Int. J. Anal. Appl. (2022), 20:34 (inf, sup)-Hesitant Fuzzy Ideals of BCK/BCI-Algebras Noppakao Ratchakhwan1, Pongpun Julatha1, Thiti Gaketem2, Pannawit Khamrot3, Rukchart Prasertpong4, Aiyared Iampan2,∗ 1Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand 2Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand 3Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok 65000, Thailand 4Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan 60000, Thailand ∗Corresponding author: aiyared.ia@up.ac.th Abstract. In this paper, we introduce the concept of (inf,sup)-hesitant fuzzy ideals, which is a gener- alization of the concept of interval-valued fuzzy ideals, in BCK/BCI-algebras and its related properties are investigated. The concept is established in terms of sets, fuzzy sets, negative fuzzy sets, interval- valued fuzzy sets, Pythagorean fuzzy sets, bipolar fuzzy sets and hesitant fuzzy sets. Moreover, characterizations of ideals, fuzzy ideals, anti-fuzzy ideals, negative fuzzy ideals, Pythagorean fuzzy ideals and bipolar fuzzy ideals of BCK/BCI-algebras are discussed in terms of (inf,sup)-hesitant fuzzy ideals and interval-valued fuzzy ideals. 1. Introduction The concept of fuzzy sets, introduced by Zadeh [3], has been widely and successfully applied in many branches: finite state machine, computer science, automata, artificial intelligence, expert, control engineering, robotics and theory of groups, semigroups, BCK/BCI-algebras and UP-algebras. Received: May 23, 2022. 2010 Mathematics Subject Classification. 03B52, 03G25, 06F35, 08A72. Key words and phrases. BCK/BCI-algebra; hesitant fuzzy set; (inf, sup)-hesitant fuzzy ideal; interval-valued fuzzy ideal; Pythagorean fuzzy ideal; bipolar fuzzy ideal. https://doi.org/10.28924/2291-8639-20-2022-34 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-34 2 Int. J. Anal. Appl. (2022), 20:34 Several general, extended and related concepts of fuzzy sets have been introduced and studied such as interval-valued fuzzy sets [4, 5], intuitionistic fuzzy sets [6, 7], Pythagorean fuzzy sets [10–12], negative fuzzy sets [13,14], bipolar fuzzy sets [15,16], hesitant fuzzy sets [17,18,20,22] and so forth. BCK and BCI-algebras are algebraic structures, introduced by Imai, Iséki and Tanaka, that describe fragments of the propositional calculus involving implication known as BCK and BCI logic (see [29– 31]). In 1991, Xi [8] applied the concept of fuzzy sets to BCK-algebras. Later, a number of authors applied and discussed concept of fuzzy sets and its some general, extended and related concepts to BCK/BCI-algebras. Hong and Jun [9] introduced anti-fuzzy ideals of BCK-algebras and investigated their some useful properties. Subha and Dhanalakshmi [12] exposed and studied Pythagorean fuzzy ideals of BCK-algebras. Jun [5] introduced interval-valued fuzzy subalgebras and ideals of BCK- algebras, and investigated their related properties and characterizations. Lee [16] introduced bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras, investigated their related properties, and considered equivalent relations on the set of all bipolar fuzzy ideals of BCK/BCI-algebras. Jun and Ahn [19] introduced hesitant fuzzy subalgebras and ideals of BCK/BCI-algebras, and investigated their related properties and important characterizations. Muhiuddin et al. [32] introduced hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set on BCK/BCI-algebras, investigated related properties, and characterized hesitant fuzzy (subalgebras) ideals. Studying hesitant fuzzy sets on algebraic structures in the meaning of the infimum or supremum of its images, Mosrijai et al. [33] introduced sup-hesitant fuzzy UP-subalgebras, UP-filters, UP-ideals, and strong UP-ideals of UP-algebras and investigated their related properties. Muhiuddin and Jun [34] Muhiuddin et al. [35] Muhiuddin et al. [38], Harizavi and Jun [37], Jun and Song [39] and Takallo et al. [36] used hesitant fuzzy sets related to the infimum or supremum of their images in study of BCK/BCI-algebras. Jittburus and Julatha [24,25], Phummee et al. [28], and Jittburus et al. [27] used hesitant fuzzy sets related to the infimum or the supremum of their images in study of semigroups. Julatha and Iampan [21–23,26] used hesitant fuzzy sets related to the infimum or the supremum of their images in study of ternary semigroups and Γ-semigroups. As previously stated, it motivated us to study hesitant fuzzy set theory based on ideals of BCK/BCI- algebras in the meaning of infimum and supremum. On BCK/BCI-algebras, we introduce (inf, sup)- hesitant fuzzy ideals, show that it is a general concept of interval-valued fuzzy ideals, and investigate its related properties. Characterizations of (inf, sup)-hesitant fuzzy ideals are established in terms of sets, fuzzy sets, negative fuzzy sets, interval-valued fuzzy sets, Pythagorean fuzzy sets, bipolar fuzzy sets and hesitant fuzzy sets. Moreover, characterizations of ideals, fuzzy ideals, anti-fuzzy ideals, negative fuzzy ideals, Pythagorean fuzzy ideals and bipolar fuzzy ideals of BCK/BCI-algebras are discussed in terms of (inf, sup)-hesitant fuzzy ideals and interval-valued fuzzy ideals. 2. Preliminaries An algebra (X ;�, 0) of type (2, 0) is called a BCI-algebra if the followings hold: Int. J. Anal. Appl. (2022), 20:34 3 (I) (∀x,y,z ∈X)(((x �y) � (x �z)) � (z �y) = 0), (II) (∀x,y ∈X)(((x � (x �y)) �y) = 0), (III) (∀x ∈X)(x �x = 0), (IV) (∀x,y ∈X)(x �y = 0 = y �x ⇒ x = y). By a BCK-algebra we mean a BCI-algebra (X ;�, 0) satisfies 0�x = 0 for all x ∈X . For any x,y ∈X , we define x ≤ y by x �y = 0. In a BCK/BCI-algebra (X ;�, 0), the following hold: (∀x ∈X)(x � 0 = x), (2.1) (∀x,y,z ∈X)((x �y) �z = (x �z) �y). (2.2) A nonempty subset A of a BCK/BCI-algebra (X ;�, 0) is called an ideal (Id) of X if it satisfies the following: 0 ∈A, (2.3) (∀x ∈X)(y ∈A,x �y ∈A⇒ x ∈A). (2.4) We refer the reader to the books [1,2] for further information regarding BCK/BCI-algebras. In what follows, let X denote a BCK/BCI-algebra (X ,�, 0) and Y denote an arbitrary nonempty set unless otherwise specified. A fuzzy set (FS) [3] in Y is an arbitrary function from Y into [0, 1]. For FSs ζ and ξ in Y, we denote ζ ≤ ξ in case that ζ(x) ≤ ξ(x) for all x ∈Y. A FS ζ in X is call a fuzzy ideal (FId) [8] of X if it satisfies the following conditions: (∀x ∈X)(ζ(0) ≥ ζ(x)), (2.5) (∀x,y ∈X)(ζ(x) ≥ min{ζ(x �y),ζ(y)}) (2.6) and called an anti-fuzzy ideal (AFId) [9] of X if it satisfies the following conditions: (∀x ∈X)(ζ(0) ≤ ζ(x)), (2.7) (∀x,y ∈X)(ζ(x) ≤ max{ζ(x �y),ζ(y)}). (2.8) Then ζ is both a FId and an AFId of X if and only if it is a constant function. A Pythagorean fuzzy set (PFS) [10, 11] on Y is an object having the form P = {(x,ζ(x),ξ(x)) |x ∈Y} when the functions ζ : Y → [0, 1] denote the degree of membership and ξ : Y → [0, 1] denote the degree of nonmembership, and 0 ≤ (ζ(x))2 + (ξ(x))2 ≤ 1 for all x ∈ Y. For the sake of simplicity, we will use the symbol (ζ,ξ) of the PFS {(x,ζ(x),ξ(x)) | x ∈ Y}. For a FS ζ in Y, we define a FS ζ 2 by ζ 2 (x) = ζ(x) 2 for all x ∈Y. Then ( ζ 2 , ξ 2 ) and ( ζ 2 , ζ 2 ) are PFSs in Y for all FSs ζ and ξ in Y. Thus the concept of PFSs is an extension of the concept of FSs. A PFS (ζ,ξ) on X is called a Pythagorean fuzzy ideal (PFId) [12] of X if ζ is a FId and ξ is an AFId of X . A bipolar fuzzy set (BFS) [15] in Y is an object having the form B = {(x,ζ(x),ξ(x)) | x ∈ Y}, where ζ : Y → [−1, 0] is a negative fuzzy set (NFS) in Y and ξ : Y → [0, 1] is a FS in Y. We’ll use 4 Int. J. Anal. Appl. (2022), 20:34 the symbol 〈ζ,ξ〉 for the BFS {(x,ζ(x),ξ(x)) | x ∈Y} for the purpose of simplicity. Let R be the set of all real numbers. For any element r of R and any function ζ from Y into R, define functions r −ζ, r + ζ, rζ and −ζ by: r −ζ : Y → R,x 7→ r −ζ(x), (2.9) r + ζ : Y → R,x 7→ r + ζ(x) (2.10) rζ : Y → R,x 7→ rζ(x) (2.11) −ζ : Y → R,x 7→−ζ(x). (2.12) Then the followings hold: (1) 〈ζ − 1,ζ〉 is a BFS in Y for any FS ζ in Y, (2) ( 1+ζ 2 , ξ 2 ) and ( ξ 2 , 1+ζ 2 ) are PFSs in Y for any BFS 〈ζ,ξ〉 in Y, (3) 〈ζ − 1,ξ〉 and 〈ξ− 1,ζ〉 are BFSs in Y for any PFS (ζ,ξ) in Y. Thus the concept of BFSs is an extension of the concept of FSs. A BFS B = 〈ζ,ξ〉 in X is called a bipolar fuzzy ideal (BFId) [16] of X if it satisfies the following conditions: (∀x ∈X)(ζ(0) ≤ ζ(x)), (2.13) (∀x ∈X)(ξ(0) ≥ ξ(x)), (2.14) (∀x,y ∈X)(ζ(x) ≤ max{ζ(x �y),ζ(y)}), (2.15) (∀x,y ∈X)(ξ(x) ≥ min{ξ(x �y),ξ(y)}). (2.16) By a negative fuzzy ideal (NFId) of X we mean a NFS ζ of X satisfies the conditions (2.13) and (2.15). Then a BFS 〈ζ,ξ〉 of X is a BFId of X if and only if ζ is a NFId and ξ is a FId of X . By an interval number ă we mean an interval [a−,a+], where 0 ≤ a− ≤ a+ ≤ 1. The set of all interval numbers is denoted by D([0, 1]). For two elements ă = [a−,a+] and b̆ = [b−,b+] in D([0, 1]), define the operations -, =, ≺ and rmin in case of two elements in D([0, 1]) as follows: (1) ă - b̆ ⇔ a+ ≤ b+ and a− ≤ b−, (2) ă = b̆ ⇔ a+ = b+ and a− = b−, (3) ă ≺ b̆ ⇔ ă - b̆ and ă 6= b̆, (4) rmin{ă, b̆} = [min{a−,b−}, min{a+,b+}]. An interval-valued fuzzy set (IvFS) [4] on Y is defined to be a function λ̆ : Y → D([0, 1]), where λ̆(x) = [λ̆L(x), λ̆U(x)] for all x ∈Y, λ̆L and λ̆U are FSs in Y such that λ̆L ≤ λ̆U. Thus the concept of IvFSs is an extension of the concept of FSs. An IvFS λ̆ on X is called an interval-valued fuzzy ideal Int. J. Anal. Appl. (2022), 20:34 5 (IvFId) [5] of X if it satisfies: (∀x ∈X)(λ̆(x) - λ̆(0)), (2.17) (∀x,y ∈X)(rmin{λ̆(x �y), λ̆(y)}- λ̆(x)). (2.18) Remark 2.1. an IvFS λ̆ on X is an IvFId of X if and only if λ̆L and λ̆U are FIds of X . A hesitant fuzzy set (HFS) [17,18] on Y is defined to be a function ω̃ : Y → ℘([0, 1]) when ℘([0, 1]) is the set of all subsets of [0, 1]. Note that every IvFS on Y is a HFS on Y. Then the concept of HFSs is a generalization of the concept of IvFSs, and the concept of HFSs is an extension of the concept of FSs. A HFS ω̃ is a hesitant fuzzy ideal (HFId) [19,20] of X if it satisfies the following: (∀x ∈X)(ω̃(x) ⊆ ω̃(0)), (2.19) (∀x,y ∈X)(ω̃(x �y) ∩ ω̃(y) ⊆ ω̃(x)). (2.20) 3. Main Results For an element ∇∈ ℘([0, 1]), define INF∇ [24,27] and SUP∇ [25,26] by INF∇ = { inf ∇ 0 if ∇ 6= ∅, otherwise, and SUP∇ = { sup∇ 0 if ∇ 6= ∅, otherwise. Definition 3.1. A HFS ω̃ on X is called an (inf, sup)-hesitant fuzzy ideal ((inf, sup)-HFId) of X if the set [X , ω̃,∇] is an Id of X for all ∇ ∈ ℘([0, 1]) when [X , ω̃,∇] := {x ∈ X | INF ω̃(x) ≥ INF∇, SUP ω̃(x) ≥ SUP∇} is not empty. Example 3.1. Let X = {0,u,v,w,x} be a BCI-algebra [1] with the following Cayley table: � 0 u v w x 0 0 0 v w x u u 0 v w x v v v 0 x w w w w x 0 v x x x w v 0 Define a HFS ω̃ on X by ω̃(0) = [0.6, 0.8], ω̃(u) = (0.5, 0.7), ω̃(v) = [0.5, 0.6] ∪ {0.7}, ω̃(w) = {0.3, 0.4}, ω̃(z) = (0.3, 0.4). It is routine to verify that ω̃ is an (inf, sup)-HFId of X . 6 Int. J. Anal. Appl. (2022), 20:34 Example 3.2. Let X = {0,w,x,y,z} be a BCK-algebra with the following Cayley table: � 0 w x y z 0 0 0 0 0 0 w w 0 0 0 0 x x x 0 0 0 y y x w 0 w z z x w w 0 Define a HFS ω̃ on X by ω̃(0) = {0.8, 0.9, 1}, ω̃(w) = (0.6, 0.8], ω̃(x) = ω̃(y) = {0}, ω̃(z) = ∅. It is routine to verify that ω̃ is an (inf, sup)-HFId of X . Moreover, we know that ω̃ is not a HFId of X because ω̃(w) * ω̃(0), and ω̃ is not an IvFId of X because it is not an IvFS. For any HFS ω̃ on Y, define the FSs Fω̃ and Fω̃ in Y by (∀x ∈Y)(Fω̃(x) = SUP ω̃(x)), (3.1) (∀x ∈Y)(Fω̃(x) = INF ω̃(x)). (3.2) A HFS ϑ̃ on Y is called an infimum complement [21, 24] of ω̃ on Y if INF ϑ̃(x) = (1 − Fω̃)(x) for all x ∈ Y and called a supremum complement of ω̃ on Y if SUP ϑ̃(x) = (1 −Fω̃)(x) for all x ∈Y. Let IC(ω̃) and SC(ω̃) be the set of all infimum complements of ω̃ and the set of all supremum complements of ω̃, respectively. Define the HFSs ω̃± and ω̃∓ on Y by ω̃±(x) = {(1 −Fω̃)(x)} and ω̃∓(x) = {(1 −Fω̃)(x)} for all x ∈Y. Then we have ω̃± ∈ IC(ω̃), Fω̃± = 1 −Fω̃ and ω̃∓ ∈ SC(ω̃), Fω̃ ∓ = 1 −Fω̃. Next, we investigate characterizations of (inf, sup)-HFIds of BCK/BCI-algebras in terms of Ids, FIds, AFIds and NFIds. Lemma 3.1. Let ω̃ be a HFS on X . Then the followings are equivalent. (1) ω̃ is an (inf, sup)-HFId of X . (2) The set [X , ω̃, ă] is an Id of X for all ă ∈D([0, 1]) when [X , ω̃, ă] is not empty. (3) Fω̃ and Fω̃ are FIds of X . (4) F ϑ̃ and F θ̃ are AFIds of X for all ϑ̃ ∈ IC(ω̃) and θ̃ ∈ SC(ω̃). (5) Fω̃± and Fω̃ ∓ are AFIds of X . (6) F ϑ̃ − 1 and F θ̃ − 1 are NFIds of X for all ϑ̃ ∈ IC(ω̃) and θ̃ ∈ SC(ω̃). (7) Fω̃± − 1 and Fω̃ ∓ − 1 are NFIds of X . Proof. (1) ⇒ (2), (4) ⇒ (5) and (6) ⇒ (7). They are clear. (2) ⇒ (3). Let x ∈ X and ă := {t ∈ [0, 1] | INF ω̃(x) ≤ t ≤ SUP ω̃(x)}. Then ă ∈ D([0, 1]) and x ∈ [X , ω̃, ă]. By the assumption (2), we get [X , ω̃, ă] is an Id of X and so 0 ∈ [X , ω̃, ă]. Thus SUP ω̃(x) = a+ ≤ SUP ω̃(0) and INF ω̃(x) = a− ≤ INF ω̃(0), which imply that Fω̃(x) ≤ Fω̃(0) and Fω̃(x) ≤Fω̃(0). Hence, Fω̃ and Fω̃ satisfy the condition (2.5). To show that Fω̃ and Fω̃ satisfy the Int. J. Anal. Appl. (2022), 20:34 7 condition (2.6), let x,y ∈X and b̆ := {t ∈ [0, 1] | min{INF ω̃(y), INF ω̃(x �y)}≤ t ≤ min{SUP ω̃(y), SUP ω̃(x �y)}}. Then b̆ ∈D([0, 1]) and y,x �y ∈ [X , ω̃, b̆]. By the assumption (2), we have x ∈ [X , ω̃, b̆]. Thus Fω̃(x) = SUP ω̃(x) ≥ b+ = min{SUP ω̃(y), SUP ω̃(x �y)} = min{Fω̃(y),Fω̃(x �y)}, Fω̃(x) = INF ω̃(x) ≥ b− = min{INF ω̃(y), INF ω̃(x �y)} = min{Fω̃(y),Fω̃(x �y)}. Hence, Fω̃ and Fω̃ satisfy the condition (2.6). Therefore, it follows from the conditions (2.5) and (2.6) that Fω̃ and Fω̃ are FIds of X . (3) ⇒ (1). Let ∇ be an element of ℘([0, 1]) such that [X , ω̃,∇] 6= ∅. Let x ∈ X and y,x � y ∈ [X , ω̃,∇]. Then SUP ω̃(y) ≥ SUP∇, INF ω̃(y) ≥ INF∇, SUP ω̃(x�y) ≥ SUP∇ and INF ω̃(x�y) ≥ INF∇. By the assumption (3), we have SUP ω̃(0) = Fω̃(0) ≥Fω̃(y) = SUP ω̃(y) ≥ SUP∇, INF ω̃(0) = Fω̃(0) ≥Fω̃(y) = INF ω̃(y) ≥ INF∇, SUP ω̃(x) = Fω̃(x) ≥ min{Fω̃(y),Fω̃(x �y)} = min{SUP ω̃(y), SUP ω̃(x �y)}≥ SUP∇, and INF ω̃(x) = Fω̃(x) ≥ min{Fω̃(y),Fω̃(x �y)} = min{INF ω̃(y), INF ω̃(x �y)}≥ INF∇. Thus 0,x ∈ [X , ω̃,∇]. Hence, [X , ω̃,∇] is an Id of X . Therefore, ω̃ is an (inf, sup)-HFId of X . (3) ⇒ (4). Let ϑ̃ ∈ IC(ω̃) and θ̃ ∈ SC(ω̃). By the assumption (3), we obtain that F ϑ̃ and F θ̃ satisfy the conditions (2.5) and (2.6). Thus, for all x,y ∈X , we have F θ̃(0) = 1 −Fω̃(0) ≤ 1 −Fω̃(x) = F θ̃(x), F ϑ̃ (0) = 1 −Fω̃(0) ≤ 1 −Fω̃(x) = Fϑ̃(x), F θ̃(x) = 1 −Fω̃(x) ≤ 1 − min{Fω̃(y),Fω̃(x �y)} = max{1 −Fω̃(y), 1 −Fω̃(x �y)} 8 Int. J. Anal. Appl. (2022), 20:34 = max{F θ̃(y),F θ̃(x �y)}, F ϑ̃ (x) = 1 −Fω̃(x) ≤ 1 − min{Fω̃(y),Fω̃(x �y)} = max{1 −Fω̃(y), 1 −Fω̃(x �y)} = max{F ϑ̃ (y),F ϑ̃ (x �y)}. Hence, F ϑ̃ and F θ̃ satisfy that conditions (2.7) and (2.8) that they are AFIds of X . (4) ⇒ (6). Let ϑ̃ ∈ IC(ω̃) and θ̃ ∈ SC(ω̃). It is clear that F ϑ̃ − 1 and F θ̃ − 1 are NFSs in X . By the assumption (4), we get that F ϑ̃ and F θ̃ satisfy the conditions (2.7) and (2.8). Thus, for all x,y ∈X , we get (F θ̃ − 1)(0) = F θ̃(0) − 1 ≤F θ̃(x) − 1 = (F θ̃ − 1)(x), (F ϑ̃ − 1)(0) = F ϑ̃ (0) − 1 ≤F ϑ̃ (x) − 1 = (F ϑ̃ − 1)(x), (F θ̃ − 1)(x) = F θ̃(x) − 1 ≤ max{F θ̃(y),F θ̃(x �y)}− 1 = max{F θ̃(y) − 1,F θ̃(x �y) − 1} = max{(F θ̃ − 1)(y), (F θ̃ − 1)(x �y)}, (F ϑ̃ − 1)(x) = F ϑ̃ (x) − 1 ≤ max{F ϑ̃ (y),F ϑ̃ (x �y)}− 1 = max{F ϑ̃ (y) − 1,F ϑ̃ (x �y) − 1} = max{(F ϑ̃ − 1)(y), (F ϑ̃ − 1)(x �y)}. Hence, F ϑ̃ − 1 and F θ̃ − 1 satisfy that conditions (2.13) and (2.15) that they are NFIds of X . (5) ⇒ (7). It is similar to prove (4) ⇒ (6). (7) ⇒ (3). Let x,y ∈X . Since Fω̃± − 1 = −Fω̃, Fω̃ ∓ − 1 = −Fω̃ and by the assumption (7), we have −Fω̃(0) ≤−Fω̃(x), −Fω̃(0) ≤−Fω̃(x), and −Fω̃(x) ≤ max{−Fω̃(y),−Fω̃(x �y)} = −(min{Fω̃(y),Fω̃(x �y)}), −Fω̃(x) ≤ max{−Fω̃(y),−Fω̃(x �y)} = −(min{Fω̃(y),Fω̃(x �y)}). Thus Fω̃(0) ≥ Fω̃(x), Fω̃(0) ≥ Fω̃(x), Fω̃(x) ≥ min{Fω̃(y),Fω̃(x � y)} and Fω̃(x) ≥ min{Fω̃(y),Fω̃(x � y)}. Hence, Fω̃ and Fω̃ satisfy the conditions (2.5) and (2.6). Therefore, Fω̃ and Fω̃ are FIds of X . � Proposition 3.1. Every IvFId of X is an (inf, sup)-HFId of X . Proof. It follows from Remark 2.1 and Lemma 3.1 � Int. J. Anal. Appl. (2022), 20:34 9 The converse of Proposition 3.1 is not generally true, which can see in Example 3.2. By Proposition 3.1 and Example 3.2, we obtain that an (inf, sup)-HFId of a BCK/BCI-algebra X is a generalization of the concept of an IvFId of X . Theorem 3.1. Let λ̆ be an IvFS on X . Then the followings are equivalent. (1) λ̆ is an IvFId of X . (2) The set [X , λ̆, ă] is an Id of X for all ă ∈D([0, 1]) when [X , λ̆, ă] is not empty. (3) λ̆ is an (inf, sup)-HFId of X . Proof. It follows from Remark 2.1, Lemma 3.1 and Proposition 3.1. � Theorem 3.2. Let ω̃ be a HFS on X . The followings are equivalent. (1) ω̃ is an (inf, sup)-HFId of X . (2) λ̆ is an IvFId of X when λ̆ is an IvFS on X such that λ̆L = Fω̃ and λ̆U = Fω̃. (3) ϑ̃ is an (inf, sup)-HFId of X for all HFS ϑ̃ on X such that F ϑ̃ = Fω̃ and Fϑ̃ = Fω̃. Proof. It follows from Lemma 3.1 and Theorem 3.1. � Proposition 3.2. Let ω̃ be an (inf, sup)-HFId of X and x,y,z ∈ X such that x � y ≤ z. Then Fω̃(x) ≥ min{Fω̃(y),Fω̃(z)} and Fω̃(x) ≥ min{Fω̃(y),Fω̃(z)}. Proof. Since x �y ≤ z, we have (x �y) �z = 0. Thus Fω̃(x) ≥ min{Fω̃(y),Fω̃(x �y)} ≥ min{Fω̃(y), min{Fω̃(z),Fω̃((x �y) �z)}} = min{Fω̃(y), min{Fω̃(z),Fω̃(0)}} = min{Fω̃(y),Fω̃(z)} and similarly, we hve Fω̃(x) ≥ min{Fω̃(y),Fω̃(z)}. � Corollary 3.1. Let λ̆ be an IvFId of X and x,y,z ∈X such that x�y ≤ z. Then rmin{λ̆(y), λ̆(z)}- λ̆(x). Proof. It follows from Proposition 3.2 and Theorem 3.1. � Proposition 3.3. Let ω̃ be an (inf, sup)-HFId of X and x,y ∈ X such that x ≤ y. Then Fω̃(x) ≥ Fω̃(y) and Fω̃(x) ≥Fω̃(y). Proof. Since x ≤ y, we have x �y = 0. Then Fω̃(x) ≥ min{Fω̃(y),Fω̃(x �y)} = min{Fω̃(y),Fω̃(0)} = Fω̃(y), Fω̃(x) ≥ min{Fω̃(y),Fω̃(x �y)} = min{Fω̃(y),Fω̃(0)} = Fω̃(y). Hence, Fω̃(x) ≥Fω̃(y) and Fω̃(x) ≥Fω̃(y). � 10 Int. J. Anal. Appl. (2022), 20:34 Corollary 3.2. Let λ̆ be an IvFId of X and x,y ∈X such that x ≤ y. Then λ̆(y) - λ̆(x). Proof. It follows from Proposition 3.3 and Theorem 3.1. � For any subset A of Y and ∇, ∆ ∈ ℘([0, 1]), define a map C(A,∇, ∆) [21,23] as follows: C(A,∇, ∆) : Y → ℘([0, 1]),x 7→ { ∆ ∇ if x ∈ A, otherwise. We denote C(A) for C(A, [0, 0], [1, 1]) and it is called the characteristic interval-valued fuzzy set of A on X . Theorem 3.3. Let A be a nonempty subset of X and ∇, ∆ ∈ ℘([0, 1]) such that SUP∇ < SUP ∆, INF∇ ≤ INF ∆ or SUP∇ ≤ SUP ∆, INF∇ < INF ∆. Then A is an Id of X if and only if C(A,∇, ∆) is an (inf, sup)-HFId of X . Proof. Since A is an Id of X , we have 0 ∈ A. Then FC(A,∇,∆)(0) = SUP ∆ = max{SUP ∆, SUP∇}≥FC(A,∇,∆)(x), FC(A,∇,∆)(0) = INF ∆ = max{INF ∆, INF∇}≥FC(A,∇,∆)(x) for all x ∈X . Thus FC(A,∇,∆) and FC(A,∇,∆) satisfy the condition (2.5). To show that FC(A,∇,∆) and FC(A,∇,∆) satisfy the condition (2.6), let x,y ∈ X . If y /∈ A or x �y /∈ A, then FC(A,∇,∆)(x) ≥ SUP∇ = min{FC(A,∇,∆)(y),FC(A,∇,∆)(x �y)}, FC(A,∇,∆)(x) ≥ INF∇ = min{FC(A,∇,∆)(y),FC(A,∇,∆)(x �y)}. On the other hand, suppose that y,x �y ∈ A. Since A is an Id of X , we have x ∈ A. Thus FC(A,∇,∆)(x) = SUP ∆ = min{FC(A,∇,∆)(y),FC(A,∇,∆)(x �y)}, FC(A,∇,∆)(x) = INF ∆ = min{FC(A,∇,∆)(y),FC(A,∇,∆)(x �y)}. Hence, FC(A,∇,∆) and FC(A,∇,∆) satisfy the condition (2.6). Therefore, FC(A,∇,∆) and FC(A,∇,∆) are Ids of X and by Lemma 3.1, we obtain that C(A,∇, ∆) is an (inf, sup)-HFId of X . Conversely, let x ∈ X and y,x � y ∈ A. Then C(A,∇, ∆)(y) = ∆ = C(A,∇, ∆)(x � y). If SUP∇ < SUP ∆ and INF∇≤ INF ∆, then by Lemma 3.1, we have FC(A,∇,∆)(0) ≥FC(A,∇,∆)(x) ≥ min{FC(A,∇,∆)(y),FC(A,∇,∆)(x �y)} = SUP ∆ > SUP∇. Thus 0,x ∈ A. In the case that SUP∇≤ SUP ∆ and INF∇ < INF ∆, then by Lemma 3.1, we get FC(A,∇,∆)(0) ≥FC(A,∇,∆)(x) ≥ min{FC(A,∇,∆)(y),FC(A,∇,∆)(x �y)} = INF ∆ > INF∇. Thus 0,x ∈ A. Therefore, A is an Id of X . � Theorem 3.4. Let A be a nonempty subset of X . The followings are equivalent. Int. J. Anal. Appl. (2022), 20:34 11 (1) A is an Id of X . (2) C(A,ă, b̆) is an IvFId of X when ă, b̆ ∈D([0, 1]) and ă ≺ b̆. (3) C(A) is an IvFId of X . Proof. It follows from Theorem 3.3 and Theorem 3.1. � For a FS ζ in Y and a positive integer n, we define the HFS H(ζ,n) and the IvFS I(ζ,n) on Y as follows: H(ζ,n) : Y → ℘([0, 1]),x 7→ { ζ 1 + n (x), n + ζ 1 + n (x)} and I(ζ,n) : Y →D([0, 1]),x 7→ {t ∈ [0, 1] | ζ 1 + n (x) ≤ t ≤ n + ζ 1 + n (x)}. Then the followings are true. (1) SUPH(ζ,n)(x) = SUPI(ζ,n)(x), INFH(ζ,n)(x) = INFI(ζ,n)(x) and H(ζ,n)(x) ⊆ I(ζ,n)(x) for all x ∈Y. (2) H(ζ, 1)(x) = {ζ 2 (x), 1+ζ 2 (x)} and I(ζ, 1)(x) = {t ∈ [0, 1] | ζ 2 (x) ≤ t ≤ 1+ζ 2 (x)} for all x ∈Y. (3) H(−ζ,n) is a HFS and I(−ζ,n) is an IvFS on Y for all NFS ζ in Y. Next, we use (inf, sup)-HFIds and IvFIds of BCK/BCI-algebras to characterize FIds in Theorem 3.5, AFIds in Theorem 3.6 and NFIds in Theorem 3.7. Theorem 3.5. Let ζ be a FS in X . The followings are equivalent. (1) ζ is a FId of X . (2) I(ζ,n) is an IvFId of X for all positive integer n. (3) H(ζ,n) is an (inf, sup)-HFId of X for all positive integer n. (4) ω̃ is an (inf, sup)-HFId of X for all HFS ω̃ on X and positive integer n such that Fω̃ = ζ 1+n and Fω̃ = n+ζ 1+n . Proof. By using Theorem 3.2, the conditions (2), (3) and (4) are equivalent. Next, we show that (1) and (4) are equivalent. Let ω̃ be a HFS on X and n be a positive integer such that Fω̃ = ζ 1+n and Fω̃ = n+ζ 1+n . By the assumption (1), we have Fω̃(0) = ζ(0) 1 + n ≥ ζ(x) 1 + n = Fω̃(x), Fω̃(0) = n + ζ(0) 1 + n ≥ n + ζ(x) 1 + n = Fω̃(x), Fω̃(x) = ζ(x) 1 + n ≥ min{ζ(y),ζ(x �y)} 1 + n = min{ ζ(y) 1 + n , ζ(x �y) 1 + n } = min{Fω̃(y),Fω̃(x �y)}, Fω̃(x) = n + ζ(x) 1 + n ≥ n + min{ζ(y),ζ(x �y)} 1 + n = min{ n + ζ(y) 1 + n , n + ζ(x �y) 1 + n } = min{Fω̃(y),Fω̃(x �y)} 12 Int. J. Anal. Appl. (2022), 20:34 for all x,y ∈ X . Hence, Fω̃ and Fω̃ are FIds of X and by using Lemma 3.1, we obtain that ω̃ is an (inf, sup)-HFId of X . Therefore, (4) is true. Conversely, assume that (4) is true. Let ω̃ be a HFS on X such that Fω̃ = ζ 2 and Fω̃ = 1+ζ 2 . By the assumption (4) and Lemma 3.1, we obtain that Fω̃ = ζ 2 is a FId of X . Then for all x,y ∈X , we get ζ(0) = 2(ζ(0) 2 ) ≥ 2(ζ(x) 2 ) = ζ(x) and ζ(x) = 2( ζ(x) 2 ) ≥ 2( min{ζ(y),ζ(x �y)} 2 ) = min{ζ(y),ζ(x �y)}. Hence, ζ is an Id of X , that is (1) is true. � Lemma 3.2. A FS ζ in X is an AFId of X if and only if 1 −ζ is a FId of X . Proof. Assume that ζ is an AFId of X . Then for all x,y ∈X , we get 1 −ζ(0) ≥ 1 −ζ(x) and 1 −ζ(x) ≥ 1 − max{ζ(y),ζ(x �y)} = min{1 −ζ(y), 1 −ζ(x �y)}. Then 1 −ζ is a FId of X . Conversely, assume that 1 −ζ is a FId of X . Then 1 − (1 −ζ)(0) ≤ 1 − (1 −ζ)(x) and 1 − (1 −ζ)(x) ≤ 1 − min{(1 −ζ)(y), (1 −ζ)(x �y)} = max{1 − (1 −ζ)(y), 1 − (1 −ζ)(x �y)} for all x,y ∈X . Since ζ = 1 − (1 −ζ), we obtain that ζ is an AFId of X . � Theorem 3.6. Let ζ be a FS in X . The followings are equivalent. (1) ζ is an AFId of X . (2) I(1 −ζ,n) is an IvFId of X for all positive integer n. (3) H(1 −ζ,n) is an (inf, sup)-HFId of X for all positive integer n. (4) ω̃ is an (inf, sup)-HFId of X for all HFS ω̃ on X and positive integer n such that Fω̃ = 1−ζ 1+n and Fω̃ = 1 + −ζ 1+n . Proof. It follows from Lemma 3.2 and Theorem 3.5. � Lemma 3.3. A NFS ζ in X is a NFId of X if and only if −ζ is a FId of X . Proof. Assume that ζ is a NFId of X . Let x,y ∈X . Then ζ(0) ≤ ζ(x) and ζ(x) ≤ max{ζ(y),ζ(x � y)}. Thus −ζ(0) ≥−ζ(x) and −ζ(x) ≥−(max{ζ(y),ζ(x �y)}) = min{−ζ(y),−ζ(x �y)}. Hence, −ζ is a FId of X . Int. J. Anal. Appl. (2022), 20:34 13 Conversely, assume that −ζ is a FId of X . Then ζ(0) = −(−ζ(0)) ≤−(−ζ(x)) = ζ(x) and ζ(x) = −(−ζ(x)) ≤−(min{−ζ(y),−ζ(x �y)}) = max{−(−ζ(y)),−(−ζ(x �y))} = max{ζ(y),ζ(x �y)} for all x,y ∈X . Hence, ζ is a NFId of X . � Theorem 3.7. Let ζ be a NFS in X . The followings are equivalent. (1) ζ is a NFId of X . (2) I(−ζ,n) is an IvFId of X for all positive integer n. (3) H(−ζ,n) is an (inf, sup)-HFId of X for all positive integer n. (4) ω̃ is an (inf, sup)-HFId of X for all HFS ω̃ on X and positive integer n such that Fω̃ = −ζ 1+n and Fω̃ = n−ζ 1+n . Proof. It follows from Lemma 3.3 and Theorem 3.5. � For any HFS ω̃ on Y and any element ∇ of ℘([0, 1]), define the HFS Hω̃∇ on Y by Hω̃∇(x) = {t ∈∇ | F ω̃± 2 (x) ≤ t ≤ 1+F ω̃ 2 (x)} for all x ∈Y. We denote Hω̃ for Hω̃ [0,1] . Then Hω̃∇(x) ⊆H ω̃ ∆(x) ⊆H ω̃(x) when x ∈Y and ∇⊆ ∆ ⊆ [0, 1]. Theorem 3.8. Let ω̃ be a HFS on X . The followings are equivalent. (1) ω̃ is an (inf, sup)-HFId of X . (2) Hω̃∇ is a HFId of X for all ∇∈ ℘([0, 1]). (3) Hω̃ is a HFId of X . Proof. (1) ⇒ (2). Let x ∈X , ∇∈ ℘([0, 1]) and t ∈Hω̃∇(x). Then t ∈∇ and F ω̃± 2 (x) ≤ t ≤ 1+F ω̃ 2 (x) . By the assumption (1) and Lemma 3.1, we get Fω̃±(x) ≥Fω̃±(0) and Fω̃(x) ≤Fω̃(0). Thus Fω̃± 2 (0) ≤ Fω̃± 2 (x) ≤ t ≤ 1 + Fω̃ 2 (x) ≤ 1 + Fω̃ 2 (0) and so t ∈Hω̃(0). Hence, Hω̃(x) ⊆Hω̃(0). Therefore, Hω̃ satisfies the condition (2.19). To show that Hω̃ satisfies the condition (2.20), let x,y ∈ X , ∇ ∈ ℘([0, 1]) and t ∈ Hω̃∇(y) ∩ Hω̃∇(x �y). Then t ∈∇, Fω̃± 2 (y) ≤ t ≤ 1+F ω̃ 2 (y) and F ω̃± 2 (x �y) ≤ t ≤ 1+F ω̃ 2 (x �y). 14 Int. J. Anal. Appl. (2022), 20:34 By the assumption (1) and Lemma 3.1, we have Fω̃±(x) ≤ max{Fω̃±(y),Fω̃±(x �y)} and Fω̃(x) ≥ min{Fω̃(y),Fω̃(x �y)}. Thus Fω̃± 2 (x) ≤ max{ Fω̃± 2 (y), Fω̃± 2 (x �y)} ≤ t ≤ min{ 1 + Fω̃ 2 (y), 1 + Fω̃ 2 (x �y)} ≤ 1 + Fω̃ 2 (x), and so t ∈Hω̃∇(x). Hence, H ω̃ ∇(y)∩H ω̃ ∇(x�y) ⊆H ω̃ ∇(x). It is showed that H ω̃ ∇ satisfies the condition (2.20). Therefore, it follows from the conditions (2.19) and (2.20) that Hω̃∇ is a HFId of X for all ∇∈ ℘([0, 1]). (2) ⇒ (3). It is clear. (3) ⇒ (1). Let x,y ∈ X . Then Fω̃± 2 (x), 1+F ω̃ 2 (x) ∈ Hω̃(x) and max{Fω̃± 2 (y), F ω̃± 2 (x � y)}, min{1+F ω̃ 2 (y), 1+F ω̃ 2 (x � y)} ∈ Hω̃(y) ∩ Hω̃(x � y). By the assumption (3), we get F ω̃± 2 (x), 1+F ω̃ 2 (x) ∈ Hω̃(0) and max{Fω̃± 2 (y), F ω̃± 2 (x � y)}, min{1+F ω̃ 2 (y), 1+F ω̃ 2 (x � y)} ∈ Hω̃(x). Thus F ω̃± 2 (x) ≥ Fω̃± 2 (0), 1+F ω̃ 2 (x) ≤ 1+F ω̃ 2 (0), max{Fω̃± 2 (y), F ω̃± 2 (x � y)} ≥ Fω̃± 2 (x) and min{1+F ω̃ 2 (y), 1+F ω̃ 2 (x � y)} ≤ 1+F ω̃ 2 (x). Since Fω̃ = 1 − 2( F ω̃± 2 ) and Fω̃ = 2( 1+F ω̃ 2 ) − 1, we have Fω̃(0) = 2( 1 + Fω̃ 2 (0)) − 1 ≥ 2( 1 + Fω̃ 2 (x)) − 1 = Fω̃(x), Fω̃(0) = 1 − 2( Fω̃± 2 (0)) ≥ 1 − 2( Fω̃± 2 (x)) = Fω̃(x), Fω̃(x) = 2( 1 + Fω̃ 2 (x)) − 1 ≥ 2(min{ 1 + Fω̃ 2 (y), 1 + Fω̃ 2 (x �y)}) − 1 = min{2( 1 + Fω̃ 2 (y)) − 1, 2( 1 + Fω̃ 2 (x �y)) − 1} = min{Fω̃(y),Fω̃(x �y)}, Fω̃(x) = 1 − 2( Fω̃± 2 (x)) ≥ 1 − 2(max{ Fω̃± 2 (y), Fω̃± 2 (x �y)}) = min{1 − 2( Fω̃± 2 (y)), 1 − 2( Fω̃± 2 (x �y))} = min{Fω̃(y),Fω̃(x �y)}. Hence, Fω̃ and Fω̃ are FIds of X and by using Lemma 3.1, we obtain that ω̃ is an (inf, sup)-HFId of X . � Int. J. Anal. Appl. (2022), 20:34 15 Theorem 3.9. Let ω̃ be a HFS on X . The followings are equivalent. (1) ω̃ is an (inf, sup)-HFId of X . (2) (Fω̃,F θ̃) is a PFId of X for all θ̃ ∈ SC(ω̃). (3) (Fω̃,Fω̃ ∓ ) is a PFId of X . (4) (F ω̃ 2 , F ϑ̃ 2 ) is a PFId of X for all ϑ̃ ∈ IC(ω̃). (5) (F ω̃ 2 , F ω̃± 2 ) is a PFId of X . Proof. (1) ⇒ (2) and (1) ⇒ (4). They follow from Lemma 3.1. (2) ⇒ (3) and (4) ⇒ (5). They are clear. (3) ⇒ (1). By the assumption (3), we obtain that Fω̃ is a FId and Fω̃ ∓ is an AFId of X . Since Fω̃ = 1 −Fω̃ ∓ and Lemma 3.2, we get Fω̃ is a FId of X . Hence, Fω̃ and Fω̃ are FIds of X and by using Lemma 3.1, we have that ω̃ is an (inf, sup)-HFId of X . (5) ⇒ (1). It is similar to prove in the case (3) ⇒ (1). � For any PFS P = (ζ,ξ) in Y, define the HFS H(P ) on Y by H(P )(x) = {t ∈ [0, 1] | 1−ξ 2 (x) ≤ t ≤ 1+ζ 2 (x)} for all x ∈Y. Theorem 3.10. Let P = (ζ,ξ) be a PFS in X . The followings are equivalent. (1) P is a PFId of X . (2) H(P ) is an (inf, sup)-HFId of X . (3) H(P ) is an IvFId of X . Proof. It follows from Theorem 3.2 and Lemmas 3.1 and 3.2. � Theorem 3.11. Let ω̃ be a HFS on X . The followings are equivalent. (1) ω̃ is an (inf, sup)-HFId of X . (2) 〈F ϑ̃ − 1,Fω̃〉 is a BFId of X for all ϑ̃ ∈ IC(ω̃). (3) 〈Fω̃± − 1,Fω̃〉 is a BFId of X . Proof. (1) ⇒ (2). It follows from Lemma 3.1. (2) ⇒ (3). It is clear. (3) ⇒ (1). By the assumption (3), we have that Fω̃ is a FId and Fω̃± − 1 is a NFId of X . Since Fω̃ = −(Fω̃± − 1) and Lemma 3.3, we get Fω̃ is a FId of X . Thus Fω̃ and Fω̃ are FIds of X and by using Lemma 3.1, we obtain that ω̃ is an (inf, sup)-HFId of X . � For any BFS B = 〈ζ,ξ〉 on Y, define the HFS H〈B〉 on Y by H〈B〉(x) = {t ∈ [0, 1] | −ζ 2 (x) ≤ t ≤ 1+ξ 2 (x)} for all x ∈Y. Theorem 3.12. Let B = 〈ζ,ξ〉 be a BFS in X . The followings are equivalent. (1) B is a BFId of X . 16 Int. J. Anal. Appl. (2022), 20:34 (2) H〈B〉 is an (inf, sup)-HFId of X . (3) H〈B〉 is an IvFId of X . Proof. It follows from Theorem 3.2 and Lemmas 3.1 and 3.3. � 4. Conclusions In present paper, we have introduced an (inf, sup)-HFId, which is one of genaral concepts of an IvFId, in BCK/BCI-algebras, and investigated its some important properties. As important study results, characterizations of (inf, sup)-HFIds have been discussed by sets, FSs, NFSs, PFSs, HFSs, IvFSs and BFSs. Also, we use concepts of (inf, sup)-HFIds and IvFIds to study characterizations of Ids, FIds, AFIds, NFIds, PFIds and BFIds. In our future study of BCK/BCI-algebras and other algebras, the following objectives considered: • to get more results of HFSs in the meaning of the infimum and supremum of its images, • to define neutrosophic sets in BCK/BCI-algebras and related structures by means of HFSs in the meaning of the infimum and supremum of its images, • to define (inf, sup)-type of HFSs baded on subalgebras, H-ideals and p-ideals of BCK/BCI- algebras, • to introduce (inf, sup)-HFIds in UP-algebras, BE-algebras, semigroups and LA-semigroups. Acknowledgment: This work was supported by (i) University of Phayao (UP), (ii) Thailand Science Research and Innovation (TSRI), and (iii) National Science, Research and Innovation Fund (NSRF) [Grant number: FF65-RIM047]. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. 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Log. 49 (2020), 53-78. https: //doi.org/10.18778/0138-0680.2020.03. https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522171 https://doi.org/10.3792/pja/1195522171 https://doi.org/10.5937/MatMor1802029M https://digitalcommons.pvamu.edu/aam/vol15/iss1/19 https://doi.org/10.1080/16168658.2021.1993668 https://doi.org/10.2298/FIL2012189H https://doi.org/10.29020/nybg.ejpam.v13i1.3575 https://doi.org/10.18778/0138-0680.2020.03 https://doi.org/10.18778/0138-0680.2020.03 1. Introduction 2. Preliminaries 3. Main Results 4. Conclusions References