Int. J. Anal. Appl. (2023), 21:27 Intuitionistic Hesitant Fuzzy UP (BCC)-Filters of UP (BCC)-Algebras Aiyared Iampan1,∗, R. Alayakkaniamuthu2, P. Gomathi Sundari2, N. Rajesh2 1Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 2Department of Mathematics, Rajah Serfoji Government College (Affiliated to Bharathidasan University), Thanjavur 613005, Tamilnadu, India ∗Corresponding author: aiyared.ia@up.ac.th Abstract. The concepts of intuitionistic hesitant fuzzy UP (BCC)-subalgebras, UP (BCC)-ideals, and UP (BCC)-filters of UP (BCC)-algebras are presented, some of their features are explained, and their extensions are demonstrated using the theory of hesitant fuzzy sets as a foundation. The necessary conditions for those intuitionistic hesitant fuzzy sets are provided and include their relation to their complement. The concept of prime and weakly prime of intuitionistic hesitant fuzzy sets was also introduced and studied. We also talk about the connections between intuitionistic hesitant fuzzy UP (BCC)-subalgebras (UP (BCC)-ideals, UP (BCC)-filters) and their level subsets. The homomorphic pre-images of intuitionistic hesitant fuzzy UP (BCC)-filters in UP (BCC)-algebras are also studied and some related properties are investigated. 1. Introduction The concept of fuzzy sets was proposed by Zadeh [15]. The theory of fuzzy sets has several applica- tions in real-life situations, and many scholars have researched fuzzy set theory. After the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. The integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [1, 2, 4]. In 2009 - 2010, Torra and Narukawa [13, 14] introduced the notion of hesitant fuzzy sets, that is a function from a reference set to a power set of the unit interval. The notion of hesitant fuzzy sets is the other generalization of the notion fuzzy sets. The Received: Feb. 11, 2023. 2010 Mathematics Subject Classification. 03G25, 03E72. Key words and phrases. UP (BCC)-algebra; intuitionistic hesitant fuzzy UP (BCC)-subalgebra; intuitionistic hesitant fuzzy UP (BCC)-ideal; intuitionistic hesitant fuzzy UP (BCC)-filter. https://doi.org/10.28924/2291-8639-21-2023-27 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-27 2 Int. J. Anal. Appl. (2023), 21:27 hesitant fuzzy set theories developed by Torra and others have found many applications in the domain of mathematics and elsewhere. After the introduction of the notion of hesitant fuzzy sets by Torra and Narukawa [13, 14], several researches were conducted on the generalizations of the notion of hesitant fuzzy sets and application to many logical algebras such as: in 2012, Zhu, Xu and Xia [16] introduced the notion of dual hesitant fuzzy sets, which is a new extension of fuzzy sets. In 2014, Jun, Ahn and Muhiuddin [7] introduced the notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals in BCK/BCI-algebras. Jun and Song [9] introduced the notions of (Boolean, prime, ultra, good) hesitant fuzzy filters and hesitant fuzzy MV-filters of MTL-algebras. Iampan [6] introduced a new algebraic structure, called a UP-algebra, and Mosrijai et. al. [11] introduced the notion of hesitant fuzzy sets on UP-algebras. The notions of hesitant fuzzy subalgebras, hesitant fuzzy filters and hesitant fuzzy UP-ideals play an important role in studying the many logical algebras. The concepts of UP-algebras (see [6]) and BCC-algebras (see [10]) are the same concept, as shown by Jun et al. [8] in 2022. In this publication and following investigations, our research team will refer to it as BCC rather than UP because of respect for Komori, who first characterized it in 1984. In this paper, the concepts of intuitionistic hesitant fuzzy BCC-subalgebras, BCC-ideals, and BCC- filters of BCC-algebras are presented, some of their features are explained, and their extensions are demonstrated using the theory of hesitant fuzzy sets as a foundation. The necessary conditions for those intuitionistic hesitant fuzzy sets are provided and include their relation to their complement. The concept of prime and weakly prime of intuitionistic hesitant fuzzy sets was also introduced and studied. We also talk about the connections between intuitionistic hesitant fuzzy BCC-subalgebras (BCC-ideals, BCC-filters) and their level subsets. The homomorphic pre-images of intuitionistic hesitant fuzzy BCC- filters in BCC-algebras are also studied and some related properties are investigated. 2. Preliminaries The concept of BCC-algebras (see [10]) can be redefined without the condition (2.6) as follows: An algebra X = (X, ·, 0) of type (2, 0) is called a BCC-algebra if it satisfies the following conditions: (∀x,y,z ∈ X)((y ·z) · ((x ·y) · (x ·z)) = 0) (2.1) (∀x ∈ X)(0 ·x = x) (2.2) (∀x ∈ X)(x · 0 = 0) (2.3) (∀x,y ∈ X)(x ·y = 0 = y ·x ⇒ x = y) (2.4) After this we assign X instead of a BCC-algebra (X, ·, 0) until otherwise specified. We define a binary relation ≤ on X as follows: (∀x,y ∈ X)(x ≤ y ⇔ x ·y = 0) (2.5) Int. J. Anal. Appl. (2023), 21:27 3 In X, the following assertions are valid (see [6]). (∀x ∈ X)(x ≤ x) (2.6) (∀x,y,z ∈ X)(x ≤ y,y ≤ z ⇒ x ≤ z) (2.7) (∀x,y,z ∈ X)(x ≤ y ⇒ z ·x ≤ z ·y) (2.8) (∀x,y,z ∈ X)(x ≤ y ⇒ y ·z ≤ x ·z) (2.9) (∀x,y,z ∈ X)(x ≤ y ·x, in particular, y ·z ≤ x · (y ·z)) (2.10) (∀x,y ∈ X)(y ·x ≤ x ⇔ x = y ·x) (2.11) (∀x,y ∈ X)(x ≤ y ·y) (2.12) (∀a,x,y,z ∈ X)(x · (y ·z) ≤ x · ((a ·y) · (a ·z))) (2.13) (∀a,x,y,z ∈ X)(((a ·x) · (a ·y)) ·z ≤ (x ·y) ·z) (2.14) (∀x,y,z ∈ X)((x ·y) ·z ≤ y ·z) (2.15) (∀x,y,z ∈ X)(x ≤ y ⇒ x ≤ z ·y) (2.16) (∀x,y,z ∈ X)((x ·y) ·z ≤ x · (y ·z)) (2.17) (∀a,x,y,z ∈ X)((x ·y) ·z ≤ y · (a ·z)) (2.18) Definition 2.1. [6] A nonempty subset S of X is called a BCC-subalgebra of X if x ·y ∈ S ∀x,y ∈ S. Definition 2.2. [6] A nonempty subset I of X is called a BCC-ideal of X if (1) 0 ∈ I, (2) (∀x,y,z ∈ X)(x · (y ·z),y ∈ I ⇒ x ·z ∈ I). Definition 2.3. [12] A nonempty subset F of X is called a BCC-filter of X if (1) 0 ∈ F, (2) (∀x,y ∈ X)(x ·y ∈ F,x ∈ F ⇒ y ∈ F ). Definition 2.4. [13] A hesitant fuzzy set on a reference set X is defined in term of a function h that when applied to X return a subset of [0, 1], that is, h : X →P([0, 1]). Definition 2.5. [3] An intuitionistic hesitant fuzzy set on a reference set X is defined in the form H = (h,k), where h and k are functions that when applied to X return a subset of [0, 1], that is, h,k : X →P([0, 1]). Definition 2.6. [11] A hesitant fuzzy set h on X is said to be a hesitant fuzzy BCC-filter of X if the following conditions are hold: (∀x ∈ X)(h(0) ⊇ h(x)) (2.19) (∀x,y ∈ X)(h(y) ⊇ h(x ·y) ∩h(x)) (2.20) 4 Int. J. Anal. Appl. (2023), 21:27 Definition 2.7. [13] The complement of a hesitant fuzzy set h in a reference set X is the hesitant fuzzy set h defined by h(x) = [0, 1] −h(x) for all x ∈ X. Definition 2.8. [13] The complement of an intuitionistic hesitant fuzzy set H = (h,k) on a reference set X is the intuitionistic hesitant fuzzy set H = (k,h) defined by h(x) = [0, 1] −h(x) and k(x) = [0, 1] −k(x) for all x ∈ X. 3. Intuitionistic hesitant fuzzy BCC-filters In this section, the concepts of intuitionistic hesitant fuzzy BCC-subalgebras, BCC-ideals, and BCC-filters of BCC-algebras are presented, some of their features are explained. Definition 3.1. An intuitionistic hesitant fuzzy set H = (h,k) on X is called an intuitionistic hesitant fuzzy BCC-subalgebra of X if it satisfies the following property: (∀x,y ∈ X) ( h(x ·y) ⊇ h(x) ∩h(y) k(x ·y) ⊆ k(x) ∪k(y) ) (3.1) Definition 3.2. The characteristic intuitionistic hesitant fuzzy set of a subset A of a set X is defined to be the structure χA = (hχA,kχA), where hχA(x) = { [0, 1] if x ∈ A ∅ otherwise and kχA(x) = { ∅ if x ∈ A [0, 1] otherwise. Lemma 3.1. The constant 0 of X is in a nonempty subset B of X if and only if hχB (0) ⊇ hχB (x) and kχB (0) ⊆ kχB (x) for all x ∈ X. Proof. If 0 ∈ B, then hB(0) = [0, 1]. Thus hB(0) = [0, 1] ⊇ hB(x) for all x ∈ X. Also, kB(0) = ∅. Then kB(0) = ∅⊆ kB(x) for all x ∈ X. Conversely, assume that hB(0) ⊇ hB(x) and kB(0) ⊆ kB(x) for all x ∈ X. Since B is a nonempty subset of X, we have a ∈ B for some a ∈ X. Then hB(0) ⊇ hB(a) = [0, 1], so hB(0Y ) = [0, 1]. Hence, 0 ∈ B. � Definition 3.3. An intuitionistic hesitant fuzzy set H = (h,k) on X is said to be an intuitionistic hesitant fuzzy BCC-filter of X if the following conditions are hold: (∀x ∈ X) ( h(0) ⊇ h(x) k(0) ⊆ k(x) ) (3.2) (∀x,y ∈ X) ( h(y) ⊇ h(x ·y) ∩h(x) k(y) ⊆ k(x ·y) ∪k(x) ) (3.3) Int. J. Anal. Appl. (2023), 21:27 5 Example 3.1. Let X = {0, 1, 2, 3} with the following Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 1 0 0 3 0 1 2 0 Then X is a BCC-algebra. We define an intuitionistic hesitant fuzzy set H = (h,k) on X as follows: h(0) = [0, 1],h(1) = {0.1},h(2) = ∅,h(3) = {0.2, 0.3}, k(0) = ∅,k(1) = {0.1, 0.2},k(2) = [0, 1],k(3) = {0.3} Then H is an intuitionistic hesitant fuzzy BCC-subalgebra of X. Definition 3.4. An intuitionistic hesitant fuzzy set H = (h,k) on X is said to be an intuitionistic hesitant fuzzy BCC-ideal of X if (3.2) and the following condition are hold: (∀x,y,z ∈ X) ( h(x ·y) ⊇ h(x · (y ·z)) ∩h(y) k(x ·y) ⊆ k(x · (y ·z)) ∪k(y) ) (3.4) Theorem 3.1. Every intuitionistic hesitant fuzzy BCC-ideal of X is an intuitionistic hesitant fuzzy BCC-filter. Proof. Let H = (h,k) be an intuitionistic hesitant fuzzy BCC-ideal of X. Then (3.2) holds. Let x,y ∈ X. Then h(y) = h(0 ·y) ⊇ h(0 · (x ·y)) ∩h(x) = h(x ·y) ∩h(x), k(y) = k(0 ·y) ⊆ k(0 · (x ·y)) ∪k(x) = k(x ·y) ∪k(x). Hence, H is an intuitionistic hesitant fuzzy BCC-filter of X. � The following example shows that the converse of Theorem 3.1 is not true in general. Example 3.2. Let X = {0, 1, 2, 3} with the following Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 3 3 2 0 1 0 0 3 0 1 2 0 Then X is a BCC-algebra. We define an intuitionistic hesitant fuzzy set H = (h,k) on X as follows: h(0) = [0, 1],h(1) = {0.7},h(2) = ∅,h(3) = {0.2, 0.5}, k(0) = ∅,k(1) = {0.1, 0.2},k(2) = {0.1, 0.2, 0.3},k(3) = [0, 1] 6 Int. J. Anal. Appl. (2023), 21:27 Then H is an intuitionistic hesitant fuzzy BCC-filter of X but not an intuitionistic hesitant fuzzy BCC-ideal of X. Theorem 3.2. Every intuitionistic hesitant fuzzy BCC-filter of X is an intuitionistic hesitant fuzzy BCC-subalgebra. Proof. Let H = (h,k) be an intuitionistic hesitant fuzzy BCC-filter of X. Then for all x,y ∈ X, h(x ·y) ⊇ h(y · (x ·y)) ∩h(y) = h(0) ∩h(y) = h(y) ⊇ h(x) ∩h(y), k(x ·y) ⊆ k(y · (x ·y)) ∪k(y) = k(0) ∪k(y) = k(y) ⊆ k(x) ∪k(y). Hence, H is an intuitionistic hesitant fuzzy BCC-subalgebra of X. � The following example shows that the converse of Theorem 3.2 is not true in general. Example 3.3. Let X = {0, 1, 2, 3} with the following Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 0 0 3 3 0 1 2 0 Then X is a BCC-algebra. We define an intuitionistic hesitant fuzzy set H = (h,k) on X as follows: h(0) = {0.1, 0.2, 0.3},h(1) = {0.1},h(2) = {0.2},h(3) = ∅, k(0) = ∅,k(1) = {0.1},k(2) = {0.1, 0.2},k(3) = [0, 1] Then H is an intuitionistic hesitant fuzzy BCC-subalgebra of X but not an intuitionistic hesitant fuzzy BCC-filter of X. Theorem 3.3. A nonempty subset F of X is a BCC-filter of X if and only if the characteristic intuitionistic hesitant fuzzy set χF = (hχF ,kχF ) is an intuitionistic hesitant fuzzy BCC-filter of X. Proof. Assume that F is a BCC-filter of X. Since 0 ∈ F, it follows from Lemma 3.1 that hχF (0) ⊇ hχF (x) for all x ∈ X. Next, let x,y ∈ X. Case 1 : If x,y ∈ F, then hχF (x) = [0, 1] and hχF (y) = [0, 1]. Hence, hχF (y) = [0, 1] ⊇ hχF (x ·y) = hχF (x ·y)∩hχF (x). Also, kχF (x) = ∅ and kχF (y) = ∅. Hence, kχF (y) = ∅⊆ kχF (x ·y) = kχF (x ·y) ∪kχF (x). Case 2 : If x /∈ F and y ∈ F, then hχF (x) = ∅ and hχF (y) = [0, 1]. Thus hχF (y) = [0, 1] ⊇ ∅ = hχF (x · y) ∩ hχF (x). Also kχF (x) = [0, 1] and kχF (y) = ∅. Thus kχF (y) = ∅ ⊆ [0, 1] = kχF (x ·y) ∪kχF (x). Case 3 : If x ∈ F and y /∈ F, then hχF (x) = [0, 1] and hχF (y) = ∅. Since F is a BCC-filter of X, we have x · y /∈ F or x /∈ F. But x ∈ F, so x · y /∈ F. Then hχF (x · y) = ∅. Thus Int. J. Anal. Appl. (2023), 21:27 7 hχF (y) = ∅ ⊇ ∅ = hχF (x · y) ∩ hχF (x). Also, kχF (x) = ∅, kχF (y) = [0, 1] and kχF (x · y) = [0, 1]. Thus kχF (y) = [0, 1] ⊆ [0, 1] = kχF (x ·y) ∪kχF (x). Case 4 : If x /∈ F and y /∈ F, then hχF (x) = ∅ and hχF (y) = ∅. Thus hχF (y) = ∅⊆∅ = hχF (x ·y)∩ hχF (x). Also, kχF (x) = [0, 1] and kχF (y) = [0, 1]. Thus kχF (y) = [0, 1] ⊆ [0, 1] = kχF (x ·y)∪kχF (x). Hence, χF = (hχF ,kχF ) is an intuitionistic hesitant fuzzy BCC-filter of X. Conversely, assume that χF = (hχF ,kχF ) is an intuitionistic hesitant fuzzy BCC-filter of X. Since hχF (0) ⊇ hχF (x) for all x ∈ X, it follows from Lemma 3.1 that 0 ∈ F. Next, let x,y ∈ X be such that x ·y ∈ F and x ∈ F. Then hχF (x ·y) = [0, 1] and hχF (x) = [0, 1]. Thus hχF (y) ⊇ hχF (x ·y)∩hχF (x) = [0, 1], so hχF (y) = [0, 1]. Therefore, y ∈ F and so F is a BCC-filter of X. � Definition 3.5. An intuitionistic hesitant fuzzy set H = (h,k) on X is called a prime intuitionistic hesitant fuzzy set on X if it satisfies the following property: (∀x,y ∈ X) ( h(x ·y) ⊆ h(x) ∪h(y) k(x ·y) ⊇ k(x) ∩k(y) ) (3.5) Definition 3.6. [5] A nonempty subset B of X is called a prime subset of X if it satisfies the following property: (∀x,y ∈ X)(x ·y ∈ B ⇒ x ∈ B or y ∈ B) Theorem 3.4. A nonempty subset B of X is a prime subset of X if and only if the characteristic intuitionistic hesitant fuzzy set χB is a prime intuitionistic hesitant fuzzy set on X. Proof. Assume that B is a prime subset of X and let x,y ∈ X. Case 1 : If x · y ∈ B, then hχB (x · y) = [0, 1]. Since B is a prime subset of X, we have x ∈ B or y ∈ B. Then hχB (x) = [0, 1] or hχB (y) = [0, 1], so hχB (x) ∪ hχB (y) = [0, 1]. Hence, hχB (x ·y) = [0, 1] ⊆ [0, 1] = hχB (x) ∪hχB (y). Also, kχB (x ·y) = ∅⊇ kχB (x) ∩kχB (y). Case 2 : If x · y /∈ B, then hχB (x · y) = ∅ ⊆ hχB (x) ∪ hχB (y). Also, kχB (x · y) = [0, 1] ⊇ kχB (x) ∩kχB (y). Hence, χB is a prime intuitionistic hesitant fuzzy set on X. Conversely, assume that χB = (hχB,kχB ) is a prime intuitionistic hesitant fuzzy set on X. Let x,y ∈ X be such that x ·y ∈ B. Then hχB (x ·y) = [0, 1], so [0, 1] = hχB (x ·y) ⊆ hχB (x) ∪hχB (y). Thus hχB (x) ∪hχB (y) = [0, 1], so hχB (x) = [0, 1] or hχB (y) = [0, 1]. Hence, x ∈ B or y ∈ B and so B is a prime subset of X. � Theorem 3.5. Let H = (h,k) be an intuitionistic hesitant fuzzy set on X. Then the following statements are equivalent: (1) H is a prime intuitionistic hesitant fuzzy BCC-filter of X, (2) H is a constant intuitionistic hesitant fuzzy set on X. 8 Int. J. Anal. Appl. (2023), 21:27 Proof. Assume that H is a prime intuitionistic hesitant fuzzy BCC-filter of X. Then h(0) ⊇ h(x) and k(0) ⊆ k(x) for all x ∈ X. By (2.6), we have h(0) = h(x · x) ⊆ h(x) ∪ h(x) = h(x) and k(0) = k(x ·x) ⊇ k(x)∪k(x) = k(x) for all x ∈ X and so h(x) = h(0) and k(x) = k(0) for all x ∈ X. Hence, H is a constant intuitionistic hesitant fuzzy set on X. Conversely, assume that H is a constant intuitionistic hesitant fuzzy set on X. Hence, we can easily show that H is a prime intuitionistic hesitant fuzzy BCC-filter of X. � Definition 3.7. [5] A nonempty subset B of X is called a weakly prime subset of X if it satisfies the following property: (∀x,y ∈ X,x 6= y)(x ·y ∈ B ⇒ x ∈ B or y ∈ B) Definition 3.8. [5] A BCC-filter B of X is called a weakly prime BCC-filter of X if B is a weakly prime subset of X. Definition 3.9. An intuitionistic hesitant fuzzy set H = (h,k) on X is called a weakly prime intuition- istic hesitant fuzzy set on X if it satisfies the following property: (∀x,y ∈ X,x 6= y) ( h(x ·y) ⊆ h(x) ∪h(y) k(x ·y) ⊇ k(x) ∩k(y) ) (3.6) Definition 3.10. An intuitionistic hesitant fuzzy BCC-filter H = (h,k) of X is called a weakly prime intuitionistic hesitant fuzzy BCC-filter of X if H is a weakly prime intuitionistic hesitant fuzzy set on X. Theorem 3.6. A nonempty subset B of X is a weakly prime subset of X if and only if the characteristic intuitionistic hesitant fuzzy set χB is a weakly prime intuitionistic hesitant fuzzy set on X. Proof. Assume that B is a weakly prime subset of X and let x,y ∈ X be such that x 6= y. Case 1 : If x · y ∈ B, then hχB (x · y) = [0, 1]. Since B is a weakly prime subset of X, we have x ∈ B or y ∈ B. Then hχB (x) = [0, 1] or hχB (y) = [0, 1], so hχB (x) ∪hχB (y) = [0, 1]. Hence, hχB (x·y) = [0, 1] ⊆ [0, 1] = hχB (x)∪hχB (y). Also, kχB (x) = ∅ or kχB (y) = ∅, so kχB (x)∩kχB (y) = ∅. Hence, kχB (x ·y) = ∅⊇∅ = kχB (x) ∩kχB (y). Case 2 : If x · y /∈ B, then hχB (x · y) = ∅ ⊆ hχB (x) ∪ hχB (y). Also, kχB (x · y) = [0, 1] ⊇ kχB (x) ∩kχB (y). Hence, χB is a weakly prime intuitionistic hesitant fuzzy set on X. Conversely, assume that hχB is a weakly prime intuitionistic hesitant fuzzy set on X. Let x,y ∈ X be such that x ·y ∈ B and x 6= y. Then hχB (x ·y) = [0, 1], so [0, 1] = hχB (x ·y) ⊆ hχB (x) ∪hχB (y). Thus hχB (x) ∪hχB (y) = [0, 1], so hχB (x) = [0, 1] or hχB (y) = [0, 1]. Hence, x ∈ B or y ∈ B and so B is a weakly prime subset of X. � Int. J. Anal. Appl. (2023), 21:27 9 Theorem 3.7. A nonempty subset F of X is a weakly prime BCC-filter of X if and only if the characteristic intuitionistic hesitant fuzzy set χF is a weakly prime intuitionistic hesitant fuzzy BCC- filter of X. Proof. It is straightforward by Theorems 3.3 and 3.6. � Theorem 3.8. An intuitionistic hesitant fuzzy set H = (h,k) is an intuitionistic hesitant fuzzy BCC- filter of X if and only if the hesitant fuzzy sets h and k are hesitant fuzzy BCC-filters of X. Proof. Assume that H = (h,k) is an intuitionistic fuzzy BCC-filter of X. Then for any x,y ∈ X, we have h(0) ⊇ h(x) and h(y) ⊇ h(x · y) ∩ h(x). Hence, h is a hesitant fuzzy BCC-filter of X. Now for any x,y ∈ X, we have k(0) ⊆ k(x) and k(y) ⊆ k(x · y) ∪ k(x). Then k(0) = [0, 1] − k(0) ⊇ [0, 1] −k(x) = k(x) and k(y) = [0, 1] −k(y) ⊇ [0, 1] − (k(x ·y) ∪k(x)) = [0, 1] −k(x ·y) ∩ [0, 1] −k(x) = k(x ·y) ∩k(x). Hence, k is a hesitant fuzzy BCC-filter of X. Conversely, assume that the hesitant fuzzy sets h and k are hesitant fuzzy BCC-filters of X. Then for any x,y ∈ X, we have h(0) ⊇ h(x) and h(y) ⊇ h(x · y) ∩h(x). Now for any x,y ∈ X, we have k(0) ⊇ k(x) and k(y) ⊇ k(x ·y)∩k(x). Then [0, 1]−k(0) ⊇ [0, 1]−k(x) and so k(0) ⊆ k(x). Now, [0, 1] −k(y) ⊇ [0, 1] −k(x ·y) ∩ [0, 1] −k(x) = [0, 1] − (k(x ·y) ∪k(x)), k(y) ⊆ k(x ·y) ∪k(x). Hence, H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X. � Theorem 3.9. An intuitionistic hesitant fuzzy set H = (h,k) is an intuitionistic hesitant fuzzy BCC- filter of X if and only if the intuitionistic hesitant fuzzy set H = (k,h) is an intuitionistic hesitant fuzzy BCC-filter of X. Proof. Assume that H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X. Then for any x,y,z ∈ X, h(0) ⊇ h(x) and h(y) ⊇ h(x ·y)∩h(x). Hence, for any x,y,z ∈ X, h(0) = [0, 1]−h(0) ⊆ [0, 1] −h(x) = h(x) and h(y) = [0, 1] −h(y) ⊆ [0, 1] − (h(x ·y) ∩h(x)) = [0, 1] −h(x ·y) ∪ [0, 1] −h(x) = h(x ·y) ∪h(x). 10 Int. J. Anal. Appl. (2023), 21:27 Now, for any x,y,z ∈ X, k(0) ⊆ k(x) and k(y) ⊆ k(x · y) ∪ k(x). Hence, for any x,y,z ∈ X, k(0) = [0, 1] −k(0) ⊇ [0, 1] −k(x) = k(x) and k(y) = [0, 1] −k(y) ⊇ [0, 1] − (k(x ·y) ∪k(x)) = [0, 1] −k(x ·y) ∩ [0, 1] −k(x) = k(x ·y) ∩k(x). Hence, H = (k,h) is an intuitionistic hesitant fuzzy BCC-filter of X. Conversely, assume that the intuitionistic hesitant fuzzy set H = (k,h) is an intuitionistic hesitant fuzzy BCC-filter of X. Then for any x,y,z ∈ X, k(0) ⊇ k(x) and k(y) ⊇ k(x · y) ∩ k(x). Then [0, 1] − k(0) ⊇ [0, 1] − k(x) and [0, 1] − k(y) ⊇ [0, 1] − (k(x · y) ∪ k(x)), so k(0) ⊆ k(x) and k(y) ⊆ k(x · y) ∪k(x). Now, for any x,y,z ∈ X, we have h(0) ⊆ h(x) and h(y) ⊆ h(x · y) ∪h(x). Then [0, 1] −h(0) ⊆ [0, 1] −h(x) and [0, 1] −h(y) ⊇ [0, 1] − (h(x · y) ∪h(x)), so h(0) ⊇ h(x) and h(y) ⊇ h(x ·y) ∩h(x). Hence, H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X. � Definition 3.11. Let H = (h,k) be an intuitionistic hesitant fuzzy set on X. The intuitionistic hesitant fuzzy sets ⊕H and ⊗H are defined as ⊕H = (h,h) and ⊗H = (k,k). Theorem 3.10. If H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X, then the sets Xh = {x ∈ X | h(x) = h(0)} and Xk = {x ∈ X | k(x) = k(0)} are BCC-filters of X. Proof. Clearly, 0 ∈ Xh∩Xk. Let x,y ∈ X be such that x ·y,x ∈ Xh. Then h(x ·y) = h(0) and h(x) = h(0). Since H is an intuitionistic hesitant fuzzy BCC-filter of X, by (3.3), h(y) ⊇ h(x·y)∩h(x) = h(0), whence h(y) = h(0), by (3.2). This means that y ∈ Xh. Hence, Xh is a BCC-filter of X. Let x,y ∈ X be such that x ·y,x ∈ Xk. Then k(x ·y) = k(0) and k(x) = k(0). Since H is an intuitionistic hesitant fuzzy BCC-filter of X, by (3.3), k(y) ⊆ k(x ·y) ∪k(x) = k(0), whence k(y) = k(0), by (3.2). This means that y ∈ Xk. Hence, Xk is a BCC-filter of X. � Theorem 3.11. An intuitionistic hesitant fuzzy set H = (h,k) is an intuitionistic hesitant fuzzy BCC- filter of X if and only if the intuitionistic hesitant fuzzy sets ⊕H and ⊗H are intuitionistic hesitant fuzzy BCC-filters of X. Proof. Assume that H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X. Let x ∈ X. Then h(0) = [0, 1] −h(0) ⊆ [0, 1] −h(x) = h(x). Let x,y ∈ X. Then h(y) = [0, 1] −h(y) ⊆ [0, 1] − (h(x ·y) ∩h(x)) = ([0, 1] −h(x ·y)) ∪ ([0, 1] −h(x)) = h(x ·y) ∪h(x). Hence, ⊕H is an intuitionistic hesitant fuzzy BCC-filter of X. Int. J. Anal. Appl. (2023), 21:27 11 Let x ∈ X. Then k(0) = [0, 1] −k(0) ⊇ [0, 1] −k(x) = k(x). Let x,y ∈ X. Then k(y) = [0, 1] −k(y) ⊇ [0, 1] − (k(x ·y) ∪k(x)) = ([0, 1] −k(x ·y)) ∩ ([0, 1] −k(x)) = k(x ·y) ∩k(x). Hence, ⊗H is an intuitionistic hesitant fuzzy BCC-filter of X. Conversely, assume that ⊕H and ⊗H are intuitionistic hesitant fuzzy BCC-filters of X. Then for any x,y ∈ X, we have h(0) ⊇ h(x) and h(y) ⊇ h(x ·y)∩h(x) and k(0) ⊆ k(x) and k(y) ⊆ k(x ·y)∪k(x). Hence, H is an intuitionistic hesitant fuzzy BCC-filter of X. � Lemma 3.2. If H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X, then (∀x,y,z ∈ X) ( z ≤ x ·y ⇒ { h(y) ⊇ h(x) ∩h(z) k(y) ⊆ k(x) ∪k(z) ) . (3.7) Proof. Let x,y,z ∈ X be such that z ≤ x ·y. Then z · (x ·y) = 0 and so h(y) ⊇ h(x ·y) ∩h(x) ⊇ h(z · (x ·y)) ∩h(z) ∩h(x) = h(0) ∩h(z) ∩h(x) = h(x) ∩h(z), k(y) ⊆ k(x ·y) ∪k(x) ⊆ k(z · (x ·y)) ∪k(z) ∪k(x) = k(0) ∪k(z) ∪k(x) = k(x) ∪k(z). � Lemma 3.3. If H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X, then (∀x,y ∈ X) ( x ≤ y ⇒ { h(y) ⊇ h(x) k(y) ⊆ k(x) ) . (3.8) Proof. Let x,y ∈ X be such that x ≤ y. Then x ·y = 0 and so h(y) ⊇ h(x ·y) ∩h(x) = h(0) ∩h(x) = h(x), k(y) ⊆ k(x ·y) ∪k(x) = k(0) ∪k(x) = k(x). � Lemma 3.4. If H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of X, then (∀x,y,z ∈ X) ( h(y ·z) ∩h(x ·y) ⊆ h(x ·z) k(y ·z) ∪k(x ·y) ⊇ k(x ·z) ) . (3.9) 12 Int. J. Anal. Appl. (2023), 21:27 Proof. Let x,y,z ∈ X. By (2.1), we have (y ·z) ≤ (x ·y) · (x ·z). Then it follows from Lemma 3.2 that h(y ·z) ∩h(x ·y) ⊆ h(x ·z), k(y ·z) ∪k(x ·y) ⊇ k(x ·z). � Definition 3.12. Let h : X →P([0, 1]). For any π ∈P([0, 1]), the sets U(h,π) = {x ∈ X | h(x) ⊇ π} and U+(h,π) = {x ∈ X | h(x) ⊃ π} are called an upper π-level subset and an upper π-strong level subset of h, respectively. The sets L(h,π) = {x ∈ X | h(x) ⊆ π} and L−(h,π) = {x ∈ X | h(x) ⊂ π} are called a lower π-level subset and a lower π-strong level subset of h, respectively. The set E(h,π) = {x ∈ X | h(x) = π} is called an equal π-level subset of h. Then U(h,π) = U+(h,π) ∪E(h,π) and L(h,π) = L−(h,π) ∪E(h,π). Theorem 3.12. An intuitionistic hesitant fuzzy set H = (h,k) on X is an intuitionistic hesitant fuzzy BCC-filter of X if and only if for all π ∈ P([0, 1]), the nonempty subsets U(h,π) and L(k,π) of X are BCC-filters. Proof. Assume that H is an intuitionistic hesitant fuzzy BCC-filter of X. Let π ∈ P([0, 1]) be such that U(h,π) 6= ∅ and let x ∈ U(h,π). Then h(x) ⊇ π. Since H is an intuitionistic hesitant fuzzy BCC-filter of X, we have h(0) ⊇ h(x) ⊇ π. Thus 0 ∈ U(h,π). Next, let x,y ∈ X be such that x,x · y ∈ U(h,π). Then h(x) ⊇ π and h(x · y) ⊇ π. Since H is an intuitionistic hesitant fuzzy BCC-filter of X, we have h(y) ⊇ h(x ·y)∩h(x) ⊇ π. So y ∈ U(h,π). Let π ∈P([0, 1]) be such that L(k,π) 6= ∅ and let x ∈ L(k,π). Then k(x) ⊆ π. Since H is an intuitionistic hesitant fuzzy BCC-filter of X, we have k(0) ⊆ k(x) ⊆ π. Thus 0 ∈ L(k,π). Next, let x,y ∈ X be such that x,x ·y ∈ L(k,π). Then k(x) ⊆ π and k(x ·y) ⊆ π. Since H is an intuitionistic hesitant fuzzy BCC-filter of X, we have k(y) ⊆ k(x ·y) ∪k(x) ⊆ π. So y ∈ L(k,π). Hence, U(h,π) and L(k,π) are BCC-filters of X. Conversely, assume that for all π ∈P([0, 1]), the nonempty subsets U(h,π) and L(k,π) of X are BCC-filters. Let x ∈ X. Then h(x) ∈ P([0, 1]). Choose π = h(x) ∈ P([0, 1]). Then h(x) ⊇ π. Thus x ∈ U(h,π). By assumption, we have U(h,π) is a BCC-filter of X and thus 0 ∈ U(h,π). So h(0) ⊇ π = h(x). Let x,y ∈ X. Then h(x),h(x·y) ∈P([0, 1]). Choose π = h(x)∩h(x·y) ∈P([0, 1]). Then h(x) ⊇ π and h(x · y) ⊇ π. Since x,x · y ∈ U(h,π) 6= ∅. By assumption, we have U(h,π) is a BCC-filter of X and then y ∈ U(h,π). Thus h(y) ⊇ π = h(x) ∩ h(x · y). Let x ∈ X. Then k(x) ∈ P([0, 1]). Choose π1 = k(x) ∈ P([0, 1]). Then k(x) ⊆ π1. Thus x ∈ L(k,π1). By assumption, we have L(k,π1) is a BCC-filter of X and thus 0 ∈ L(k,π1). So k(0) ⊆ π1 = k(x). Let x,y ∈ X. Then k(x),k(x ·y) ∈P([0, 1]). Choose π1 = k(x) ∪k(x ·y) ∈P([0, 1]). Then k(x) ⊆ π1 and k(x ·y) ⊆ π1. Since x,x ·y ∈ L(k,π1) 6= ∅. By assumption, we have L(k,π1) is a BCC-filter of X and then y ∈ L(k,π1). Thus k(y) ⊆ π1 = k(x) ∪k(x · y). Hence, H is an intuitionistic hesitant fuzzy BCC-filter of X. � Int. J. Anal. Appl. (2023), 21:27 13 The following theorem can be proved similarly to Theorem 3.12. Theorem 3.13. An intuitionistic hesitant fuzzy set H = (h,k) on X is an intuitionistic hesitant fuzzy BCC-subalgebra (BCC-ideal) of X if and only if for all π ∈ P([0, 1]), the nonempty subsets U(h,π) and L(k,π) of X are BCC-subalgebras (BCC-ideals). Definition 3.13. Let {Hα | α ∈ ∆} be a family of intuitionistic hesitant fuzzy sets on a reference set X. We define the intuitionistic hesitant fuzzy set ⋂ α∈∆ Hα = ( ⋂ α∈∆ hα, ⋃ α∈∆ kα) by ( ⋂ α∈∆ hα)(x) = ⋂ α∈∆ hα(x) and ( ⋃ α∈∆ kα)(x) = ⋃ α∈∆ kα(x) for all x ∈ X, which is called the intuitionistic hesitant intersection of intuitionistic hesitant fuzzy sets. Proposition 3.1. If {Hα | α ∈ ∆} is a family of intuitionistic hesitant fuzzy BCC-filters of X, then⋂ α∈∆ Hα is an intuitionistic hesitant fuzzy BCC-filter of X. Proof. Let {Hα | α ∈ ∆} be a family of intuitionistic hesitant fuzzy BCC-filter of X. Let x ∈ X. Then ( ⋂ α∈∆ hα)(0) = ⋂ α∈∆ hα(0) ⊇ ⋂ α∈∆ hα(x) = ( ⋂ α∈∆ hα)(x), ( ⋃ α∈∆ kα)(0) = ⋃ α∈∆ kα(0) ⊆ ⋃ α∈∆ kα(x) = ( ⋃ α∈∆ kα)(x). Let x,y ∈ X. Then ( ⋂ α∈∆ hα)(y) = ⋂ α∈∆ hα(y) ⊇ ⋂ α∈∆ (hα(x ·y) ∩hα(x)) = ( ⋂ α∈∆ hα(x ·y)) ∩ ( ⋂ α∈∆ hα(x)) = ( ⋂ α∈∆ hα)(x ·y) ∩ ( ⋂ α∈∆ hα)(x), ( ⋃ α∈∆ kα)(y) = ⋃ α∈∆ kα(y) ⊆ ⋃ α∈∆ (kα(x ·y) ∪kα(x)) = ⋃ α∈∆ kα(x ·y) ∪ ⋃ α∈∆ kα(x) = ( ⋃ α∈∆ kα)(x ·y) ∪ ( ⋃ α∈∆ kα)(x). Hence, ⋂ α∈∆ Hα is an intuitionistic hesitant fuzzy BCC-filter of X. � Definition 3.14. Let A = (hA,kA) and B = (hB,kB) be intuitionistic hesitant fuzzy sets on sets X and Y , respectively. The Cartesian product A×B = (h,k) defined by h(x,y) = hA(x) ∩hB(y) and k(x,y) = kA(x) ∪kB(y), where h : X ×Y → P([0, 1]) and k : X ×Y → P([0, 1]) for all x ∈ X and y ∈ Y . 14 Int. J. Anal. Appl. (2023), 21:27 Remark 3.1. Let (X, ·, 0X) and (Y,?, 0Y ) be BCC-algebras. Then (X × Y,�, (0X, 0Y )) is a BCC- algebra defined by (x,y) � (u,v) = (x ·u,y ? v) for every x,u ∈ X and y,v ∈ Y . Proposition 3.2. If A = (hA,kA) and B = (hB,kB) are two intuitionistic hesitant fuzzy BCC-filters of BCC-algebras X and Y , respectively, then the Cartesian product A×B is also an intuitionistic hesitant fuzzy BCC-filter of X ×Y . Proof. Let (x,y) ∈ X ×Y . Then h(0X, 0Y ) = hA(0X) ∩hB(0Y ) ⊇ hA(x) ∩hB(y) = h(x,y), k(0X, 0Y ) = kA(0X) ∪kB(0Y ) ⊆ kA(x) ∪kB(y) = k(x,y). Let (x1,x2), (y1,y2) ∈ X ×Y . Then h(y1,y2) = hA(y1) ∩hB(y2) ⊇ (hA(x1 ·y1) ∩hA(x1)) ∩ (hB(x2 ? y2) ∩hB(x2)) = hA(x1 ·y1) ∩hB(x2 ? y2) ∩hA(x1) ∩hB(x2) = h(x1 ·y1,x2 ? y2) ∩h(x1,x2) = h((x1,x2) � (y1,y2)) ∩h(x1,x2), k(y1,y2) = kA(y1) ∪kB(y2) ⊆ (kA(x1 ·y1) ∪kA(x1)) ∪ (kB(x2 ? y2) ∪kB(x2)) = kA(x1 ·y1) ∪kB(x2 ? y2) ∪kA(x1) ∪kB(x2) = k(x1 ·y1,x2 ? y2) ∪k(x1,x2) = k((x1,x2) � (y1,y2)) ∪k(x1,x2). Hence, A×B is an intuitionistic hesitant fuzzy BCC-filter of X ×Y . � Theorem 3.14. Two intuitionistic hesitant fuzzy sets A = (hA,kA) and B = (hB,kB) are intuitionistic hesitant fuzzy BCC-filters of BCC-algebras X and Y , respectively if and only if the intuitionistic hesitant fuzzy sets ⊕(A×B) and ⊗(A×B) are intuitionistic hesitant fuzzy BCC-filters of X ×Y . Proof. It follows from Proposition 3.2 and Theorem 3.11. � A mapping f : (X, ·, 0X) → (Y,?, 0Y ) of BCC-algebras is called a homomorphism if f (x · y) = f (x) ? f (y) for all x,y ∈ X. Note that if f : X → Y is a homomorphism of BCC-algebras, then f (0X) = 0Y . Int. J. Anal. Appl. (2023), 21:27 15 Definition 3.15. Let f be a function from a nonempty set X to a nonempty set Y . If H = (h,k) is an intuitionistic hesitant fuzzy set on Y , then the intuitionistic hesitant fuzzy set f−1(H) = (h◦ f ,k ◦ f ) in X is called the pre-image of H under f . Theorem 3.15. Let f : (X, ·, 0X) → (Y,?, 0Y ) be a homomorphism of BCC-algebras. If H = (h,k) is an intuitionistic hesitant fuzzy BCC-filter of Y , then f−1(H) = (h◦f ,k◦f ) is an intuitionistic hesitant fuzzy BCC-filter of X. Proof. By assumption, h(f (0X)) = h(0Y ) ⊇ h(y) for every y ∈ Y . In particular, (h ◦ f )(0X) = h(f (0X)) ⊇ h(f (x)) = (h ◦ f )(x) for all x ∈ X. Also, k(f (0X)) = k(0Y ) ⊆ k(y) for every y ∈ Y . In particular, (k ◦ f )(0X) = k(f (0X)) ⊆ k(f (x)) = (k ◦ f )(x) for all x ∈ X. Let x,y ∈ X. Then (h◦ f )(y) = h(f (y)) ⊇ h(f (x) ? f (y)) ∩h(f (x)) = h(f (x ·y)) ∩h(f (x)) = (h◦ f )(x ·y) ∩ (h◦ f )(x), (k ◦ f )(y) = k(f (y)) ⊆ k(f (x) ? f (y)) ∪k(f (x)) = k(f (x ·y)) ∪k(f (x)) = (k ◦ f )(x ·y) ∪ (k ◦ f )(x). Hence, f−1(H) is an intuitionistic hesitant fuzzy BCC-filter of X. � 4. Conclusion In the present paper, we have introduced the concepts of intuitionistic hesitant fuzzy BCC- subalgebras, BCC-ideals, and BCC-filters of BCC-algebras. The relationship between intuitionistic hesitant fuzzy BCC-subalgebras (BCC-ideals, BCC-filters) and their level subsets is described. More- over, the homomorphic pre-images of intuitionistic hesitant fuzzy BCC-filters in BCC-algebras are also studied and some related properties are investigated. Acknowledgment: This research project was supported by the Thailand Science Research and Inno- vation Fund and the University of Phayao (Grant No. FF66-UoE017). Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] B. Ahmad, A. Kharal, On Fuzzy Soft Sets, Adv. Fuzzy Syst. 2009 (2009), 586507. https://doi.org/10.1155/ 2009/586507. [2] M. Atef, M.I. Ali, T.M. Al-shami, Fuzzy Soft Covering-Based Multi-Granulation Fuzzy Rough Sets and Their Applications, Comput. Appl. 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