EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 4, 2021, 1388-1401 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fuzzy sets in Hyper UP-algebras Rohaima M. Amairanto1,∗, Rowena T. Isla2 1 Department of Mathematics, Mindanao State University-University Training Center, 9700 Marawi 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. In this paper, we apply the concept of fuzzy set to hyper UP-subalgebras. We introduce the notions of fuzzy hyper UP-subalgebra and fuzzy hyper UP-filter and establish some of their properties. Furthermore, some properties of hyper homomorphism in relation to fuzzy hyper UP- subalgebras and fuzzy hyper UP-filters are also presented 2020 Mathematics Subject Classifications: 08A30, 08A72 Key Words and Phrases: Hyper UP-algebra, hyper UP-filter, fuzzy set, fuzzy hyper UP- subalgebra, fuzzy hyper UP-filter 1. Introduction The concept of hypergraphs, which is a generalization of the notion of classical algebraic groups, was introduced by F. Marty [14] in 1934. Since then, hyperstructure theory has seen tremendous development. For basic notions and results on hyperstructure theory and some of its applications, see P. Corcini and V. Leoreanu [4]. L. Zadeh [21] defined a fuzzy set as a class of objects with a continuum of grades of membership, as inspired by the process of human perception and recognition. From its inception in 1965, fuzzy set theory became a phenomenon since its logic can deal with information that is imprecise, vague, partially true, or without sharp boundaries. The reader may refer to [19] for a compilation of articles on fuzzy sets, fuzzy logic and their applications. Fuzzy hyperstructures is an application of the notion of fuzzy sets and their variants to algebra. Many articles and several books are available on fuzzy hyperstructures, such as: In 1997, P. Corcini and I. Tofan [5] introduced and investigated fuzzy hypergroups. In 2001, Y.B. Jun and X.L. Xin [9] considered the fuzzification of the notion of a (weak, strong, reflexive) hyper BCK-ideal, gave relations among them, and investigated some related properties. V. Leoreanu-Fotea and B. Davvaz [11] introduced and investigated ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4143 Email addresses: rohaima87@yahoo.com (R. Amairanto), rowena.isla@g.msuiit.edu.ph (R. Isla) http://www.ejpam.com 1388 © 2021 EJPAM All rights reserved. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1389 fuzzy hyperrings in 2009. They analyzed fuzzy substructures and homomorphisms between fuzzy hyperrings. In 2011, R. Ameri and T. Nozari [2] introduced the concept of fuzzy regular (resp., fuzzy strongly regular) relations of hyperalgebras and obtained their basic properties. They also proved that with every fuzzy hyper algebra, a unique hyperalgebra can be associated via a regular (resp., strongly regular) relation. In 2012, F. Nisar et al. [7] introduced distributive hyper BCI-ideals and applied the concept of fuzzy set to investigate the relations between fuzzy distributive hyper BCI-ideals and distributive hyper BCI-ideals of a hyper BCI-algebra. In 2015, B. Davvaz and I. Cristea [6] summarized the research progress of fuzzy hyperstructures. In 2017, P. Corcini [3] gave a brief excursus on some results on hyperstructures, their connections with fuzzy sets, and extensions to weak structures. In 2019, X. Xin et al. [20] introduced the notions of intuitionistic fuzzy soft (resp., weak, s-weak, strong) hyper BCK-ideal and investigated related properties and relations. Moreover, G. Tabaranza and J. Vilela [18] applied the fuzzy set to hyper B-algebras, while A. Macodi-Ringia and G. Petalcorin Jr. [12] introduced the implicative hyper GR-ideal and the fuzzy implicative hyper GR-ideal of type 1 (resp., of type 2), and investigated several properties. Characterizations of fuzzy implicative hyper GR-ideals of type 1 are also given. Ringia and Petalcorin [13] also investigated intuitionistic fuzzy hyper GR-ideals in hyper GR-algebras in 2020. In 2017, A. Iampan [8] defined a new algebraic structure called UP-algebra and showed that the notion of UP-algebra is a generalization of KU-algebra that was introduced by C. Prabpayak and U. Leerawat [15]. In the same year, S. Mostafa et al. [17] applied the hyper structure theory to KU-algebras. In 2019, D. Romano [16] introduced the concept of hyper UP-algebra, presented some related properties, and considered homomorphisms between hyper UP-algebras. In 2020, R. Amairanto and R. Isla [1] investigated the concept of regular congruence relation on hyper UP-algebras and established some homomorphism theorems on such algebras. They also examined the notion of hyper product of hyper UP-algebras. In this paper, we apply the concept of fuzzy set to hyper UP-algebras. We introduce the notions of fuzzy hyper UP-subalgebra and fuzzy hyper UP-filter and investigate their basic properties. Some properties of hyper homomorphism in relation to fuzzy hyper UP-subalgebra and fuzzy hyper UP-filter are also provided. 2. Preliminaries Let H be a nonempty set and P∗(H) be the set of all nonempty subsets of H. A hyperoperation on H is a mapping from H ×H into P∗(H). Definition 1. [16] A hyper UP-algebra is a set H with constant 0 and hyperoperation ⊛ satisfying the following axioms: for all x, y, z ∈ H, (HUP1) y ⊛ z ≪ [(x⊛ y)⊛ (x⊛ z)], (HUP2) x⊛ 0 = {0}, (HUP3) 0⊛ x = {x}, R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1390 (HUP4) x ≪ y and y ≪ x imply x = y, where x ≪ y is defined by 0 ∈ x ⊛ y and for every A,B ⊆ P∗(H), A ≪ B is defined by: for all a ∈ A, there exists b ∈ B such that a ≪ b. In such case, we call “≪” the hyperorder in H. A hyper UP-algebra H with constant 0 and hyperoperation ⊛ is denoted by (H;⊛, 0). By (HUP2) or (HUP3), x⊛ y ̸= ∅ for all x, y ∈ H. Example 1. Let H = {0, a, b, c, d} be a set with a binary operation ⊛ defined by the following Cayley table: ⊛ 0 a b c d 0 {0} {a} {b} {c} {d} a {0} {0,a} {0,b} {c} {d} b {0} {a} {0,b} {c} {d} c {0} {0,a} {0,b} {0,a,c} {d} d {0} {0,a} {0,b} {0,a,c} {0,d} By routine calculations, (H;⊛, 0) is a hyper UP-algebra. Definition 2. [16] Let (H,⊛, 0) be a hyper UP-algebra and let I be a subset of H containing 0. If I is a hyper UP-algebra with respect to the hyper operation “ ⊛ ” on H, we say that I is a hyper UP-subalgebra of H. Example 2. In Exampe 1, it can be verified that the set {0, a, b, c} is a hyper UP-algebra. Thus, {0, a, b, c} is a hyper UP-subalgebra of H. Example 3. Let K = {0, 1, 2} be a set with a binary operation ⊛ defined by the following Cayley table: ⊛ 0 1 2 0 {0} {1} {2} 1 {0} {0,2} {0,2} 2 {0} {1} {0,2} By routine calculations, (K;⊛, 0) is a hyper UP-algebra. Note that if I = {0, 1}, I is not a hyper UP-subalgebra since 1⊛ 1 = {0, 2} ⊈ I. Proposition 1. [16] Let (H;⊛, 0) be a hyper UP-algebra. Then the following hold for all x, y ∈ H and for every nonempty subsets A,B ⊆ H: (i) 0⊛ 0 = {0} (ii) x ≪ 0 (iii) x ≪ x (iv) y ≪ x⊛ y (v) A ⊆ B implies A ≪ B (vi) 0⊛A = A (vii) A⊛ 0 = {0} R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1391 Proposition 2. [16] Let I be a non-empty subset of a hyper UP-algebra (H,⊛, 0). Then I is a hyper UP-subalgebra of H if and only if for all x, y ∈ I, x⊛ y ⊆ I holds. Definition 3. [16] Let (H;⊛, 0) and (H ′;⊛′, 0′) be hyper UP-algebras. A mapping f : H → H ′ is called a hyper homomorphism if (HH1) f(0) = 0′, and (HH2) f(x⊛ y) = f(x)⊛′ f(y) for all x, y ∈ H. Definition 4. [21] A fuzzy set in a nonempty set H (or a fuzzy subset of H) is an arbitrary function µ : H −→ [0, 1], where [0, 1] is the unit segment of the real line. If I ⊆ H, the characteristic function µI of H is a function of H into {0, 1} defined as follows: µI(x) = { 1, if x ∈ I 0, if x /∈ I. By the definition of characteristic function, µI is a function of H into {0, 1} ⊂ [0, 1]. Then, µI is a fuzzy set in H. Definition 5. [10] Let X and Y be two nonempty sets, µ a fuzzy set of Y and f : X −→ Y a mapping. The preimage of µ under f , denoted by µf , is the fuzzy set of X defined by µf (x) = µ(f(x)) for all x ∈ X, that is, µf = µ ◦ f. Definition 6. [10] Let µ be a fuzzy set of X and f : X −→ Y a mapping. The mapping f(µ) : Y −→ [0, 1] defined by f(µ)(y) = { supx∈f−1(y){µ(x)}, if f−1(y) ̸= ∅, 0, if f−1(y) = ∅, is called the image of µ under f , where f−1(y) = {x ∈ X : f(x) = y}. Definition 7. [10] Let {µα : α ∈ A} be a nonempty family of fuzzy sets of X, where A is an arbitrary index set. The intersection of µα, denoted by ∧α∈Aµα, is defined by ∧α∈Aµα(x) = infα∈A{µα(x)} for all x ∈ X. 3. Fuzzy hyper UP-subalgebras In this section, we introduce the notion of fuzzy hyper UP-subalgebra and study some of its basic properties. For brevity, we denote a hyper UP-algebra (H;⊛, 0) by H, from here onwards. Definition 8. A fuzzy set µ in a hyper UP-algebraH is called a fuzzy hyper UP-subalgebra of H if for any x, y ∈ H, infa∈x⊛y {µ(a)} ≥ min{µ(x), µ(y)}. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1392 Example 4. 1. By Examples 1 and 2, H = {0, a, b, c, d} is a hyper UP-algebra and the set {0, a, b, c} is a hyper UP-subalgebra. It can be easily verified that µ(x) = { 1, if x ∈ {0, a, b, c} 0, if x ∈ {d} is a fuzzy hyper UP-subalgebra of H. 2. Let H = {0, 1, 2} be a set with a binary operation ⊛ defined by the following Cayley table: ⊛ 0 1 2 0 {0} {1} {2} 1 {0} {0,2} {0,2} 2 {0} {1} {0,2} Define a fuzzy subset µ : H → [0, 1] by µ(0) = 0.9, µ(1) = 0.5, µ(2) = 0.3. Then µ is not a fuzzy hyper UP-subalgebra of H since 0.3 = µ(2) = infa∈1⊛1 {µ(a)} < min{µ(1), µ(1)} = 0.5. Lemma 1. Let µ be a fuzzy hyper UP-subalgebra of a hyper UP-algebra H. Then µ(0) ≥ µ(x) for all x ∈ H. Moreover, if µ is onto, then µ(0) = 1. Proof. Let x ∈ H. By Proposition 1(iii), x ≪ x, that is, 0 ∈ x ⊛ x. Then µ(0) ≥ infa∈x⊛x {µ(a)} ≥ min{µ(x), µ(x)} = µ(x). If µ is onto, then µ(y) = 1, for some y ∈ H. Thus, 1 = µ(y) ≤ µ(0) ≤ 1. Hence, µ(0) = 1. Theorem 1. Let µ be a fuzzy hyper UP-subalgebra of a hyper UP-algebra H. Then there exists a sequence ⟨xn⟩ ⊆ H such that limn→∞ µ(xn) = 1 if and only if µ(0) = 1. Proof. Suppose that there exists ⟨xn⟩ ⊆ H such that limn→∞ µ(xn) = 1. By Lemma 1, it will follow that µ(0) ≥ µ(xn) for all n. This implies that 1 ≥ µ(0) = limn→∞ µ(0) ≥ limn→∞ µ(xn) = 1, that is, µ(0) = 1. Conversely, assume that µ(0) = 1. Consider the se- quence ⟨xn⟩ = ⟨0, 0, 0, · · · , 0, · · · ⟩ in H. Thus, ⟨µ(xn)⟩ = ⟨µ(0), µ(0), µ(0), · · · , µ(0), · · · ⟩ = ⟨1, 1, 1, · · · , 1, · · · ⟩ . Hence, limn→∞ µ(xn) = 1. Definition 9. Let µ be a fuzzy subset of a hyper UP-algebra H and t ∈ [0, 1] . Then the upper level set µt is the set µt = {x ∈ H : µ(x) ≥ t}. Theorem 2. Let µ be a fuzzy subset of a hyper UP-algebra H. Then µ is a fuzzy hyper UP-subalgebra of H if and only if for all t ∈ [0, 1] ,∅ ̸= µt is a hyper UP-subalgebra of H. Proof. Suppose that µ is a fuzzy hyper UP-subalgebra of H and µt ̸= ∅. Let x, y ∈ µt and let a ∈ x ⊛ y. Then µ(x), µ(y) ≥ t and so min{µ(x), µ(y)} ≥ t. Since µ is a fuzzy hyper UP-subalgebra of H, by Definition 8, µ(a) ≥ infa′∈x⊛y {µ(a′)} ≥ t. It follows that a ∈ µt. Since a is arbitrary, x⊛ y ⊆ µt. By Proposition 2, µt is a hyper UP-subalgebra of H. Conversely, suppose µt is a hyper UP-subalgebra of H for each t ∈ [0, 1]. Let x, y ∈ H. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1393 Then µ(x) ≥ min{µ(x), µ(y)} and µ(y) ≥ min{µ(x), µ(y)}. Take t0 = min{µ(x), µ(y)}. Then x, y ∈ µt0 and so x ⊛ y ⊆ µt0 . Let a ∈ x ⊛ y. Then µ(a) ≥ t0 which implies that infa∈x⊛y {µ(a)} ≥ t0 = min{µ(x), µ(y)}. Thus, µ is a fuzzy hyper UP-subalgebra of H. Theorem 3. Let µ be a fuzzy hyper UP-subalgebra of a hyper UP-algebra H. Then the set Hµ = {x ∈ H : µ(x) = µ(0)} is a hyper UP-subalgebra of H. Proof. Note that 0 ∈ Hµ so that Hµ ̸= ∅. Let x, y ∈ Hµ and let a ∈ x ⊛ y. Then µ(x) = µ(y) = µ(0). Since µ is a fuzzy hyper UP-subalgebra, µ(a) ≥ µ(0) by Definition 8. Thus, by Lemma 1, µ(a) = µ(0) for all a ∈ x⊛ y, that is, a ∈ Hµ. Since a is an arbitrary element of x⊛y, it follows that x⊛y ⊆ Hµ. Thus, Hµ is a hyper UP-subalgebra of H. Lemma 2. Let {µα : α ∈ A} be a nonempty family of fuzzy hyper UP-algebras of a hyper UP-algebra H. Then (i) inf a∈x⊛y { inf α∈A {µα(a)}} ≥ inf α∈A { inf a∈x⊛y {µα(a)}}. (ii) infα∈A{min{µα(x), µα(y)}} ≥ min{ inf α∈A {µα(x)}, inf α∈A {µα(y)}}. Proof. Let {µα : α ∈ A} be a nonempty family of fuzzy hyper UP-subalgebras of H and let x, y ∈ H. (i) For all a ∈ x ⊛ y, we have µα(a) ≥ infa∈x⊛y{µα(a)} for all α ∈ A. Thus, for all a ∈ x⊛ y, infα∈A{µα(a)} ≥ infa∈x⊛y{µα(a)} ≥ infα∈A{infa∈x⊛y{µα(a)}}. Hence, infa∈x⊛y{infα∈A{µα(a)}} ≥ infα∈A{infa∈x⊛y{µα(a)}}. (ii) Note that µα(x) ≥ infα∈A{µα(x)} and µα(y) ≥ infα∈A{µα(y)}. Then, {min{µα(x), µα(y)}} ≥ min{ inf α∈A {µα(x)}, inf α∈A {µα(y)}. Thus, infα∈A{min{µα(x), µα(y)}} ≥ min{ inf α∈A {µα(x)}, inf α∈A {µα(y)}. Hence, the conclusion follows. Example 5. Let H = {0, 1, 2, 3} be a set. Define the hyperoperation ⊛ by the following Cayley table: ⊛ 0 1 2 3 0 {0} {1} {2} {3} 1 {0} {0, 1} {1, 2, 3} {0, 3} 2 {0} {0, 1} {0, 2} {0, 3} 3 {0} {1} {1, 2, 3} {0, 3} By routine calculations, (H;⊛, 0) is a hyper UP-algebra. Define a fuzzy subset µ : H → [0, 1] by µ(0) = 0.9, µ(1) = 0.7, µ(2) = 0.4, µ(3) = 0.2. Let A = {0, 1} and B = {0, 2}. By routine calculations, µA and µB are fuzzy hyper UP-subalgebras of H. But µA∪B is not a fuzzy hyper UP-subalgebra of H since 0.2 = infa∈1⊛2{µ(a)} < min{µ(1), µ(2)} = 0.4. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1394 Remark 1. The union of two fuzzy hyper UP-subalgebras of a hyper UP-algebra H is not necessarily a fuzzy hyper UP-subalgebra of H. Theorem 4. The intersection of any nonempty family of fuzzy hyper UP-subalgebras of a hyper UP-algebra H is also a fuzzy hyper UP-subalgebra of H. Proof. Let {µα : α ∈ A} be a nonempty family of fuzzy hyper UP-subalgebras of H. Let x, y ∈ H. Then by Definitions 7 and 8 and Lemma 2(i) and (ii), inf a∈x⊛y {∧α∈Aµα(a)} = inf a∈x⊛y { inf α∈A {µα(a)}} ≥ infα∈A{infa∈x⊛y{µα(a)}} ≥ infα∈A{min{µα(x), µα(y)}} ≥ min{infα∈A{µα(x)}, infα∈A{µα(y)}} = min{∧α∈Aµα(x),∧α∈Aµα(y)}. Hence, the conclusion follows. Theorem 5. Let K be a nonempty subset of a hyper UP-algebra H and let α, β ∈ [0, 1] with α > β. Let µK be a fuzzy subset of H defined by µK(x) = { α, if x ∈ K β, if x /∈ K for all x ∈ H. Then µK is a fuzzy hyper UP-subalgebra of H if and only if K is a hyper UP-subalgebra of H. Proof. Suppose that µK is a fuzzy hyper UP-subalgebra of H and let x, y ∈ K. Then µK(x) = α = µK(y). By Definition 8, infa∈x⊛y{µK(a)} ≥ α. Since β < α, µK(a) = α for all a ∈ x⊛ y. This implies that x⊛ y ⊆ K. Thus, K is a hyper UP-subalgebra of H. Conversely, suppose K is a hyper UP-subalgebra of H. Let x, y ∈ H. If x, y ∈ K, then x ⊛ y ∈ K since K is a hyper UP-subalgebra of H. Hence, infa∈x⊛y(µK(a)) = α = min{µK(x), µK(y)}. Suppose x /∈ K or y /∈ K. Then min{µK(x), µK(y)} = β. Thus, infa∈x⊛y µK(a) ≥ min{µK(x), µK(y)}. This shows that µK is a fuzzy hyper UP-subalgebra of H. Theorem 6. Let H be a hyper UP-algebra. Then every hyper UP-subalgebra of H is an upper level hyper UP-subalgebra of a fuzzy hyper UP-subalgebra of H. Proof. Let K be a hyper UP-subalgebra of H. For a fixed t ∈ (0, 1] , we consider the fuzzy subset µ defined by µ(x) = { t, if x ∈ K 0, if x /∈ K. By Theorem 5, µ is a fuzzy hyper UP-subalgebra of H. Since 0 ∈ K,µ(0) = t. Hence, 0 ∈ µt = {x ∈ H : µ(x) = t}. By Theorem 2, µt is a hyper UP-subalgebra of H. Let R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1395 x ∈ K. Then µ(x) = t which implies that x ∈ µt. Thus, K ⊆ µt. On the other hand, suppose that x ∈ µt. Then µ(x) = t which means that x ∈ K. Thus, µt ⊆ K. Hence, K = µt. Recall that each hyper UP-subalgebra of a hyper UP-algebra H contains the element 0. Thus, for any family {Kn} of hyper UP-subalgebras of H, 0 ∈ ⋂∞ n=1Kn and so ⋂∞ n=1Kn ̸= ∅. Theorem 7. Let H be a hyper UP-algebra and let {Kn : n = 1, 2, · · · } be a family of hyper UP-subalgebras of H such that H = K1 ⊇ K2 ⊇ · · · . Let µ be a fuzzy set in H defined by µ(x) = { n n+1 , if x ∈ Kn\Kn+1, 1, if x ∈ ⋂∞ n=1Kn. Then µ is a fuzzy hyper UP-subalgebra of H. Proof. Let x, y ∈ H. Consider the following cases. Case 1: x, y ∈ Kn\Kn+1. Then µ(x) = n n+1 = µ(y). Since Kn is a hyper UP- subalgebra, x ⊛ y ⊆ Kn. Then x ⊛ y ⊆ Kn+1 or x ⊛ y ⊈ Kn+1. If x ⊛ y ⊈ Kn+1, then x ⊛ y ⊆ Kn\Kn+1 and for all a ∈ x ⊛ y, µ(a) = n n+1 . Thus, infa∈x⊛y{µ(a)} = n n+1 = min{µ(x), µ(y)}. Suppose x ⊛ y ⊈ Kn\Kn+1. If x ⊛ y ⊆ ∩∞ m=1Km, then infa∈x⊛y{µ(a)} = 1 > n n+1 = min{µ(x), µ(y)}. Suppose x ⊛ y ⊈ Km. Then there exists a ∈ x ⊛ y such that a ∈ Km\Km+1 for some m ≥ n. By the Well-Ordering Principle, there exists a smallest positive integer p ≥ n such that z ∈ Kp\Kp+1 for some z ∈ x⊛ y. Hence, infa∈x⊛y{µ(a)} = p p+1 ≥ n n+1 . Case 2: x ∈ Ks\Ks+1, y ∈ Kr\Kr+1. Without loss of generality, assume that s < r. The s s+1 < r r+1 and Ks ⊇ Kr. Since Ks is a hyper UP-subalgebra, x⊛ y ⊆ Ks. As in the case of a portion of Case 1, infa∈x⊛y{µ(a)} = s+j s+j+1 > s s+1 = min{µ(x), µ(y)}. Case 3: x, y ∈ ⋂∞ n=1Kn. Then x⊛ y ⊆ ⋂∞ n=1Kn so that for all a ∈ x⊛ y, µ(a) = 1. Thus, infa∈x⊛y{µ(a)} = 1 = min{µ(x), µ(y)}. Case 4: x ∈ ⋂∞ n=1Kn, and y /∈ ⋂∞ n=1Kn or (x /∈ ⋂∞ n=1Kn and y ∈ ⋂∞ n=1Kn.) Without loss of generality, assume that x ∈ ⋂∞ n=1Kn and y /∈ ⋂∞ n=1Kn. This means that there exists r such that y /∈ Kr. Thus, the set S = {q : y /∈ Kq} ̸= ∅. By Well-Ordering Principle, there exists a smallest element t ∈ S. This means that y ∈ Kt−1\Kt so that µ(y) = t−1 t . So, min{µ(x), µ(y)} = min{1, t−1 t } = t−1 t . As in the case of a part of Case 1, infa∈x⊛y{µ(a)} = t+j t+j+1 > t−1 t = min{µ(x), µ(y)}. Corollary 1. Let H be a hyper UP-algebra. Then for any family of hyper UP-subalgebras {Kn : n = 1, 2, · · · } of H such that H = K1 ⊇ K2 ⊇ · · · , there exists fuzzy hyper UP- subalgebras of H whose upper level sets are exactly the hyper UP-subalgebras in the family. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1396 Proof. Define a fuzzy set µ of H by µ(x) = { n n+1 , if x ∈ Kn\Kn+1, 1, if x ∈ ⋂∞ n=1Kn. Then by Theorem 7, µ is a fuzzy hyper UP-subalgebra of H. Let x ∈ H. For each n, consider µ n n+1 = {x ∈ H : µ(x) ≥ n n+1}. Let x ∈ Kn. If x ∈ Kn\Kn+1, then µ(x) = n n+1 . If x ∈ ∩∞ m=1Km, then µ(x) = 1 > n n+1 . Hence, x ∈ µ n n+1 , showing that Kn ⊆ µ n n+1 . If y ∈ µ n n+1 , then µ(y) ≥ n n+1 . If µ(y) = 1, then y ∈ ∩∞ m=1Km. Hence, y ∈ Kn. Suppose that y /∈ ∩∞ m=1Km. Choose the smallest integer p such that y /∈ Kp. Then y ∈ Kp−1\Kp and µ(y) = p−1 p ≥ n n+1 . Hence, p − 1 ≥ n. This implies that y ∈ Kn and µ n n+1 ⊆ Kn. Thus, n n+1 = Kn. 4. Fuzzy hyper UP-filter In this section, we introduce the notion of fuzzy hyper UP-filter of a hyper UP-algebra and study some of its basic properties. Definition 10. A subset G of a hyper UP-algebra H is called a hyper UP-filter of H if it satisfies the following properties: (i) 0 ∈ G (ii) for any x, y ∈ H,x ∈ G and x⊛ y ⊆ G imply y ∈ G. In Example 1, it can be verified by routine calculations that the set {0, a, c} is a hyper UP-filter of H. Definition 11. A fuzzy set µ in a hyper UP-algebra H is called a fuzzy hyper UP-filter of H if it satifies the following properties: for any x, y ∈ H, (i) µ(0) ≥ µ(x), (ii) µ(y) ≥ min{µ(x), infa∈x⊛y{µ(a)}}. Remark 2. A fuzzy hyper UP-filter need not be a fuzzy hyper UP-subalgebra. Example 6. Consider the hyper UP-filter {0, a, c} of H in Example 1. It can be easily verified that µ(x) = { 1, if x ∈ {0, a, c} 0, if x ∈ {b, d} is a fuzzy hyper UP-filter of H. Example 7. Consider the fuzzy set µ in Example 4(2). Note that µ is not a fuzzy hyper UP-subalgebra of H but by routine calculations, µ is a fuzzy hyper UP-filter of H. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1397 Proposition 3. Let µ be a fuzzy subset of a hyper UP-algebra H. If µ is a fuzzy hyper UP-filter of H, then µt is a hyper UP-filter of H for each t ∈ [0, 1] with µt ̸= ∅. Proof. Suppose that µ is a fuzzy hyper UP-filter of H and µt ̸= ∅, where t ∈ [0, 1]. Then there exists a ∈ µt and so µ(a) ≥ t. By Definition 11, µ(0) ≥ µ(a) ≥ t, that is, 0 ∈ µt. Let x, y ∈ H with x ∈ µt and x ⊛ y ⊆ µt. Then µ(x) ≥ t and for all a ∈ x ⊛ y, µ(a) ≥ t. Hence, infa∈x⊛y{µ(a)} ≥ t. Since µ is a fuzzy hyper UP-filter, µ(y) ≥ min{µ(x), infa∈x⊛y{µ(a)}} ≥ t. Hence, y ∈ µt and so µt is a hyper UP-filter of H. For any nonempty subset K of a hyper UP-algebra H, we define a fuzzy set µK in H by µK(x) = { α, if x ∈ K β, if x /∈ K for all x ∈ H and α, β ∈ [0, 1] with α > β. Lemma 3. Let K be a nonempty subset of a hyper UP-algebra H. If µK is a fuzzy hyper UP-filter of H, then 0 ∈ K. Proof. Let µK be a fuzzy hyper UP-filter of H. Since K ̸= ∅, it follows that there exists a ∈ K with µK(a) = α. By Definition 11(i), µK(0) ≥ µK(a) = α. Since α > β, we have µK(0) = α. Hence, 0 ∈ K. Theorem 8. Let K be a nonempty subset of a hyper UP-algebra H. If µK is a fuzzy hyper UP-filter of H, then K is a hyper UP-filter of H. Proof. Suppose that µK is a fuzzy hyper UP-filter of H. Let x, y ∈ H with x ∈ K and x ⊛ y ⊆ K. Then for all a ∈ x ⊛ y, we have a ∈ K, that is, µK(a) = α. It follows that infa∈x⊛y{µ(a)} = α. By Definition 11(ii), µ(y) ≥ min{µ(x), infa∈x⊛y{µ(a)}} = α. Since α > β, it follows that µK(y) = α. Hence, y ∈ µK and since 0 ∈ K by Lemma 3, µK is a hyper UP-filter of H. Lemma 4. Let {µα : α ∈ A} be a nonempty family of fuzzy subsets of a hyper UP-algebra H. Then inf α∈A { min { µα(x), inf a∈x⊛y {µα(a)} }} ≥ min { inf α∈A {µα(x)}, inf α∈A { inf a∈x⊛y {µα(a)}} } . Proof. Let x, y ∈ H. Note that ∀α ∈ A, µα(x) ≥ infα∈A{µα(x)} and infa∈x⊛y{µα(a)} ≥ infα∈A{infa∈x⊛y{µα(a)}}. Thus, ∀ α ∈ A, we have min{µα(x), inf a∈x⊛y {µα(a)}} ≥ min { inf α∈A {µα(x)}, inf α∈A { inf a∈x⊛y {µα(a)}} } . Hence, inf α∈A { min { µα(x), inf a∈x⊛y {µα(a)} }} ≥ min { inf α∈A {µα(x)}, inf α∈A { inf a∈x⊛y {µα(a)}} } . R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1398 Theorem 9. Let {µα : α ∈ A} be a nonempty family of fuzzy subsets of a hyper UP- algebra H. If µα is a fuzzy hyper UP-filter of H for all α ∈ A, then so is ∧α∈Aµα. Proof. Let x, y ∈ H. Since each µα is a fuzzy hyper UP-filter of H,µα(0) ≥ µα(x) for all α ∈ A. Thus for all α ∈ A, infα∈A{µα(0)} ≥ µα(x). Hence, infα∈A{µα(0)} ≥ infα∈A{µα(x)}, or equivalently, ∧α∈Aµα(0) ≥ ∧α∈Aµα(x). Now, by Definitions 11 and 7 and Lemma 4, ∧α∈Aµα(y) = inf α∈A {µα(y)} ≥ inf α∈A { min{µα(x), inf a∈x⊛y {µα(a)}} } ≥ min { inf α∈A {µα(x), inf α∈A { inf a∈x⊛y {µα(a)}} } ≥ min { inf α∈A {µα(x), inf a∈x⊛y { inf α∈A {µα(a)}} } = min { ∧α∈Aµα(x), inf a∈x⊛y {∧α∈A{µα(a)}} } . Hence, the conclusion follows. 5. Hyper Homomorphism of Fuzzy Hyper UP-algebras This section provides some properties of hyper homomorphism in relation to the con- cepts of fuzzy hyper UP-subalgebra and fuzzy hyper UP-filter. By Definitions 3, 5, 7 and Lemma 1, we deduce the following proposition: Proposition 4. Let f : G → H be a hyper homomorphism of hyper UP-algebras G and H. (i) If µ is a fuzzy hyper UP-subalgebra of G, then f(µ)(0H) = µ(0G). (ii) If µ is a fuzzy hyper UP-subalgebra of H, then µf (0G) = 0H . Proof. Let f : G → H be a hyper homomorphism. (i) By Definition 3, 0G ∈ f−1(0H) and so f−1(0H) ̸= ∅. By Definition 6 and Lemma 1, f(µ)(0H) = supx∈f−1(0H){µ(x)} = µ(0G). (ii) By Definitions 3 and 5, µf (0G) = µ(f(0G)) = µ(0H). Lemma 5. Let f : G → H be a hyper homomorphism of hyper UP-algebras G and H, µ be a fuzzy subset of H and µf be a fuzzy hyper UP-subalgebra of G. If A = {µ(f(a)) = µf (a) : a ∈ x⊛G y} and B = {µ(f(a)) = µf (a) : f(a) ∈ f(x)⊛H f(y)}, then A=B. R. Amairanto, R. Isla / Eur. J. Pure Appl. Math, 14 (4) (2021), 1388-1401 1399 Proof. Let z ∈ A. Then z = µ(f(a)) for some a ∈ x ⊛G y. Since f is a hyper homomorphism, we have f(a) ∈ f(x ⊛G y) = f(x) ⊛H f(y). Thus z ∈ B and so A ⊆ B. Conversely, suppose that z ∈ B. Then z = µ(f(a)) for some f(a) ∈ f(x)⊛H f(y). Since f is a hyper homomorphism, f(a) ∈ f(x) ⊛H f(y) = f(x ⊛G y). Thus, f(a) = f(a′), where a′ ∈ x⊛G y. So, z = µ(f(a′)), where a′ ∈ x⊛G y. Hence, z ∈ A which means that B ⊆ A. Therefore, A = B. Theorem 10. Let f : G → H be a hyper epimorphism of hyper UP-algebras G and H and µ be a fuzzy subset of H. If µf is a fuzzy hyper UP-subalgebra of G, then µ is a fuzzy hyper UP-subalgebra of H. Proof. Let f be a hyper epimorphism of hyper UP-algebras G and H and suppose µf is a fuzzy hyper UP-subalgebra of G. Let x, y ∈ H. Since f is onto, there exist x′, y′ ∈ G such that f(x′) = x and f(y′) = y. Hence, µ(x) = µf (x′) and µ(y) = µf (y′). Let z ∈ x ⊛H y. Then f is onto implies that there exists z′ ∈ G such that f(z′) = z, that is, µf (z′) = µ(z). Note that f(z′) = z ∈ x ⊛H y = f(x′) ⊛H f(y′). Hence, by Lemma 5, z′ ∈ x′ ⊛G y′. By Definition 8 and the assumption that µf is a fuzzy hyper UP-subalgebra of G, we have µ(z) = µf (z′) ≥ infa∈x′⊛Gy′{µf (a)} ≥ min{µf (x′), µf (y′)} = min{µ(f(x′)), µ(f(y′))} = min{µ(x), µ(y)}. Thus, infz∈x⊛Hy{µ(z)} ≥ min{µ(x), µ(y)}. Hence, µ is a fuzzy hyper UP-subalgebra of H. Theorem 11. Let f : G → H be a hyper homomorphism of hyper UP-algebras G and H. If µ is a fuzzy hyper UP-filter of H, then µf is a fuzzy hyper UP-filter of G. Proof. Let x ∈ G. Then f(x) ∈ H. Since µ : H −→ [0, 1] is a fuzzy hyper UP-filter of H,µ(0H) = µ(f(0G)) ≥ µ(f(x)), that is, µf (0G) ≥ µf (x) for all x ∈ G. Next, let x, y ∈ G. Then f(x), f(y) ∈ H. Again, by our assumption on µ, and by Lemma 5, µ(f(y)) = µf (y) ≥ min{µ(f(x)), inf f(z)∈f(x)⊛Hf(y) {µ(f(z))}} = min{µf (x), inf z∈x⊛Gy {µf (z)}}. Hence, µf is a fuzzy hyper UP-filter of G. Acknowledgements This research is funded by the Philippine Department of Science and Technology- Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP) and the Mindanao State University-Iligan Institute of Technology. The au- thors would like to thank the reviewers for their invaluable comments and suggestions that led to this improved version of the paper. REFERENCES 1400 References [1] R. Amairanto and R. Isla. Hyper Homomorphism and hyper product of hyper UP- algebras. European Journal of Pure and Applied Mathematics, 13(3):483–497, 2020. [2] R. Ameri and T. Nozari. Fuzzy hyperalgebras. Computers and Mathematics with Applications, 61:149–154, 2011. [3] P. Corcini. Some remarks on hyperstructures their connections with fuzzy sets and extensions to weak structures. Ratio Mathematika, 33:61–76, 2017. [4] P. Corcini and V. Leoreanu. Applications of Hyperstructure Theory. Springer-US, 2003. [5] P. Corsini and I. Tofan. On fuzzy hypergroups. Pure Mathematics and Applications, 8(1):29–37, 1997. [6] B. Davvaz and I. Cristea. Fuzzy Algebraic Hyperstructures. Springer: Cham, Switzer- land, 2015. [7] S.A. Bhatti F. Nisar, R.S. Tariq. Fuzzy ideals in hyper BCI-algebras. World Applied Sciences Journal, 16(12):1771–1777, 2012. [8] A. Iampan. A new branch of the logical algebra: UP-algebras. Journal of Algebra and Related Topics, 5(1):35–54, 2017. [9] Y.B. Jun and X.L. Xin. Fuzzy hyper BCK-ideals of hyper BCK-algebras. Scientia Mathematicae Japonicae, 53(2):353–360, 2001. [10] K.H. Lee. First Course on Fuzzy Theory and Applications. Springer-Verlag Berlin Heidelberg, 2005. [11] V. Leoreanu-Fotea and B. Davvaz. Fuzzy hyperrings. Fuzzy Sets Syst., 160:2366–2378, 2009. [12] A. Macodi-Ringia and Jr G. Petalcorin. Some results on fuzzy implicative hyper GR-ideals. European Journal of Pure and Applied Mathematics, 12(2):409–417, 2019. [13] A. Macodi-Ringia and Jr G. Petalcorin. On intuitionistic fuzzy hyper GR-ideals in hyper GR-algebras. European Journal of Pure and Applied Mathematics, 13(2):246– 257, 2020. [14] F. Marty. Sur une generalisation de la notion de groupe. In 8th Congress des Math- ematician Scandinaves, pages 45–49, Stockholm, 1934. [15] C. Prabpayak and U. Leerawat. On ideals and congruence in KU-algebras. Scientia Magna Journal, 5(1):54–57, 2009. [16] D. Romano. Hyper UP-algebras. Journal of Hyperstructures, 8(2):112–122, 2019. REFERENCES 1401 [17] F. Kareem S. Mostafa and B. Davvaz. Hyper structure theory applied to KU-algebras. Journal of Hyperstructures, 6(2):82–95, 2017. [18] G. Tabaranza and J. Vilela. Fuzzy Hyper B-algebras. JP Journal of Algebra, Number Theory and Applications., 41(2):205–218, 2019. [19] M. Voskoglou. Fuzzy sets, Fuzzy Logic and Their Applications. Mdpi AG, 2020. [20] M. Bakhshi X. Xin, R.A. Borzooie and Y.B. Jun. Intuitionistic fuzzy soft hyper BCK-algebras. Symmetry, 11(3):399 (article code), 2019. [21] L. Zadeh. Fuzzy sets. Information and Control, 41:338–353, 1965.