) x∗x = 1X ; (BE2) x∗1X = 1X ; (BE3) 1X ∗x = x; and (BE4) x∗ (y ∗z) = y ∗ (x∗z). A relation “≤” on X, called BE-ordering, is defined by x ≤ y if and only if x ∗ y = 1X . Throughout this paper, we denote a BE-algebra (X, ∗, 1X) simply by X if no confusion arises. A non-empty subset S of X is said to be a subalgebra of X if x ∗ y ∈ S for all ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3966 Email addresses: jralbaracin@up.edu.ph (J. Albaracin), jocelyn.vilela@g.msuiit.edu.ph (J. Vilela) http://www.ejpam.com 423 c© 2021 EJPAM All rights reserved. J. Albaracin, J. Vilela / Eur. J. Pure Appl. Math, 14 (2) (2021), 423-430 424 x, y ∈ S. A non-empty subset F of X is said to be a filter of X if: (F1) 1X ∈ F ; and (F2) x ∗ y ∈ F and x ∈ F imply y ∈ F . A filter F of X is said to be normal if it satisfies the following condition: for all x, y, z ∈ X, x ∗ y ∈ F implies (z ∗ x) ∗ (z ∗ y) ∈ F and (y∗z)∗(x∗z) ∈ F . A BE-algebra X is said to be self distributive if x∗(y∗z) = (x∗y)∗(x∗z) for all x, y, z ∈ X. It is called commutative if satisfies (x∗y)∗y = (y∗x)∗x for all x, y ∈ X. It is said to be a transitive BE-algebra if it satisfies the condition: y∗z ≤ (x∗y)∗(x∗z) for all x, y, z ∈ X. If X is a transitive BE-algebra, then the relation “≤” is transitive. Let X and Y be BE-algebras. A mapping f : X → Y is a homomorphism if f(x∗y) = f(x)∗f(y) for all x, y ∈ X, see [4]. Theorem 1. [4] Let X and Y be BE-algebras. If f : X → Y is a homomorphism, then f(1X) = 1Y . Theorem 2. [4] Every commutative BE-algebra X is transitive. Proposition 1. [6] If X is a transitive BE-algebra, then every filter of X is normal. Theorem 3. [6] Let X be a commutative BE-algebra. There is a bijection between con- gruence relations and filters of X. Definition 1. [5] Let I be a set and ≤ be a binary operation on I. We call I = (I,≤) a directed partially ordered set or directed poset if it satisfies the following conditions: (i) i ≤ i, for i ∈ I; (ii) i ≤ j and j ≤ k imply i ≤ k, for i, j, k ∈ I; (iii) i ≤ j and j ≤ i imply i = j, for i, j ∈ I; and (iv) if i, j ∈ I, there exists some k ∈ I such that i, j ≤ k. Definition 2. An inverse or projective system of BE-algebras over a directed poset I, consists of a collection {Xi | i ∈ I} of BE-algebras indexed by I, and a collection of homomorphisms ϕij : Xi → Xj, defined whenever i ≥ j, such that ϕjkϕij = ϕik whenever i, j, k ∈ I and i ≥ j ≥ k. In addition, we assume that ϕii is the identity mapping idXi on Xi. We shall denote such a system by {Xi, ϕij , I}, or by {Xi, ϕij} if the index set I is clearly understood. Definition 3. Let Y be a BE-algebra, {Xi, ϕij , I} an inverse system of BE-algebras over a directed poset I, and let ψi : Y → Xi be a homomorphism for each i ∈ I. These mappings ψi are said to be compatible if ϕijψi = ψj whenever j ≤ i. Definition 4. Let {Xi, ϕij , I} be an inverse system of BE-algebras over a directed poset I. A subalgebra X of ∏ i∈I Xi together with compatible homomorphisms ϕi : X → Xi where i ∈ I is an inverse limit or a projective limit of the inverse system {Xi, ϕij , I} if the following universal property is satisfied: whenever Y is a BE-algebra and {ψi : Y → Xi (i ∈ I)} is a set of compatible homomorphisms, then there is a unique homomorphism ψ : Y → X such that ϕiψ = ψi for all i ∈ I. We say that ψ is “induced” or “determined” by the compatible homomorphisms ψi. The maps ϕi : X → Xi are called projections. We shall denote the inverse limit of the inverse system {Xi, ϕij , I} by lim←−i∈IXi, lim←−Xi, (lim←−Xi, ϕi) or (X,ϕi). J. Albaracin, J. Vilela / Eur. J. Pure Appl. Math, 14 (2) (2021), 423-430 425 2. Some Properties of Inverse Limit of BE-algebras Theorem 4. Let {Xi, ϕij , I} be an inverse system of BE-algebras over a directed poset I. Then (X,ϕi) is an inverse limit of the inverse system {Xi, ϕij , I}, where X = { (xi) ∈ ∏ i∈I Xi | for all i, j ∈ I such that i ≥ j, ϕij(xi) = xj } and ϕi : X → Xi is the restriction of the natural projection ρi : ∏ i∈I Xi → Xi, that is, ϕi = ρi|X . Proof. Let (xi), (yi) ∈ X. Then ϕij(xi) = xj and ϕij(yi) = yj whenever i ≥ j. Thus, ϕij(xi ∗ yi) = ϕij(xi) ∗ ϕij(yi) = xj ∗ yj whenever i ≥ j. Since (xi) ∗ (yi) = (xi ∗ yi), (xi) ∗ (yi) ∈ X. Hence, X is a subalgebra of ∏ i∈I Xi. Let ϕi : X → Xi be the restriction of the natural projection ρi : ∏ i∈I Xi → Xi. We will show that the BE-algebra X together with ϕi is the inverse limit of {Xi, ϕij , I}. Let (xi) ∈ X. Then ϕijϕi((xi)) = ϕij(xi) = xj = ϕj((xi)) whenever i ≥ j. Thus, ϕijϕi = ϕj whenever i ≥ j. This implies that ϕi’s are compatible. Let Y be a BE-algebra and {ψi : Y → Xi} be a set of compatible homomorphisms. Consider the mapping ψ : Y → X defined by ψ(y) = (ψi(y)) for each y ∈ Y . Since ψi’s are compatible, ϕijψi = ψj whenever i ≥ j. Thus, ϕij(ψi(y)) = ψj(y) whenever i ≥ j and for y ∈ Y . Hence, (ψi(y)) ∈ X. Let y ∈ Y . Then ϕiψ(y) = ϕi((ψi(y)) = ψi(y) for all i ∈ I. Thus, ϕiψ = ψi for all i ∈ I. Now, we will show that ψ is unique. Suppose that φ : Y → X is another homomorphism such that ϕiφ = ψi for all i ∈ I. Suppose further that there exists y ∈ Y such that ψ(y) 6= φ(y). By the definition of X, there exists i ∈ I such that ϕi(ψ(y)) = ψi(y) 6= ϕi(φ(y)). This is a contradiction. Hence, ψ(y) = φ(y) for all y ∈ Y . Therefore, ψ is unique. Consequently, X is an inverse limit of {Xi, ϕij , I}. Let (X,ϕi) be an inverse limit of the inverse system of BE-algebras {Xi, ϕij , I}. By definition, the maps ϕi : X → Xi are compatible. Thus, the universal property of the inverse limit shows that there exists a unique homomorphism ϕ : X → X such that ϕiϕ = ϕi for all i ∈ I. Since ϕiidX = ϕi for all i ∈ I and ϕ is unique, ϕ = idX . This observation is stated in the following remark. Remark 1. Let {Xi, ϕij , I} be an inverse system of BE-algebras over a directed poset I and let (X,ϕi) be an inverse limit of {Xi, ϕij , I}. Then the homomorphism idX : X → X satisfies ϕiidX = ϕi for all i ∈ I and is the only homomorphism with this property. Theorem 5. Let {Xi, ϕij , I} be an inverse system of BE-algebras over a directed poset I. Then the inverse limit is unique up to isomorphism, that is, if (X,ϕi) and (Y, ψi) are two limits of the inverse system {Xi, ϕij , I}, then there is a unique isomorphism ϕ : X → Y such that ψiϕ = ϕi for each i ∈ I. Proof. Suppose that (X,ϕi) and (Y, ψi) are two inverse limits of the inverse system {Xi, ϕij , I}. Since the maps ψi : Y → Xi are compatible, the universal property of the inverse limit (X,ϕi) shows that there exists a unique homomorphism ψ : Y → X such that ϕiψ = ψi for all i ∈ I. Similarly, there exists a unique homomorphism ϕ : Y → X such that ψiϕ = ϕi for all i ∈ I. It follows that ψi = ϕiψ = ψiϕψ for all i ∈ I. Thus, by Remark 1, ϕψ = idY . Similarly, ψϕ = idX . Therefore, ϕ is an isomorphism. J. Albaracin, J. Vilela / Eur. J. Pure Appl. Math, 14 (2) (2021), 423-430 426 Note that if S is a subalgebra of a transitive BE-algebra X, then for all x, y, z ∈ S, y ∗ z ≤ (x ∗ y) ∗ (x ∗ z). Thus, S is also transitive. Also, in [4], the direct product of transitive BE-algebras is transitive and every commutative and self-distributive BE- algebra is transitive. Thus, we have the following result. Proposition 2. Let {Xi, ϕij , I} be an inverse system of BE-algebras. If each Xi is tran- sitive (resp. commutative, self-distributive) for all i ∈ I, then lim←−Xi is transitive (resp. commutative, self-distributive). Proposition 3. Let {Xi, ϕij , I} be an inverse system of BE-algebras and let X = lim←−Xi be its corresponding inverse limit. Suppose that i0 ∈ I such that i0 ≥ i1, . . . , it and ϕi0ik(Ai0) ⊆ Aik where Aik ⊆ Xik for all k = 0, 1, . . . , t. Then X ∩ [( ∏ i 6=i0 Xi ) ×Ai0 ] = X ∩ [( ∏ i 6=i0,...,it Xi ) ×Ai0 × · · · ×Ait ] . Proof. Let (xi) ∈ X ∩ [( ∏ i 6=i0 Xi ) ×Ai0 ] . Then (xi) ∈ X and (xi) ∈ ( ∏ i 6=i0 Xi ) ×Ai0 . Thus, ϕij(xi) = xj for all i ≥ j and xi0 ∈ Ai0 . Hence, ϕi0j(xi0) = xj for all j ≤ i0. Since i0 ≥ i1, . . . , it, ϕi0ik(xi0) = xik for all k = 1, . . . , t. Since ϕi0ik(Ai0) ⊆ Aik for all k = 1, . . . , t, we have xik ∈ Aik for all k = 1, . . . , t. This implies that (xi) ∈ ( ∏ i 6=i0,...,it Xi ) × Ai0 × · · · ×Ait . Since (xi) ∈ X, we get (xi) ∈ X ∩ [( ∏ i 6=i0,...,it Xi ) ×Ai0 × · · · ×Ait ] . The other inclusion is easy to show. Corollary 1. Let {Xi, ϕij , I} be an inverse system of BE-algebras and let X = lim←−Xi be its corresponding inverse limit. Then X ∩ [( ∏ i 6=i0 Xi ) × {1Xi0 } ] = X ∩ [( ∏ i 6=i0,...,it Xi ) × {1Xi0 } × · · · × {1Xit } ] where i0 ≥ i1, . . . , it. Proof. Since ϕi0ik is a homomorphism for all i0 ≥ i1, . . . , it and for all k = 1, . . . , t, by Theorem 1, ϕi0ik(1Xi0 ) = 1Xik for all k = 1, . . . , t. Thus, ϕi0ik({1Xi0 }) = {1Xik } for all i0 ≥ i1, . . . , it and for all k = 1, . . . , t. By Proposition 3, the result holds. Theorem 6. Let {Xi, ϕij , I} be an inverse system of BE-algebras and let (X = lim←−Xi, ϕi) be its corresponding inverse limit where ϕi : X → Xi is the restriction of the natural projection ρi : ∏ i∈I Xi → Xi. Then for all j ∈ I, X ∩ [(∏ i 6=j Xi ) × {1Xj} ] = Kerϕj. J. Albaracin, J. Vilela / Eur. J. Pure Appl. Math, 14 (2) (2021), 423-430 427 Proof. For any j ∈ I, we have Kerϕj = {(xi) ∈ X | ϕj((xi)) = ρj((xi)) = xj = 1Xj} = {(xi) ∈ X | xj = 1Xj} = X ∩ ∏ i 6=j Xi × {1Xj}  . 3. Completion of a BE-algebra Proposition 4. Let X be a BE-algebra and I be the set of congruences of X. Then (I,≤) is a directed poset where the binary operation ≤ on I is defined by θ ≤ φ whenever φ ⊆ θ for all θ, φ ∈ I. Theorem 7. Let X be a BE-algebra and I be the set of congruences of X. Then {X/φ, ϕφθ, I} is an inverse system where θ ≤ φ whenever φ ⊆ θ for all θ, φ ∈ I and ϕφθ : X/φ→ X/θ is the epimorphism defined by ϕφθ([x]φ) = [x]θ. Proof. By Proposition 4, (I,≤) is a directed poset. We will show that {X/φ, ϕφθ, I} is an inverse system, that is, ϕθλϕφθ = ϕφλ whenever φ ≥ θ ≥ λ. Now, ϕθλϕφθ([x]φ) = ϕθλ(ϕφθ([x]φ)) = ϕθλ([x]θ) = [x]λ = ϕφλ([x]φ). Therefore, ϕθλϕφθ = ϕφλ whenever φ ≥ θ ≥ λ. Note that ϕθθ : X/θ → X/θ is defined by ϕθθ([x]θ) = [x]θ. Thus, ϕθθ = idX/θ. Therefore, {X/φ, ϕφθ, I} is an inverse system. Theorem 8. Let X be a BE-algebra and I be the set of congruences on X. Consider the inverse system {X/φ, ϕφθ, I}. Then X̂ = { ([x]φ) ∈ ∏ φ∈I X/φ | for all φ, θ ∈ I such that φ ≥ θ, ϕφθ([x]φ) = [x]θ } together with the projections ϕφ : X̂ → X/φ for all φ ∈ I is an inverse limit of {X/φ, ϕφθ, I}. Proof. The result follows from Theorem 4. We call X̂ of Theorem 8 as the completion of the BE-algebra X. Lemma 1. Let X be a BE-algebra and {X/φ, ϕφθ, I} be the defining inverse system of the completion X̂ of X. Then the canonical epimorphisms ϕθ : X → X/θ are compatible where θ ∈ I. Proof. Let x ∈ X. Then ϕφθ(ϕφ(x)) = ϕφθ([x]φ) = [x]θ = ϕθ(x) whenever φ ≥ θ. Therefore, the canonical epimorphisms ϕθ’s are compatible. Theorem 9. Let X be a BE-algebra and {X/φ, ϕφθ, I} be the defining inverse system of the completion X̂ of X. Then the canonical epimorphisms ϕθ : X → X/θ induce a homomorphism γ : X → X̂ defined by γ(x) = (ϕθ(x)) = ([x]θ). J. Albaracin, J. Vilela / Eur. J. Pure Appl. Math, 14 (2) (2021), 423-430 428 Proof. By Lemma 1, {ϕθ : X → X/θ} is a set of compatible homomorphism. Since X is a BE-algebra and X̂ = lim←−X/θ, by Definition 4 and from the proof of Theorem 4, there exists a homomorphism γ : X → X̂ with γ(x) = (ϕθ(x)) = ([x]θ) for each x ∈ X. Theorem 10. Let X be a BE-algebra and {X/φ, ϕφθ, I} be the defining inverse system of the completion X̂ of X. Then γ−1(([x]θ)) = ⋂ θ∈I [x]θ for all ([x]θ) ∈ X̂. Proof. Let y ∈ γ−1(([x]θ)). Then γ(y) = ([x]θ). By the definition of γ, γ(y) = ([y]θ). Thus, ([y]θ) = ([x]θ). Hence, [y]θ = [x]θ for all θ ∈ I. This implies that y ∈ ⋂ θ∈I [x]θ. It follows that γ−1(([x]θ)) ⊆ ⋂ θ∈I [x]θ. Let y ∈ ⋂ θ∈I [x]θ. Then y ∈ [x]θ for all θ ∈ I. Thus, [y]θ = [x]θ for all θ ∈ I. Hence, ([y]θ) = ([x]θ). Since γ(y) = ([y]θ), it follows that γ(y) = ([x]θ). Hence, y ∈ γ−1(([x]θ)). Therefore, ⋂ θ∈I [x]θ ⊆ γ−1(([x]θ)). Suppose X is commutative BE-algebra. By Theorem 3, there is a one to one corre- spondence between the set of congruences of X and the set of filters of X. For every congruence θ, there is a corresponding ≡F , where F is a filter, such that θ =≡F . With this, we can just write every congruence θ as a filter F in X. Theorem 11. Let X be a commutative BE-algebra and let {Fi | i ∈ I} be the family of filters of X such that Fi ⊆ Fj whenever i ≥ j. Then Kerγ = ⋂ i∈I Fi. Proof. Let X be a commutative BE-algebra. Then Kerγ = {x ∈ X | γ(x) = ([1X ]Fi) ∀i ∈ I} = {x ∈ X | ([x]Fi) = ([1X ]Fi) ∀i ∈ I} = {x ∈ X | x ∈ [1X ]Fi = Fi ∀i ∈ I} = ⋂ i∈I Fi. Lemma 2. Let X be a BE-algebra. If F and G are normal filters of X, then F ∩ G is also a normal filter of X. Theorem 12. Let X be a BE-algebra and let I be a family of normal filters of X such that F ∩ G ∈ I for every F,G ∈ I. Then {X/F,ϕFG, I} is an inverse system of BE-algebras where ϕFG : X/F → X/G is the epimorphism defined by ϕFG([x]F ) = ([x]G) whenever F ⊆ G. J. Albaracin, J. Vilela / Eur. J. Pure Appl. Math, 14 (2) (2021), 423-430 429 Proof. We will denote by F ≥ G whenever F ⊆ G. We will show that (I,≤) is a directed poset. Let F,G,H ∈ I. Since F ⊆ F for all F ∈ I, F ≤ F for all F ∈ I. Suppose that F ≤ G and G ≤ H. Then G ⊆ F and H ⊆ G. By set inclusion, H ⊆ F . Hence, F ≤ H. Suppose that F ≤ G and G ≤ F . Then G ⊆ F and F ⊆ G. By set inclusion, F = G. Note that F ∩G ⊆ F , F ∩G ⊆ G and F ∩G ∈ I. Thus, F ≤ F ∩G and G ≤ F ∩G. Hence, for every F,G ∈ I, there exists H = F ∩ G ∈ I such that F,G ≤ H. Therefore, (I,≤) is a directed poset. We will show that {X/F,ϕFG, I} is an inverse system, that is, ϕGHϕFG = ϕFH whenever F ≥ G ≥ H. Now, ϕGHϕFG([x]F ) = ϕGH(ϕFG([x]F )) = ϕGH([x]G) = [x]H = ϕFH([x]F ). Therefore, ϕGHϕFG = ϕFH whenever F ≥ G ≥ H. Note that ϕFF : X/F → X/F is defined by ϕFF ([x]F ) = [x]F . Thus, ϕFF = idX/F . Therefore, {X/F,ϕFG, I} is an inverse system. If I is the family of all normal filters of X such that X/F is finite for all F ∈ I, then we call the inverse limit of the inverse system {X/F,ϕFG, I} the normal completion of X. Example 1. Consider the commutative BE-algebra X = {1, a, b, c, } with the operation ∗ defined by the Cayley table shown below. ∗ 1 a b c 1 1 a b c a 1 1 b c b 1 a 1 c c 1 a b 1 By Theorem 3, there is a bijection between the congruence relations and filters of X. Thus, the set I of congruence relations of X is completely determined by the set of all filters of X. The filters of X are F1 = {1}, F2 = {1, a}, F3 = {1, b}, F4 = {1, c}, F5 = {1, a, b}, F6 = {1, a, c}, F7 = {1, b, c} and F8 = X. Since X is commutative, by Theorem 2, X is transitive. By Proposition 1, all filters of X are normal. Hence, ≡Fi is a congruence relation on X where i = 1, . . . , 8. Thus, I = {≡Fi | i = 1, . . . , 8}. We will just denote ≡Fi by Fi for all i ∈ I. Note that ≡Fi⊆≡Fj if Fi ⊆ Fj. Now, X/F1 = {[1]F1 , [a]F1 , [b]F1 , [c]F1} where [1]F1 = {1}, [a]F1 = {a}, [b]F1 = {b} and [c]F1 = {c}; X/F2 = {[1]F2 , [b]F2 , [c]F2} where [1]F2 = F2, [b]F2 = {b} and [c]F2 = {c}; X/F3 = {[1]F3 , [a]F3 , [c]F3} where [1]F3 = F3, [a]F3 = {a} and [c]F3 = {c}; X/F4 = {[1]F4 , [a]F4 , [b]F4} where [1]F4 = F4, [a]F4 = {a} and [b]F4 = {b}; X/F5 = {[1]F5 , [c]F5} where [1]F5 = F5 and [c]F5 = {c}; X/F6 = {[1]F6 , [b]F6} where [1]F6 = F6 and [b]F6 = {b}; X/F7 = {[1]F7 , [a]F7} where [1]F7 = F7 and [a]F7 = {a}; and REFERENCES 430 X/F8 = {[1]F8} where [1]F8 = X. Therefore, {X/Fi, ϕFiFj , I} is an inverse system of BE-algebras where ϕFiFj : X/Fi → X/Fj is the canonical epimorphism whenever Fi ≥ Fj, that is, Fi ⊆ Fj. In particular, {X/Fi, ϕFiFj , I} is a normal completion of X. Example 2. Consider the inverse system {X/Fi, ϕFiFj , I} in Example 1. Then lim←−X/Fi = {([x]Fi) ∈ ∏8 i=1X/Fi | ∀ Fi, Fj ∈ I such that Fi ≥ Fj , ϕFiFj ([x]Fi) = [x]Fj}. Now, the elements α1 = ([1]F1 , [1]F2 , [1]F3 , [1]F4 , [1]F5 , [1]F6 , [1]F7 , [1]F8), α2 = ([a]F1 , [1]F2 , [a]F3 , [a]F4 , [1]F5 , [1]F6 , [a]F7 , [1]F8), α3 = ([b]F1 , [b]F2 , [1]F3 , [b]F4 , [1]F5 , [b]F6 , [1]F7 , [1]F8), and α4 = ([c]F1 , [c]F2 , [c]F3 , [1]F4 , [c]F5 , [1]F6 , [1]F7 , [1]F8) of ∏8 i=1X/Fi are the only elements that satisfies the condition of lim←−X/Fi. Thus, lim←−X/Fi = {α1, α2, α3, α4}. References [1] Y. Imai and K. Iséki. On Axiom systems of Propositional Calculi XIV. Proceedings of the Japan Academy, 42:19–22, 1996. [2] H.S. Kim and Y.H. Kim. On BE-algebras. Scientiae Mathematicae Japonicae Online, pages 1299–1302, 2004. [3] K.H. Kim and Y.H. Yon. Dual BCK-Algebra and MV-algebra. Scientiae Mathematicae Japonicae Online, pages 393–399, 2007. [4] S.R. Mukkamala. A Course in BE-algebra. Springer Nature Singapore Pte Ltd., Singapore, 2018. [5] L. Ribes and P. Zalesskii. Profinite Groups. Springer, Verlag Berlin Heidelberg, 2010. [6] A. Walendziak. 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