EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 548-576 ISSN 1307-5543 – ejpam.com Published by New York Business Global Single-valued Neutrosophic Soft sets in Hyper UP-Algebra Allan N. Cano1,∗, Gaudencio C. Petalcorin, Jr.1 1 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, the notions of SVN hyper UP -algebra and SVNS hyper UP -algebra are introduced, and some of their structural properties are investigated. Moreover, the Cartesian product of SVNS hyper UP -algebra is discussed and proved to be a SVNS hyper UP - algebra. Finally, the homomorphic image and preimage of SVNS hyper UP -algebra under SVNS functions are studied and showed also to be SVNS hyper UP -algebra. 2020 Mathematics Subject Classifications: 08A30 Key Words and Phrases: Hyper UP -algebra, single-valued neutrosophic set, single-valued neu- trosophic soft set 1. Introduction The concept of fuzzy sets and fuzzy logic has been used widely in many applications involving uncertainties. Such concept was initiated by L. Zadeh [10]. Resulting from vagueness or partial belongingness of an element in a set, fuzzy set is successful in han- dling uncertainties. However, there are still some situations which it cannot cover like problems involving incomplete information. Motivated by this, a lot of researchers ex- tended this concept and presented a different theories regarding uncertainty which include intuitionistic fuzzy set theory [3], interval-valued intuitionistic fuzzy set theory [9] and so on. Later on, Smarandache [18] generalized intuitionistic fuzzy set theory by introduc- ing the concept of neutrosophic set in 1998. Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. It is a powerful general formal framework that has been recently proposed. To have its real life application in engineering and science, neutro- sophic set needs to be specified from a technical point of view. That is why single valued neutrosophic set was introduced by Wang et al. [22] together with its various proper- ties. Single-valued neutrosophic set has been developing rapidly due to its wide range of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4637 Email addresses: allan.cano@g.msuiit.edu.ph (A. Cano), gaudencio.petalcorin@g.msuiit.edu.ph (G. Petalcorin) https://www.ejpam.com 548 © 2023 EJPAM All rights reserved. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 549 theoretical elegance and application areas. The reader may refer to the following articles [7, 8, 16, 17, 19, 20] as references. In 1999, Molodtsov [14] studied another mathematical theory called soft set theory by giving parameterized approach to uncertainties. On the other hand, Maji [12] unified the fundamental theories of neutrosophic set and soft set, and came up with the concept of neutrosophic soft set. Some theoretical advancement and applications have been reported in the following literatures [1, 4, 5, 11]. The hyper algebraic structure theory was introduced in 1934 by F. Marty [13] at the 8th congress of Scandinavian Mathematicians. This theory is then applied by Y. B. Jun et al. [21] to BCK-algebras to produce the notion of hyper BCK-algebras as a generalization of the BCK-algebras. After that, many researchers have been inspired to generalize some existing algebras and one of them is D. Romano [15]. He has come up with the concept of hyper UP -algebras to generalize UP -algebras. In this paper, we utilize the notions of single-valued neutrosophic sets and single-valued neutrosophic soft sets to hyper UP -algebra to generate SVN hyper UP -algebra and SVNS hyper UP -algebra. Several of their basic properties are studied. In addition, we define the Cartesian product of SVNS hyper UP -algebra, and image and preimage of SVNS hyper UP -algebra under SVNS function. Each of them is discussed and illustrated with corresponding examples. 2. Preliminary Concepts Definition 1. [15] Let P(H) to be the power set of H. Consider P∗(H) = P(H) \ {∅}. A hyperoperation on a nonempty set H is a function ◦ : H ×H −→ P∗(H). The image of (x, y) ∈ H ×H under ◦ is denoted by x ◦ y. If x ∈ H and A,B are nonempty subsets of H, then we define (i) A ◦B = ⋃ a∈A,b∈B a ◦ b; (ii) A ◦ x = A ◦ {x}; and (iii) x ◦B = {x} ◦B. Definition 2. [15] Let x, y ∈ H and A,B ⊆ H. Then (i) x ≪ y if and only if 0 ∈ x ◦ y; and (ii) A ≪ B if and only if for any a ∈ A, there exists b ∈ B such that a ≪ b. We call ≪ a hyperorder on H. Remark 1. [15] For all A,B ⊆ H, A ≪ B implies 0 ∈ A ◦B. Definition 3. [15] Let X be a nonempty set such that 0 ∈ X and (X, ◦,≪, 0) be a hyperstructure. Then (X, ◦,≪, 0) is called a hyper UP-algebra if the following formulas are valid: ∀x, y, z ∈ X, (HUP1) y ◦ z ≪ (x ◦ y) ◦ (x ◦ z), A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 550 (HUP2) x ◦ 0 = {0}, (HUP3) 0 ◦ x = {x}, and (HUP4) x ≪ y ∧ y ≪ x =⇒ x = y. Example 1. Let X = {0, r, s, t} be a set. If we define a hyper operation “◦” as following: ◦ 0 r s t 0 {0} {r} {s} {t} r {0} {0, r} {0, s} {r, s} s {0} {r, s} {0, s} {r} t {0} {0, r, s, t} {s, t} {0} , then the routine calculation will show that (X, ◦,≪, 0) is a hyper UP -algebra. Example 2. Let X = {0, u, v}. Define a hyper operation “◦” as follows: ◦ 0 u v 0 {0} {u} {v} u {0} {0, u} {0, v} v {0} {u, v} {0, v} . By routine calculation, (X, ◦,≪, 0) is a hyper UP -algebra. Example 3. Let X = {0, a, b}. Define a hyper operation “◦” as follows: ◦ 0 a b 0 {0} {a} {b} a {0} {0, a, b} {0, b} b {0} {0, a, b} {0} . Observe that a ≪ b and b ≪ a. But a ̸= b. Thus, (X, ◦,≪, 0) does not satisfy (HUP4) and so it is not a hyper UP -algebra. Proposition 1. [15] Let (H, ◦,≪, 0) be a hyper UP-algebra. Then the following hold for all x, y, z ∈ H and for every nonempty subsets A,B,C ⊆ H: (i) A ⊆ B implies A ≪ B (v) z ≪ x ◦ z (ii) 0 ◦ 0 = {0} (vi) A ◦ 0 = {0} (iii) x ≪ 0 (vii) 0 ◦A = A (iv) x ≪ x (viii) (0 ◦ 0) ◦ x = {x} Proposition 2. [15] Let S be a nonempty subset of a hyper UP-algebra (X, ◦,≪, 0). Then S is a hyper UP-subalgebra of X if and only if ∀x, y ∈ S, x ◦ y ⊆ S. Definition 4. [15] Let (X1, ◦1,≪1, 01) and (X2, ◦2,≪2, 02) be hyper UP -algebras. A mapping f : X1 −→ X2 is called a hyper homomorphism if for all a, b ∈ X1, (i) f(01) = 02 and A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 551 (ii) f(a ◦1 b) = f(a) ◦2 f(b). Definition 5. [2] Let f : (X1, ◦1,≪1, 01) −→ (X2, ◦2,≪2, 02) be a hyper homomorphism. We say that f is a hyper monomorphism if f is one-to-one and f is a hyper epimorphism if f is onto. We also say that f is a hyper isomorphism if f is both one-to-one and onto. In this case, X1 and X2 are hyper isomorphic which is denoted as X1 ∼=H X2. Definition 6. [2] Let (X1, ◦1,≪1, 01) and (X2, ◦2,≪2, 02) be hyper UP -algebras. Define a set X1 ×X2 by X1 ×X2 = {(a, b) : a ∈ X1 and b ∈ X2}. with a hyperoperation “ ◦ ” on X1 ×X2 given by (a, b) ◦ (c, d) = (a ◦1 c, b ◦2 d) and a hyperorder “≪” given by (a, b) ≪ (c, d) ⇐⇒ a ≪1 c and b ≪2 d for all (a, b), (c, d) ∈ X1 ×X2.. Then (X1 ×X2, ◦,≪, (01, 02)) is called the hyper product of X1 and X2. Definition 7. [18] Let U be the universe. A neutrosophic set A is characterized by a truth membership function TA, an indeterminacy membership function IA, and a falsity membership function FA where TA, IA,FA are real standard or non-standard elements of ]−0, 1+[ with −0 = 0− ϵ and 1+ = 1 + ϵ for any infinitesimal number ϵ. It can be written as A = {⟨x, (TA(x), IA(x),FA(x))⟩ |x ∈ U} where TA, IA,FA : U −→]−0, 1+[ and −0 ≤ TA(x) + IA(x) + FA(x) ≤ 3+. However, it is difficult to use a neutrosophic set with values from real standard or non- standard subsets of ]−0, 1+[ in real life application especially scientific and engineering problem [22]. So, this paper considers the neutrosophic set which takes values from the interval [0, 1]. Definition 8. [22] Let X be a space of points (objects), with a generic element in X denoted by x. A single valued neutrosophic set (SVNS ) A in X is characterized by truth- membership function TA, indeterminacy-membership function IA and falsity-membership function FA. For each point x ∈ X, TA(x), IA(x),FA(x) ∈ [0, 1]. Definition 9. [14] Given an initial universe set U and set E of parameters or attributes with respect to U , let P(U) denote the power set of U and A ⊆ E. A pair (F,A) is called a soft set over U , where F is a mapping given by F : A −→ P(U). For any ϵ ∈ A, F (ϵ) may be considered as the set of ϵ-approximate elements of the soft set (F,A). A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 552 The concept of neutrosophic soft set was first defined by Maji [12] and later on, it was modified by Deli and Broumi [6] as given below: Definition 10. Let U be an initial universe set and E be a set of parameters. Let N (U) denote the set of all neutrosophic sets of U . Then a neutrosophic soft set (F,E) over U is a set defined by a set valued function F representing a mapping F : E −→ N (U) where F is called approximate function of the neutrosophic soft set (F,E). In other words, the neutrosophic soft set is a parameterized family of some elements of the set N (U) and therefore it can be written as a set of ordered pairs (F,E) = {(e, { 〈 x, (TF (e)(x), IF (e)(x),FF (e)(x)) 〉 })|x ∈ U, e ∈ E} where TF (e)(x), IF (e)(x),FF (e)(x) ∈ [0, 1], respectively called the truth-membership, indeterminacy- membership, falsity-membership function of F (e). Since supremum of each T , I,F is 1 so the inequality 0 ≤ TF (e)(x) + IF (e)(x) + FF (e)(x) ≤ 3 is obvious. Definition 11. [6] The complement of a neutrosophic soft set (F,E) over U is denoted by (F,E)c and is defined by (F,E)c = {(e, { 〈 x, (FF (e)(x), 1− IF (e)(x), TF (e)(x)) 〉 })|x ∈ U, e ∈ E}. Definition 12. [6] Let (H,E) and (G,E) be two neutrosophic soft sets over the common universe U . Then (H,E) is said to be neutrosophic soft subset of (G,E) if ∀e ∈ E and ∀x ∈ U , TH(e)(x) ≤ TG(e)(x), IH(e)(x) ≥ IG(e)(x),FH(e)(x) ≥ FG(e)(x). We write (H,E) ⊆ (G,E) and (G,E) is a neutrosophic soft superset of (H,E). Definition 13. [5] A binary operation ∗ : [0, 1] × [0, 1] −→ [0, 1] is continuous t-norm if ∗ satisfies the following conditions : (i) ∗ is commutative and associative. (ii) ∗ is continuous. (iii) a ∗ 1 = 1 ∗ a = a, ∀a ∈ [0, 1]. (iv) a ∗ b ≤ c ∗ d if a ≤ c, b ≤ d with a, b, c, d ∈ [0, 1]. A few examples of continuous t-norm are a ∗ b = ab, a ∗ b = min{a, b}, a ∗ b = max{a+ b− 1, 0}. Definition 14. [5] A binary operation ⋄ : [0, 1] × [0, 1] −→ [0, 1] is continuous t-conorm (s− norm) if ⋄ satisfies the following conditions : (i) ⋄ is commutative and associative. (ii) ⋄ is continuous. (iii) a ⋄ 0 = 0 ⋄ a = a, ∀a ∈ [0, 1]. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 553 (iv) a ⋄ b ≤ c ⋄ d if a ≤ c, b ≤ d with a, b, c, d ∈ [0, 1]. A few examples of continuous s-norm are a ⋄ b = a + b − ab, a ⋄ b = max{a, b}, a ⋄ b = min{a+ b, 1}. Definition 15. [6] Let (H,E) and (G,E) be two neutrosophic soft sets over the common universe U . (i) Then the union of (H,E) and (G,E) is denoted by (H,E)∪ (G,E) = (K,E) and is defined by: (K,E) = {(e, { 〈 x, (TK(e)(x), IK(e)(x),FK(e)(x)) 〉 })|x ∈ U, e ∈ E} where TK(e)(x) = TH(e)(x) ⋄ TG(e)(x) IK(e)(x) = IH(e)(x) ∗ IG(e)(x) FK(e)(x) = FH(e)(x) ∗ FG(e)(x). (ii) Then the intersection of (H,E) and (G,E) is denoted by (H,E) ∩ (G,E) = (F,E) and is defined by: (F,E) = {(e, { 〈 x, (TF (e)(x), IF (e)(x),FF (e)(x)) 〉 })|x ∈ U, e ∈ E} where TF (e)(x) = TH(e)(x) ∗ TG(e)(x) IF (e)(x) = IH(e)(x) ⋄ IG(e)(x) FF (e)(x) = FH(e)(x) ⋄ FG(e)(x). In this paper, we use the minimality and maximality as binary operations ∗ and ⋄ respectively to define the union and intersection of two NSS sets. Example 4. Consider U = {s1, s2, s3} be the set of all students and E = {a1, a2} be the set of parameters where a1 stands for the parameter ‘brilliant’, a2 stands for the parameter ‘healthy’. Define a mapping H : E −→ N (U) by H(a1) = {⟨s1, (0.1, 0.5, 0.4)⟩ , ⟨s2, (0.6, 0.6, 0.7)⟩ , ⟨s3, (0.5, 0.6, 0.4)⟩} H(a2) = {⟨s1, (0.8, 0.4, 0.5)⟩ , ⟨s2, (0.7, 0.7, 0.3)⟩ , ⟨s3, (0.7, 0.5, 0.6)⟩}. and a mapping G : E −→ N (U) by G(a1) = {⟨s1, (0.8, 0.5, 0.6)⟩ , ⟨s2, (0.5, 0.7, 0.6)⟩ , ⟨s3, (0.4, 0.7, 0.5)⟩}, A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 554 G(a2) = {⟨s1, (0.7, 0.6, 0.5)⟩ , ⟨s2, (0.6, 0.8, 0.4)⟩ , ⟨s3, (0.5, 0.8, 0.6)⟩}. Then the neutrosophic soft sets (H,E) and (G,E) are collections of approximations as below: (H,E) = {(a1, {⟨s1, (0.1, 0.5, 0.4)⟩ , ⟨s2, (0.6, 0.6, 0.7)⟩ , ⟨s3, (0.5, 0.6, 0.4)⟩}) (a2, {⟨s1, (0.8, 0.4, 0.5)⟩ , ⟨s2, (0.7, 0.7, 0.3)⟩ , ⟨s3, (0.7, 0.5, 0.6)⟩})} and (G,E) = {(a1, {⟨s1, (0.8, 0.5, 0.6)⟩ , ⟨s2, (0.5, 0.7, 0.6)⟩ , ⟨s3, (0.4, 0.7, 0.5)⟩}) (a2, {⟨s1, (0.7, 0.6, 0.5)⟩ , ⟨s2, (0.6, 0.8, 0.4)⟩ , ⟨s3, (0.5, 0.8, 0.6)⟩})}. Thus, their union and intersection are (H,E) ∪ (G,E) = {(a1, {⟨s1, (0.8, 0.5, 0.4)⟩ , ⟨s2, (0.6, 0.6, 0.6)⟩ , ⟨s3, (0.5, 0.6, 0.4)⟩}) (a2, {⟨s1, (0.8, 0.4, 0.5)⟩ , ⟨s2, (0.7, 0.7, 0.3)⟩ , ⟨s3, (0.7, 0.5, 0.6)⟩})} and (H,E) ∩ (G,E) = {(a1, {⟨s1, (0.1, 0.5, 0.6)⟩ , ⟨s2, (0.5, 0.7, 0.7)⟩ , ⟨s3, (0.4, 0.7, 0.5)⟩}) (a2, {⟨s1, (0.7, 0.6, 0.5)⟩ , ⟨s2, (0.6, 0.8, 0.4)⟩ , ⟨s3, (0.5, 0.8, 0.6)⟩})}, respectively. 3. Main Results 3.1. Single-Valued Neutrosophic Hyper UP-subalgebra In this section, we introduce the concept of single-valued neutrosophic hyper UP - subalgebra and prove some of its basic properties. From here onwards, we simply denote a hyper UP -algebra (X, ◦,≪, 0) by X. Given a single-valued neutrosophic set A = (TA, IA,FA) in a hyper UP -algebra X and a subset S of X, we denote the following: ∗TA(S) = sup y∈S TA(y) and ∗TA(S) = inf y∈S TA(y); ∗IA(S) = sup y∈S IA(y) and ∗IA(S) = inf y∈S IA(y); ∗FA(S) = sup y∈S FA(y) and ∗FA(S) = inf y∈S FA(y). Definition 16. Let A = (TA, IA,FA) be a single-valued neutrosophic set in a hyper UP - algebra X. Then A is said to be a single-valued neutrosophic (SVN) hyper UP-subalgebra of X if for all x, y ∈ X, ∗TA(x ◦ y) ≥ min{TA(x), TA(y)}, A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 555 ∗IA(x ◦ y) ≤ max{IA(x), IA(y)}, and ∗FA(x ◦ y) ≤ max{FA(x),FA(y)}. Example 5. Consider a hyperUP -algebra (X, ◦,≪, 0) of Example 1 whereX = {0, r, s, t}. Also, define a single-valued neutrosophic set A = (TA, IA,FA) in X by the following: TA(x) = ( 0 r s t 0.87 0.42 0.56 0.29 ) , IA(x) = ( 0 r s t 0.39 0.79 0.76 0.94 ) , and FA(x) = ( 0 r s t 0.49 0.83 0.53 0.95 ) . By routine calculation, A is a SVN hyper UP -subalgebra of X. Example 6. Consider X = N ∪ {0} and a hyperoperation “◦” on X defined by x ◦ y =  {0} if y = 0, {0, y} if y = x, y ̸= 0, {y} otherwise. By thorough inspection, X is a hyper UP -algebra. Define a single-valued neutrosophic set A = (TA, IA,FA) in X by TA(x) = { 1 if x = 0, 0.5 if x ̸= 0. IA(x) = { 0 if x = 0, 0.5 if x ̸= 0. FA(x) = { 0 if x = 0, 0.5 if x ̸= 0. Again, by thorough inspection, A is a SVN hyper UP -subalgebra of X. Proposition 3. Let A = (TA, IA,FA) be a SVN hyper UP-subalgebra of X. Then for all x, y ∈ X, (i) TA(0) ≥ TA(x) IA(0) ≤ IA(x) FA(0) ≤ FA(x). (ii) ∗TA(0 ◦ x) = TA(x) ∗IA(0 ◦ x) = IA(x) ∗FA(0 ◦ x) = FA(x) A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 556 (iii) ∗TA(x ◦ 0) = TA(0) ∗IA(x ◦ 0) = IA(0) ∗FA(x ◦ 0) = FA(0) (iv) If TA(x) = TA(0) IA(x) = IA(0) FA(x) = FA(0) , then ∗TA(x ◦ y) ≥ TA(y) ∗IA(x ◦ y) ≤ IA(y) ∗FA(x ◦ y) ≤ FA(y) . (v) If TA(y) = TA(0) IA(y) = IA(0) FA(y) = FA(0) , then ∗TA(x ◦ y) ≥ TA(x) ∗IA(x ◦ y) ≤ IA(x) ∗FA(x ◦ y) ≤ FA(x) . (vi) If ∗TA(x ◦ y) = TA(x) ∗IA(x ◦ y) = IA(x) ∗FA(x ◦ y) = FA(x) , then TA(x) = TA(0) IA(x) = IA(0) FA(x) = FA(0) and TA(y) = TA(0) IA(y) = IA(0) FA(y) = FA(0) . Proof. Let A = (TA, IA,FA) be a SVN hyper UP -subalgebra in X and let x, y ∈ X (i) Note that x ≪ x. Then 0 ∈ x ◦ x. By hypothesis, TA(0) ≥ ∗TA(x ◦ x) ≥ min{TA(x), TA(x)} = TA(x), IA(0) ≤ ∗IA(x ◦ x) ≤ max{IA(x), IA(x)} = IA(x), and FA(0) ≤ ∗FA(x ◦ x) ≤ max{FA(x),FA(x)} = FA(x). (ii-iii) The proofs are straightforward since 0 ◦ x = {x} and x ◦ 0 = {0}. (iv) Assume that TA(x) = TA(0). By hypothesis and by (i), ∗TA(x◦y) ≥ min{TA(0), TA(y)} = TA(y). Using similar routine, IA(x) = IA(0) implies that ∗IA(x ◦ y) ≤ IA(y) and FA(x) = FA(0) implies that ∗FA(x ◦ y) ≤ FA(y). (v) Using similar arguments from (iv), the claim is true. (vi) Assume that ∗TA(x◦y) = TA(x). Taking x = 0, we have ∗TA(0◦y) = TA(0). By (ii), TA(y) =∗ TA(0◦y) = TA(0). Similarly, ∗IA(x◦y) = IA(x) implies that IA(y) = IA(0) and ∗FA(x ◦ y) = FA(x) implies that FA(y) = FA(0). On the other hand, if we take y = 0, we get ∗TA(x ◦ 0) = TA(x). By (iii), TA(0) =∗ TA(x ◦ 0) = TA(x). Also, IA(x) = IA(0) and FA(x) = FA(0) will follow. Proposition 4. If A = (TA, IA,FA) is a SVN hyper UP-subalgebra of X, then the set K = {x ∈ X|TA(x) = TA(0), IA(x) = IA(0),FA(x) = FA(0)} is a hyper UP-subalgebra of X. Proof. Let A = (TA, IA,FA) be a SVN hyper UP -subalgebra of X and let K = {x ∈ X|TA(x) = TA(0), IA(x) = IA(0),FA(x) = FA(0)}. Note that K ̸= ∅ since 0 ∈ K. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 557 Now, suppose x, y ∈ K and z ∈ x ◦ y. Then TA(x) = TA(0) = TA(y), IA(x) = IA(0) = IA(y),FA(x) = FA(0) = FA(y). By Proposition 3(i) and by hypothesis, we get TA(z) ≥ ∗TA(x ◦ y) ≥ min{TA(x), TA(y)} = min{TA(0), TA(0)} = TA(0), IA(z) ≤ ∗IA(x ◦ y) ≤ max{IA(x), IA(y)} = max{IA(0), IA(0)} = IA(0), and similarly, FA(z) ≤ FA(0). Thus, TA(z) = TA(0), IA(z) = IA(0), and FA(z) = FA(0). That is, z ∈ K and so x◦y ⊆ K. By Proposition 2, K is a hyper UP -subalgebra of X. We define the following α, β, γ-level subsets of X and their intersection: Tα A = {x ∈ X : TA(x) ≥ α}, IβA = {x ∈ X : IA(x) ≤ β}, F γ A = {x ∈ X : FA(x) ≤ γ}, and A(α,β,γ) = Tα A ∩ IβA ∩ F γ A. where A = (TA, IA,FA) is a SVN set in X and α, β, γ ∈ [0, 1]. Theorem 1. Let A = (TA, IA,FA) be a SVN set in X. Then A is a SVN hyper UP- subalgebra of X if and only if A(α,β,γ) is a hyper UP-subalgebra of X for all α, β, γ ∈ [0, 1]. Proof. Let A = (TA, IA,FA) be a SVN set in X. (⇒) Assume thatA is a SVN hyperUP -subalgebra ofX. Note that TA(x), IA(x),FA(x) ∈ [0, 1] ∀x ∈ X. Then take α = TA(x), β = IA(x) and γ = FA(x). By Proposition 3(i), TA(0) ≥ TA(x) = α, IA(0) ≤ IA(x) = β, and FA(0) ≤ FA(x) = γ. Hence, 0 ∈ Tα A ∩ IαA ∩ Fα A = A(α,β,γ) and so A(α,β,γ) ̸= ∅. Now, we let x, y ∈ A(α,β,γ) for all α, β, γ ∈ [0, 1]. Dealing first with Tα A , we have x, y ∈ Tα A . Let z ∈ x ◦ y. Then TA(x) ≥ α, TA(y) ≥ α, and TA(z) ≥∗ TA(x ◦ y). By assumption, TA(z) ≥ ∗TA(x ◦ y) ≥ min{TA(x), TA(y)} = α. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 558 Thus, z ∈ Tα A and so x ◦ y ⊆ Tα A . For I β A, we have x, y ∈ IβA. Then IA(x) ≤ β, IA(y) ≤ β, and IA(z) ≤∗ IA(x ◦ y). By assumption, IA(z) ≤ ∗IA(x ◦ y) ≤ max{IA(x), IA(y)} = β. Thus, z ∈ IβA and so x ◦ y ⊆ IβA. Using similar arguments, x ◦ y ⊆ F γ A for x, y ∈ F γ A. Now, it follows that x ◦ y ⊆ Tα A ∩ IβA ∩ F γ A = A(α,β,γ). By Proposition 2, A(α,β,γ) is a hyper UP -subalgebra. (⇐) Assume that A(α,β,γ) is a hyper UP -subalgebra of X for all α, β, γ ∈ [0, 1] and let x, y ∈ X. Note that TA(x), TA(y), IA(x), IA(y),FA(x),FA(y) ∈ [0, 1]. Then take α = min{TA(x), TA(y)}, β = max{IA(x), IA(y)}, and γ = max{TA(x), TA(y)} and so we have TA(x) ≥ α, TA(y) ≥ α, IA(x) ≤ β, IA(y) ≤ β, FA(x) ≤ γ, and FA(y) ≤ γ. Thus, x, y ∈ Tα A ∩ IβA ∩ F γ A = A(α,β,γ). By assumption, x ◦ y ⊆ A(α,β,γ). This means that ∗TA(x ◦ y) ≥ α = min{TA(x), TA(y)}, ∗IA(x ◦ y) ≤ β = max{IA(x), IA(y)}, and ∗FA(x ◦ y) ≤ γ = max{FA(x),FA(y)}. Hence, A is a SVN hyper UP -subalgebra of X. Corollary 1. Let A = (TA, IA,FA) be a SVN hyper UP-subalgebra of X. If 0 ≤ α ≤ α ′ ≤ 1, 0 ≤ β ≤ β ′ ≤ 1, and 0 ≤ γ ≤ γ ′ ≤ 1, then A(α ′ ,β,γ) is a hyper UP-subalgebra of A(α,β ′ ,γ ′ ). Proof. Let A = (TA, IA,FA) be a SVN hyper UP -subalgebra of X and let 0 ≤ α ≤ α ′ ≤ 1, 0 ≤ β ≤ β ′ ≤ 1, and 0 ≤ γ ≤ γ ′ ≤ 1. By Theorem 1, A(α ′ ,β,γ) and A(α,β ′ ,γ ′ ) are both hyper UP -subalgebra of X. We are left to show that A(α ′ ,β,γ) ⊆ A(α,β ′ ,γ ′ ). Let y ∈ Tα ′ A . Then TA(y) ≥ α ′ ≥ α. Thus, y ∈ Tα A and so Tα ′ A ⊆ Tα A . Next, let z ∈ IβA. Then IA(z) ≤ β ≤ β ′ . Thus, z ∈ Iβ ′ A and so IβA ⊆ Iβ ′ A . Similarly, F γ A ⊆ F γ ′ A . Hence, A(α ′ ,β,γ) = Tα ′ A ∩ IβA ∩ F γ A ⊆ Tα A ∩ Iβ ′ A ∩ F γ ′ A = A(α,β ′ ,γ ′ ). Consequently, A(α ′ ,β,γ) is a hyper UP -subalgebra of A(α,β ′ ,γ ′ ). For fixed numbers α1, α2, β1, β2, γ1, γ2 ∈ [0, 1] such that α1 ≥ α2, β1 ≥ β2,γ1 ≥ γ2, and a nonempty subset G of X, we define a SVN set AG [ α1, β2, γ2 α2, β1, γ1 ] = ( TAG [ α1 α2 ] , IAG [ β2 β1 ] ,FAG [ γ2 γ1 ]) where TAG [ α1 α2 ] (x) = { α1 if x ∈ G α2 otherwise , A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 559 IAG [ β2 β1 ] (x) = { β2 if x ∈ G β1 otherwise , and FAG [ γ2 γ1 ] (x) = { γ2 if x ∈ G γ1 otherwise . Theorem 2. Let G be a nonempty subset of X and AG [ α1, β2, γ2 α2, β1, γ1 ] be a SVN set in X. Then AG [ α1, β2, γ2 α2, β1, γ1 ] satisfies Proposition 3(i) if and only if 0 ∈ G. Proof. Let G be a nonempty subset of X and AG [ α1, β2, γ2 α2, β1, γ1 ] be a SVN set in X. (⇒) Assume that AG [ α1, β2, γ2 α2, β1, γ1 ] satisfies Proposition 3(i). Since G ̸= ∅, there exists g ∈ G. Thus, TAG [ α1 α2 ] (g) = α1. Now, TAG [ α1 α2 ] (0) ≥ TAG [ α1 α2 ] (g) = α1 ≥ TAG [ α1 α2 ] (0). That is, TAG [ α1 α2 ] (0) = α1. Hence, 0 ∈ G. (⇐) Assume that 0 ∈ G. Then TAG [ α1 α2 ] (0) = α1, IAG [ β2 β1 ] (0) = β2 and FAG [ γ2 γ1 ] (0) = γ2. For all x ∈ X, TAG [ α1 α2 ] (0) = α1 ≥ TAG [ α1 α2 ] (x), IAG [ β2 β1 ] (0) = β2 ≤ IAG [ β2 β1 ] (x) and FAG [ γ2 γ1 ] (0) = γ2 ≤ FAG [ γ2 γ1 ] (x). Hence, AG [ α1, β2, γ2 α2, β1, γ1 ] satisfies Proposition 3(i). A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 560 Theorem 3. Let AG [ α1, β2, γ2 α2, β1, γ1 ] be a SVN set in X. Then AG [ α1, β2, γ2 α2, β1, γ1 ] is a SVN hyper UP-subalgebra of X if and only if a nonempty subset of G is a hyper UP-subalgebra of X. Proof. Let AG [ α1, β2, γ2 α2, β1, γ1 ] be a SVN set in X. (⇒) Assume that AG [ α1, β2, γ2 α2, β1, γ1 ] is a SVN hyper UP -subalgebra of X. Since G ̸= ∅, we let x, y ∈ G and z ∈ x ◦ y. Then TAG [ α1 α2 ] (x) = α1 = TAG [ α1 α2 ] (y). By assumption, TAG [ α1 α2 ] (z) ≥ ∗TAG [ α1 α2 ] (x ◦ y) ≥ min { TAG [ α1 α2 ] (x), TAG [ α1 α2 ] (y) } = α1 ≥ TAG [ α1 α2 ] (z). That is, TAG [ α1 α2 ] (z) = α1. Thus, z ∈ G and so x ◦ y ⊆ G. By Proposition 2, G is a hyper UP -subalgebra of X. (⇐) Assume that G is a hyper UP -subalgebra of X and suppose x, y ∈ X. Consider the following cases: Case 1. x, y ∈ G By assumption, x ◦ y ⊆ G. Thus, ∗TAG [ α1 α2 ] (x ◦ y) = α1 ≥ α1 = min { TAG [ α1 α2 ] (x), TAG [ α1 α2 ] (y) } , ∗IAG [ β2 β1 ] (x ◦ y) = β2 ≤ β2 = max { IAG [ β2 β1 ] (x), IAG [ β2 β1 ] (y) } , and ∗FAG [ γ2 γ1 ] (x ◦ y) = γ2 ≤ γ2 = max { FAG [ γ2 γ1 ] (x),FAG [ γ2 γ1 ] (y) } . Case 2. x ∈ G and y /∈ G So we have ∗TAG [ α1 α2 ] (x ◦ y) ≥ α2 A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 561 = min{α1, α2} = min { TAG [ α1 α2 ] (x), TAG [ α1 α2 ] (y) } , ∗IAG [ β2 β1 ] (x ◦ y) ≤ β1 = max{β2, β1} = max { IAG [ β2 β1 ] (x), IAG [ β2 β1 ] (y) } , and ∗FAG [ γ2 γ1 ] (x ◦ y) ≤ γ1 = max{γ2, γ1} = max { FAG [ γ2 γ1 ] (x),FAG [ γ2 γ1 ] (y) } . Case 3. x /∈ G and y ∈ G Using similar routine done in case 2, we have ∗TAG [ α1 α2 ] (x ◦ y) ≥ min { TAG [ α1 α2 ] (x), TAG [ α1 α2 ] (y) } , ∗IAG [ β2 β1 ] (x ◦ y) ≤ max { IAG [ β2 β1 ] (x), IAG [ β1 β2 ] (y) } , and ∗FAG [ γ2 γ1 ] (x ◦ y) ≤ max { FAG [ γ2 γ1 ] (x),FAG [ γ2 γ1 ] (y) } . Case 4. x /∈ G and y /∈ G Now, ∗TAG [ α1 α2 ] (x ◦ y) ≥ α2 = min{α2, α2} = min { TAG [ α1 α2 ] (x), TAG [ α1 α2 ] (y) } , ∗IAG [ β2 β1 ] (x ◦ y) ≤ β1 = max{β1, β1} A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 562 = max { IAG [ β2 β1 ] (x), IAG [ β2 β1 ] (y) } , and ∗FAG [ γ2 γ1 ] (x ◦ y) ≤ γ1 = max{γ1, γ1} = max { FAG [ γ2 γ1 ] (x),FAG [ γ2 γ1 ] (y) } . Hence, AG [ α1, β2, γ2 α2, β1, γ1 ] is a SVN hyper UP -subalgebra of X. Theorem 4. Let G be a hyper UP-subalgebra of X. Then there exists a SVN hyper UP-subalgebra A = (TA, IA,FA) of X such that A(α,β,γ) = G for α, β, γ ∈ [0, 1]. Proof. Let G be a hyper UP -subalgebra of X. For fixed α, β, γ ∈ (0, 1], consider A = AG [ α, 0, 0 0, β, γ ] . Since G be a hyper UP -subalgebra of X, AG [ α, 0, 0 0, β, γ ] is a SVN hyper UP -subalgebra of X by Theorem 3. Now, let x ∈ G. Then TA(x) = TAG [ α 0 ] (x) = α ≥ α, IA(x) = IAG [ 0 β ] (x) = 0 ≤ β, and FA(x) = FAG [ 0 γ ] (x) = 0 ≤ γ. Thus, x ∈ Tα A ∩ IβA ∩ F γ A = A(α,β,γ) and so G ⊆ A(α,β,γ). Also, let y ∈ A(α,β,γ). Then TA(y) ≥ α, IA(y) ≤ β, and FA(y) ≤ γ. Suppose that y /∈ G. Then 0 = TAG [ α 0 ] (y) = TA(y) ≥ α. It follows that α = 0. This is a contradiction since α ∈ (0, 1]. Thus, y ∈ G and so A(α,β,γ) ⊆ G. Consequently, A(α,β,γ) = G. Theorem 5. Given a chain of hyper UP-subalgebras of X: A0 ⊂ A1 ⊂ A2 ⊂ A3 ⊂ . . . ⊂ An = X. Then there exists a SVN hyper UP- subalgebra A = (TA, IA,FA) of X such that A(αk,βk,γk) = Ak where αk, βk, γk ∈ [0, 1] for 0 ≤ k ≤ n. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 563 Proof. Let {αk|k = 0, 1, . . . , n} be a finite decreasing sequence and {βk|k = 0, 1, . . . , n}, {γk|k = 0, 1, . . . , n} be finite increasing sequences such that αk, βk, γk ∈ [0, 1] for 1 ≤ k ≤ n. Define a SVN set A = (TA, IA,FA) in X by TA(A0) = α0, IA(A0) = β0, FA(A0) = γ0,TA(Ak \ Ak−1) = αk, IA(Ak \ Ak−1) = βk, and FA(Ak \ Ak−1) = γk for 1 ≤ k ≤ n. We will show that A is a SVN hyper UP -subalgebra of X. Let a, b ∈ X. Consider the following cases: Case 1. a, b ∈ Ak \Ak−1 Then TA(a) = αk = TA(b), IA(a) = βk = IA(b), and FA(a) = γk = FA(b). Since Ak is a hyper UP -subalgebra of X, a ◦ b ⊆ Ak. Subcase 1.1. a ◦ b ⊆ Ak \Ak−1 ∗TA(a ◦ b) = αk ≥ αk = min{TA(a), TA(b)}, ∗IA(a ◦ b) = βk ≤ βk = max{IA(a), IA(b)}, and ∗FA(a ◦ b) = γk ≤ γk = max{FA(a),FA(b)}. Subcase 1.2. a ◦ b ⊆ Ak−1 For some r ∈ [0, k − 1], we have ∗TA(a ◦ b) = αk−1−r ≥ αk = min{TA(a), TA(b)}, ∗IA(a ◦ b) = βk−1−r ≤ βk = max{IA(a), IA(b)}, and ∗FA(a ◦ b) = γk−1−r ≤ γk = max{FA(a),FA(b)}. Subcase 1.3. a◦b = [(a◦b)∩(Ak\Ak−1)]∪[(a◦b)∩Ak−1] where (a◦b)∩(Ak\Ak−1) ̸= ∅ and (a ◦ b) ∩Ak−1 ̸= ∅ ∗TA(a ◦ b) = αk ≥ αk = min{TA(a), TA(b)}, ∗IA(a ◦ b) = βk ≤ βk = max{IA(a), IA(b)}, and ∗FA(a ◦ b) = γk ≤ γk = max{FA(a),FA(b)}. Case 2. a ∈ Ai \Ai−1 and b ∈ Aj \Aj−1 for i > j > 0 Then TA(a) = αi, TA(b) = αj , IA(a) = βi, IA(b) = βj , FA(a) = γi, and FA(b) = γj . Since Ai is a hyper UP -subalgebra of X and Aj ⊂ Ai, we have a ◦ b ⊆ Ai. Subcase 2.1. a ◦ b ⊆ Ai \Aj For some r ∈ [0, i− j − 1], we have ∗TA(a ◦ b) = αi−r ≥ αi = min{TA(a), TA(b)}, A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 564 ∗IA(a ◦ b) = βi−r ≤ βi = max{IA(a), IA(b)}, and ∗FA(a ◦ b) = γi−r ≤ γi = max{FA(a),FA(b)}. Subcase 2.2. a ◦ b ⊆ Aj For some r ∈ [0, j], we have ∗TA(a ◦ b) = αj−r ≥ αi = min{TA(a), TA(b)}, ∗IA(a ◦ b) = βj−r ≤ βi = max{IA(a), IA(b)}, and ∗FA(a ◦ b) = γj−r ≤ γi = max{FA(a),FA(b)}. Subcase 2.3. a ◦ b = [(a ◦ b) ∩ (Ai \Aj)] ∪ [(a ◦ b) ∩Aj ] where (a ◦ b) ∩ (Ai \Aj) ̸= ∅ and (a ◦ b) ∩Aj ̸= ∅ For some r ∈ [0, i− j − 1], we have ∗TA(a ◦ b) = αi−r ≥ αi = min{TA(a), TA(b)}, ∗IA(a ◦ b) = βi−r ≤ βi = max{IA(a), IA(b)}, and ∗FA(a ◦ b) = γi−r ≤ γi = max{FA(a),FA(b)}. Thus, A is a SVN hyper UP -subalgebra of X. Furthermore, note that Tα0 A = {s ∈ X|TA(s) ≥ α0} = A0, Iβ0 A = {s ∈ X|IA(s) ≤ β0} = A0, and F γ0 A = {s ∈ X|FA(s) ≤ γ0} = A0. Thus, A0 = Tα0 A ∩ Iβ0 A ∩ F γ0 A = A(α0,β0,γ0). For 0 < k ≤ n, let x ∈ Ak. Then x ∈ Ak−i \Ak−i−1 ∃0 ≤ i ≤ k − 1. Thus, TA(x) = αk−i ≥ αk, IA(x) = βk−i ≤ βk and FA(x) = γk−i ≤ γk ∃0 ≤ i ≤ k − 1. So we have x ∈ Tαk A ∩ Iβk A ∩ F γk A = A(αk,βk,γk) and Ak ⊆ A(αk,βk,γk). Also, let y ∈ A(αk,βk,γk). Then TA(y) ≥ αk, IA(y) ≤ βk, and FA(y) ≤ γk. The values of TA(y), IA(y), and FA(y) that will make the three inequalities true are TA(y) = αt, IA(y) = βt, and FA(y) = γt ∃0 < t ≤ k. This implies that y ∈ Ak−i \Ak−i−1 ∃0 ≤ i ≤ k − 1. That is, y ∈ Ak since (Ak−i \Ak−i−1) ⊆ Ak ∃0 ≤ i ≤ k−1. Hence, A(αk,βk,γk) ⊆ Ak. Consequently, A(αk,βk,γk) = Ak. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 565 3.2. Single-Valued Neutrosophic Soft Hyper UP-subalgebra In this section, we define the single-valued neutrosophic soft hyper UP -subalgebra and prove some related properties. Definition 17. Let (∆, E) be a single-valued neutrosophic soft set over a hyper UP - algebra X. Then (∆, E) is said to be single-valued neutrosophic soft (SV NS) hyper UP-subalgebra of X if for all x, y ∈ X and e ∈ E, ∗T∆(e)(x ◦ y) ≥ min{T∆(e)(x), T∆(e)(y)}, ∗I∆(e)(x ◦ y) ≤ max{T∆(e)(x), T∆(e)(y)}, and ∗F∆(e)(x ◦ y) ≤ max{T∆(e)(x), T∆(e)(y)}; that is, ∆(e) is a SVN hyper UP -subalgebra of X. Example 7. Consider the hyper UP -algebra (X, ◦,≪, 0) of Example 2 where X = {0, u, v}. Let E = {e1, e2} be the set of parameters and let ∆ : E −→ N (X) be de- fined by ∆(e1) = {⟨0, (0.9, 0.2, 0.45)⟩ , ⟨u, (0.74, 0.57, 0.7)⟩ , ⟨v, (0.8, 0.42, 0.52)⟩} and ∆(e2) = {⟨0, (0.8, 0.2, 0.4)⟩ , ⟨u, (0.4, 0.45, 0.5)⟩ , ⟨v, (0.67, 0.3, 0.5)⟩}. Then (∆, E) = {(e1, {⟨0, (0.9, 0.2, 0.45)⟩ , ⟨u, (0.74, 0.57, 0.7)⟩ , ⟨v, (0.8, 0.42, 0.52)⟩}), (e2, {⟨0, (0.8, 0.2, 0.4)⟩ , ⟨u, (0.4, 0.45, 0.5)⟩ , ⟨v, (0.67, 0.3, 0.5)⟩})}. is a SVNS set over X. By routine calculation, (∆, E) is a SVNS hyper UP -subalgebra of X. Proposition 5. Let (∆, E) be a SVNS hyper UP-subalgebra of X. Then for all x, y ∈ X and e ∈ E, (i) T∆(e)(x) ≤ T∆(e)(0) I∆(e)(x) ≥ I∆(e)(0) F∆(e)(x) ≥ F∆(e)(0) , (ii) ∗T∆(e)(0 ◦ x) = T∆(e)(x). ∗I∆(e)(0 ◦ x) = I∆(e)(x) ∗F∆(e)(0 ◦ x) = F∆(e)(x). (iii) ∗T∆(e)(x ◦ 0) = T∆(e)(0) ∗I∆(e)(x ◦ 0) = I∆(e)(0) ∗F∆(e)(x ◦ 0) = F∆(e)(0) (iv) If T∆(e)(x) = T∆(e)(0) I∆(e)(x) = I∆(e)(0) F∆(e)(x) = F∆(e)(0) , then ∗T∆(e)(x ◦ y) ≥ T∆(e)(y) ∗I∆(e)(x ◦ y) ≤ I∆(e)(y) ∗F∆(e)(x ◦ y) ≤ F∆(e)(y) . A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 566 (v) If T∆(e)(y) = T∆(e)(0) I∆(e)(y) = I∆(e)(0) F∆(e)(y) = F∆(e)(0) , then ∗T∆(e)(x ◦ y) ≥ T∆(e)(x) ∗I∆(e)(x ◦ y) ≤ I∆(e)(x) ∗F∆(e)(x ◦ y) ≤ F∆(e)(x) . (vi) If ∗T∆(e)(x ◦ y) = T∆(e)(x) ∗I∆(e)(x ◦ y) = I∆(e)(x) ∗F∆(e)(x ◦ y) = F∆(e)(x) , then T∆(e)(x) = T∆(e)(0) I∆(e)(x) = I∆(e)(0) F∆(e)(x) = F∆(e)(0) and T∆(e)(y) = T∆(e)(0) I∆(e)(y) = I∆(e)(0) F∆(e)(y) = F∆(e)(0) . Proof. Using similar arguments from Proposition 3, this proposition is valid. Theorem 6. Let (∆1, E) and (∆2, E) be two SVNS hyper UP-subalgebras of X. Then (i) (∆1, E) ∩ (∆2, E) is a SVNS hyper UP-subalgebra of X. (ii) (∆1, E) ∪ (∆2, E) is not generally a SVNS hyper UP-subalgebra of X. Proof. Let (∆1, E) and (∆2, E) be two SVNS hyper UP -subalgebras of X. (i) Let (∆, E) = (∆1, E) ∩ (∆2, E), x, y ∈ X and e ∈ E. Then we have ∗T∆(e)(x ◦ y) = inf a∈x◦y T∆(e)(a) = inf a∈x◦y min{T∆1(e)(a), T∆2(e)(a)} ≥ min{ inf a∈x◦y T∆1(e)(a), inf a∈x◦y T∆2(e)(a)} = min{∗T∆1(e)(x ◦ y),∗ T∆2(e)(x ◦ y)} ≥ min{min{T∆1(e)(x), T∆1(e)(y)},min{T∆2(e)(x), T∆2(e)(y)}} = min{min{T∆1(e)(x), T∆2(e)(x)},min{T∆1(e)(y), T∆2(e)(y)}} = min{T∆(e)(x), T∆(e)(y)}. Also, ∗I∆(e)(x ◦ y) = sup a∈x◦y I∆(e)(a) = sup a∈x◦y max{I∆1(e)(a), I∆2(e)(a)} ≤ max{ sup a∈x◦y I∆1(e)(a), sup a∈x◦y I∆2(e)(a)} = max{∗I∆1(e)(x ◦ y),∗ I∆2(e)(x ◦ y)} ≤ max{max{I∆1(e)(x), I∆1(e)(y)},max{I∆2(e)(x), I∆2(e)(y)}} = max{max{I∆1(e)(x), I∆2(e)(x)},max{I∆1(e)(y), I∆2(e)(y)}} = max{I∆(e)(x), I∆(e)(y)}. Similarly, ∗F∆(e)(x ◦ y) ≤ max{F∆(e)(x),F∆(e)(y)}. Thus, (∆, E) is a SVNS hyper UP -subalgebra of X. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 567 (ii) Using the hyper UP -algebra (X, ◦,≪, 0) of Example 1 where X = {0, r, s, t}, we consider the two SVNS hyper UP -subalgebras (∆1, E) and (∆2, E) of X given by: for e ∈ E, T∆1(e)(x) = { 0.5 if x ∈ {0, s}, 0 otherwise. I∆1(e)(x) = { 0 if x ∈ {0, s}, 0.5 otherwise. F∆1(e)(x) = { 0 if x ∈ {0, s}, 0.5 otherwise. and T∆2(e)(x) = { 0.7 if x ∈ {0, t}, 0 otherwise. I∆2(e)(x) = { 0 if x ∈ {0, t}, 0.7 otherwise. F∆2(e)(x) = { 0 if x ∈ {0, t}, 0.7 otherwise. , respectively. Let (∆, E) = (∆1, E) ∪ (∆2, E). For all e ∈ E, taking x = s and y = t gives ∗T∆(e)(x ◦ y) = ∗T∆(e)(s ◦ t) = ∗T∆(e)({r}) = T∆(e)(r) = max{T∆1(e)(r), T∆2(e)(r)} = max{0, 0} = 0 and min{T∆(e)(x), T∆(e)(y)} = min{T∆(e)(s), T∆(e)(t)} = min{max{T∆1(e)(s), T∆2(e)(s)},max{T∆1(e)(t), T∆2(e)(t)}} = min{max{0.5, 0},max{0, 0.7}} = min{0.5, 0.7} = 0.5. That is, ∗T∆(e)(x ◦ y) = 0 < 0.5 = min{T∆(e)(x), T∆(e)(y)}. Thus, (∆, E) is not a SVNS hyper UP -subalgebra of X. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 568 3.3. Cartesian Product of SVNS Hyper UP-subalgebra In this section, we define the Cartesian product of SVNS hyper UP -subalgebra and prove that it is also a SVNS hyper UP -subalgebra. Definition 18. Let (∆1, E) and (∆2, E) be two SVNS hyper UP -subalgebras of X1 and X2, respectively. Then their Cartesian product is (∆, E ×E) = (∆1, E)× (∆2, E), where ∆(a, b) = ∆1(a)×∆2(b) for (a, b) ∈ E × E. Analytically, ∆(a, b) = { 〈 (x, y), (T∆(a,b)(x, y), I∆(a,b)(x, y),F∆(a,b)(x, y)) 〉 |(x, y) ∈ X1 ×X2} where T∆(a,b)(x, y) = min{T∆1(a)(x), T∆2(b)(y)}, I∆(a,b)(x, y) = max{I∆1(a)(x), I∆2(b)(y)}, and F∆(a,b)(x, y) = max{F∆1(a)(x),F∆2(b)(y)}. for (a, b) ∈ E × E. Example 8. Using E = {e1, e2} as the set of parameters, consider the hyper UP -algebra X = {01, r, s, t} of Example 1 as X1 with its hyperoperation “◦1” and its SVNS hyper UP -subalgebra (∆1, E) given by T∆1(e)(x) = { 0.5 if x ∈ {01, s}, 0 otherwise. I∆1(e)(x) = { 0 if x ∈ {01, s}, 0.5 otherwise. F∆1(e)(x) = { 0 if x ∈ {01, s}, 0.5 otherwise. for e ∈ E. Consider also hyper UP -algebra X = {02, u, v} of Example 2 as X2 with its hyper operation “ ◦2 ” and its SVNS hyper UP -subalgebra (∆2, E) given by (∆2, E) = {(e1, {⟨02, (0.9, 0.2, 0.45)⟩ , ⟨u, (0.74, 0.57, 0.7)⟩ , ⟨v, (0.8, 0.42, 0.52)⟩}), (e2, {⟨02, (0.8, 0.2, 0.4)⟩ , ⟨u, (0.4, 0.45, 0.5)⟩ , ⟨v, (0.67, 0.3, 0.5)⟩})}. Then the Cartesian product of (∆1, E) and (∆2, E) is (∆, E) = {((e1, e1), {⟨(01, 02), (0.5, 0.2, 0.45)⟩ , ⟨(01, u), (0.5, 0.57, 0.7)⟩ , ⟨(01, v), (0.5, 0.42, 0.52)⟩ , ⟨(r, 02), (0, 0.5, 0.5)⟩ , ⟨(r, u), (0, 0.57, 0.7)⟩ , ⟨(r, v), (0, 0.5, 0.52)⟩ , ⟨(s, 02), (0.5, 0.2, 0.45)⟩ , ⟨(s, u), (0.5, 0.57, 0.7)⟩ , ⟨(s, v), (0.5, 0.42, 0.52)⟩ , ⟨(t, 02), (0, 0.5, 0.5)⟩ , ⟨(t, u), (0, 0.57, 0.7)⟩ , ⟨(t, v), (0, 0.5, 0.52)⟩}), ((e1, e2), {⟨(01, 02), (0.5, 0.2.0.4)⟩ , A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 569 ⟨(01, u), (0.4, 0.45, 0.5)⟩ , ⟨(01, v), (0.5, 0.3, 0.5)⟩ , ⟨(r, 02), (0, 0.5, 0.5)⟩ , ⟨(r, u), (0, 0.5, 0.5)⟩ , ⟨(r, v), (0, 0.5, 0.5)⟩ , ⟨(s, 02), (0.5, 0.2, 0.4)⟩ , ⟨(s, u), (0.4, 0.45, 0.5)⟩ , ⟨(s, v), (0.5, 0.3, 0.5)⟩ , ⟨(t, 02), (0, 0.5, 0.5)⟩ , ⟨(t, u), (0, 0.5, 0.5)⟩ , ⟨(t, v), (0, 0.5, 0.5)⟩}), ((e2, e1), {⟨(01, 02), (0.5, 0.2, 0.45)⟩ , ⟨(01, u), (0.5, 0.57, 0.7)⟩ , ⟨(01, v), (0.5, 0.42, 0.52)⟩ , ⟨(r, 02), (0, 0.5, 0.5)⟩ , ⟨(r, u), (0, 0.57, 0.7)⟩ , ⟨(r, v), (0, 0.5, 0.52)⟩ , ⟨(s, 02), (0.5, 0.2, 0.45)⟩ , ⟨(s, u), (0.5, 0.57, 0.7)⟩ , ⟨(s, v), (0.5, 0.42, 0.52)⟩ , ⟨(t, 02), (0, 0.5, 0.5)⟩ , ⟨(t, u), (0, 0.57, 0.7)⟩ , ⟨(t, v), (0, 0.5, 0.52)⟩}), ((e2, e2), {⟨(01, 02), (0.5, 0.2.0.4)⟩ , ⟨(01, u), (0.4, 0.45, 0.5)⟩ , ⟨(01, v), (0.5, 0.3, 0.5)⟩ , ⟨(r, 02), (0, 0.5, 0.5)⟩ , ⟨(r, u), (0, 0.5, 0.5)⟩ , ⟨(r, v), (0, 0.5, 0.5)⟩ , ⟨(s, 02), (0.5, 0.2, 0.4)⟩ , ⟨(s, u), (0.4, 0.45, 0.5)⟩ , ⟨(s, v), (0.5, 0.3, 0.5)⟩ , ⟨(t, 02), (0, 0.5, 0.5)⟩ , ⟨(t, u), (0, 0.5, 0.5)⟩ , ⟨(t, v), (0, 0.5, 0.5)⟩})} Theorem 7. Let (∆1, E) and (∆2, E) be two SVNS hyper UP-subalgebras of (X1, ◦1,≪1 , 01) and (X2, ◦2,≪2, 02), respectively. Then their Cartesian product (∆1, E)× (∆2, E) is a SVNS hyper UP-subalgebra of (X1 ×X2, ◦,≪, (01, 02)). Proof. Let (∆1, E) and (∆2, E) be two SVNS hyper UP -subalgebras of X1 and X2, respectively and let (∆, E × E) = (∆1, E) × (∆2, E), where ∆(a, b) = ∆1(a) ×∆2(b) for (a, b) ∈ E × E. For (u, v), (x, y) ∈ X1 ×X2, we have ∗T∆(a,b)((u, v) ◦ (x, y)) = ∗T∆(a,b)(u ◦1 x, v ◦2 y) = inf (r,t)∈(u◦1x)×(v◦2y) T∆(a,b)(r, t) = inf (r,t)∈(u◦1x)×(v◦2y) min{T∆1(a)(r), T∆2(b)(t)} ≥ min{ inf r∈u◦1x T∆1(a)(r), inf t∈v◦2y T∆2(b)(t)} = min{∗T∆1(a)(u ◦1 x),∗ T∆2(b)(v ◦2 y)} ≥ min{min{T∆1(a)(u), T∆1(a)(x)},min{T∆2(b)(v), T∆2(b)(y)}} = min{min{T∆1(a)(u), T∆2(b)(v)},min{T∆1(a)(x), T∆2(b)(y)}} = min{T∆(a,b)(u, v), T∆(a,b)(x, y)}. Also, ∗I∆(a,b)((u, v) ◦ (x, y)) = ∗I∆(a,b)(u ◦1 x, v ◦2 y) = sup (r,t)∈(u◦1x)×(v◦2y) I∆(a,b)(r, t) = sup (r,t)∈(u◦1x)×(v◦2y) max{I∆1(a)(r), I∆2(b)(t)} ≤ max{ sup r∈u◦1x I∆1(a)(r), sup t∈v◦2y I∆2(b)(t)} = max{∗I∆1(a)(u ◦1 x),∗ I∆2(b)(v ◦2 y)} ≤ max{max{I∆1(a)(u), I∆1(a)(x)},max{I∆2(b)(v), I∆2(b)(y)}} = max{max{I∆1(a)(u), I∆2(b)(v)},max{I∆1(a)(x), I∆2(b)(y)}} A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 570 = max{I∆(a,b)(u, v), I∆(a,b)(x, y)}. Similarly, ∗F∆(a,b)((u, v) ◦ (x, y)) = max{F∆(a,b)(u, v),F∆(a,b)(x, y)}. Hence, (∆, E × E) is a SVNS hyper UP -subalgebra of X1 ×X2. 3.4. Homomorphism of SVNS Hyper UP-subalgebra In this section, we define the image and preimage of SVNS hyper UP -subalgebra and prove that they are SVNS hyper UP -subalgebra under SVNS homomorphic function. Definition 19. Let (X1, ◦1,≪1, 01) and (X2, ◦2,≪2, 02) be two hyper UP -algebras and (∆1, E), (∆2, E) be two SVNS hyper UP -subalgebra of X1 and X2, respectively. Then the pair (φ, ρ) is called a SVNS function from X1 to X2 where φ : X1 −→ X2 and ρ : E −→ E. Definition 20. Under the SVNS function (φ, ρ), (i) The image of (∆1, E) is denoted by (φ, ρ)(∆1, E) and is defined by (φ, ρ)(∆1, E) = (φ(∆1), ρ(E)) = {(b, φ(∆1)(b))|b ∈ ρ(E)} where for all b ∈ ρ(E) and y ∈ X2, Tφ(∆1)(b)(y) =  max φ(x)=y max ρ(a)=b T∆1(a)(x) if x ∈ φ−1(y), 0 otherwise, Iφ(∆1)(b)(y) =  min φ(x)=y min ρ(a)=b I∆1(a)(x) if x ∈ φ−1(y), 1 otherwise, and Fφ(∆1)(b)(y) =  min φ(x)=y min ρ(a)=b F∆1(a)(x) if x ∈ φ−1(y), 1 otherwise. (ii) The preimage (∆2, E) is denoted by (φ, ρ)−1(∆2, E) and defined by (φ, ρ)−1(∆2, E) = (φ−1(∆2), ρ −1(E)) = {(a, φ−1(∆2)(a))|a ∈ ρ−1(E)} where for all a ∈ ρ−1(E) and x ∈ X1, Tφ−1(∆2)(a)(x) = T∆2(ρ(a))(φ(x)), Iφ−1(∆2)(a)(x) = I∆2(ρ(a))(φ(x)), and Fφ−1(∆2)(a)(x) = F∆2(ρ(a))(φ(x)). A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 571 Definition 21. Let the pair (φ, ρ) be a SVNS function from X1 into X2, then (φ, ρ) is called a SVNS homomorphism if φ is a hyper homomorphism from X1 to X2 and is said to be a SVNS isomorphism if φ is a hyper isomorphism from X1 to X2 and ρ is an injective map from E to E. Example 9. Let X1 = {01, r, s} with hyperoperation given by ◦ 01 r s 01 {01} {r} {s} r {01} {01} {s} s {01} {r} {01} . Then X1 is hyper UP -algebra by thorough inspection. Considering X2 = {02, u, v} as the second hyper UP -algebra of Example 2 and E = N as the set of parameters, we define mappings φ : X1 −→ X2 by φ(01) = 02 φ(r) = v φ(s) = u; and ρ : E −→ E by ρ(a) = 2a. Let (∆1, E) be a SVNS set over X1 given by T∆1(a)(x) = { 1 2a if x ∈ {01, s}, 0 otherwise. I∆1(a)(x) = { 0 if x ∈ {01, s}, 1− 1 a otherwise. F∆1(a)(x) = { 0 if x ∈ {01, s}, 1 2a+1 otherwise. for a ∈ E. By inspection, (∆1, E) is a SVNS hyper UP -subalgebra of X1. Thus, the image of (∆1, E) under (φ, ρ) is Tφ(∆1)(b)(y) = { 0 if y ∈ {v}, 1 b otherwise. Iφ(∆1)(b)(y) = { 1− 2 b if y ∈ {v}, 0 otherwise. Fφ(∆1)(b)(y) = { 1 b+1 if y ∈ {v}, 0 otherwise. for b ∈ 2N and y ∈ X2. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 572 Example 10. Consider the two hyper UP -algebras X1 and X2 of Example 9. Define E = {e1, e2} as set of parameters and mappings φ : X1 −→ X2 by φ(01) = 02 φ(r) = v φ(s) = u; and ρ : E −→ E by ρ(e1) = e2 ρ(e2) = e1. Also, consider the SVNS hyper UP -subalgebra (∆, E) of X2 from Example 7 which is given by ∆(e1) = {⟨02, (0.9, 0.2, 0.45)⟩ , ⟨u, (0.74, 0.57, 0.7)⟩ , ⟨v, (0.8, 0.42, 0.52)⟩} and ∆(e2) = {⟨02, (0.8, 0.2, 0.4)⟩ , ⟨u, (0.4, 0.45, 0.5)⟩ , ⟨v, (0.67, 0.3, 0.5)⟩}. Thus, the preimage of (∆, E) under (φ, ρ) is given by Tφ−1(∆)(e1)(01) = T∆(ρ(e1))(φ(01)) = T∆(e2)(02) = 0.8 Iφ−1(∆)(e1)(01) = I∆(ρ(e1))(φ(01)) = I∆(e2)(02) = 0.2 Fφ−1(∆)(e1)(01) = F∆(ρ(e1))(φ(01)) = F∆(e2)(02) = 0.4 Tφ−1(∆)(e1)(r) = T∆(ρ(e1))(φ(r)) = T∆(e2)(v) = 0.67 Iφ−1(∆)(e1)(r) = I∆(ρ(e1))(φ(r)) = I∆(e2)(v) = 0.3 Fφ−1(∆)(e1)(r) = F∆(ρ(e1))(φ(r)) = F∆(e2)(v) = 0.5 Tφ−1(∆)(e1)(s) = T∆(ρ(e1))(φ(s)) = T∆(e2)(u) = 0.4 Iφ−1(∆)(e1)(s) = I∆(ρ(e1))(φ(s)) = I∆(e2)(u) = 0.45 Fφ−1(∆)(e1)(s) = F∆(ρ(e1))(φ(s)) = F∆(e2)(u) = 0.5 Tφ−1(∆)(e2)(01) = T∆(ρ(e2))(φ(01)) = T∆(e1)(02) = 0.9 Iφ−1(∆)(e2)(01) = I∆(ρ(e2))(φ(01)) = I∆(e1)(02) = 0.2 Fφ−1(∆)(e2)(01) = F∆(ρ(e2))(φ(01)) = F∆(e1)(02) = 0.45 Tφ−1(∆)(e2)(r) = T∆(ρ(e2))(φ(r)) = T∆(e1)(v) = 0.8 Iφ−1(∆)(e2)(r) = I∆(ρ(e2))(φ(r)) = I∆(e1)(v) = 0.42 Fφ−1(∆)(e2)(r) = F∆(ρ(e2))(φ(r)) = F∆(e1)(v) = 0.52 Tφ−1(∆)(e2)(s) = T∆(ρ(e2))(φ(s)) = T∆(e1)(u) = 0.74. Iφ−1(∆)(e2)(s) = I∆(ρ(e2))(φ(s)) = I∆(e1)(u) = 0.57. Fφ−1(∆)(e2)(s) = F∆(ρ(e2))(φ(s)) = F∆(e1)(u) = 0.7. Theorem 8. Let (φ, ρ) be a SVNS homomorphism from (X1, ◦1,≪1, 01) to (X2, ◦2,≪2 , 02). If (∆1, E) is a SVNS hyper UP-subalgebra of X1, then (φ, ρ)(∆1, E) is a SVNS hyper UP-subalgebra of X2. A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 573 Proof. Let (φ, ρ) be a SVNS homomorphism from X1 to X2, (∆1, E) is a SVNS hyper UP -subalgebra of X1, b ∈ ρ(E), and x, y ∈ X2. (i) For φ−1(x) = ∅ or φ−1(y) = ∅, the proof is straightforward. (ii) Assume that there exist x0, y0 ∈ X1 such that φ(x0) = x and φ(y0) = y. Then x ◦2 y = φ(x0) ◦2 φ(y0) = φ(x0 ◦1 y0). Now, ∗Tφ(∆1)(b)(x ◦2 y) = inf z∈x◦2y Tφ(∆1)(b)(z) = inf z0∈x0◦1y0 [ max φ(z0)=z max ρ(a)=b T∆1(a)(z0) ] ≥ inf z0∈x0◦1y0 [ max ρ(a)=b T∆1(a)(z0) ] = max ρ(a)=b [ inf z0∈x0◦1y0 T∆1(a)(z0) ] = max ρ(a)=b [ ∗T∆1(a)(x0 ◦1 y0) ] ≥ max ρ(a)=b [ min{T∆1(a)(x0), T∆1(a)(y0)} ] = min{max ρ(a)=b T∆1(a)(x0), max ρ(a)=b T∆1(a)(y0)} Since the inequality is satisfied for all x0, y0 ∈ X1 satisfying φ(x0) = x and φ(y0) = y, it follows that ∗Tφ(∆1)(b)(x ◦2 y) ≥ min{ max φ(x0)=x max ρ(a)=b T∆1(a)(x0), max φ(y0)=y max ρ(a)=b T∆1(a)(y0)} = min{Tφ(∆1)(b)(x), Tφ(∆1)(b)(y)}. Also, ∗Iφ(∆1)(b)(x ◦2 y) = sup z∈x◦2y Iφ(∆1)(b)(z) = sup z0∈x0◦1y0 [ min φ(z0)=z min ρ(a)=b I∆1(a)(z0) ] ≤ sup z0∈x0◦1y0 [ min ρ(a)=b I∆1(a)(z0) ] = min ρ(a)=b [ sup z0∈x0◦1y0 I∆1(a)(z0) ] = min ρ(a)=b [ ∗I∆1(a)(x0 ◦1 y0) ] ≤ min ρ(a)=b [ max{I∆1(a)(x0), I∆1(a)(y0)} ] = max{ min ρ(a)=b I∆1(a)(x0), min ρ(a)=b I∆1(a)(y0)} A. Cano, G. Petalcorin / Eur. J. Pure Appl. Math, 16 (1) (2023), 548-576 574 Since the inequality is satisfied for all x0, y0 ∈ X1 satisfying φ(x0) = x and φ(y0) = y, it follows that ∗Iφ(∆1)(b)(x ◦2 y) ≤ max{ min φ(x0)=x min ρ(a)=b I∆1(a)(x0), max φ(y0)=y max ρ(a)=b I∆1(a)(y0)} = max{Iφ(∆1)(b)(x), Iφ(∆1)(b)(y)}. Similarly, ∗Fφ(∆1)(b)(x ◦2 y) ≤ max{Fφ(∆1)(b)(x),Fφ(∆1)(b)(y)}. Hence, (φ, ρ)(∆1, E) is a SVNS hyper UP -subalgebra of X2. Theorem 9. Let (φ, ρ) be a SVNS homomorphism from (X1, ◦1,≪1, 01) to (X2, ◦2,≪2 , 02). If (∆2, E) is a SVNS hyper UP-subalgebra of X2, then (φ, ρ)−1(∆2, E) is a SVNS hyper UP-subalgebra of X1. Proof. Let (φ, ρ) be a SVNS homomorphism from X1 to X2, (∆2, E) be a SVNS hyper UP -subalgebra of X2, a ∈ ρ−1(E), x, y ∈ X1. Now, ∗Tφ−1(∆2)(a)(x ◦1 y) = ∗T∆2(ρ(a))(φ(x ◦1 y)) = ∗T∆2(ρ(a))(φ(x) ◦2 φ(y)) ≥ min{T∆2(ρ(a))(φ(x)), T∆2(ρ(a))(φ(y))} = min{Tφ−1(∆2)(a)(x), Tφ−1(∆2)(a)(y)}, ∗Iφ−1(∆2)(a)(x ◦1 y) = ∗I∆2(ρ(a))(φ(x ◦1 y)) = ∗I∆2(ρ(a))(φ(x) ◦2 φ(y)) ≤ max{I∆2(ρ(a))(φ(x)), I∆2(ρ(a))(φ(y))} = max{Iφ−1(∆2)(a)(x), Iφ−1(∆2)(a)(y)}, and similarly, ∗Fφ−1(∆2)(a)(x ◦1 y) = max{Fφ−1(∆2)(a)(x),Fφ−1(∆2)(a)(y)}. Thus, (φ, ρ)−1(∆2, S2) is a SVNS hyper UP -subalgebra of X1. 4. Conclusions In this paper, we have introduced the SVN and SVNS hyper UP -subalgebra together with their properties. Aside from that, the concept of Cartesian product of SVNS hyper UP -subalgebra and the homomorphic image and preimage of SVNS hyper UP -subalgebra have been investigated. This study contributes to the development of the notion of hyper UP -algebra under neutrosophic soft environment. It also opens a door for further study by establishing SVNS hyper UP -filter, SVNS hyper UP -ideals and some of their variations. 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