Anti-type of hesitant fuzzy sets on UP-algebras EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 4, 2018, 976-1002 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Anti-type of hesitant fuzzy sets on UP-algebras∗ Phakawat Mosrijai1, Aiyared Iampan1,† 1 Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand Abstract. This paper aims to introduce the notions of anti-hesitant fuzzy UP-subalgebras of UP-algebras, anti-hesitant fuzzy UP-filters, anti-hesitant fuzzy UP-ideals, and anti-hesitant fuzzy strongly UP-ideals, and prove some results. Furthermore, we discuss the relationships between anti-hesitant fuzzy UP-subalgebras (resp., anti-hesitant fuzzy UP-filters, anti-hesitant fuzzy UP- ideals, anti-hesitant fuzzy strongly UP-ideals) and some level subsets of hesitant fuzzy sets on UP-algebras. 2010 Mathematics Subject Classifications: 03G25 Key Words and Phrases: UP-algebra, anti-hesitant fuzzy strongly UP-ideal, anti-hesitant fuzzy UP-ideal, anti-hesitant fuzzy UP-filter and anti-hesitant fuzzy UP-subalgebra 1. Introduction The branch of the logical algebra, UP-algebras was introduced by Iampan [2] in 2017, and it is known that the class of KU-algebras [8] is a proper subclass of the class of UP-algebras. It have been examined by several researchers, for example, Somjanta et al. [14] introduced the notion of fuzzy sets in UP-algebras, the notion of intuitionistic fuzzy sets in UP-algebras was introduced by Kesorn et al. [5], Kaijae et al. [4] introduced the notions of anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras of UP-algebras, the notion of Q-fuzzy sets in UP-algebras was introduced by Tanamoon et al. [17], Sripaeng et al. [16] introduced the notion anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras, the notion of N -fuzzy sets in UP-algebras was introduced by Songsaeng and Iampan [15], Senapati et al. [12, 13] applied cubic set and interval-valued intuitionistic fuzzy structure in UP-algebras, Romano [9] introduced the notion of proper UP-filters in UP-algebras, etc. A hesitant fuzzy set on a set is a function from a reference set to a power set of the unit interval. The notion of a hesitant fuzzy set on a set was first considered by Torra [18] in 2010. The hesitant fuzzy set, which can be perfectly described in terms of the opinions ∗This work was financially supported by the University of Phayao. †Corresponding Author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3335 Email addresses: phakawat.mo@gmail.com (P. Mosrijai), aiyared.ia@up.ac.th (A. Iampan) http://www.ejpam.com 976 c© 2018 EJPAM All rights reserved. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 977 of decision makers is a very useful tool to deal with uncertainty. The hesitant fuzzy set theories developed by Torra and others have found many applications in the domain of mathematics and elsewhere. In UP-algebras, Mosrijai et al. [6] extended the notion of fuzzy sets in UP-algebras to hesitant fuzzy sets on UP-algebras, and Satirad et al. [11] considered level subsets of a hesitant fuzzy set on UP-algebras in 2017. The notion of partial constant hesitant fuzzy sets on UP-algebras was introduced by Mosrijai et al. [7] afterwards. In this paper, the notion of anti-hesitant fuzzy UP-subalgebras (resp., anti-hesitant fuzzy UP-filters, anti-hesitant fuzzy UP-ideals and anti-hesitant fuzzy strongly UP-ideals) of UP-algebras are introduced and proved some results. Further, we discuss the relation between anti-hesitant fuzzy UP-subalgebras (resp., anti-hesitant fuzzy UP-filters, anti- hesitant fuzzy UP-ideals and anti-hesitant fuzzy strongly UP-ideals) and level subsets of a hesitant fuzzy set. 2. Basic Results on UP-Algebras Before we begin our study, we will introduce the definition of a UP-algebra. Definition 1. [2] An algebra A = (A, ·, 0) of type (2, 0) is called a UP-algebra where A is a nonempty set, · is a binary operation on A, and 0 is a fixed element of A (i.e., a nullary operation) if it satisfies the following axioms: for any x, y, z ∈ A, (UP-1) (y · z) · ((x · y) · (x · z)) = 0, (UP-2) 0 · x = x, (UP-3) x · 0 = 0, and (UP-4) x · y = 0 and y · x = 0 imply x = y. From [2], we know that the notion of UP-algebras is a generalization of KU-algebras. Example 1. [10] Let X be a universal set and let Ω ∈ P(X). Let PΩ(X) = {A ∈ P(X) | Ω ⊆ A}. Define a binary operation · on PΩ(X) by putting A · B = B ∩ (A′ ∪ Ω) for all A,B ∈ PΩ(X). Then (PΩ(X), ·,Ω) is a UP-algebra and we shall call it the generalized power UP-algebra of type 1 with respect to Ω. Example 2. [10] Let X be a universal set and let Ω ∈ P(X). Let PΩ(X) = {A ∈ P(X) | A ⊆ Ω}. Define a binary operation ∗ on PΩ(X) by putting A ∗ B = B ∪ (A′ ∩ Ω) for all A,B ∈ PΩ(X). Then (PΩ(X), ∗,Ω) is a UP-algebra and we shall call it the generalized power UP-algebra of type 2 with respect to Ω. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 978 Example 3. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 4 0 0 1 2 3 4 1 0 0 1 3 3 2 0 0 0 3 3 3 0 0 0 0 3 4 0 0 0 0 0 Then (A, ·, 0) is a UP-algebra which is not a KU-algebra because (0 · 2)((2 · 4) · (0 · 4)) = 2 · (3 · 4) = 2 · 3 = 3 6= 0 (see the definition in [8]). In what follows, let A denote UP-algebras unless otherwise specified. The following proposition is very important for the study of UP-algebras. Proposition 1. [2, 3] In a UP-algebra A = (A, ·, 0), the following properties hold: (1) (∀x ∈ A)(x · x = 0), (2) (∀x, y, z ∈ A)(x · y = 0, y · z = 0⇒ x · z = 0), (3) (∀x, y, z ∈ A)(x · y = 0⇒ (z · x) · (z · y) = 0), (4) (∀x, y, z ∈ A)(x · y = 0⇒ (y · z) · (x · z) = 0), (5) (∀x, y ∈ A)(x · (y · x) = 0), (6) (∀x, y ∈ A)((y · x) · x = 0⇔ x = y · x), (7) (∀x, y ∈ A)(x · (y · y) = 0), (8) (∀a, x, y, z ∈ A)((x · (y · z)) · (x · ((a · y) · (a · z))) = 0), (9) (∀a, x, y, z ∈ A)((((a · x) · (a · y)) · z) · ((x · y) · z) = 0), (10) (∀x, y, z ∈ A)(((x · y) · z) · (y · z) = 0), (11) (∀x, y, z ∈ A)(x · y = 0⇒ x · (z · y) = 0), (12) (∀x, y, z ∈ A)(((x · y) · z) · (x · (y · z)) = 0), and (13) (∀a, x, y, z ∈ A)(((x · y) · z) · (y · (a · z)) = 0). On a UP-algebra A = (A, ·, 0), we define a binary relation ≤ on A [2] as follows: for any x, y ∈ A, x ≤ y if and only if x · y = 0. Definition 2. [1, 2, 14] A nonempty subset S of a UP-algebra (A, ·, 0) is called (1) a UP-subalgebra of A if for any x, y ∈ S, x · y ∈ S. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 979 (2) a UP-filter of A if (i) the constant 0 of A is in S, and (ii) for any x, y ∈ A, x · y ∈ S and x ∈ S imply y ∈ S. (3) a UP-ideal of A if (i) the constant 0 of A is in S, and (ii) for any x, y, z ∈ A, x · (y · z) ∈ S and y ∈ S imply x · z ∈ S. (4) a strongly UP-ideal of A if (i) the constant 0 of A is in S, and (ii) for any x, y, z ∈ A, (z · y) · (z · x) ∈ S and y ∈ S imply x ∈ S. Guntasow et al. [1] proved the generalization that the notion of UP-subalgebras is a generalization of UP-filters, the notion of UP-filters is a generalization of UP-ideals, and the notion of UP-ideals is a generalization of strongly UP-ideals. Moreover, they also proved that a UP-algebra A is the only one strongly UP-ideal of itself. 3. Basic Results on Hesitant Fuzzy Sets Definition 3. [18] Let X be a reference set. A hesitant fuzzy set on X is defined in term of a function hH that when applied to X return a subset of [0, 1], that is, hH : X → P([0, 1]). A hesitant fuzzy set hH can also be viewed as the following mathematical representation: H := {(x, hH(x)) | x ∈ X} where hH(x) is a set of some values in [0, 1], denoting the possible membership degrees of the elements x ∈ X to the set H. We say that a hesitant fuzzy set H on X is a constant hesitant fuzzy set if its function hH is constant. Definition 4. [6] Let H be a hesitant fuzzy set on A. The hesitant fuzzy set H defined by hH(x) = [0, 1]− hH(x) for all x ∈ A is said to be the complement of H on A. Remark 1. [6] For all hesitant fuzzy set H on A, we have H = H. Theorem 1. A hesitant fuzzy set H is a constant hesitant fuzzy set on A if and only if the complement of H is a constant hesitant fuzzy set on A. Proof. Let H be a constant hesitant fuzzy set on A. Then hH(x) = hH(0) for all x ∈ A. Thus [0, 1]− hH(x) = [0, 1]− hH(0) for all x ∈ A. Therefore, hH(x) = hH(0) for all x ∈ A. Hence, H is a constant hesitant fuzzy set on A. Conversely, let H be a constant hesitant fuzzy set on A. Then hH(x) = hH(0) for all x ∈ A. Thus [0, 1]− hH(x) = [0, 1]− hH(0) for all x ∈ A. Therefore, hH(x) = hH(0) for all x ∈ A. Hence, H is a constant hesitant fuzzy set on A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 980 Definition 5. [6] A hesitant fuzzy set H on a A is called (1) a hesitant fuzzy UP-subalgebra of A if it satisfies the following property: for any x, y ∈ A, hH(x · y) ⊇ hH(x) ∩ hH(y). (2) a hesitant fuzzy UP-filter of A if it satisfies the following properties: for any x, y ∈ A, (1) hH(0) ⊇ hH(x), and (2) hH(y) ⊇ hH(x · y) ∩ hH(x). (3) a hesitant fuzzy UP-ideal of A if it satisfies the following properties: for any x, y, z ∈ A, (1) hH(0) ⊇ hH(x), and (2) hH(x · z) ⊇ hH(x · (y · z)) ∩ hH(y). (4) a hesitant fuzzy strongly UP-ideal of A if it satisfies the following properties: for any x, y, z ∈ A, (1) hH(0) ⊇ hH(x), and (2) hH(x) ⊇ hH((z · y) · (z · x)) ∩ hH(y). Mosrijai et al. [6] proved that the notion of hesitant fuzzy UP-subalgebras of UP- algebras is a generalization of hesitant fuzzy UP-filters, the notion of hesitant fuzzy UP- filters of UP-algebras is a generalization of hesitant fuzzy UP-ideals, and the notion of hesitant fuzzy UP-ideals of UP-algebras is a generalization of hesitant fuzzy strongly UP- ideals. Theorem 2. [6] A hesitant fuzzy set H on A is a hesitant fuzzy strongly UP-ideal of A if and only if it is a constant hesitant fuzzy set on A. 4. Anti-Type of Hesitant Fuzzy Sets In this section, we introduce the notions of anti-hesitant fuzzy UP-subalgebras, anti- hesitant fuzzy UP-filters, anti-hesitant fuzzy UP-ideals and anti-hesitant fuzzy strongly UP-ideals of UP-algebras, provide the necessary examples and prove its generalizations. Definition 6. A hesitant fuzzy set H on a A is called an anti-hesitant fuzzy UP-subalgebra of A if it satisfies the following property: for any x, y ∈ A, hH(x · y) ⊆ hH(x) ∪ hH(y). By Proposition 1 (1), we have hH(0) = hH(x ·x) ⊆ hH(x)∪hH(x) = hH(x) for all x ∈ A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 981 Example 4. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by 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 0 0 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = ∅, hH(1) = {0.5},hH(2) = {0.6}, and hH(3) = [0.5, 0.6]. Using this data, we can show that H is an anti-hesitant fuzzy UP-subalgebra of A. Definition 7. A hesitant fuzzy set H on a A is called an anti-hesitant fuzzy UP-filter of A if it satisfies the following properties: for any x, y ∈ A, (1) hH(0) ⊆ hH(x), and (2) hH(y) ⊆ hH(x · y) ∪ hH(x). Example 5. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 4 2 0 0 0 3 3 3 0 1 2 0 3 4 0 1 2 0 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = {0.8},hH(1) = [0.8, 0.9),hH(2) = [0.8, 0.9],hH(3) = [0.6, 0.9], and hH(4) = [0.6, 0.9]. Using this data, we can show that H is an anti-hesitant fuzzy UP-filter of A. Definition 8. A hesitant fuzzy set H on a A is called an anti-hesitant fuzzy UP-ideal of A if it satisfies the following properties: for any x, y, z ∈ A, (1) hH(0) ⊆ hH(x), and (2) hH(x · z) ⊆ hH(x · (y · z)) ∪ hH(y). Example 6. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 1 0 3 3 0 1 2 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 982 hH(0) = {1}, hH(1) = {1},hH(2) = {0, 1}, and hH(3) = [0, 1]. Using this data, we can show that H is an anti-hesitant fuzzy UP-ideal of A. Definition 9. A hesitant fuzzy set H on a A is called an anti-hesitant fuzzy strongly UP-ideal of A if it satisfies the following properties: for any x, y, z ∈ A, (1) hH(0) ⊆ hH(x), and (2) hH(x) ⊆ hH((z · y) · (z · x)) ∪ hH(y). Example 7. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 2 2 0 1 0 1 3 0 0 0 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = {0, 0.2}, hH(1) = {0, 0.2},hH(2) = {0, 0.2}, and hH(3) = {0, 0.2}. Using this data, we can show that H is an anti-hesitant fuzzy strongly UP-ideal of A. Theorem 3. A hesitant fuzzy set H on A is an anti-hesitant fuzzy strongly UP-ideal of A if and only if it is a constant hesitant fuzzy set on A. Proof. Assume that H is an anti-hesitant fuzzy strongly UP-ideal of A. Then hH(0) ⊆ hH(x) and hH(x) ⊆ hH((z · y) · (z ·x))∪ hH(y) for all x, y, z ∈ A. For any x ∈ A, we choose z = x and y = 0. Then hH(x) ⊆ hH((x · 0) · (x · x)) ∪ hH(0) = hH(0 · 0) ∪ hH(0) ((UP-3) and Proposition 1 (1)) = hH(0) ∪ hH(0) ((UP-2)) = hH(0) ⊆ hH(x), so hH(0) = hH(x). Hence, H is a constant hesitant fuzzy set on A. Conversely, assume that H is a constant hesitant fuzzy set on A. Then, for any x ∈ A,hH(0) = hH(x), so hH(0) ⊆ hH(x). For any x, y, z ∈ A,hH(x) = hH((z·y)·(z·x)) = hH(y), so hH(x) = hH((z · y) · (z · x)) ∪ hH(y). Thus hH(x) ⊆ hH((z · y) · (z · x)) ∪ hH(y). Hence, H is an anti-hesitant fuzzy strongly UP-ideal of A. Corollary 1. For UP-algebras, we can conclude that the notions of anti-hesitant fuzzy strongly UP-ideals and hesitant fuzzy strongly UP-ideals coincide. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 983 Proof. It is straightforward by Theorem 2 and 3. Corollary 2. A hesitant fuzzy set H on A is an anti-hesitant fuzzy strongly UP-ideal of A if and only if H on A is an anti-hesitant fuzzy strongly UP-ideal of A. Proof. It is straightforward by Theorem 1 and 3. By Using Corollary 1, we can show that a hesitant fuzzy set H on A is an anti-hesitant fuzzy strongly UP-ideal of A if and only if H on A is an anti-hesitant fuzzy strongly UP-ideal of A. Theorem 4. Every anti-hesitant fuzzy UP-filter of A is an anti-hesitant fuzzy UP-subalgebra of A. Proof. Assume that H is an anti-hesitant fuzzy UP-filter of A. Then for any x, y ∈ A, hH(x · y) ⊆ hH(y · (x · y)) ∪ hH(y) (Definition 7 (2)) = hH(0) ∪ hH(y) (Proposition 1 (5)) = hH(y) (Definition 7 (1)) ⊆ hH(x) ∪ hH(y). Hence, H is an anti-hesitant fuzzy UP-subalgebra of A. The converse of Theorem 4 is not true in general. By Example 4, we obtain H is an anti-hesitant fuzzy UP-subalgebra of A. Since hH(1) = {0.5} * {0.6} = ∅∪{0.6} = hH(0)∪ hH(2) = hH(2·1)∪hH(2), we have H is not an anti-hesitant fuzzy UP-filter of A. Therefore, the notion of anti-hesitant fuzzy UP-subalgebras of UP-algebras is generalization of anti- hesitant fuzzy UP-filters. Theorem 5. Every anti-hesitant fuzzy UP-ideal of A is an anti-hesitant fuzzy UP-filter of A. Proof. Assume that H is an anti-hesitant fuzzy UP-ideal of A. Then for any x, y ∈ A, hH(0) ⊆ hH(x) and hH(y) = hH(0 · y) ((UP-2)) ⊆ hH(0 · (x · y)) ∪ hH(x) (Definition 8 (2)) = hH(x · y) ∪ hH(x). ((UP-2)) Hence, H is an anti-hesitant fuzzy UP-filter of A. The converse of Theorem 5 is not true in general. By Example 5, we obtain H is an anti-hesitant fuzzy UP-filter of A. Since hH(3 · 4) = hH(3) = [0.6, 0.9] * [0.8, 0.9) = {0.8}∪ [0.8, 0.9) = hH(0)∪hH(2) = hH(3 · (2 ·4))∪hH(2), we have H is not an anti-hesitant fuzzy UP-ideal of A. Therefore, the notion of anti-hesitant fuzzy UP-filters of UP-algebras is generalization of anti-hesitant fuzzy UP-ideals. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 984 Theorem 6. Every anti-hesitant fuzzy strongly UP-ideal of A is an anti-hesitant fuzzy UP-ideal of A. Proof. Assume that H is an anti-hesitant fuzzy strongly UP-ideal of A. Then for any x, y ∈ A, hH(0) ⊆ hH(x) and hH(x · z) ⊆ hH((z · y) · (z · (x · z))) ∩ hH(y) (Definition 9 (2)) = hH((z · y) · 0) ∩ hH(y) (Proposition 1 (5)) = hH(0) ∩ hH(y) ((UP-3)) = hH(y) (Definition 9 (1)) = hH(x · (y · z)) ∩ hH(y). Hence, H is an anti-hesitant fuzzy UP-ideal of A. The converse of Theorem 6 is not true in general. By Theorem 3, we obtain an anti- hesitant fuzzy strongly UP-ideal is a constant hesitant fuzzy set. But anti-hesitant fuzzy UP-ideal is not a constant hesitant fuzzy set in general. Therefore, the notion of anti- hesitant fuzzy UP-ideals of UP-algebras is generalization of anti-hesitant fuzzy strongly UP-ideals. Proposition 2. Let H be an anti-hesitant fuzzy UP-filter (and also anti-hesitant fuzzy UP-ideal, anti-hesitant fuzzy strongly UP-ideal) of A. Then for any x, y ∈ A, x ≤ y implies hH(x) ⊇ hH(y) ⊇ hH(x · y). Proof. Let x, y ∈ A be such that x ≤ y. Then x · y = 0. Since H is an anti-hesitant fuzzy UP-filter (resp., anti-hesitant fuzzy UP-ideal, anti-hesitant fuzzy strongly UP-ideal) of A, we have hH(y) ⊆ hH(x · y) ∪ hH(x) = hH(0) ∪ hH(x) = hH(x). By Proposition 1 (5), we obtain y ≤ x · y and thus hH(y) ⊇ hH(x · y). 5. Level Subsets of a Hesitant Fuzzy Set Definition 10. [11] Let H be a hesitant fuzzy set on A. For any ε ∈ P([0, 1]), the sets U(H; ε) = {x ∈ A | hH(x) ⊇ ε} and U+(H; ε) = {x ∈ A | hH(x) ⊃ ε} are called an upper ε-level subset and an upper ε-strong level subset of H, respectively. The sets L(H; ε) = {x ∈ A | hH(x) ⊆ ε} and L−(H; ε) = {x ∈ A | hH(x) ⊂ ε} are called a lower ε-level subset and a lower ε-strong level subset of H, respectively. The set P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 985 E(H; ε) = {x ∈ A | hH(x) = ε} is called an equal ε-level subset of H. Then U(H; ε) = U+(H; ε) ∪ E(H; ε) and L(H; ε) = L−(H; ε) ∪ E(H; ε). Proposition 3. Let H be a hesitant fuzzy set on A and let ε ∈ P([0, 1]). Then the following statements hold : (1) U(H; ε) = L(H; [0, 1]− ε), (2) U+(H; ε) = L−(H; [0, 1]− ε), (3) L(H; ε) = U(H; [0, 1]− ε), and (4) L−(H; ε) = U+(H; [0, 1]− ε). Proof. (1) Let x ∈ A and let ε ∈ P([0, 1]). Then x ∈ U(H; ε) if and only if hH(x) ⊇ ε if and only if [0, 1] − hH(x) ⊆ [0, 1] − ε if and only if hH(x) ⊆ [0, 1] − ε if and only if x ∈ L(H; [0, 1]− ε). Therefore, U(H; ε) = L(H; [0, 1]− ε). (2) Let x ∈ A and let ε ∈ P([0, 1]). Then x ∈ U+(H; ε) if and only if hH(x) ⊃ ε if and only if [0, 1] − hH(x) ⊂ [0, 1] − ε if and only if hH(x) ⊂ [0, 1] − ε if and only if x ∈ L−(H; [0, 1]− ε). Therefore, U+(H; ε) = L−(H; [0, 1]− ε). (3) Let x ∈ A and let ε ∈ P([0, 1]). Then x ∈ L(H; ε) if and only if hH(x) ⊆ ε if and only if [0, 1] − hH(x) ⊇ [0, 1] − ε if and only if hH(x) ⊇ [0, 1] − ε if and only if x ∈ U(H; [0, 1]− ε). Therefore, L(H; ε) = U(H; [0, 1]− ε). (4) Let x ∈ A and let ε ∈ P([0, 1]). Then x ∈ L−(H; ε) if and only if hH(x) ⊂ ε if and only if [0, 1] − hH(x) ⊃ [0, 1] − ε if and only if hH(x) ⊃ [0, 1] − ε if and only if x ∈ U+(H; [0, 1]− ε). Therefore, L−(H; ε) = U+(H; [0, 1]− ε). Lemma 1. [11] Let H be a hesitant fuzzy set on A. Then the following statements hold: for any x, y ∈ A, (1) [0, 1]− (hH(x) ∪ hH(y)) = ([0, 1]− hH(x)) ∩ ([0, 1]− hH(y)), and (2) [0, 1]− (hH(x) ∩ hH(y)) = ([0, 1]− hH(x)) ∪ ([0, 1]− hH(y)). 5.1. Lower ε-Level Subsets Theorem 7. A hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-subalgebra of A if and only if for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a UP-subalgebra of A. Proof. Assume that H is an anti-hesitant fuzzy UP-subalgebra of A. Let ε ∈ P([0, 1]) be such that L(H; ε) 6= ∅, and let x, y ∈ A be such that x ∈ L(H; ε) and y ∈ L(H; ε). Then hH(x) ⊆ ε and hH(y) ⊆ ε. Since H is an anti-hesitant fuzzy UP-subalgebra of A, we have hH(x · y) ⊆ hH(x)∪ hH(y) ⊆ ε and thus x · y ∈ L(H; ε). Hence, L(H; ε) is a UP-subalgebra of A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 986 Conversely, assume that for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a UP- subalgebra of A. Let x, y ∈ A. Then hH(x),hH(y) ∈ P([0, 1]). Choose ε = hH(x)∪hH(y) ∈ P([0, 1]). Then hH(x) ⊆ ε and hH(y) ⊆ ε. Thus x, y ∈ L(H; ε) 6= ∅. By assumption, L(H; ε) is a UP-subalgebra of A and thus x · y ∈ L(H; ε). Therefore, hH(x · y) ⊆ ε = hH(x) ∪ hH(y). Hence, H is an anti-hesitant fuzzy UP-subalgebra of A. Theorem 8. A hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-filter of A if and only if for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a UP-filter of A. Proof. Assume that H is an anti-hesitant fuzzy UP-filter of A. Let ε ∈ P([0, 1]) be such that L(H; ε) 6= ∅ and let x ∈ A be such that x ∈ L(H; ε). Then hH(x) ⊆ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(0) ⊆ hH(x) ⊆ ε and thus 0 ∈ L(H; ε). Next, let x, y ∈ A be such that x · y ∈ L(H; ε) and x ∈ L(H; ε). Then hH(x · y) ⊆ ε and hH(x) ⊆ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(y) ⊆ hH(x · y) ∪ hH(x) ⊆ ε and thus y ∈ L(H; ε). Hence, L(H; ε) is a UP-filter of A. Conversely, assume that for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a UP-filter of A. Let x ∈ A. Then hH(x) ∈ P([0, 1]). Choose ε = hH(x) ∈ P([0, 1]). Then hH(x) ⊆ ε. Thus x ∈ L(H; ε). By assumption, we have L(H; ε) is a UP-filter of A and so 0 ∈ L(H; ε). Therefore, hH(0) ⊆ ε = hH(x). Next, let x, y ∈ A. Then hH(x · y), hH(x) ∈ P([0, 1]). Choose ε = hH(x · y) ∪ hH(x) ∈ P([0, 1]). Then hH(x·y) ⊆ ε and hH(x) ⊆ ε. Thus x·y, x ∈ L(H; ε) 6= ∅. By assumption, we have L(H; ε) is a UP-filter of A and so y ∈ L(H; ε). Therefore, hH(y) ⊆ ε = hH(x·y)∪hH(x). Hence, H is an anti-hesitant fuzzy UP-filter of A. Theorem 9. A hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-ideal of A if and only if for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a UP-ideal of A. Proof. Assume that H is an anti-hesitant fuzzy UP-ideal of A. Let ε ∈ P([0, 1]) be such that L(H; ε) 6= ∅ and let x ∈ A be such that x ∈ L(H; ε). Then hH(x) ⊆ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(0) ⊆ hH(x) ⊆ ε and thus 0 ∈ L(H; ε). Next, let x, y, z ∈ A be such that x · (y · z) ∈ L(H; ε) and y ∈ L(H; ε). Then hH(x · (y · z)) ⊆ ε and hH(y) ⊆ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(x · z) ⊆ hH(x · (y · z))∪hH(y) ⊆ ε and thus x · z ∈ L(H; ε). Hence, L(H; ε) is a UP-ideal of A. Conversely, assume that for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a UP-ideal of A. Let x ∈ A. Then hH(x) ∈ P([0, 1]). Choose ε = hH(x) ∈ P([0, 1]). Then hH(x) ⊆ ε. Thus x ∈ L(H; ε) 6= ∅. By assumption, we have L(H; ε) is a UP-ideal of A and so 0 ∈ L(H; ε). Therefore, hH(0) ⊆ ε = hH(x). Next, let x, y, z ∈ A. Then hH(x · (y · z)),hH(y) ∈ P([0, 1]). Choose ε = hH(x · (y · z))∪ hH(y) ∈ P([0, 1]). Then hH(x · (y · z)) ⊆ ε and hH(y) ⊆ ε. Thus x · (y · z), y ∈ L(H; ε) 6= ∅. By assumption, we have L(H; ε) is a UP-ideal of A and so x · z ∈ L(H; ε). Therefore, hH(x · z) ⊆ ε = hH(x · (y · z)) ∪ hH(y). Hence, H is an anti-hesitant fuzzy UP-ideal of A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 987 Theorem 10. Let H be a hesitant fuzzy set on A. Then the following statements are equivalent : (1) H is an anti-hesitant fuzzy strongly UP-ideal of A, (2) a nonempty subset L(H; ε) of A is a strongly UP-ideal of A for all ε ∈ P([0, 1]), and (3) a nonempty subset U(H; ε) of A is a strongly UP-ideal of A for all ε ∈ P([0, 1]). Proof. (1)⇒(2) Assume that H is an anti-hesitant fuzzy strongly UP-ideal of A. By Theorem 3, we obtain H is a constant hesitant fuzzy set on A and so hH(x) = hH(y) for all x, y ∈ A. Let ε ∈ P([0, 1]) be such that L(H; ε) 6= ∅. There exists a ∈ L(H; ε) be such that hH(a) ⊆ ε. Thus hH(x) = hH(a) ⊆ ε for all x ∈ A and so x ∈ L(H; ε) for all x ∈ A. Therefore, L(H; ε) = A. Hence, L(H; ε) is a strongly UP-ideal of A. (2)⇒(3) Assume that for all ε ∈ P([0, 1]), a nonempty subset L(H; ε) of A is a strongly UP-ideal of A. Let ε ∈ P([0, 1]) be such that U(H; ε) 6= ∅. If U(H; ε) 6= A, then there exist x ∈ U(H; ε) and y /∈ U(H; ε). So hH(x) ⊇ ε and hH(y) + ε. Consider, εy = hH(y) ∈ P([0, 1]). Then y ∈ L(H; εy) and εy + ε. By assumption, we have L(H; εy) is a strongly UP-ideal of A and so L(H; εy) = A. Thus hH(x) ⊆ εy. Since hH(x) ⊇ ε, we have εy ⊇ ε, a contradiction. Therefore, U(H; ε) = A. Hence, U(H; ε) is a strongly UP-ideal of A. (3)⇒(1) Assume that for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a strongly UP-ideal of A. Assume that H is not a constant hesitant fuzzy set on A. There exist x, y ∈ A be such that hH(x) 6= hH(y). Now, x ∈ U(H; hH(x)) 6= ∅ and y ∈ U(H; hH(y)) 6= ∅. By assumption, we have U(H; hH(x)) and U(H; hH(y)) are strongly UP-ideals of A and thus U(H; hH(x)) = A = U(H; hH(y)). Then x ∈ U(H; hH(y)) and y ∈ U(H; hH(x)). Thus hH(x) ⊇ hH(y) and hH(y) ⊇ hH(x). So hH(x) = hH(y), a contradiction. Therefore, H is a constant hesitant fuzzy set on A. By Theorem 3, we obtain H is an anti-hesitant fuzzy strongly UP-ideal of A. 5.2. Lower ε-Strong Level Subsets Theorem 11. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy UP-subalgebra of A, then for all ε ∈ P([0, 1]), L−(H; ε) is a UP-subalgebra of A if L−(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a UP-subalgebra of A, then H is an anti-hesitant fuzzy UP-subalgebra of A. Proof. (1) Assume that H is an anti-hesitant fuzzy UP-subalgebra of A. Let ε ∈ P([0, 1]) be such that L−(H; ε) 6= ∅, and let x, y ∈ A be such that x ∈ L−(H; ε) and y ∈ L−(H; ε). Then hH(x) ⊂ ε and hH(y) ⊂ ε. Since H is an anti-hesitant fuzzy UP- subalgebra of A, we have hH(x · y) ⊆ hH(x)∪ hH(y) ⊂ ε and thus x · y ∈ L−(H; ε). Hence, L−(H; ε) is a UP-subalgebra of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a UP-subalgebra of A. Assume that there exist x, y ∈ A such that hH(x · y) * P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 988 hH(x) ∪ hH(y). Since Im(H) is a chain, we have hH(x · y) ⊃ hH(x) ∪ hH(y). Choose ε = hH(x · y) ∈ P([0, 1]). Then hH(x) ⊂ ε and hH(y) ⊂ ε. Thus x, y ∈ L−(H; ε) 6= ∅. By assumption, we have L−(H; ε) is a UP-subalgebra of A and so x · y ∈ L−(H; ε). Thus hH(x · y) ⊂ ε = hH(x · y), a contradiction. Therefore, hH(x · y) ⊆ hH(x) ∪ hH(y) for all x, y ∈ A. Hence, H is an anti-hesitant fuzzy UP-subalgebra of A. Example 8. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 4 0 0 1 2 3 4 1 0 0 0 0 0 2 0 2 0 0 0 3 0 2 2 0 0 4 0 2 2 4 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = (0, 1),hH(1) = [0, 1), hH(2) = (0, 1], hH(3) = [0, 1), and hH(4) = [0, 1]. Then Im(H) is not a chain. If ε ⊆ (0, 1), then L−(H; ε) = ∅. If ε = [0, 1) or ε = (0, 1], then L−(H; ε) = {0}. If ε = [0, 1], then. L−(H; ε) = {0, 1, 2, 3}. Using this data, we can show that all nonempty subset L−(H; ε) of A is a UP-subalgebra of A. Since hH(3 ·1) = hH(2) = (0, 1] * [0, 1) = hH(3) ∪ hH(1), we have H is not an anti-hesitant fuzzy UP-subalgebra of A. Theorem 12. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy UP-filter of A, then for all ε ∈ P([0, 1]), L−(H; ε) is a UP-filter of A if L−(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a UP-filter of A, then H is an anti-hesitant fuzzy UP-filter of A. Proof. (1) Assume that H is an anti-hesitant fuzzy UP-filter of A. Let ε ∈ P([0, 1]) be such that L−(H; ε) 6= ∅ and let x ∈ A be such that x ∈ L−(H; ε). Then hH(x) ⊂ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(0) ⊆ hH(x) ⊂ ε and thus 0 ∈ L−(H; ε). Next, let x, y ∈ A be such that x · y ∈ L−(H; ε) and x ∈ L−(H; ε). Then hH(x · y) ⊂ ε and hH(x) ⊂ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(y) ⊆ hH(x · y) ∪ hH(x) ⊂ ε and thus y ∈ L−(H; ε). Hence, L−(H; ε) is a UP-filter of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a UP-filter of A. Assume that there exists x ∈ A such that hH(0) * hH(x). Since Im(H) is a chain, we have hH(0) ⊃ hH(x). Choose ε = hH(0) ∈ P([0, 1]). Then hH(x) ⊂ hH(0) = ε. Thus x ∈ L−(H; ε) 6= ∅. By assumption, we have L−(H; ε) is a UP- filter of A and so 0 ∈ L−(H; ε). Therefore, hH(0) ⊂ ε = hH(0), a contradiction. Hence, hH(0) ⊆ hH(x) for all x ∈ A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 989 Next, assume that there exist x, y ∈ A such that hH(y) * hH(x · y) ∪ hH(x). Since Im(H) is a chain, we have hH(y) ⊃ hH(x · y) ∪ hH(x). Choose ε = hH(y) ∈ P([0, 1]). Then hH(x · y) ⊂ ε and hH(x) ⊂ ε. Thus x · y, x ∈ L−(H; ε) 6= ∅. By assumption, we have L−(H; ε) is a UP-filter of A and so y ∈ L−(H; ε). Thus hH(y) ⊂ ε = hH(y), a contradiction. Therefore, hH(y) ⊆ hH(x · y) ∪ hH(x) for all x, y ∈ A. Hence, H is an anti-hesitant fuzzy UP-filter of A. Example 9. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 4 0 0 1 2 3 4 1 0 0 1 3 4 2 0 0 0 3 4 3 0 0 0 0 4 4 0 0 0 0 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = (0, 1),hH(1) = [0, 1), hH(2) = (0, 1], hH(3) = [0, 1], and hH(4) = [0, 1]. Then Im(H) is not a chain. If ε ⊆ (0, 1), then L−(H; ε) = ∅. If ε = [0, 1) or ε = (0, 1], then L−(H; ε) = {0}. If ε = [0, 1], then L−(H; ε) = {0, 1, 2}. Using this data, we can show that all nonempty subset L−(H; ε) of A is a UP-filter of A. Since hH(2) = (0, 1] * [0, 1) = hH(1) ∪ hH(1) = hH(1 · 2) ∪ hH(1), we have H is not an anti-hesitant fuzzy UP-filter of A. Theorem 13. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy UP-ideal of A, then for all ε ∈ P([0, 1]), L−(H; ε) is a UP-ideal of A if L−(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a UP-ideal of A, then H is an anti-hesitant fuzzy UP-ideal of A. Proof. (1) Assume that H is an anti-hesitant fuzzy UP-ideal of A. Let ε ∈ P([0, 1]) be such that L−(H; ε) 6= ∅ and let x ∈ A be such that x ∈ L−(H; ε). Then hH(x) ⊂ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(0) ⊆ hH(x) ⊂ ε and thus 0 ∈ L−(H; ε). Next, let x, y, z ∈ A be such that x · (y · z) ∈ L−(H; ε) and y ∈ L−(H; ε). Then hH(x · (y · z)) ⊂ ε and hH(y) ⊂ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(x · z) ⊆ hH(x · (y · z)) ∪ hH(y) ⊂ ε and thus x · z ∈ L−(H; ε). Hence, L−(H; ε) is a UP-ideal of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a UP-ideal of A. Assume that there exists x ∈ A such that hH(0) * hH(x). Since Im(H) is a chain, we have hH(0) ⊃ hH(x). Choose ε = hH(0) ∈ P([0, 1]). Then hH(x) ⊂ hH(0) = ε. Thus x ∈ L−(H; ε) 6= ∅. By assumption, we have L−(H; ε) is a UP- ideal of A and so 0 ∈ L−(H; ε). Therefore, hH(0) ⊂ ε = hH(0), a contradiction. Hence, hH(0) ⊆ hH(x) for all x ∈ A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 990 Next, assume that there exist x, y, z ∈ A such that hH(x·z) * hH(x·(y·z))∪hH(y). Since Im(H) is a chain, we have hH(x·z) ⊃ hH(x·(y ·z))∪hH(y). Choose ε = hH(x·z) ∈ P([0, 1]). Then hH(x · (y · z)) ⊂ ε and hH(y) ⊂ ε. Thus x · (y · z), y ∈ L−(H; ε) 6= ∅. By assumption, we have L−(H; ε) is a UP-ideal of A and so x ·z ∈ L−(H; ε). Thus hH(x ·z) ⊂ ε = hH(x ·z), a contradiction. Therefore, hH(x · z) ⊆ hH(x · (y · z)) ∪ hH(y) for all x, y, z ∈ A. Hence, H is an anti-hesitant fuzzy UP-ideal of A. Example 10. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 4 0 0 1 2 3 4 1 0 0 2 3 4 2 0 0 0 3 4 3 0 0 2 0 4 4 0 0 0 0 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = (0, 1),hH(1) = [0, 1), hH(2) = [0, 1], hH(3) = (0, 1], and hH(4) = [0, 1]. Then Im(H) is not a chain. If ε ⊆ (0, 1), then L−(H; ε) = ∅. If ε = [0, 1) or ε = (0, 1], then L−(H; ε) = {0}. If ε = [0, 1], then L−(H; ε) = {0, 1, 3}. Using this data, we can show that all nonempty subset L−(H; ε) of A is a UP-ideal of A. Since hH(0 · 1) = hH(1) = [0, 1) * (0, 1] = hH(0) ∪ hH(3) = hH(0 · (3 · 1)) ∪ hH(3), we have H is not an anti-hesitant fuzzy UP-ideal of A. Theorem 14. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy strongly UP-ideal of A, then for all ε ∈ P([0, 1]), L−(H; ε) is a strongly UP-ideal of A if L−(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a strongly UP-ideal of A, then H is an anti-hesitant fuzzy strongly UP-ideal of A. Proof. (1) Assume that H is an anti-hesitant fuzzy strongly UP-ideal of A. By Theorem 3, we obtain H is a constant hesitant fuzzy set on A and so hH(x) = hH(y) for all x, y ∈ A. Let ε ∈ P([0, 1]) be such that L−(H; ε) 6= ∅. There exists a ∈ L−(H; ε) be such that hH(a) ⊂ ε. Thus hH(x) = hH(a) ⊂ ε for all x ∈ A and so x ∈ L−(H; ε) for all x ∈ A. Therefore, L−(H; ε) = A. Hence, L−(H; ε) is a strongly UP-ideal of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset L−(H; ε) of A is a strongly UP-ideal of A. Assume that H is not a constant hesitant fuzzy set on A. There exist x, y ∈ A be such that hH(x) 6= hH(y). Since Im(H) is a chain, we have hH(x) ⊂ hH(y) or hH(x) ⊃ hH(y). Without loss of generality, assume that hH(x) ⊂ hH(y), then x ∈ L−(H; hH(y)) 6= ∅. By assumption, we have L−(H; hH(y)) is a strongly UP-ideal of A and so L−(H; hH(y)) = A. Thus y ∈ A = L−(H; hH(y)) and so hH(y) ⊂ hH(y), a contradiction. Therefore, H is a constant hesitant fuzzy set on A. By Theorem 3, we obtain H is an anti-hesitant fuzzy strongly UP-ideal of A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 991 Example 11. Let A = {0, 1} be a set with a binary operation · defined by the following Cayley table: · 0 1 0 0 1 1 0 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = (0, 1], and hH(1) = [0, 1). Then Im(H) is not a chain. If ε ⊆ [0, 1) or ε ⊆ (0, 1], then L−(H; ε) = ∅. If ε = [0, 1], then L−(H; ε) = A. Thus a nonempty subset L−(H; ε) of A is a strongly UP-ideal of A. By Theorem 3 and H is not a constant hesitant fuzzy set on A, we have H is not an anti-hesitant fuzzy strongly UP-ideal of A. 5.3. Upper ε-Level Subsets Theorem 15. A hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-subalgebra of A if and only if for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a UP-subalgebra of A. Proof. Assume that H is an anti-hesitant fuzzy UP-subalgebra of A. Let ε ∈ P([0, 1]) be such that U(H; ε) 6= ∅, and let x, y ∈ A be such that x ∈ U(H; ε) and y ∈ U(H; ε). Then hH(x) ⊇ ε and hH(y) ⊇ ε. Since H is an anti-hesitant fuzzy UP-subalgebra of A, we obtain hH(x · y) ⊆ hH(x) ∪ hH(y). By Lemma 1 (2), we have [0, 1] − hH(x · y) ⊆ ([0, 1]−hH(x))∪([0, 1]−hH(y)) = [0, 1]−(hH(x)∩hH(y)). Thus hH(x·y) ⊇ hH(x)∩hH(y) ⊇ ε. Therefore, x · y ∈ U(H; ε). Hence, U(H; ε) is a UP-subalgebra of A. Conversely, assume that for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a UP- subalgebra of A. Let x, y ∈ A. Choose ε = hH(x)∩hH(y) ∈ P([0, 1]). Then hH(x) ⊇ ε and hH(y) ⊇ ε. Thus x, y ∈ U(H; ε) 6= ∅. By assumption, we have U(H; ε) is a UP-subalgebra of A and so x · y ∈ U(H; ε). Therefore, hH(x · y) ⊇ ε = hH(x) ∩ hH(y). By Lemma 1 (2), we have hH(x · y) = [0, 1]− hH(x · y) ⊆ [0, 1]− (hH(x) ∩ hH(y)) = ([0, 1]− hH(x)) ∪ ([0, 1]− hH(y)) = hH(x) ∪ hH(y). Hence, H is an anti-hesitant fuzzy UP-subalgebra of A. Theorem 16. A hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-filter of A if and only if for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a UP-filter of A. Proof. Assume that H is an anti-hesitant fuzzy UP-filter of A. Let ε ∈ P([0, 1]) be such that U(H; ε) 6= ∅, and let x ∈ A be such that x ∈ U(H; ε). Then hH(x) ⊇ ε. Since P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 992 H is an anti-hesitant fuzzy UP-filter of A, we have hH(0) ⊆ hH(x). Thus [0, 1]− hH(0) ⊆ [0, 1]− hH(x). Therefore, hH(0) ⊇ hH(x) ⊇ ε. Hence, 0 ∈ U(hH; ε). Next, let x, y ∈ A be such that x ·y ∈ U(H; ε) and x ∈ U(H; ε). Then hH(x ·y) ⊇ ε and hH(x) ⊇ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(y) ⊆ hH(x · y) ∪ hH(x). By Lemma 1 (2), we have [0, 1] − hH(y) ⊆ ([0, 1] − hH(x · y)) ∪ ([0, 1] − hH(x)) = [0, 1] − (hH(x · y) ∩ hH(x)). Thus hH(y) ⊇ hH(x · y) ∩ hH(x) ⊇ ε. Therefore, y ∈ U(H; ε). Hence, U(H; ε) is a UP-filter of A. Conversely, assume that for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a UP-filter of A. Let x ∈ A. Choose ε = hH(x) ∈ P([0, 1]). Then hH(x) ⊇ ε. Thus x ∈ U(H; ε) 6= ∅. By assumption, we have U(H; ε) is a UP-filter of A and so 0 ∈ U(H; ε). Therefore, hH(0) ⊇ ε = hH(x). Hence, hH(0) = [0, 1]− hH(0) ⊆ [0, 1]− hH(x) = hH(x). Next, let x, y ∈ A. Choose ε = hH(x · y) ∩ hH(x) ∈ P([0, 1]). Then hH(x · y) ⊇ ε and hH(x) ⊇ ε. Thus x · y, x ∈ U(H; ε) 6= ∅. By assumption, we have U(H; ε) is a UP-filter of A and so y ∈ U(H; ε). Therefore, hH(y) ⊇ ε = hH(x · y) ∩ hH(x). By Lemma 1 (2), we have hH(y) = [0, 1]− hH(y) ⊆ [0, 1]− (hH(x · y) ∩ hH(x)) = ([0, 1]− hH(x · y)) ∪ ([0, 1]− hH(x)) = hH(x · y) ∪ hH(x). Hence, H is an anti-hesitant fuzzy UP-filter of A. Theorem 17. A hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-ideal of A if and only if for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a UP-ideal of A. Proof. Assume that H is an anti-hesitant fuzzy UP-ideal of A. Let ε ∈ P([0, 1]) be such that U(H; ε) 6= ∅, and let x ∈ A be such that x ∈ U(H; ε). Then hH(x) ⊇ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(0) ⊆ hH(x). Thus [0, 1]− hH(0) ⊆ [0, 1]− hH(x). Therefore, hH(0) ⊇ hH(x) ⊇ ε. Hence, 0 ∈ U(H; ε). Next, let x, y, z ∈ A be such that x · (y · z) ∈ U(H; ε) and y ∈ U(H; ε). Then hH(x · (y · z)) ⊇ ε and hH(y) ⊇ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we obtain hH(x · z) ⊆ hH(x · (y · z)) ∪ hH(y). By Lemma 1 (2), we have [0, 1] − hH(x · z) ⊆ ([0, 1] − hH(x · (y · z))) ∪ ([0, 1] − hH(y)) = [0, 1] − (hH(x · (y · z)) ∩ hH(y)). Thus hH(x · z) ⊇ hH(x · (y · z)) ∪ hH(y) ⊇ ε. Therefore, x · z ∈ U(H; ε). Hence, U(H; ε) is a UP-ideal of A. Conversely, assume that for all ε ∈ P([0, 1]), a nonempty subset U(H; ε) of A is a UP-ideal of A. Let x ∈ A. Choose ε = hH(x) ∈ P([0, 1]). Then hH(x) ⊇ ε. Thus x ∈ U(H; ε) 6= ∅. By assumption, we have U(H; ε) is a UP-ideal of A and so 0 ∈ U(H; ε). Therefore, hH(0) ⊇ ε = hH(x). Hence, hH(0) = [0, 1]− hH(0) ⊆ [0, 1]− hH(x) = hH(x). Next, let x, y, z ∈ A. Choose ε = hH(x·(y·z))∩hH(y) ∈ P([0, 1]). Then hH(x·(y·z)) ⊇ ε and hH(y) ⊇ ε. Thus x · (y · z), y ∈ U(H; ε) 6= ∅. By assumption, we have U(H; ε) is a UP-ideal of A and so x · z ∈ U(H; ε). Therefore, hH(x · z) ⊇ ε = hH(x · (y · z))∩ hH(y). By Lemma 1 (2), we have hH(x · z) = [0, 1]− hH(x · z) P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 993 ⊆ [0, 1]− (hH(x · (y · z)) ∩ hH(y)) = ([0, 1]− hH(x · (y · z))) ∪ ([0, 1]− hH(y)) = hH(x · (y · z)) ∪ hH(y). Hence, H is an anti-hesitant fuzzy UP-ideal of A. Theorem 18. Let H be a hesitant fuzzy set on A. Then the following statements are equivalent : (1) H is an anti-hesitant fuzzy strongly UP-ideal of A, (2) a nonempty subset U(H; ε) of A is a strongly UP-ideal of A for all ε ∈ P([0, 1]), and (3) a nonempty subset L(H; ε) of A is a strongly UP-ideal of A for all ε ∈ P([0, 1]). Proof. It is straightforward by Theorem 10 and Corollary 2. 5.4. Upper ε-Strong Level Subsets Theorem 19. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy UP-subalgebra of A, then for all ε ∈ P([0, 1]), U+(H; ε) is a UP-subalgebra of A if U+(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a UP-subalgebra of A, then H is an anti-hesitant fuzzy UP-subalgebra of A. Proof. (1) Assume that H is an anti-hesitant fuzzy UP-subalgebra of A. Let ε ∈ P([0, 1]) be such that U+(H; ε) 6= ∅, and let x, y ∈ A be such that x ∈ U+(H; ε) and y ∈ U+(H; ε). Then hH(x) ⊃ ε and hH(y) ⊃ ε. Since H is an anti-hesitant fuzzy UP-subalgebra of A, we obtain hH(x · y) ⊆ hH(x) ∪ hH(y). By Lemma 1 (2), we have [0, 1]− hH(x · y) ⊆ ([0, 1]−hH(x))∪([0, 1]−hH(y)) = [0, 1]−(hH(x)∩hH(y)). Thus hH(x·y) ⊇ hH(x)∩hH(y) ⊃ ε. Therefore, x · y ∈ U+(H; ε). Hence, U+(H; ε) is a UP-subalgebra of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a UP-subalgebra of A. Assume that there exist x, y ∈ A such that hH(x · y) * hH(x) ∪ hH(y). Since Im(H) is a chain, we have hH(x · y) ⊃ hH(x) ∪ hH(y). By Lemma 1 (2), we have [0, 1]−hH(x · y) ⊃ ([0, 1]−hH(x))∪ ([0, 1]−hH(y)) = [0, 1]− (hH(x)∩hH(y)). Thus hH(x · y) ⊂ hH(x) ∩ hH(y). Choose ε = hH(x · y) ∈ P([0, 1]). Then hH(x) ⊃ ε and hH(y) ⊃ ε. Thus x, y ∈ U+(H; ε) 6= ∅. By assumption, we have U+(H; ε) is a UP- subalgebra of A and so x · y ∈ U+(H; ε). Thus hH(x · y) ⊃ ε = hH(x · y), a contradiction. Therefore, hH(x · y) ⊆ hH(x) ∪ hH(y) for all x, y ∈ A. Hence, H is an anti-hesitant fuzzy UP-subalgebra of A. Example 12. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the Cayley table from Example 8. Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 994 hH(0) = {0, 1},hH(1) = {1}, hH(2) = {0},hH(3) = {1}, and hH(4) = ∅. Then Im(H) is not a chain. If ε = {1} or ε = {0}, then U+(H; ε) = {0}. If ε = ∅, then U+(H; ε) = {0, 1, 3}. Otherwise, U+(H; ε) = ∅. Using this data, we can show that all nonempty subset U+(H; ε) of A is a UP-subalgebra of A. By Definition 4, we have hH(0) = (0, 1),hH(1) = [0, 1), hH(2) = (0, 1], hH(3) = [0, 1), and hH(3) = [0, 1]. Since hH(3 ·1) = hH(2) = (0, 1] * [0, 1) = hH(3)∪hH(1), we have H is not an anti-hesitant fuzzy UP-subalgebra of A. Theorem 20. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy UP-filter of A, then for all ε ∈ P([0, 1]), U+(H; ε) is a UP-filter of A if U+(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a UP-filter of A, then H is an anti-hesitant fuzzy UP-filter of A. Proof. (1) Assume that H is an anti-hesitant fuzzy UP-filter of A. Let ε ∈ P([0, 1]) be such that U+(H; ε) 6= ∅, and let x ∈ A be such that x ∈ U+(H; ε). Then hH(x) ⊃ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(0) ⊆ hH(x). Thus [0, 1]− hH(0) ⊆ [0, 1]− hH(x). Therefore, hH(0) ⊇ hH(x) ⊃ ε. Hence, 0 ∈ U+(hH; ε). Next, let x, y ∈ A be such that x · y ∈ U+(H; ε) and x ∈ U+(H; ε). Then hH(x · y) ⊃ ε and hH(x) ⊃ ε. Since H is an anti-hesitant fuzzy UP-filter of A, we have hH(y) ⊆ hH(x · y)∪hH(x). By Lemma 1 (2), we have [0, 1]−hH(y) ⊆ ([0, 1]−hH(x ·y))∪ ([0, 1]−hH(x)) = [0, 1]− (hH(x · y) ∩ hH(x)). Thus hH(y) ⊇ hH(x · y) ∩ hH(x) ⊃ ε. Therefore, y ∈ U+(H; ε). Hence, U+(H; ε) is a UP-filter of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a UP-filter of A. Assume that there exists x ∈ A such that hH(0) * hH(x). Since Im(H) is a chain, we have hH(0) ⊃ hH(x). and thus [0, 1] − hH(0) ⊃ [0, 1] − hH(x). So hH(0) ⊂ hH(x). Choose ε = hH(0) ∈ P([0, 1]). Then hH(x) ⊃ ε. Thus x ∈ U+(H; ε) 6= ∅. By assumption, we have U+(H; ε) is a UP-filter of A and so 0 ∈ L−(H; ε). Therefore, hH(0) ⊃ ε = hH(0), a contradiction. Hence, hH(0) ⊆ hH(x) for all x ∈ A. Next, assume that there exist x, y ∈ A such that hH(y) * hH(x·y)∪hH(x). Since Im(H) is a chain, we have hH(y) ⊃ hH(x · y) ∪ hH(x). By Lemma 1 (2), we have [0, 1]− hH(y) ⊃ ([0, 1]−hH(x·y))∪([0, 1]−hH(x)) = [0, 1]−(hH(x·y)∩hH(x)). Thus hH(y) ⊂ hH(x·y)∩hH(x). Choose ε = hH(y) ∈ P([0, 1]). Then hH(x · y) ⊃ ε and hH(x) ⊃ ε. Thus x · y, x ∈ U+(H; ε) 6= ∅. By assumption, we have U+(H; ε) is a UP-filter of A and so y ∈ U+(H; ε). Thus hH(y) ⊃ ε = hH(y), a contradiction. Therefore, hH(y) ⊆ hH(x · y) ∪ hH(x) for all x, y ∈ A. Hence, H is an anti-hesitant fuzzy UP-filter of A. Example 13. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the Cayley table from Example 9. Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 995 hH(0) = {0, 1}, hH(1) = {1},hH(2) = {0},hH(3) = ∅, and hH(4) = ∅. Then Im(H) is not a chain. If ε = {1} or ε = {0}, then U+(H; ε) = {0}. If ε = ∅, then U+(H; ε) = {0, 1, 2}. Otherwise, U+(H; ε) = ∅. Using this data, we can show that all nonempty subset U+(H; ε) of A is a UP-filter of A. By Definition 4, we have hH(0) = (0, 1),hH(1) = [0, 1), hH(2) = (0, 1], hH(3) = [0, 1], and hH(4) = [0, 1]. Since hH(2) = (0, 1] * [0, 1) = hH(1) ∪ hH(1) = hH(1 · 2) ∪ hH(1), we have H is not an anti-hesitant fuzzy UP-filter of A. Theorem 21. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy UP-ideal of A, then for all ε ∈ P([0, 1]), U+(H; ε) is a UP-ideal of A if U+(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a UP-ideal of A, then H is an anti-hesitant fuzzy UP-ideal of A. Proof. (1) Assume that H is an anti-hesitant fuzzy UP-ideal of A. Let ε ∈ P([0, 1]) be such that U+(H; ε) 6= ∅, and let x ∈ A be such that x ∈ U(H; ε). Then hH(x) ⊃ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(0) ⊆ hH(x). Thus [0, 1]− hH(0) ⊆ [0, 1]− hH(x). Therefore, hH(0) ⊇ hH(x) ⊃ ε. Hence, 0 ∈ U+(H; ε). Next, let x, y, z ∈ A be such that x · (y · z) ∈ U+(H; ε) and y ∈ U+(H; ε). Then hH(x · (y · z)) ⊃ ε and hH(y) ⊃ ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we obtain hH(x · z) ⊆ hH(x · (y · z)) ∪ hH(y). By Lemma 1 (2), we have [0, 1] − hH(x · z) ⊆ ([0, 1] − hH(x · (y · z))) ∪ ([0, 1] − hH(y)) = [0, 1] − (hH(x · (y · z)) ∩ hH(y)). Thus hH(x · z) ⊇ hH(x · (y · z)) ∩ hH(y) ⊃ ε. Therefore, x · z ∈ U+(H; ε). Hence, U+(H; ε) is a UP-ideal of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a UP-ideal of A. Assume that there exists x ∈ A such that hH(0) * hH(x). Since Im(H) is a chain, we have hH(0) ⊃ hH(x). Then [0, 1] − hH(0) ⊃ [0, 1] − hH(x). Thus hH(0) ⊂ hH(x). Choose ε = hH(0) ∈ P([0, 1]). Then hH(x) ⊃ ε. Thus x ∈ U+(H; ε) 6= ∅. By assumption, we have U+(H; ε) is a UP-ideal of A and so 0 ∈ U+(H; ε). Therefore, hH(0) ⊃ ε = hH(0), a contradiction. Hence, hH(0) ⊆ hH(x) for any x ∈ A. Next, assume that there exist x, y, z ∈ A such that hH(x · z) * hH(x · (y · z)) ∪ hH(y). Since Im(H) is a chain, we have hH(x ·z) ⊃ hH(x ·(y ·z))∪hH(y). By Lemma 1 (2), we have [0, 1]−hH(x·z) ⊃ ([0, 1]−hH(x·(y·z)))∪([0, 1]−hH(y)) = [0, 1]−(hH(x·(y·z))∩hH(y)). Thus hH(x · z) ⊂ hH(x · (y · z))∩hH(y). Choose ε = hH(x · z) ∈ P([0, 1]). Then hH(x · (y · z)) ⊃ ε and hH(y) ⊃ ε. Thus x · (y · z), y ∈ U+(H; ε) 6= ∅. By assumption, we have U+(H; ε) is a UP-ideal of A and so x · z ∈ L−(H; ε). Thus, hH(x · z) ⊃ ε = hH(x · z), a contradiction. Therefore, hH(x ·z) ⊆ hH(x · (y ·z))∪hH(y) for all x, y, z ∈ A. Hence, H is an anti-hesitant fuzzy UP-ideal of A. Example 14. Let A = {0, 1, 2, 3, 4} be a set with a binary operation · defined by the Cayley table from Example 10. Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 996 hH(0) = {0, 1}, hH(1) = {1},hH(2) = ∅, hH(3) = {0}, and hH(4) = ∅. Then Im(H) is not a chain. If ε = {1} or ε = {0}, then U+(H; ε) = {0}. If ε = ∅, then U+(H; ε) = {0, 1, 3}. Otherwise, U+(H; ε) = ∅. Using this data, we can show that all nonempty subset U+(H; ε) of A is a UP-ideal of A. By Definition 4, we have hH(0) = (0, 1),hH(1) = [0, 1), hH(2) = [0, 1], hH(3) = (0, 1], and hH(4) = [0, 1]. Since hH(0 · 1) = hH(1) = [0, 1) * (0, 1] = hH(0) ∪ hH(3) = hH(0 · (3 · 1)) ∪ hH(3), we have H is not an anti-hesitant fuzzy UP-ideal of A. Theorem 22. Let H be a hesitant fuzzy set on A. Then the following statements hold : (1) if H is an anti-hesitant fuzzy strongly UP-ideal of A, then for all ε ∈ P([0, 1]), U+(H; ε) is a strongly UP-ideal of A if U+(H; ε) is nonempty, and (2) if Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a strongly UP-ideal of A, then H is an anti-hesitant fuzzy strongly UP-ideal of A. Proof. (1) Assume that H is an anti-hesitant fuzzy strongly UP-ideal of A. By Theorem 3, we obtain H is a constant hesitant fuzzy set on A. By Corollary 2, we have H is a constant hesitant fuzzy set on A and so hH(x) = hH(y) for all x, y ∈ A. Let ε ∈ P([0, 1]) be such that U+(H; ε) 6= ∅. There exists a ∈ U+(H; ε) be such that hH(a) ⊃ ε. Thus hH(x) = hH(a) ⊃ ε for all x ∈ A and so x ∈ U+(H; ε) for all x ∈ A. Therefore, U+(H; ε) = A. Hence, U+(H; ε) is a strongly UP-ideal of A. (2) Assume that Im(H) is a chain and for all ε ∈ P([0, 1]), a nonempty subset U+(H; ε) of A is a strongly UP-ideal of A. Assume that H is not a constant hesitant fuzzy set on A. By Corollary 2, we have H is not a constant hesitant fuzzy set on A. There exist x, y ∈ A be such that hH(x) 6= hH(y). Since Im(H) is a chain, we have hH(x) ⊂ hH(y) or hH(x) ⊃ hH(y). Without loss of generality, assume that hH(x) ⊂ hH(y), then y ∈ U+(H; hH(x)) 6= ∅. By assumption, we have U+(H; hH(x)) is a strongly UP-ideal of A and so U+(H; hH(x)) = A. Thus x ∈ A = U+(H; hH(x)) and so hH(x) ⊂ hH(x), a contradiction. Therefore, H is a constant hesitant fuzzy set on A. By Theorem 3, we obtain H is an anti-hesitant fuzzy strongly UP-ideal of A. Example 15. Let A = {0, 1} be a set with a binary operation · defined by the Cayley table from Example 11. Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = {0}, and hH(1) = {1}. Then Im(H) is not a chain. If ε = ∅, then U+(H; ε) = A. Otherwise, U+(H; ε) = ∅. Thus a nonempty subset U+(H; ε) of A is a strongly UP-ideal of A. By Definition 4, we have hH(0) = (0, 1], and hH(1) = [0, 1). By Theorem 3 and because H is not a constant hesitant fuzzy set on A, we have H is not an anti-hesitant fuzzy strongly UP-ideal of A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 997 5.5. Equal ε-Level Subsets Theorem 23. If a hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-subalgebra of A, then for all ε ∈ P([0, 1]), a nonempty subset E(H; ε) of A is a UP-subalgebra of A where L−(H; ε) is empty. Proof. Assume that H is an anti-hesitant fuzzy UP-subalgebra of A. Let ε ∈ P([0, 1]) be such that E(H; ε) 6= ∅ but L−(H; ε) = ∅, and let x, y ∈ A be such that x ∈ E(H; ε) and y ∈ E(H; ε). Then hH(x) = ε and hH(y) = ε. Because H is an anti-hesitant fuzzy UP-subalgebra of A, we have hH(x · y) ⊆ hH(x) ∪ hH(y) = ε. Thus x · y ∈ L(H; ε). Since L−(H; ε) is empty, we obtain L(H; ε) = L−(H; ε)∪E(H; ε) = ∅ ∪E(H; ε) = E(H; ε). Therefore, x · y ∈ E(H; ε). Hence, E(H; ε) is a UP-subalgebra of A. The following example show that the converse of Theorem 23 is not true in general. Example 16. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 1 3 2 0 0 0 3 3 0 1 1 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = ∅, hH(1) = [0, 0.6], hH(2) = [0, 0.3], and hH(3) = [0, 0.3]. If ε 6= ∅, then L−(H; ε) 6= ∅. If ε = ∅, then L−(H; ε) = ∅ and E(H; ε) = {0}. Thus E(H; ε) is clearly a UP-subalgebra of A. Since hH(3·2) = hH(1) = [0, 0.6] * [0, 0.3] = hH(3)∪hH(2), we have H is not an anti-hesitant fuzzy UP-subalgebra of A. Theorem 24. If a hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-filter of A, then for all ε ∈ P([0, 1]), a nonempty subset E(H; ε) of A is a UP-filter of A where L−(H; ε) is empty. Proof. Assume that H is an anti-hesitant fuzzy UP-filter of A. Let ε ∈ P([0, 1]) be such that E(H; ε) 6= ∅ but L−(H; ε) = ∅, and let x ∈ A be such that x ∈ E(H; ε). Then hH(x) = ε. Because H is an anti-hesitant fuzzy UP-filter of A, we obtain hH(0) ⊆ hH(x) = ε and thus 0 ∈ L(H; ε). Since L−(H; ε) is empty, we have 0 ∈ L(H; ε) = E(H; ε). Next, let x, y ∈ A be such that x · y ∈ E(H; ε) and x ∈ E(H; ε). Then hH(x · y) = ε and hH(x) = ε. Because H is an anti-hesitant fuzzy UP-filter of A, we have hH(y) ⊆ hH(x · y) ∪ hH(x) = ε. Thus y ∈ L(H; ε). Since L−(H; ε) is empty, we obtain L(H; ε) = E(H; ε). Therefore, y ∈ E(H; ε). Hence, E(H; ε) is a UP-filter of A. The converse of Theorem 24 is not true in general. By Example 16, we still have E(H; ε) = {0} is a UP-filter of A. Since hH(1) = [0, 0.6] * [0, 0.3] = hH(0) ∪ hH(2) = hH(2 · 1) ∪ hH(2), we have H is not an anti-hesitant fuzzy UP-filter of A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 998 Theorem 25. If a hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-ideal of A, then ε ∈ P([0, 1]), a nonempty subset E(H; ε) of A is a UP-ideal of A where L−(H; ε) is empty. Proof. Assume that H is an anti-hesitant fuzzy UP-ideal of A. Let ε ∈ P([0, 1]) be such that E(H; ε) 6= ∅ but L−(H; ε) = ∅, and let x ∈ A be such that x ∈ E(H; ε). Then hH(x) = ε. Because H is an anti-hesitant fuzzy UP-ideal of A, we obtain hH(0) ⊆ hH(x) = ε and thus 0 ∈ L(H; ε). Since L−(H; ε) is empty, we have 0 ∈ L(H; ε) = E(H; ε). Next, let x, y, z ∈ A be such that x · (y · z) ∈ E(H; ε) and y ∈ E(H; ε). Then hH(x · (y · z)) = ε and hH(y) = ε. Because H is an anti-hesitant fuzzy UP-ideal of A, we have hH(x · z) ⊆ hH(x · (y · z)) ∪ hH(y) = ε. Thus x · z ∈ L(H; ε). Since L−(H; ε) is empty, we obtain L(H; ε) = E(H; ε). Therefore, x · z ∈ E(H; ε). Hence, E(H; ε) is a UP-ideal of A. The converse of Theorem 25 is not true in general. By Example 16, we still have E(H; ε) = {0} is a UP-ideal of A. Since hH(0 · 1) = hH(1) = [0, 0.6] * [0, 0.3] = hH(0) ∪ hH(2) = hH(0 · (2 · 1)) ∪ hH(2), we have H is not an anti-hesitant fuzzy UP-ideal of A. Theorem 26. A hesitant fuzzy set H on A is an anti-hesitant fuzzy strongly UP-ideal of A if and only if E(H; hH(0)) is a strongly UP-ideal of A. Proof. Assume that H is an anti-hesitant fuzzy strongly UP-ideal of A. By Theorem 3, we obtain H is a constant hesitant fuzzy set on A and so hH(x) = hH(0) for all x ∈ A. Then E(H; hH(0)) = A. Hence, E(H; hH(0)) is a strongly UP-ideal of A. Conversely, assume that E(H; hH(0)) is a strongly UP-ideal of A. Then E(H; hH(0)) = A and so hH(x) = hH(0) for all x ∈ A. Therefore, H is a constant hesitant fuzzy set on A. By Theorem 3, H is an anti-hesitant fuzzy strongly UP-ideal of A. Moreover, we still obtain theorems of equal ε-level subsets with a hesitant fuzzy UP-subalgebra. (resp., hesitant fuzzy UP-filter, hesitant fuzzy UP-ideal, hesitant fuzzy strongly UP-ideal) Theorem 27. If a hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-subalgebra of A, then for all ε ∈ P([0, 1]), a nonempty subset E(H; ε) of A is a UP-subalgebra of A where U+(H; ε) is empty. Proof. Assume that H is an anti-hesitant fuzzy UP-subalgebra of A. Let ε ∈ P([0, 1]) be such that E(H; ε) 6= ∅ but U+(H; ε) = ∅, and let x, y ∈ A be such that x ∈ E(H; ε) and y ∈ E(H; ε). Then hH(x) = ε and hH(y) = ε. Since H is an anti-hesitant fuzzy UP-subalgebra of A, we have hH(x · y) ⊇ hH(x) ∩ hH(y) = ε. Thus x · y ∈ U(H; ε). Since U+(H; ε) is empty, we obtain U(H; ε) = U+(H; ε) ∪ E(H; ε) = ∅ ∪ E(H; ε) = E(H; ε). Therefore, x · y ∈ E(H; ε). Hence, E(H; ε) is a UP-subalgebra of A. The following example show that the converse of Theorem 27 is not true in general. Example 17. Let A = {0, 1, 2, 3} be a set with a binary operation · defined by the following P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 999 Cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 0 0 0 3 0 0 1 0 Then (A, ·, 0) is a UP-algebra. We define a hesitant fuzzy set H on A as follows: hH(0) = [0, 1],hH(1) = {0},hH(2) = [0, 0.1], and hH(3) = [0, 0.1]. If ε 6= [0, 1], then U+(H; ε) 6= ∅. If ε = [0, 1], then U+(H; ε) = ∅ and E(H; ε) = {0}. Thus E(H; ε) is clearly a UP-subalgebra of A. Since hH(3 · 2) = hH(1) = {0} + [0, 0.1] = hH(3) ∩ hH(2), we have H is not an anti-hesitant fuzzy UP-subalgebra of A. Theorem 28. If a hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-filter of A, then for all ε ∈ P([0, 1]), a nonempty subset E(H; ε) of A is a UP-filter of A where U+(H; ε) is empty. Proof. Assume that H is an anti-hesitant fuzzy UP-filter of A. Let ε ∈ P([0, 1]) be such that E(H; ε) 6= ∅ but U+(H; ε) = ∅, and let x ∈ A be such that x ∈ E(H; ε). Then hH(x) = ε. Because H is an anti-hesitant fuzzy UP-filter of A, we obtain hH(0) ⊇ hH(x) = ε and thus 0 ∈ U(H; ε). Since U+(H; ε) is empty, we have 0 ∈ U(H; ε) = E(H; ε). Next, let x, y ∈ A be such that x · y ∈ E(H; ε) and x ∈ E(H; ε). Then hH(x · y) = ε and hH(x) = ε. Because H is an anti-hesitant fuzzy UP-filter of A, we have hH(y) ⊇ hH(x · y) ∩ hH(x) = ε. Thus y ∈ L(H; ε). Since U+(H; ε) is empty, we obtain U(H; ε) = E(H; ε). Therefore, y ∈ E(H; ε). Hence, E(H; ε) is a UP-filter of A. The converse of Theorem 28 is not true in general. By Example 17, we still have E(H; ε) = {0} is a UP-filter of A. Since hH(1) = {0} + [0, 0.1] = hH(0) ∩ hH(3) = hH(3 · 1) ∩ hH(3), we have H is not an anti-hesitant fuzzy UP-filter of A. Theorem 29. If a hesitant fuzzy set H on A is an anti-hesitant fuzzy UP-ideal of A, then for all ε ∈ P([0, 1]), a nonempty subset E(H; ε) of A is a UP-ideal of A where U+(H; ε) is empty. Proof. Assume that H is an anti-hesitant fuzzy UP-ideal of A. Let ε ∈ P([0, 1]) be such that E(H; ε) 6= ∅ but U+(H; ε) = ∅, and let x ∈ A be such that x ∈ E(H; ε). Then hH(x) = ε. Because H is an anti-hesitant fuzzy UP-filter of A, we obtain hH(0) ⊇ hH(x) = ε and thus 0 ∈ U(H; ε). Since U+(H; ε) is empty, we have 0 ∈ U(H; ε) = E(H; ε). Next, let x, y, z ∈ A be such that x · (y · z) ∈ E(H; ε) and y ∈ E(H; ε). Then hH(x · (y · z)) = ε and hH(y) = ε. Since H is an anti-hesitant fuzzy UP-ideal of A, we have hH(x · z) ⊇ hH(x · (y · z)) ∩ hH(y) = ε. Thus x · z ∈ U(H; ε). Since L−(H; ε) is empty, we obtain U(H; ε) = E(H; ε). Therefore, x · z ∈ E(H; ε). Hence, E(H; ε) is a UP-ideal of A. The converse of Theorem 29 is not true in general. By Example 17, we still have E(H; ε) = {0} is a UP-ideal of A. Since hH(3·2) = hH(1) = {0} + [0, 0.1] = hH(0)∩hH(2) = hH(3 · (2 · 2)) ∩ hH(2), we have H is not an anti-hesitant fuzzy UP-ideal of A. P. Mosrijai, A. Iampan / Eur. J. Pure Appl. Math, 11 (4) (2018), 976-1002 1000 Theorem 30. A hesitant fuzzy set H on A is an anti-hesitant fuzzy strongly UP-ideal of A if and only if E(H; hH(0)) is a strongly UP-ideal of A. Proof. It is straightforward by Theorem 26 and 3. 6. Conclusions and Future Work In this paper, we have introduced the notion of anti-hesitant fuzzy UP-subalgebras (resp., anti-hesitant fuzzy UP-filters, anti-hesitant fuzzy UP-ideals and anti-hesitant fuzzy strongly UP-ideals) of UP-algebras and investigated some of its important properties. Then we have the diagram of anti-type of hesitant fuzzy sets on UP-algebras below. In our future study of UP-algebras, may be the following topics should be considered: • To get more results in anti-hesitant fuzzy UP-subalgebras, anti-hesitant fuzzy UP- filters, anti-hesitant fuzzy UP-ideals, and anti-hesitant fuzzy strongly UP-ideals of UP-algebras. • To define anti-hesitant fuzzy soft UP-subalgebras, anti-hesitant fuzzy soft UP-filters, anti-hesitant fuzzy soft UP-ideals, and anti-hesitant fuzzy soft strongly UP-ideals over UP-algebras. • To define operations of hesitant fuzzy soft sets over UP-algebras. Acknowledgements The authors wish to express their sincere thanks to the referees for the valuable sug- gestions which lead to an improvement of this paper. REFERENCES 1001 References [1] T. Guntasow, S. Sajak, A. Jomkham, and A. Iampan. Fuzzy translations of a fuzzy set in UP-algebras. J. Indones. Math. Soc., 23(2):1–19, 2017. [2] A. Iampan. A new branch of the logical algebra: UP-algebras. J. Algebra Relat. Top., 5(1):35–54, 2017. [3] A. Iampan. Introducing fully UP-semigroups. Manuscript accepted for publication in Discuss. Math., Gen. Algebra Appl., September 2018. [4] W. Kaijae, P. Poungsumpao, S. Arayarangsi, and A. Iampan. UP-algebras character- ized by their anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras. Ital. J. Pure Appl. Math., 36:667–692, 2016. [5] B. Kesorn, K. Maimun, W. Ratbandan, and A. Iampan. Intuitionistic fuzzy sets in UP-algebras. Ital. J. Pure Appl. Math., 34:339–364, 2015. [6] P. Mosrijai, W. Kamti, A. Satirad, and A. Iampan. Hesitant fuzzy sets on UP- algebras. Konuralp J. Math., 5(2):268–280, 2017. [7] P. Mosrijai, A. Satirad, and A. Iampan. Partial constant hesitant fuzzy sets on UP- algebras. J. New Theory, 22:39–50, 2018. [8] C. Prabpayak and U. Leerawat. On ideals and congruences in KU-algebras. Sci. Magna, 5(1):54–57, 2009. [9] D. A. Romano. Proper UP-filters of UP-algebra. Univ. J. Math. Appl., 1(2):98–100, 2018. [10] A. Satirad, P. Mosrijai, and A. Iampan. Generalized power UP-algebras. Manuscript accepted for publication in Int. J. Math. Comput. Sci., May 2018. [11] A. Satirad, P. Mosrijai, W. Kamti, and A. Iampan. Level subsets of a hesitant fuzzy set on UP-algebras. Ann. Fuzzy Math. Inform., 14(3):279–302, 2017. [12] T. Senapati, Y. B. Jun, and K. P. Shum. Cubic set structure applied in UP-algebras. Discrete Math. Algorithms Appl., 10(4):1850049, 2018. [13] T. Senapati, G. Muhiuddin, and K. P. Shum. Representation of UP-algebras in interval-valued intuitionistic fuzzy environment. Ital. J. Pure Appl. Math., 38:497– 517, 2017. [14] J. Somjanta, N. Thuekaew, P. Kumpeangkeaw, and A. Iampan. Fuzzy sets in UP- algebras. Ann. Fuzzy Math. Inform., 12(6):739–756, 2016. [15] M. Songsaeng and A. Iampan. N -fuzzy UP-algebras and its level subsets. J. Algebra Relat. Top., 6(1):1–24, 2018. REFERENCES 1002 [16] S. Sripaeng, K. Tanamoon, and A. Iampan. On anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras. J. Inf. Optim. Sci., 39(5):1095–1127, 2018. [17] K. Tanamoon, S. Sripaeng, and A. Iampan. Q-fuzzy sets in UP-algebras. Songk- lanakarin J. Sci. Technol., 40(1):9–29, 2018. [18] V. Torra. Hesitant fuzzy sets. Int. J. Intell. Syst., 25(6):529–539, 2010.