EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 2221-2234 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Intuitionistic Fuzzy Implicative Hyper GR-ideals Amila P. Macodi1,∗, Archie G. Dorig2 1 Mathematics Department, Faculty, Mindanao State University, Marawi City, Lanao del Sur, Philippines 2 Department of Fisheries, Marine Biology and Environmental Sciences, Faculty, North Eastern Mindanao State University - Lianga Campus, Lianga, Surigao del Sur, Philippines Abstract. The influx of research on hyperstructure theory has encouraged many researchers to introduce new algebras. Notable among these is the work of Indangan and Petalcorin [5] who introduced hyper GR-algebras and that of Macodi and Petalcorin [12] who studied a fuzzification of hyper GR-algebras. Macodi extended the fuzzification of hyper GR-algebras into intuitionistic fuzzification and introduced the concept of an intuitionistic fuzzy hyper GR-ideals [13]. Following the works of Macodi and Petalcorin, this paper established an intuitionistic fuzzy implicative hyper GR-ideals and obtained their characterization using level subsets. Moreover, some properties of intuitionistic fuzzy implicative hyper GR-ideals are presented. 2020 Mathematics Subject Classifications: 20N20, 06F35, 03G25, 03E72, 03B52, 08A72 Key Words and Phrases: Hyper GR-algebras, implicative hyper GR-ideals, fuzzy implicative hyper GR-ideals, intuitionistic fuzzy implicative hyper GR-ideals 1. Introduction Hyperstructure theory, also called multi-algebras, was introduced in 1934 by Marty [14] at the 8th Congress of Scandinavian Mathematicians. Hyperstructures have many applications in several sectors of both pure and applied sciences. It is for this reason that many researchers work on this subject. Jun, et al. [20] applied the hyperstructures to BCK-algebras and introduced the concept of generalizing a BCK-algebra into a hyper BCK-algebra. They also investigated some properties of hyper BCK-algebra. After the introduction of the concept of hyper BCK-algebras, several studies were conducted. Amongst these studies, the most notable are those made by Jun and Long [7], Jun and Shim [8], Borzooei and Bakhsi [2], Borzooei and Jun [3], and Jun and Song [9]. Based on this hyperstructure, Indangan and Petalcorin [5, 6] introduced a new hyperstructure which is called a hyper GR-algebra. They established some results on ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5344 Email addresses: amila.macodi@msumain.edu.ph (A. Macodi), archie.dorig@msumain.edu.ph (A. Dorig) https://www.ejpam.com 2221 © 2024 EJPAM All rights reserved. A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2222 hyper GR-ideals of a hyper GR-algebra and some hyper homomorphic properties of hyper GR-algebras. In 2019, Macodi and Petalcorin [12] published some results on fuzzy implicative hyper GR-ideals of hyper GR-algebras. They obtained more results on fuzzy structures in Hyper GR-algebras [11] and also established intuitionistic fuzzy hyper GR-ideals in hyper GR-algebras [13]. The concept of intuitionistic fuzzy sets was developed by Atanassov [1] as an extension of Zadeh’s fuzzy set [21]. A prominent characteristic of intuitionistic fuzzy sets is that they assign to each element a membership degree and a non-membership degree. Intutionistic fuzzy set theory basically defies the claim that an element x belongs to a given degree say µ(x) to a fuzzy set A, it naturally follows that x should not belong to A to the extent 1 − µ(x). Consequently, many authors have paid attention to the intuitionistic fuzzy set theory since it has been successfully applied in different areas, such as logic programming [10], decision making problems [4] and medical diagnosis [15], to name a few, and related studies are made in [17, 18]. The concept is also applied in various applications, such as intuitionistic fuzzy neural networks, intuitionistic fuzzy decision making, intuitionistic fuzzy machine learning, and intuitionistic fuzzy semantic representations [19]. Recently, many researchers have applied intuitionistic fuzzy sets to hyper algebras such as hyper BCK-algebra [3], BCI-algebra [16] and hyper GR-algebra [13]. This paper is particularly interested in the concept of intuitionistic fuzzy sets applied to implicative hyper GR-ideal of hyper GR-algebra. The definition of intuitionistic fuzzy implicative hyper GR-ideal is somewhat parallel with the definition of intuitionistic fuzzy hyper GR-ideal which was introduced by Macodi [13]. 2. Prelimenaries This section presents some preliminary concepts and known properties that are needed in this study. Definition 1. [5] Let H be a nonempty set and ⊛ be a hyperoperation on H. Then (H;⊛, 0) is called a hyper GR-algebra if 0 ∈ H and if for all x, y, z ∈ H, the following conditions are satisfied: (i) (x⊛ z)⊛ (y ⊛ z) ≪ x⊛ y; [HGR1] (ii) (x⊛ y)⊛ z = (x⊛ z)⊛ y; [HGR2] (iii) x ≪ x; [HGR3] (iv) 0⊛ (0⊛ x) ≪ x, x ̸= 0; and, [HGR4] (v) (x⊛ y)⊛ z ≪ y ⊛ z. [HGR5] Definition 2. [12] A nonempty subset I of a hyper GR-algebra H is called an implicative hyper GR-ideal of H if for any x, y, z ∈ H, (i) 0 ∈ I; and, [IH1] (ii) (x⊛ z)⊛ (y ⊛ x) ⊆ I and z ∈ I imply that x ∈ I. [IH2] A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2223 Definition 3. [21] Let M be a nonempty set. A fuzzy set µ in M is the function µ : M → [0, 1]. The complement of a fuzzy set µ, denoted by µ̄, is the fuzzy set in M given by µ̄(x) = 1− µ(x) for all x ∈ M . Definition 4. [12] A fuzzy set µ in a hyper GR-algebra H is called fuzzy implicative hyper GR-ideal of Type 1 if for x, y, z ∈ H, µ(0) ≥ µ(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µ(u), µ(z) } [FIM1] Definition 5. [1] An intuitionistic fuzzy set A in a nonempty set H is an object having the form A = {(x, µA(x), γA(x) : x ∈ H}, where the functions µA : H → [0, 1] and γA : H → [0, 1] denote the degree of membership and degree of nonmembership, respectively, and for all x ∈ H, 0 ≤ µA(x) + γA(x) ≤ 1. Furthermore, πA(x) = 1 − µA(x) − γA(x) is called the intuitionistic fuzzy set index or hesitation margin of x in A. πA(x) is the degree of indeterminancy of x ∈ H to intuitionistic fuzzy set A and πA(x) ∈ [0, 1]. πA(x) expresses the lack of knowledge of whether x belongs to intuitionistic fuzzy set A or not. For the sake of simplicity, we shall use the symbol A = (µA(x), γA(x)) to denote the intuitionistic fuzzy set A = {(x, µA(x), γA(x)) : x ∈ H}. Definition 6. [16] For an intuitionistic fuzzy set A = (µA , γA) in H and s, t ∈ [0, 1], the set A⟨t,s⟩ = {x ∈ H : µA(x) ≥ t, γA(x) ≤ s} is called a level subset of A. Definition 7. [16] An intuitionistic fuzzy relation on a nonempty set H is an intuitionistic fuzzy set B = (µB , γB ) where µB : H ×H → [0, 1] and γB : H ×H → [0, 1]. Definition 8. [16] If B is an intuitionistic fuzzy relation on a nonempty set H and A is an intuitionistic fuzzy set in H, then B is an intuitionistic fuzzy relation on a set A if µB (x, y) ≤ min{µA(x), µA(y)} and γB (x, y) ≥ max{γA(x), γA(y)} for all x, y ∈ H. Definition 9. [16] If A is an intuitionistic fuzzy set in a nonempty set H, the strongest intuitionistic fuzzy relation on H is an intuitionistic fuzzy relation on H, denoted by BA = ((µB )µA , (γB )γA ), given by (µB )µA (x, y) = min{µA(x), µA(y)} and (γB )γA (x, y) = max{γA(x), γA(y)} for all x, y ∈ H. A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2224 Definition 10. [16] Let A and B be an intuitionistic fuzzy sets in a nonempty set H. The Cartesian product of A and B is defined for all x, y ∈ H as follows (µA × µB )(x, y) = min{µA(x), µB (y)} and (γA × γB )(x, y) = max{γA(x), γB (y)} Definition 11. [13] An intuitionistic fuzzy set A = (µA(x), γA(x)) in a hyper GR-algebra H is an intuitionistic fuzzy hyper GR-ideal of H if for all x, y ∈ H the following hold: (i) µA(0) ≥ µA(x) and γA(0) ≤ γA(x); [IFGR1] (ii) µA(x) ≥ min { inf u∈x⊛y µA(u), µA(y) } ; and, [IFGR2] (iii) γA(x) ≤ max { sup v∈x⊛y γA(v), γA(y) } . [IFGR3] Theorem 1. [12] A fuzzy set µ in a hyper GR-algebra H is a fuzzy implicative hyper GR-ideal of Type 1 if and only if µt is an implicative hyper GR-ideal of H whenever µt ̸= ∅ and t ∈ [0, 1]. Lemma 1. [13] Let µ : H → [0, 1] be a fuzzy set and S ⊆ H. Then (a) 1− sup x∈S µ(x) = inf x∈S (1− µ(x)) and (b) 1− inf x∈S µ(x) = sup x∈S (1− µ(x)). Corollary 1. [13] Let µ : H → [0, 1] be a fuzzy set and S ⊆ H. Then (a) 1−max x∈S µ(x) = min x∈S (1− µ(x)) and (b) 1−min x∈S µ(x) = max x∈S (1− µ(x)). 3. Intuitionistic Fuzzy Implicative Hyper GR-ideals Definition 12. An intuitionistic fuzzy set A = (µA , γA) in a hyper GR-algebra H is an intuitionistic fuzzy implicative hyper GR-ideal of H if for all x, y, z ∈ H the following hold: (i) µA(x) ≤ µA(0) and γA(x) ≥ γA(0); [IFIGR1] (ii) µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } ; and, [IFIGR2] (iii) γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . [IFIGR3] A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2225 The following examples illustrate the dissimilarity of intuitionistic fuzzy implicative hyper GR-ideal from intuitionistic fuzzy hyper GR-ideal. Example 1. Consider the hyper GR-algebra H = {0, 1, 2} and the Cayley table in Table 1. ⊛ 0 1 2 0 {0,1} {0,1} {0,1} 1 {1} {0,1} {0,1} 2 {0,2} {0,2} {0,1,2} Table 1: Hyper GR-algebra Define the fuzzy sets µA and γA , respectively by, µA(x) =  0.5, if x = 0 0.3, if x = 1 0.2 if x = 2 and γA(x) =  0.4, if x = 0 0.6, if x = 1 0.7 if x = 2 . (i) Clearly, µA(0) ≥ µA(x) and γA(0) ≤ γA(x) for all x ∈ H. (ii) Table 2 and Table 3 show that µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } . x z x⊛ z y ⊛ x A = (x⊛ z)⊛ (y ⊛ x) y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 0 0 {0,1} {0,1} {1} {0,2} {0,1} {0,1} {0,1} 0 1 {0,1} {0,1} {1} {0,2} {0,1} {0,1} {0,1} 0 2 {0,1} {0,1} {1} {0,2} {0,1} {0,1} {0,1} 1 0 {1} {0,1} {0,1} {0,2} {0,1} {0,1} {0,1} 1 1 {0,1} {0,1} {0,1} {0,2} {0,1} {0,1} {0,1} 1 2 {0,1} {0,1} {0,1} {0,2} {0,1} {0,1} {0,1} 2 0 {0,2} {0,1} {0,1} H H H H 2 1 {0,2} {0,1} {0,1} H H H H 2 2 H {0,1} {0,1} H H H H Table 2: IFGR2 A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2226 x z S = inf u∈(x⊛z)⊛(y⊛x) µA(u) min{S, µA(z)} µA(x) ≥ min{S, µA(z)} y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 0 0 0.3 0.3 0.3 0.3 0.3 0.3 � � � 0 1 0.3 0.3 0.3 0.3 0.3 0.3 � � � 0 2 0.3 0.3 0.3 0.2 0.2 0.2 � � � 1 0 0.3 0.3 0.3 0.3 0.3 0.3 � � � 1 1 0.3 0.3 0.3 0.3 0.3 0.3 � � � 1 2 0.3 0.3 0.3 0.2 0.2 0.2 � � � 2 0 0.2 0.2 0.2 0.2 0.2 0.2 � � � 2 1 0.2 0.2 0.2 0.2 0.2 0.2 � � � 2 2 0.2 0.2 0.2 0.2 0.2 0.2 � � � Table 3: IFIGR2(continuation) (iii) Table 2 and Table 4 show that γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . x z T = sup v∈(x⊛z)⊛(y⊛x) γA(v) max{T, γA(z)} γA(x) ≤ max{T, γA(z)} y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 0 0 0.6 0.6 0.6 0.6 0.6 0.6 � � � 0 1 0.6 0.6 0.6 0.6 0.6 0.6 � � � 0 2 0.6 0.6 0.6 0.7 0.7 0.7 � � � 1 0 0.6 0.6 0.6 0.6 0.6 0.6 � � � 1 1 0.6 0.6 0.6 0.6 0.6 0.6 � � � 1 2 0.6 0.6 0.6 0.7 0.7 0.7 � � � 2 0 0.7 0.7 0.7 0.7 0.7 0.7 � � � 2 1 0.7 0.7 0.7 0.7 0.7 0.7 � � � 2 2 0.7 0.7 0.7 0.7 0.7 0.7 � � � Table 4: IFIGR3 Thus, A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of hyper GR-algebra H. Example 2. Consider the hyper GR-algebra H = {0, 1, 2} and the Cayley table in Table 5. ⊛ 0 1 2 0 {0,1} {0,1} {0,1} 1 {0} {0,1} {0,1} 2 {0,2} {0,2} {0,1,2} Table 5: Hyper GR-algebra A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2227 Define the fuzzy sets µA and γA , respectively by, µA(x) =  0.5, if x = 0 0.3, if x = 1 0.2 if x = 2 and γA(x) =  0.4, if x = 0 0.6, if x = 1 0.7 if x = 2 . (i) Clearly, µA(0) ≥ µA(x) and γA(0) ≤ γA(x) for all x ∈ H. (ii) Table 6 and Table 7 show that µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } . x z x⊛ z y ⊛ x A = (x⊛ z)⊛ (y ⊛ x) y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 0 0 {0,1} {0,1} {0} {1,2} {0,1} {0,1} {0,1} 0 1 {0,1} {0,1} {0} {1,2} {0,1} {0,1} {0,1} 0 2 {0,1} {0,1} {0} {1,2} {0,1} {0,1} {0,1} 1 0 {0} {0,1} {0,1} {0,2} {0,1} {0,1} {0,1} 1 1 {0,1} {0,1} {0,1} {0,2} {0,1} {0,1} {0,1} 1 2 {0,1} {0,1} {0,1} {0,2} {0,1} {0,1} {0,1} 2 0 {1,2} {0,1} {0,1} H H H H 2 1 {0,2} {0,1} {0,1} H H H H 2 2 H {0,1} {0,1} H H H H Table 6: IFGR2 x z S = inf u∈(x⊛z)⊛(y⊛x) µA(u) min{S, µA(z)} µA(x) ≥ min{S, µA(z)} y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 0 0 0.3 0.3 0.3 0.3 0.3 0.3 � � � 0 1 0.3 0.3 0.3 0.3 0.3 0.3 � � � 0 2 0.3 0.3 0.3 0.2 0.2 0.2 � � � 1 0 0.3 0.3 0.3 0.3 0.3 0.3 � � � 1 1 0.3 0.3 0.3 0.3 0.3 0.3 � � � 1 2 0.3 0.3 0.3 0.2 0.2 0.2 � � � 2 0 0.2 0.2 0.2 0.2 0.2 0.2 � � � 2 1 0.2 0.2 0.2 0.2 0.2 0.2 � � � 2 2 0.2 0.2 0.2 0.2 0.2 0.2 � � � Table 7: IFIGR2(continuation) (iii) Table 6 and Table 8 show that γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2228 x z T = sup v∈(x⊛z)⊛(y⊛x) γA(v) max{T, γA(z)} γA(x) ≤ max{T, γA(z)} y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 y = 0 y = 1 y = 2 0 0 0.6 0.6 0.6 0.6 0.6 0.6 � � � 0 1 0.6 0.6 0.6 0.6 0.6 0.6 � � � 0 2 0.6 0.6 0.6 0.7 0.7 0.7 � � � 1 0 0.6 0.6 0.6 0.6 0.6 0.6 � � � 1 1 0.6 0.6 0.6 0.6 0.6 0.6 � � � 1 2 0.6 0.6 0.6 0.7 0.7 0.7 � � � 2 0 0.7 0.7 0.7 0.7 0.7 0.7 � � � 2 1 0.7 0.7 0.7 0.7 0.7 0.7 � � � 2 2 0.7 0.7 0.7 0.7 0.7 0.7 � � � Table 8: IFIGR3 Thus, A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of H but is not an intuitionistic fuzzy hyper GR-ideal in H since µA(1) = 0.3 < 0.5 = min { inf u∈1⊛0 µA(u), µA(0) } and γA(1) = 0.6 > 0.4 = max { sup v∈1⊛0 γA(v), γA(0) } . Remark 1. Example 2 shows that not all intuitionistic fuzzy implicative hyper GR-ideal are intuitionistic fuzzy hyper GR-ideal of H. The following theorems exhibit a characterization of intuitionistic fuzzy implicative hyper GR-ideal of hyper GR-algebra. Theorem 2. An intuitionistic fuzzy set A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a Hyper GR-algebra H if and only if A⟨t,s⟩ is an implicative hyper GR-ideal of H whenever A⟨t,s⟩ ̸= ∅ and t, s ∈ [0, 1]. Proof : Suppose A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal in H. By IFIGR1, µA(x) ≤ µA(0) and γA(x) ≥ γA(0) for all x ∈ H. Let t, s ∈ [0, 1]. By assumption, A⟨t,s⟩ ̸= ∅, that is there exists x ∈ A⟨t,s⟩ such that t ≤ µA(x) ≤ µA(0) and s ≥ γA(x) ≥ γA(0). Thus 0 ∈ A⟨t,s⟩. Let x, y, z ∈ H such that (x ⊛ z) ⊛ (y ⊛ x) ⊆ A⟨t,s⟩ and z ∈ A⟨t,s⟩. Then µA(z) ≥ t and γA(z) ≤ s. Also, µA(u) ≥ t and γA(v) ≤ s for all u, v,∈ (x⊛ z)⊛ (y⊛x). It implies that t is a lowerbound for {µA(u)|u ∈ (x⊛ z)⊛ (y⊛x)} and s is an upperbound for {γA(v)|v ∈ (x⊛ z)⊛ (y ⊛ x)}. Then, inf u∈(x⊛z)⊛(y⊛x) µA(u) ≥ t and sup v∈(x⊛z)⊛(y⊛x) γA(v) ≤ s. By IFIGR2 and IFIGR3, A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2229 µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } = min{t, t} = t and γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } = max{s, s} = s. Hence, x ∈ A⟨t,s⟩. Thus, A⟨t,s⟩ is a hyper GR-ideal of H. Conversely, suppose A⟨t,s⟩ is a hyper GR-ideal of H where t, s ∈ [0, 1]. Let x ∈ H and k, l ∈ [0, 1] such that k = µA(x) and l = γA(x). Note that A⟨k,l⟩ is a hyper GR-ideal of H. Thus 0 ∈ A⟨k,l⟩. Then µA(0) ≥ µA(x) = k and γA(0) ≤ γA(x) = l. Let x, y, z ∈ H and p, q ∈ [0, 1] such that p = min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } and q = max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . Let w ∈ (x⊛ z)⊛ (y ⊛ x). Then µA(w) ≥ inf u∈(x⊛z)⊛(y⊛x) µA(u) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } = p and γA(w) ≤ sup v∈(x⊛z)⊛(y⊛x) γA(v) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } = q. This implies that w ∈ A⟨p,q⟩. It follows that (x⊛ z)⊛ (y ⊛ x) ⊆ A⟨p,q⟩. Clearly, µA(z) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } = p and γA(z) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } = q. Hence, z ∈ A⟨p,q⟩. Since A⟨p,q⟩ is a implicative hyper GR-ideal of H, x ∈ A⟨p,q⟩. It implies that µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } and γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . Thus, A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of H. Theorem 3. If A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H, then the set J = {x ∈ H : µA(x) = µA(0) and γA(x) = γA(0)} is an implicative hyper GR-ideal of H. Proof : Suppose A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H. Let x, y, z ∈ H such that (x⊛ z)⊛ (y⊛x) ⊆ J and z ∈ J . It follows that 0 ∈ J . Then, µA(z) = µA(0), γA(z) = γA(0), µA(u) = µA(0) and γA(v) = γA(0) for any u, v ∈ (x⊛ z)⊛ (y ⊛ x). By IFIGR1, µA(0) ≥ µA(x) and γA(0) ≤ γA(x). By IFIGR2 and IFIGR3, A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2230 µA(0) ≥ µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } = µA(0) and γA(0) ≤ γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } = γ(0). This implies that µA(x) = µA(0) and γA(x) = γA(0) and so x ∈ J . Thus, J is an implicative hyper GR-ideal of H. Theorem 4. If A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H, then µA(x) ≥ inf u∈(x⊛0)⊛(y⊛x) µA(u) and γA(x) ≤ sup v∈(x⊛0)⊛(y⊛x) γA(v) for all x, y ∈ H. Proof : Suppose A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H. Let x, y ∈ H. By IFIGR2 and IFIGR1, µA(x) ≥ min { inf u∈(x⊛0)⊛(y⊛x) µA(u), µA(0) } = inf u∈(x⊛0)⊛(y⊛x) µA(u). Also by IFIGR3 and IFIGR1, γA(x) ≤ max { sup v∈(x⊛0)⊛(y⊛x) γA(v), γA(0) } = sup v∈(x⊛0)⊛(y⊛x) γA(v). Lemma 3.1. An intuitionistic fuzzy sets A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H if and only if the fuzzy sets µA and γ̄A are fuzzy implicative hyper GR-ideals of type 1 in H. Proof : Suppose A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of H. By IFIGR1 and IFIGR2, for any x, y, z ∈ H, µA(x) ≤ µA(0) and µA(x) ≥ min { inf u∈(x⊛z)⊛(y⊛x) µA(u), µA(z) } Hence, µA is a fuzzy implicative hyper GR-ideal of type 1 in H. Let x, y, z ∈ H. By IFIGR1 and IFIGR3, γA(x) ≥ γA(0) and γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2231 Then γ̄A(x) = 1− γA(x) ≤ 1− γA(0) = γ̄A(0), that is, γ̄A(x) ≤ γ̄A(0). By Corollary 1(a) and Lemma 1(a), γ̄A(x) = 1− γA(x) ≥ 1−max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } = min { 1− sup v∈(x⊛z)⊛(y⊛x) γA(v), 1− γA(z) } = min { inf v∈(x⊛z)⊛(y⊛x) (1− γA(v)), 1− γA(z) } = min { inf v∈(x⊛z)⊛(y⊛x) γ̄A(v), γ̄A(z) } . Implies that, γ̄A(x) ≥ min { inf v∈(x⊛z)⊛(y⊛x) γ̄A(v), γ̄A(z) } . Hence γ̄A(x) is a fuzzy implicative hyper GR-ideal of type 1 in H. Conversely, suppose that µA and γ̄A are fuzzy implicative hyper GR-ideals of type 1 in H. Let x, y, z ∈ H. By FIM1, γ̄A(x) ≤ γ̄A(0) and γA(x) = 1− γ̄A(x) ≥ 1− γ̄A(0) = γA(0). By Corollary 1(b) and Lemma 1(b), γA(x) = 1− γ̄A(x) ≤ 1−min { inf v∈(x⊛z)⊛(y⊛x) γ̄A(v), γ̄A(z) } = max { 1− inf v∈(x⊛z)⊛(y⊛x) γ̄A(v), 1− γ̄A(z) } = max { sup v∈(x⊛z)⊛(y⊛x) (1− γ̄A(v)), γA(z) } = max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . Thus, γA(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) γA(v), γA(z) } . A. Macodi, A. Dorig / Eur. J. Pure Appl. Math, 17 (3) (2024), 2221-2234 2232 Hence, the fuzzy set A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H. The following results establish some characterization and properties of intuitionistic fuzzy implicative hyper GR-ideal of hyper GR-algebra. Theorem 5. Let A = (µA , γA) be an intuitionistic fuzzy set in a hyper GR-algebra H. Then, A is an intuitionistic fuzzy implicative hyper GR-ideal in H if and only if  = (µA , µ̄A) and Ă = (γ̄A , γA) are intuitionistic fuzzy implicative hyper GR-ideals of H. Proof : Suppose that A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of H. By Lemma 3.1, µA and γ̄A are fuzzy implicative hyper GR-ideals of type 1 in H. We are left to show that µ̄A will satisfy IFIGR1 and IFIGR3. Let x, y, z ∈ H. Then by FIM1, µA(x) ≤ µA(0) and µ̄A(x) = 1− µA(x) ≥ 1− µA(0) = µ̄A(0). Thus, IFIGR1 is satisfied. By Lemma 1(b) and Corollary 1(b), µ̄A(x) = 1− µA(x) ≤ 1−min { inf v∈(x⊛z)⊛(y⊛x) µA(v), µA(z) } = max { 1− inf v∈(x⊛z)⊛(y⊛x) µA(v), 1− µA(z) } = max { sup v∈(x⊛z)⊛(y⊛x) (1− µA(v)), 1− µA(z) } = max { sup v∈(x⊛z)⊛(y⊛x) µ̄A(v), µ̄A(z) } . Implies that µ̄A(x) ≤ max { sup v∈(x⊛z)⊛(y⊛x) µ̄A(v), µ̄A(z) } . Thus, IFIGR3 is satisfied. Hence,  = (µA , µ̄A) and Ă = (γ̄A , γA) are intuitionistic fuzzy implicative hyper GR-ideals of H. Conversely, suppose  = (µA , µ̄A) and Ă = (γ̄A , γA) are intuitionistic fuzzy implicative hyper GR-ideals in H. Then by Lemma 3.1, µA and γ̄A are fuzzy implicative hyper GR-ideals of type 1 in H and A = (µA , γA) is an intuitionistic fuzzy implicative hyper GR-ideal of a hyper GR-algebra H. Corollary 3.2. For any subset I of a hyper GR-algebra H, let A(I) = (µ A(I) , γ A(I) ) be an intuitionistic fuzzy set in H where µ A(I) (x) = { m1, if x ∈ I m2, otherwise REFERENCES 2233 and γ A(I) (x) = { n1, if x ∈ I n2, otherwise for all x ∈ H, where m1,m2, n1, n2 ∈ [0, 1] with m1 > m2 and n1 < n2, mi + ni ≤ 1 for i = 1, 2. Then I is an implicative hyper GR-ideal of H if and only if A(I) = (µ A(I) , γ A(I) ) is an intuitionistic fuzzy implicative hyper GR-ideal H. Proof : Note that the level subsets of µA(I) is (µ A(I) )t1 =  ∅, if m1 < t1 ≤ 1 I, if m2 < t1 ≤ m1 H, if 0 ≤ t1 ≤ m2 . 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