Int. J. Anal. Appl. (2022), 20:58 New Generalizations of sup-Hesitant Fuzzy Ideals of Semigroups Uraiwan Jittburus1, Pongpun Julatha1,∗, Attaphol Pumila1, Napaporn Chunsee2, Aiyared Iampan3, Rukchart Prasertpong4 1Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand 2Faculty of Science and Technology, Uttaradit Rajabhat University, Uttaradit 53000, Thailand 3Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand 4Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan 60000, Thailand ∗Corresponding author: pongpun.j@psru.ac.th Abstract. As general concepts of sup-hesitant fuzzy right (resp., left, interior, two-sided) ideals of semigroups, the concepts of sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideals and sup − β - hesitant fuzzy right (resp., left, interior, two-sided) ideals are introduced and their properties are investigated. Then, the concepts are established by fuzzy sets, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets, Pythagorean fuzzy sets, hesitant fuzzy sets, hybrid sets, interval-valued fuzzy sets and cubic sets. Finally, we characterize which is intra-regular, completely regular, simple semigroups or another type of semigroups in terms of sup+α-type and sup − β -type of hesitant fuzzy sets. 1. Introduction The fuzzy set theory presented by Zadeh [46] has been successfully and widely applied in many areas such as robotics, expert, computer science, finite state machine, control engineering, logic theory, automata theory, group theory, graph theory and semigroup theory. Furthermore, in the literature, a number of concepts of fuzzy sets and their generalizations and extensions have been Received: Aug. 24, 2022. 2010 Mathematics Subject Classification. 03E72, 08A72, 20M12. Key words and phrases. semigroup; sup-hesitant fuzzy ideal; generalized sup-hesitant fuzzy ideal; Łukasiewicz fuzzy set; fuzzy ideal; hesitant fuzzy ideal; interval-valued fuzzy ideal. https://doi.org/10.28924/2291-8639-20-2022-58 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-58 2 Int. J. Anal. Appl. (2022), 20:58 introduced and studied, for instance, Łukasiewicz fuzzy sets [23], Łukasiewicz anti-fuzzy sets [22], anti-type of fuzzy sets [25, 37], negative fuzzy sets [18], bipolar fuzzy sets [29, 48], interval-valued fuzzy sets [47], intuitionistic fuzzy sets [4], Pythagorean fuzzy sets [44,45], rough sets [2,34], hesitant fuzzy sets [41,42], cubic sets [19] and hybrid sets [1,24]. On semigroups, Kuroki [27, 28] applied fuzzy sets to semigroups. Mordeson et al. [30] explained semigroup theory to fuzzy semigroup theory and showed their applications in coding theory, languages and fuzzy finite state machines. Shabir and Nawaz [37], Khan and Asif [25], Julatha and Siripitukdet [17] studied anti-type of fuzzy sets based on ideal theory in semigroups. Chinnadurai and Arulselvam [7] introduced Pythagorean fuzzy sets based on ideal theory in semigroups and investigated their properties. Narayanan and Manikantan [33], and Thillaigovindan and Chinnadurai [40] studied interval- valued fuzzy sets in semigroups. Jun and Khan [20], Umar et al. [43], and Muhiuddin [31] studied cubic sets in semigroups. Anis et al. [1], Elavarasan et al. [9] studied hybrid sets in semigroups. Jun et al. [21] and Talee et al. [39] studied hesitant fuzzy sets in semigroup. Studying hesitant fuzzy sets, in the meaning of the supremum of their images, on semigroups, Jittburus and Julatha [12] introduced sup-hesitant fuzzy ideals of semigroups and investigated properties via sets, fuzzy sets, interval-valued fuzzy sets and hesitant fuzzy sets. Phummee et al. [35] introduced sup-hesitant fuzzy interior ideals of semigroups and studied its properties by sets, fuzzy sets, interval-valued fuzzy sets and hesitant fuzzy sets. Julatha et al. [13] introduced sup-hesitant fuzzy right (left) ideals of semigroups and studied their characterizations in terms of sets, fuzzy sets, Pythagorean fuzzy sets, interval-valued fuzzy sets, hesitant fuzzy sets, cubic sets and hybrid sets. Many researchers have taken intense and eager interest in the novel area of hesitant fuzzy sets on algebraic structures in the meaning of the supremum of their images (see [1,10,12,14–16,32,35,36,38]). As previously stated, it motivated us to study hesitant fuzzy sets on semigroups in the meaning of the supremum of their images. We will introduce concepts of sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideals and sup− β -hesitant fuzzy right (resp., left, interior, two-sided) ideals and investigate their properties. Also, we will show that every sup-hesitant fuzzy right (resp., left, interior, two-sided) ideal of a semigroup is both a sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideal and a sup− β -hesitant fuzzy right (resp., left, interior, two-sided) ideal, but the converse is not true. Later, the concepts will be established by fuzzy sets, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets, Pythagorean fuzzy sets, hesitant fuzzy sets, hybrid sets, interval-valued fuzzy sets and cubic sets. Finally, we will characterize which is intra-regular, left (right) regular, completely regular, left (right) simple and simple semigroups in terms of sup+α-type and sup − β -type of hesitant fuzzy sets. 2. Preliminaries In this section we first give some basic definitions and results which will be used in this paper. Int. J. Anal. Appl. (2022), 20:58 3 In what follows, unless otherwise specified, let A be a semigroup, B be a nonempty set, ℘(B) be the power set of B and H,N ∈ ℘([0, 1]). A nonempty subset B of A is called a right ideal (resp., a left ideal, an interior ideal) of A if BA⊆B (resp., AB ⊆B, ABA⊆B) and an ideal of A if B is both a right ideal and left ideal of A. A fuzzy subset (FS) [46] of B is defined to be a function ξ : B → [0, 1] where [0, 1] is the unit interval. For FSs ξ and η of B, define ξ ≤ η if ξ(p) ≤ η(p) for all p ∈B. A FS ξ of A is called (1) a fuzzy right ideal (FRI) [30] of A if ξ(p) ≤ ξ(pq) for all p,q ∈A, (2) a fuzzy left ideal (FLI) [30] of A if ξ(q) ≤ ξ(pq) for all p,q ∈A, (3) a fuzzy ideal (FI) [30] of A if it is both a FRL and a FLI of A, that is, max{ξ(p),ξ(q)}≤ ξ(pq) for all p,q ∈A, (4) a fuzzy interior ideal (FII) [30] of A if ξ(w) ≤ ξ(pwq) for all p,q,w ∈A, (5) an anti-fuzzy right ideal (AFRI) [37] of A if ξ(pq) ≤ ξ(p) for all p,q ∈A, (6) an anti-fuzzy left ideal (AFLI) [37] of A if ξ(pq) ≤ ξ(q) for all p,q ∈A, (7) an anti-fuzzy ideal (AFI) [37] of A if it is both an AFRI and an AFLI of A, that is, ξ(pq) ≤ min{ξ(p),ξ(q)} for all p,q ∈A, and (8) an anti-fuzzy interior ideal (AFII) [25] of A if ξ(pwq) ≤ ξ(w) for all p,q,w ∈A. A Pythagorean fuzzy set (PFS) P [44, 45] in B is an object having the form P = {(p,ξ(p),η(p)) |p ∈B} where the functions ξ : B → [0, 1] and η : B → [0, 1] denote the degree of membership and the degree of nonmembership, respectively, and 0 ≤ (ξ(p))2 + (η(p))2 ≤ 1 for all p ∈B. For the sake of simplicity, we shall use the symbol (ξ,η) of the PFS {(p,ξ(p),η(p)) | p ∈B}. A PFS (ξ,η) in A is called (1) a Pythagorean fuzzy right ideal (PFRI) [7] of A if ξ is a FRI and η is an AFRI of A, (2) a Pythagorean fuzzy left ideal (PFLI) [7] of A if ξ is a FLI and η is an AFLI of A, (3) a Pythagorean fuzzy ideal (PFI) [7] of A if it is both a PFRI and a PFLI of A, and (4) a Pythagorean fuzzy interior ideal (PFII) [7] of A if ξ is a FII and η is an AFII of A. By an interval number ă we mean an interval [a−,a+], where a−,a+ ∈ [0, 1] and a− ≤ a+. We denote D([0, 1]) for the set of all interval numbers. Then, we obtain D([0, 1]) ⊆ ℘([0, 1]). For ă = [a−,a+], b̆ = [b−,b+] ∈D([0, 1]), the operations -, = and ≺ in case of two elements in D([0, 1]) are defined by: (1) ă - b̆ ⇔ a− ≤ b− and a+ ≤ b+, (2) ă = b̆ ⇔ a− = b− and a+ = b+, and (3) ă ≺ b̆ ⇔ ă - b̆ and ă 6= b̆. An interval-valued fuzzy set (IvFS) [47] on B is defined to be a function λ̆ : B →D([0, 1]), λ̆(p) 7→ [λ̆L(p), λ̆U(p)] where λ̆L and λ̆U are FSs of B such that λ̆L ≤ λ̆U. For FSs ξ and η of B with ξ ≤ η, 4 Int. J. Anal. Appl. (2022), 20:58 we define the IvFS [ξ,η] on B by [ξ,η](p) = [ξ(p),η(p)] for all p ∈B. An IvFS λ̆ = [λ̆L, λ̆U] on A is called (1) an interval-valued fuzzy right ideal (IvFRI) [33,40] of A if λ̆(p) - λ̆(pq) for all p,q ∈A, that is, λ̆L and λ̆U are FRIs of A, (2) an interval-valued fuzzy left ideal (IvFLI) [33,40] of A if λ̆(q) - λ̆(pq) for all p,q ∈A, that is, λ̆L and λ̆U are FLIs of A, (3) an interval-valued fuzzy ideal (IvFI) [33,40] of A if it is both an IvFRI and an IvFLI of A, that is, λ̆L and λ̆U are FIs of A, and (4) an interval-valued fuzzy interior ideal (IvFII) [40] of A if λ̆(w) - λ̆(pwq) for all p,q,w ∈A, that is, λ̆L and λ̆U are FIIs of A. A cubic set [19] in B is defined to be a function 〈λ̆,η〉 : B →D([0, 1])×[0, 1],p 7→ (λ̆(p),η(p)) where λ̆ : B →D([0, 1]) and η : B → [0, 1]. A cubic set 〈λ̆,η〉 in A is called (1) a cubic right ideal (CuRI) [20] of A if λ̆ is an IvFRI and η is an AFRI of A, (2) a cubic left ideal (CuLI) [20] of A if λ̆ is an IvFLI and η is an AFLI of A, (3) a cubic ideal (CuI) [20] of A if it is both a CuRI and a CuLI of A, and (4) a cubic interior ideal (CuII) [31] of A if λ̆ is an IvFII and η is an AFII of A. A hesitant fuzzy set (HFS) [41,42] on B is defined to be a function ε̂ : B → ℘([0, 1]). Note that every IvFS on B is a HFS on B. A HFS ε̂ on A is called (1) a hesitant fuzzy right ideal (HFRI) [21] of A if ε̂(p) ⊆ ε̂(pq) for all p,q ∈A, (2) a hesitant fuzzy left ideal (HFLI) [21] of A if ε̂(q) ⊆ ε̂(pq) for all p,q ∈A, (3) a hesitant fuzzy ideal (HFI) [12, 21] of A if it is both a HFRL and a HFLI of A, that is, ε̂(p) ∪ ε̂(q) ⊆ ε̂(pq) for all p,q ∈A, (4) a hesitant fuzzy interior ideal (HFII) [35,39] of A if ε̂(w) ⊆ ε̂(pwq) for all p,q,w ∈A. A hybrid set in A over a set B is defined to be a function (ε̂,η) : A→ ℘(B)×[0, 1],p 7→ (ε̂(p),η(p)) where ε̂ : A → ℘(B) and η : A → [0, 1]. Note that every cubic set in A is a hybrid set in A over [0, 1]. A hybrid set (ε̂,η) in A over [0, 1] is called (1) a hybrid right ideal (HyRI) [1] of A over [0, 1] if ε̂ is a HFRI and η is an AFRI of A, (2) a hybrid left ideal (HyLI) [1] of A over [0, 1] if ε̂ is a HFLI and η is an AFLI of A, (3) a hybrid ideal (HyI) [1] of A over [0, 1] if it is both a HyRI and a HyLI of A over [0, 1], and (4) a hybrid interior ideal (HyII) [13] of A over [0, 1] if ε̂ is a HFII and η is an AFII of A. For a HFS ε̂ on B, a nonempty subset Z of B, k ∈ [0, 1] and H∈ ℘([0, 1]), we define (1) the element SUPH [12,35] of [0, 1] by SUPH = { supH 0 if H 6= ∅, otherwise, (2) the subset S[ε̂;H] [12,35] of B by S[ε̂;H] = {p ∈B|SUPε̂(p) ≥ SUPH}, Int. J. Anal. Appl. (2022), 20:58 5 (3) the HFS H(ε̂,H) SUP [14,16] on B by H(ε̂,H) SUP (p) = {k ∈H | SUPε̂(p) ≥ k} for all p ∈B, (4) the characteristic hesitant fuzzy set (CHFS) χ̂Z of Z on B by χ̂Z : B → ℘([0, 1]),p 7→ { [0, 1] ∅ if p ∈Z, otherwise, (5) the FS fε̂ of B by fε̂(p) = SUPε̂(p) for all p ∈B, (6) a supremum complement ω̂ [36] of ε̂ onB if ω̂ is a HFS onB such that SUPω̂(p) = 1−SUPε̂(p) for all p ∈B, (7) the HFS ε̂∗ by ε̂∗(p) = {1 − SUPε̂(p)} for all p ∈B. Let SC(ε̂) be the set of all supremum complements of ε̂. Then, we obtain that (1) ε̂∗ ∈SC(ε̂), (2) fω̂(p) = 1 − SUPε̂(p) for all ω̂ ∈SC(ε̂) and p ∈B, (3) SUP(ε̂∗)∗(p) = SUPε̂(p) = fε̂(p) for all p ∈B, (2) SUPε̂(p) = 1 − (1 − SUPε̂(p)) = 1 − SUPω̂(p) for all ω̂ ∈SC(ε̂) and p ∈B, (5) SUPλ̆(p) = sup λ̆(p) = λ̆U(p) for every IvFS λ̆ on B and for all p ∈B, (6) H(ε̂,[0,1]) SUP is both a HFS and an IvFS on B. Jittburus and Julatha [12] introduced a sup-hesitant fuzzy ideal, which is a generalization of the concepts of an IvFI and a HFI, of a semigroup and studied its properties via sets, FSs, HFSs and IvFSs in the following. Definition 2.1. [12] A HFS ε̂ on A is called a sup-hesitant fuzzy ideal of A related to H (briefly, H-sup-hesitant fuzzy ideal) of A if the set S[ε̂;H] is an ideal of A. We say that ε̂ is a sup-hesitant fuzzy ideal (sup-HFI) of A if ε̂ is a H-sup-hesitant fuzzy ideal of A for all H∈ ℘([0, 1]) when S[ε̂;H] 6= ∅. Theorem 2.1. [12] Every HFI of A is a sup-HFI of A. Theorem 2.2. [12] Every IvFI of A is a sup-HFI of A. Theorem 2.3. [12] Let ε̂ be a HFS on A. The followings are equivalent: (1) ε̂ is a sup-HFI of A, (2) fε̂ is a FI of A, (3) SUPε̂(pq) ≥ max{SUPε̂(p), SUPε̂(q)} for all p,q ∈A, (4) H(ε̂,H) SUP is a HFI of A for all H∈ ℘([0, 1]). Theorem 2.4. [12] Let B be a nonempty subset of A. Then B is an ideal of A if and only if the CHFS χ̂B is a sup-HFI of A. Phummee et al. [35] introduced a sup-hesitant fuzzy interior ideal, shown a generalization of the concepts of a sup-HFI, an IvFII and a HFII, of a semigroup and studied its properties via sets, FSs, HFSs and IvFSs. 6 Int. J. Anal. Appl. (2022), 20:58 Definition 2.2. [35] A HFS ε̂ on A is called a sup-hesitant fuzzy interior ideal of A related to H (briefly, H-sup-hesitant fuzzy interior ideal) of A if the set S[ε̂;H] is an interior ideal of A. We say that ε̂ is a sup-hesitant fuzzy interior ideal (sup-HFII) of A if ε̂ is a H-sup-hesitant fuzzy interior ideal of A for all H∈ ℘([0, 1]) when S[ε̂;H] 6= ∅. Theorem 2.5. [35] Every sup-HFI of A is a sup-HFII of A. Theorem 2.6. [35] Every HFII of A is a sup-HFII of A. Theorem 2.7. [35] Every IvFII of A is a sup-HFII of A. Theorem 2.8. [35] Let ε̂ be a HFS on A. The followings are equivalent: (1) ε̂ is a sup-HFII of A, (2) fε̂ is a FII of A, (3) SUPε̂(pwq) ≥ SUPε̂(w) for all p,q,w ∈A, (4) H(ε̂,H) SUP is a HFII of A for all H∈ ℘([0, 1]). Theorem 2.9. [35] If B is a nonempty subset of A, then B is an interior ideal of A if and only if χ̂B is a sup-HFII of A. Julatha et al. [13] introduced a sup-hesitant fuzzy right (left) ideal, shown a generalization of the concept of a HFRI (HFLI) and an IvFRI (IvFLI), of a semigroup and studied its properties via sets, FSs, PFSs, HFSs, IvFSs, cubic sets and hybrid sets. Definition 2.3. [13] Let ε̂ be a HFS on A. (1) ε̂ is called a sup-hesitant fuzzy left ideal (sup-HFLI) of A if (∀p,q ∈ A)(SUPε̂(q) ≤ SUPε̂(pq)). (2) ε̂ is called a sup-hesitant fuzzy right ideal (sup-HFRI) of A if (∀p,q ∈ A)(SUPε̂(p) ≤ SUPε̂(pq)). Theorem 2.10. [13] Let ε̂ be a HFS on A. The followings are equivalent: (1) ε̂ is a sup-HFRI (sup-HFLI) of A, (2) fε̂ is a FRI (FLI) of A, (3) H(ε̂,H) SUP is a HFRI (HFLI) of A for all H∈ ℘([0, 1]). Theorem 2.11. [13] If B is a nonempty subset of A, then B is a right ideal (resp., left ideal) of A if and only if χ̂B is a sup-HFRI (resp., sup-HFLI) of A. 3. Generalized sup-hesitant fuzzy ideals In what follows, let α and β be elements of [0, 1], unless otherwise specified. We introduce the concepts of sup+α-hesitant fuzzy left (resp., right, interior, two-sided) ideals and sup − β -hesitant fuzzy left Int. J. Anal. Appl. (2022), 20:58 7 (resp., right, interior, two-sided) ideals of semigroups and investigate their properties. The concepts are established by FSs, PFSs, HFSs, IvFSs, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets, hybrid sets and cubic sets. 3.1. sup+α-hesitant fuzzy ideals. In this part, we introduce a sup + α-hesitant fuzzy right ideal, a sup + α- hesitant fuzzy left ideal, a sup+α-hesitant fuzzy interior ideal and a sup + α-hesitant fuzzy two-sided ideal of a semigroup, and investigate some of their properties. Also, it is shown that a sup+α-hesitant fuzzy left (resp., right, interior, two-sided) ideal of a semigroup is a generalization of the concept of a sup-hesitant fuzzy left (resp., right, interior, two-sided) ideal. For a HFS ε̂ on A, and H,N∈ ℘([0, 1]), we define (1) SUP+αH = min{SUPH + α, 1}, (2) Hv+α N if and only if SUP + αH≤ SUP + αN, (3) H@+α N if and only if SUP + αH < SUP + αN, (4) H∼=+α N if and only if SUP + αH = SUP + αN. We denote HvN (resp., H@N, H∼= N) for Hv+0 N (resp., H@ + 0 N, H ∼=+0 N). Then we have (1) SUP+0H = SUPH, (2) HvN if and only if SUPH≤ SUPN, (3) H@N if and only if SUPH < SUPN, (4) H∼= N if and only if SUPH = SUPN, (5) H∼=+α N if and only if Hv+α N and Nv+α H. For elements ă = [a−,a+] and b̆ = [b−,b+] in D([0, 1]), then the following are true: (1) if ă - b̆, then ă v b̆, and (2) if ă = b̆, then ă ∼= b̆. Definition 3.1. A HFS ε̂ on A is called (1) a sup+α-hesitant fuzzy right ideal (sup + α-HFRI) of A if (∀p,q ∈A)(ε̂(p) v+α ε̂(pq)), (2) a sup+α-hesitant fuzzy left ideal (sup + α-HFLI) of A if (∀p,q ∈A)(ε̂(q) v+α ε̂(pq)), (3) a sup+α-hesitant fuzzy two-sided ideal (or a sup + α-hesitant fuzzy ideal (sup + α-HFI)) of A if it is both a sup+α-HFRI and a sup + α-HFLI of A, (4) a sup+α-hesitant fuzzy interior ideal (sup + α-HFII) of A if (∀p,q,w ∈A)(ε̂(w) v+α ε̂(pwq)). Example 3.1. Let A = {(1, 1), (0, 1), (0, 0), (1, 0)}. Then A is a semigroup with respect to multipli- cation defined as follows: (p1,p2)(p3,p4) = (p1,p4) for all p1,p2,p3, p4 ∈{0, 1}. (1) A HFS ε̂1 of A is defined by ε̂1((0, 0)) = (0, 0.8), ε̂1((0, 1)) = {0.2, 0.4, 0.9}, ε̂1((1, 0)) = ∅ and ε̂1((1, 1)) = {0}. Then ε̂1 is a sup + 0.2-HFRI of A but not a sup + 0.2-HFLI of A because ε̂1((1, 0)(0, 0)) @ + 0.2 ε̂1((0, 0)). 8 Int. J. Anal. Appl. (2022), 20:58 (2) A HFS ε̂2 of A is defined by ε̂2((0, 0)) = {0.2, 0.4, 0.5}, ε̂2((0, 1)) = (0.3, 0.7), ε̂2((1, 0)) = [0, 0.5] and ε̂2((1, 1)) = {0.7, 0.8, 0.9}. Then ε̂2 is a sup + 0.3-HFLI of A but not a sup + 0.3-HFRI of A because ε̂2((1, 1)(1, 0)) @ + 0.3 ε̂2((1, 1)). Example 3.2. Let A = {p1,p2,p3,p4} and define the binary operation “ ·” on A as follows: · p1 p2 p3 p4 p1 p1 p1 p1 p1 p2 p1 p1 p1 p1 p3 p1 p1 p2 p1 p4 p1 p1 p2 p2 Then A is a be the semigroup under the binary operation “ ·” [30]. Now, define HFSs ε̂1 and ε̂2 on A by ε̂1(p1) = [0.3, 0.6], ε̂1(p2) = {0.3, 0.5}, ε̂1(p3) = ∅ and ε̂1(p4) = {1}, ε̂2(p1) = [0.3, 0.7], ε̂2(p2) = {0.3, 0.5, 0.8}, ε̂2(p3) = ∅ and ε̂2(p4) = [0.2, 0.5]. Thus (1) ε̂1 is a sup + 0.4-HFII but not a sup + 0.4-HFI of A because ε̂1(p4p4) @ + 0.4 ε̂1(p4), (2) ε̂2 is a sup + 0.3-HFI of A. Proposition 3.1. Every sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A is a sup+α-HFRI (resp., sup+α-HFLI, sup + α-HFII, sup + α-HFI) of A for all α ∈ [0, 1]. Proof. Assume that ε̂ is a sup-HFRI of A, α ∈ [0, 1] and p,q ∈A. Then SUPε̂(pq) ≥ SUPε̂(p) and so SUP+αε̂(pq) = min{SUPε̂(pq) + α, 1}≥ min{SUPε̂(p) + α, 1} = SUP + αε̂(p). Hence ε̂(p) v+α ε̂(pq). Therefore, we obtain that ε̂ is a sup+α-HFRI of A. Similarly, we can prove the other results. � Example 3.3. Let A = {p1,p2,p3,p4} be the semigroup defined in Example 3.2. We define a HFS ε̂ on A by ε̂(p1) = {0.1, 0.5, 0.8}, ε̂(p2) = [0, 0.9], ε̂(p3) = [0, 0.5] and ε̂(p4) = ∅. Then ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A for all α ∈ [0.2, 1] but ε̂ is not a sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A. Indeed, ε̂ is not a sup-HFRI and sup-HFI of A because SUPε̂(p2p1) < SUPε̂(p2), ε̂ is not a sup-HFLI of A because SUPε̂(p1p2) < SUPε̂(p2), and ε̂ is not a sup-HFII of A because SUPε̂(p1p2p3) < SUPε̂(p2). Int. J. Anal. Appl. (2022), 20:58 9 From Proposition 3.1 and Example 3.3, we have that the concept of a sup+α-HFRI (resp., sup + α- HFLI, sup+α-HFII, sup + α-HFI) of a semigroup A is a generalization of the concept of a sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A. Proposition 3.2. Let ε̂ be a HFS on A and k ∈ [0, 1]. If ε̂ is a sup+α-HFRI (resp., sup+α-HFLI, sup+α- HFII, sup+α-HFI) of A for all α ∈ [0,k], then ε̂ is a sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A. Proof. Let ε̂ be a sup+α-HFRI of A for all α ∈ [0,k]. Suppose ε̂ is not a sup-HFRI of A, that is, there exist p,q ∈A such that SUPε̂(pq) < SUPε̂(p). Choose α = min{ SUPε̂(p) − SUPε̂(pq) 2 ,k}. Then α ∈ [0,k] and SUPε̂(pq) + α ≤ SUPε̂(pq) + ( SUPε̂(p) − SUPε̂(pq) 2 ) < SUPε̂(pq) + (SUPε̂(p) − SUPε̂(pq)) = SUPε̂(p) ≤ 1. Thus SUP+αε̂(p) = min{SUPε̂(p) + α, 1} > SUPε̂(pq) + α = min{SUPε̂(pq) + α, 1} = SUP+αε̂(pq). Hence ε̂(pq) @+α ε̂(p). Since ε̂ is a sup + α-HFRI of A, we hvae ε̂(pq) @+α ε̂(p) v + α ε̂(pq). This is a contradiction. Therefore, ε̂ is a sup-HFRI of A. Similarly, we can prove the other results. � Proposition 3.3. If ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, then ε̂ is a sup+ k -HFRI (resp., sup+ k -HFLI, sup+ k -HFII, sup+ k -HFI) of A for all k ∈ [α, 1]. Proof. Assume that ε̂ is a sup+α-HFRI of A, k ∈ [α, 1] and p,q ∈ A. Then ε̂(p) v+α ε̂(pq), that is, min{SUPε̂(pq) + α, 1}≥ min{SUPε̂(p) + α, 1}. If SUPε̂(pq) + α ≥ 1, then SUPε̂(pq) + k ≥ SUPε̂(pq) + α ≥ 1 ≥ SUP+ k ε̂(p) 10 Int. J. Anal. Appl. (2022), 20:58 and so ε̂(p) v+ k ε̂(pq). On the other hand, suppose that SUPε̂(pq) + α ≥ SUPε̂(p) + α. Then SUPε̂(pq) ≥ SUPε̂(p) and so SUPε̂(pq) + k ≥ SUPε̂(p) + k ≥ SUP+ k ε̂(p). Thus ε̂(p) v+ k ε̂(pq). Therefore, ε̂ is a sup+ k -HFRI of A. Similarly, we can prove the other results. � Proposition 3.4. Every sup+α-HFI of A is a sup+α-HFII of A. Proof. Assume that ε̂ is a sup+α-HFI of A. Then ε̂(w) v+α ε̂(wq) v+α ε̂(pwq) for all p,q,w ∈ A. Therefore, ε̂ is a sup+α-HFII of A. � From Proposition 3.4 and Example 3.2, we have that the concept of a sup+α-HFII of a semigroup A is a generalization of the concept of a sup+α-HFI of A. 3.2. sup− β -hesitant fuzzy ideals. In this part, we introduce a sup− β -hesitant fuzzy right ideal, a sup− β - hesitant fuzzy left ideal, a sup− β -hesitant fuzzy interior ideal and a sup− β -hesitant fuzzy two-sided ideal of a semigroup, and investigate some of their properties. Moreover, it is shown that a sup− β -hesitant fuzzy left (resp., right, interior, two-sided) ideal of a semigroup is a generalization of the concept of a sup-hesitant fuzzy left (resp., right, interior, two-sided) ideal. For a HFS ε̂ on A and H,N∈ ℘([0, 1]), we define (1) SUP− β H = max{SUPH−β, 0}, (2) Hv− β N if and only if SUP− β H≤ SUP− β N, (3) H@− β N if and only if SUP− β H < SUP− β N, (4) H∼=−β N if and only if SUP − β H = SUP− β N. Then we have (1) SUP−0H = SUPH, (2) HvN if and only if Hv−0 N, (3) H@N if and only if H@−0 N, (4) H∼= N if and only if H∼=−0 N. Definition 3.2. Let ε̂ be a HFS on A. (1) ε̂ is called a sup− β -hesitant fuzzy right ideal (sup− β -HFRI) of A if (∀p,q ∈A)(ε̂(p) v− β ε̂(pq)), (2) ε̂ is called a sup− β -hesitant fuzzy left ideal (sup− β -HFLI) of A if (∀p,q ∈A)(ε̂(q) v− β ε̂(pq)), (3) ε̂ is called a sup− β -hesitant fuzzy two-sided ideal (or a sup− β -hesitant fuzzy ideal (sup− β -HFI)) of A if it is both a sup− β -HFRI and a sup− β -HFLI of A, (4) ε̂ is called a sup− β -hesitant fuzzy interior ideal (sup− β -HFII) of A if (∀p,q,w ∈ A)(ε̂(w) v− β ε̂(pwq)). Example 3.4. Let A = {(1, 1), (0, 1), (0, 0), (1, 0)} be a semigroup defined in Example 3.1. Int. J. Anal. Appl. (2022), 20:58 11 (1) A HFS ε̂1 of A is defined by ε̂1((0, 0)) = (0, 0.8], ε̂1((0, 1)) = {0.4, 0.8}, ε̂1((1, 0)) = ∅ and ε̂1((1, 1)) = [0, 0.4]. Then ε̂1 is a sup − 0.4-HFRI of A but not a sup − 0.4-HFLI of A because ε̂1((1, 0)(0, 1)) @ − 0.4 ε̂1((0, 1)). (2) A HFS ε̂2 of A is defined by ε̂2((0, 0)) = {0.2, 0.4, 0.5}, ε̂2((0, 1)) = (0.3, 0.6), ε̂2((1, 0)) = ∅ and ε̂2((1, 1)) = {0.4, 0.5, 0.6}. Then ε̂2 is a sup − 0.5-HFLI of A but not a sup − 0.5-HFRI of A because ε̂2((0, 1)(1, 0)) @ − 0.5 ε̂2((0, 1)). Example 3.5. Let A = {p1,p2,p3,p4} be the semigroup defined in Example 3.2. We define a HFS ε̂ on A by ε̂(p1) = [0.3, 0.7], ε̂(p2) = {0.3, 0.5}, ε̂(p3) = ∅, and ε̂(p4) = (0, 0.7). Thus (1) ε̂ is a sup− β -HFII of A for all β ∈ [0, 1] but not a sup− k -HFI of A for all k ∈ [0, 0.7) because ε̂(p4p3) @ − k ε̂(p4) for all k ∈ [0, 0.7). (2) ε̂ is a sup− β -HFI of A for all β ∈ [0.7, 1]. Proposition 3.5. Every sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A for all β ∈ [0, 1]. Proof. Assume that ε̂ is a sup-HFRI of A, β ∈ [0, 1] and p,q ∈A. Then SUPε̂(pq) ≥ SUPε̂(p) and thus SUP− β ε̂(pq) = max{SUPε̂(pq) −β, 0}≥ max{SUPε̂(p) −β, 0} = SUP− β ε̂(p). Hence ε̂(p) v− β ε̂(pq). Therefore, ε̂ is a sup− β -HFRI of A. Similarly, we can prove the other results. � Example 3.6. From Example 3.3, we get that the HFS ε̂ is a sup− β -HFRI, sup− β -HFLI, sup− β -HFII and sup− β -HFI of A for all β ∈ [0.9, 1]. However, ε̂ is not a sup-HFRI, sup-HFLI, sup-HFII and sup-HFI of A. From Example 3.6 and Proposition 3.5, we obtain that the concept of a sup− β -HFRI (resp., sup− β - HFLI, sup− β -HFII, sup− β -HFI) of a semigroup A is a generalization of the concept of a sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A. Proposition 3.6. Let ε̂ be a HFS on A and k ∈ [0, 1]. If ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β - HFII, sup− β -HFI) of A for all β ∈ [0,k], then ε̂ is a sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A. 12 Int. J. Anal. Appl. (2022), 20:58 Proof. Let ε̂ be a sup− β -HFRI of A for all β ∈ [0,k]. Suppose that SUPε̂(pq) < SUPε̂(p) for some p,q ∈A. Choose β ∈ [0,k] such that SUPε̂(p) −β > 0. Then SUP− β ε̂(p) = max{SUPε̂(p) −β, 0} = SUPε̂(p) −β. Since ε̂ is a sup− β -HFRI of A, we have SUP− β ε̂(pq) ≥ SUP− β ε̂(p) = SUPε̂(p) −β > 0. Thus SUP− β ε̂(pq) = SUPε̂(pq) −β < SUPε̂(p) −β = SUP− β ε̂(p) ≤ SUP− β ε̂(pq), which is a contradiction. Hence ε̂ is a sup-HFRI of A. Similarly, we can prove the other results. � Proposition 3.7. If ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, then ε̂ is a sup− k -HFRI (resp., sup− k -HFLI, sup− k -HFII, sup− k -HFI) of A for all k ∈ [β, 1]. Proof. Assume that ε̂ is a sup− β -HFRI of A, k ∈ [β, 1] and p,q ∈A. Then ε̂(p) v− β ε̂(pq), that is, max{SUPε̂(pq) −β, 0}≥ max{SUPε̂(p) −β, 0}≥ SUPε̂(p) −β. If 0 ≥ SUPε̂(p) −β, then SUP− k ε̂(pq) ≥ 0 ≥ SUPε̂(p) −β ≥ SUPε̂(p) −k and so SUP− k ε̂(pq) ≥ SUP− k ε̂(p), which implies that ε̂(p) v− k ε̂(pq). On the other hand, suppose that SUPε̂(pq) −β ≥ SUPε̂(p) −β. Then SUPε̂(pq) ≥ SUPε̂(p). Thus SUPε̂(pq) −k ≥ SUPε̂(p) −k. Hence SUP− k ε̂(pq) ≥ SUP− k ε̂(p), which implies that ε̂(p) v− k ε̂(pq). Therefore, ε̂ is a sup− k -HFRI of A. Similarly, we can prove the other results. � Proposition 3.8. Every sup− β -HFI of A is a sup− β -HFII of A. Proof. Assume that ε̂ is a sup− β -HFI of A. Then ε̂(w) v− β ε̂(wq) v− β ε̂(pwq) for all p,q,w ∈ A. Therefore, ε̂ is a sup− β -HFII of A. � From Proposition 3.8 and Example 3.5, we have that the concept of a sup− β -HFII of a semigroup A is a generalization of the concept of a sup− β -HFI of A. Proposition 3.9. Let ε̂ be a HFS on A. Then the followings are true: Int. J. Anal. Appl. (2022), 20:58 13 (1) ε̂ is a sup-HFRI of A if and only if ε̂(p) v ε̂(pq) for all p,q ∈A, (2) ε̂ is a sup-HFLI of A if and only if ε̂(q) v ε̂(pq) for all p,q ∈A, (3) ε̂ is a sup-HFII of A if and only if ε̂(w) v ε̂(pwq) for all p,q,w ∈A. Proof. It follows from Proposition 3.6. � Proposition 3.10. Let ε̂ be a HFS on A. Then the followings are equivalent: (1) ε̂ is a sup-HFRI (resp., sup-HFLI, sup-HFII, sup-HFI) of A, (2) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A for all β ∈ [0, 1], (3) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A for all α ∈ [0, 1]. Proof. It follows from Propositions 3.1, 3.2, 3.5 and 3.6. � 3.3. Fuzzy sets, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets and Pythagorean fuzzy sets. In this part, we characterize sup+α-HFRIs, sup + α-HFLIs, sup + α-HFIIs, sup + α-HFIs, sup − β -HFRIs, sup− β -HFLIs, sup− β -HFIIs, and sup− β -HFIs of semigroups in terms of FSs, PFSs, Łukasiewicz fuzzy sets and Łukasiewicz anti-fuzzy sets. For a FS ξ of A, consider the FS ξ+α : A→ [0, 1],p 7→ min{ξ(p) + α, 1}, which is called an α-Łukasiewicz anti-fuzzy set [22] of ξ in A. In case that 0 ≤ α ≤ 1 − sup{ξ(p) | p ∈A}, the Łukasiewicz anti-fuzzy set ξ+α is called a fuzzy α-translation [8] of ξ of type I. For a FS ξ of A, consider the FS ξ− β : A→ [0, 1],p 7→ max{ξ(p) −β, 0}. Then ξ− β (p) = max{ξ(p) + (1−β)−1, 0} for all p ∈A and so ξ− β is called an 1−β-Łukasiewicz fuzzy set [23] of ξ in A. In case that 0 ≤ β ≤ inf{ξ(p) | p ∈ A}, the Łukasiewicz fuzzy set ξ− β is called a fuzzy β-translation [8] of ξ of type II. Then we have the following results: (1) ξ−0 = ξ = ξ + 0 , (2) ξ− β ≤ ξ ≤ ξ+α, (3) ξ− k ≤ ξ− β for all k ∈ [β, 1], (4) ξ+ k ≤ ξ+α for all k ∈ [0,α], (5) the FS (fε̂)+α is an α-Łukasiewicz anti-fuzzy set of f ε̂ in A and (fε̂)+α (p) = SUP + αε̂(p) for each HFS ε̂ on A and p ∈A, (6) the FS (fε̂)− β is an 1−β-Łukasiewicz fuzzy set of fε̂ in A and (fε̂)− β (p) = SUP− β ε̂(p) for each HFS ε̂ on A and p ∈A. Theorem 3.1. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, (2) (fε̂)+α is a FRI (resp., FLI, FII, FI) of A, and 14 Int. J. Anal. Appl. (2022), 20:58 (3) (fε̂)+ k is a FRI (resp., FLI, FII, FI) of A for all k ∈ [α, 1]. Proof. (1) ⇒ (3). Assume that ε̂ is a sup+α-HFRI of A, k ∈ [α, 1] and p,q ∈A. Then ε̂(p) v+α ε̂(pq) and so min{SUPε̂(p) + α, 1} ≤ SUPε̂(pq) + α. If min{SUPε̂(p) + α, 1} = SUPε̂(p) + α, then SUPε̂(p) ≤ SUPε̂(pq). Thus (fε̂)+ k (pq) = SUP+ k ε̂(pq) ≥ SUP+ k ε̂(p) = (fε̂)+ k (p) and so (fε̂)+ k (pq) ≥ (fε̂)+ k (p). On the other hand, suppose that min{SUPε̂(p) + α, 1} = 1. Then SUPε̂(pq) + k ≥ SUPε̂(pq) + α ≥ 1 and so (fε̂)+ k (pq) = SUP+ k ε̂(pq) = 1 ≥ (fε̂)+ k (p). Hence (fε̂)+ k (pq) ≥ (fε̂)+ k (p). Therefore, (fε̂)+ k is a FRI of A for all k ∈ [α, 1]. (3) ⇒ (2). It is directly obtained from taking k = α. (2) ⇒ (1). Assume that (fε̂)+α is a FRI of A. Then (fε̂)+α (pq) ≥ (fε̂)+α (p) for all p,q ∈A. Thus SUP+αε̂(pq) = (f ε̂)+α (pq) ≥ (f ε̂)+α (p) = SUP + αε̂(p) for all p,q ∈A. Hence ε̂(p) v+α ε̂(pq) for all p,q ∈A, which implies that ε̂ is a sup+α-HFRI of A. � Theorem 3.2. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, (2) (fε̂)− β is a FRI (resp., FLI, FII, FI) of A, and (3) (fε̂)− k is a FRI (resp., FLI, FII, FI) of A for all k ∈ [β, 1]. Proof. (1) ⇒ (3). Assume that ε̂ is a sup− β -HFRI of A, k ∈ [β, 1] and p,q ∈A. If 0 ≥ SUPε̂(p)−β, then 0 ≥ SUPε̂(p) −k and so (fε̂)− k (pq) ≥ 0 = SUP− k ε̂(p) = (fε̂)− k (p). Thus (fε̂)− k (pq) ≥ (fε̂)− k (p). On the other hand, suppose that SUPε̂(p) −β > 0. Since ε̂ is a sup− β - HFRI of A, we have ε̂(p) v− β ε̂(pq). Then SUPε̂(pq) − β ≥ SUPε̂(p) − β and so SUPε̂(pq) ≥ SUPε̂(p). Thus (fε̂)− k (pq) = SUP− k ε̂(pq) ≥ SUP− k ε̂(p) = (fε̂)− k (p). Hence (fε̂)− k (pq) ≥ (fε̂)− k (p). Therefore, (fε̂)− k is a FRI of A for all k ∈ [β, 1]. (3) ⇒ (2). It is directly obtained from taking k = β. (2) ⇒ (1). Assume that (fε̂)− β is a FRI of A and p,q ∈A. Then (fε̂)− β (pq) ≥ (fε̂)− β (p) and so SUP− β ε̂(pq) = (fε̂)− β (pq) ≥ (fε̂)− β (p) = SUP− β ε̂(p). Hence ε̂(p) v− β ε̂(pq). Therefore, ε̂ is a sup− β -HFRI of A. � Int. J. Anal. Appl. (2022), 20:58 15 Lemma 3.1. Let ε̂ be a HFS on A. Then SUP+ k ω̂(p) = 1−SUP− k ε̂(p) and SUP− k ω̂(p) = 1−SUP+ k ε̂(p) for all p ∈A, ω̂ ∈SC(ε̂) and k ∈ [0, 1]. Proof. Let p ∈A, ω̂ ∈SC(ε̂) and k ∈ [0, 1]. Then SUP+ k ω̂(p) = min{SUPω̂(p) + k, 1} = min{(1 − SUPε̂(p)) + k, 1} = min{1 − (SUPε̂(p) −k), 1} = 1 − max{SUPε̂(p) −k, 0} = 1 − SUP− k ε̂(p) and SUP− k ω̂(p) = max{SUPω̂(p) −k, 0} = max{(1 − SUPε̂(p)) −k, 1 − 1} = max{1 − (SUPε̂(p) + k), 1 − 1} = 1 − min{SUPε̂(p) + k, 1} = 1 − SUP+ k ε̂(p). Therefore, SUP+ k ω̂(p) = 1 − SUP− k ε̂(p) and SUP− k ω̂(p) = 1 − SUP+ k ε̂(p). � Theorem 3.3. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, (2) (fε̂ ∗ )−α is an AFRI (resp., AFLI, AFII, AFI) of A, (3) (fω̂)−α is an AFRI (resp., AFLI, AFII, AFI) of A for all ω̂ ∈SC(ε̂), and (4) (fω̂)− k is an AFRI (resp., AFLI, AFII, AFI) of A for all ω̂ ∈SC(ε̂) and k ∈ [α, 1]. Proof. (1) ⇒ (4). Assume that ε̂ is a sup+α-HFRI of A. By Proposition 3.3, we have that ε̂ is a sup+ k -HFRI of A for all k ∈ [α, 1]. By Lemma 3.1, we get (fω̂)− k (p) = SUP− k ω̂(p) = 1 − SUP+ k ε̂(p) ≥ 1 − SUP+ k ε̂(pq) = SUP− k ω̂(pq) = (fω̂)− k (pq) for all ω̂ ∈ SC(ε̂), k ∈ [α, 1] and p,q ∈ A. Hence (fω̂)− k is an AFRI of A for all ω̂ ∈ SC(ε̂) and k ∈ [α, 1]. (4) ⇒ (3) and (3) ⇒ (2). They are clear. 16 Int. J. Anal. Appl. (2022), 20:58 (2) ⇒ (1). Assume that (fε̂ ∗ )−α is an AFRI of A and p,q ∈ A. Then (fε̂ ∗ )−α (p) ≥ (fε̂ ∗ )−α (pq) and by Lemma 3.1, we have SUP+αε̂(p) = 1 − SUP − αε̂ ∗(p) = 1 − (fε̂ ∗ )−α (p) ≤ 1 − (fε̂ ∗ )−α (pq) = 1 − SUP−αε̂ ∗(pq) = SUP+αε̂(pq). Hence ε̂(p) v+α ε̂(pq). Therefore, ε̂ is a sup+α-HFRI of A. � Theorem 3.4. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, (2) (fε̂ ∗ )+ β is an AFRI (resp., AFLI, AFII, AFI) of A, (3) (fω̂)+ β is an AFRI (resp., AFLI, AFII, AFI) of A for all ω̂ ∈SC(ε̂), and (4) (fω̂)+ k is an AFRI (resp., AFLI, AFII, AFI) of A for all ω̂ ∈SC(ε̂) and for all k ∈ [β, 1]. Proof. (1) ⇒ (4). Assume that ε̂ is a sup− β -HFRI of A. By Proposition 3.7 and Lemma 3.1, we get (fω̂)+ k (pq) = SUP+ k ω̂(pq) = 1 − SUP− k ε̂(pq) ≤ 1 − SUP− k ε̂(p) = SUP+ k ω̂(p) = (fω̂)+ k (p) for all ω̂ ∈ SC(ε̂), k ∈ [β, 1] and p,q ∈ A. Hence (fω̂)+ k is an AFRI of A for all ω̂ ∈ SC(ε̂) and k ∈ [β, 1]. (4) ⇒ (3) and (3) ⇒ (2). They are clear. (2) ⇒ (1). Assume that (fε̂ ∗ )− β is an AFRI of A. By Lemma 3.1, we get SUP− β ε̂(pq) = 1 − SUP+ β ε̂∗(pq) = 1 − (fε̂ ∗ )+ β (pq) ≥ 1 − (fε̂ ∗ )+ β (p) = 1 − SUP+ β ε̂∗(p) = SUP− β ε̂(p) for all p,q ∈A. Thus ε̂(p) v− β ε̂(pq) for all p,q ∈A, which implies that ε̂ is a sup− β -HFRI of A. � Theorem 3.5. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, (2) ((fε̂)+α, (f ε̂∗)−α ) is a PFRI (resp., PFLI, PFII, PFI) of A, (3) ((fε̂)+α, (f ω̂)−α ) is a PFRI (resp., PFLI, PFII, PFI) of A for all ω̂ ∈SC(ε̂), and (4) ((fε̂)+ k , (fω̂)− k ) is a PFRI (resp., PFLI, PFII, PFI) of A for all ω̂ ∈SC(ε̂) and k ∈ [α, 1]. Int. J. Anal. Appl. (2022), 20:58 17 Proof. It follows from Theorems 3.1 and 3.3. � Theorem 3.6. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, (2) ((fε̂)− β , (fε̂ ∗ )+ β ) is a PFRI (resp., PFLI, PFII, PFI) of A, (3) ((fε̂)− β , (fω̂)+ β ) is a PFRI (resp., PFLI, PFII, PFI) of A for all ω̂ ∈SC(ε̂), and (4) ((fε̂)− k , (fω̂)+ k ) is a PFRI (resp., PFLI, PFII, PFI) of A for all ω̂ ∈SC(ε̂) and k ∈ [β, 1]. Proof. It follows from Theorems 3.2 and 3.4. � 3.4. Hesitant fuzzy sets and hybrid sets. In this part, we characterize sup+α-HFRIs, sup + α-HFLIs, sup+α-HFIIs, sup + α-HFIs, sup − β -HFRIs, sup− β -HFLIs, sup− β -HFIIs and sup− β -HFIs of semigroups in terms of HFSs and hybrid sets. Theorem 3.7. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, and (2) H(ε̂,H) SUP is a HFRI (resp., HFLI, HFII, HFI) of A for all H∈ ℘([0, 1 −α]). Proof. (1) ⇒ (2). Assume that ε̂ is a sup+α-HFRI of A. Let H ∈ ℘([0, 1 − α]), p,q ∈ A and k ∈H(ε̂,H) SUP (p). Then ε̂(p) v+α ε̂(pq) and SUPε̂(p) ≥ k ∈H. Thus SUPε̂(pq) = (SUPε̂(pq) + α) −α ≥ SUP+αε̂(pq) −α ≥ SUP+αε̂(p) −α = min{SUPε̂(p), 1 −α} ≥ k which implies that k ∈H(ε̂,H) SUP (pq). Hence H(ε̂,H) SUP (p) ⊆H(ε̂,H) SUP (pq). Therefore, H(ε̂,H) SUP is a HFRI of A for all H∈ ℘([0, 1 −α]). (2) ⇒ (1). Assume that H(ε̂,H) SUP is a HFRI of A for all H ∈ ℘([0, 1 − α]). Let p,q ∈ A and H = [0, 1 −α]. Then SUP+αε̂(p) −α = min{SUPε̂(p), 1 −α}∈H (ε̂,H) SUP (p) ⊆H(ε̂,H) SUP (pq). Thus SUP+αε̂(pq) −α = min{SUPε̂(pq), 1 −α}≥ SUP + αε̂(p) −α. Hence SUP+αε̂(pq) ≥ SUP + αε̂(p) which implies that ε̂(p) v+α ε̂(pq). Therefore, ε̂ is a sup+α-HFRI of A. � Theorem 3.8. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, 18 Int. J. Anal. Appl. (2022), 20:58 (2) H(ε̂,H) SUP is a HFRI (resp., HFLI, HFII, HFI) of A for all H∈ ℘((β, 1]). Proof. (1) ⇒ (2). Assume that ε̂ is a sup− β -HFRI ofA. LetH∈ ℘((β, 1]), p,q ∈Aand k ∈H(ε̂,H) SUP (p). Then SUPε̂(p) ≥ k > β, k ∈H and ε̂(p) v− β ε̂(pq). Thus max{SUPε̂(pq),β} = SUP− β ε̂(pq) + β ≥ SUP− β ε̂(p) + β = max{SUPε̂(p),β} ≥ k > β, that is, SUPε̂(pq) ≥ k. Hence k ∈H(ε̂,H) SUP (pq) and so H(ε̂,H) SUP (p) ⊆H(ε̂,H) SUP (pq). Therefore, H(ε̂,H) SUP is a HFRI of A for all H∈ ℘((β, 1]). (2) ⇒ (1). Assume that H(ε̂,H) SUP is a HFRI of A for all H∈ ℘((β, 1]) and p,q ∈A. If SUPε̂(p) ≤ β, then SUP− β ε̂(p) = 0 ≤ SUP− β ε̂(pq) and so ε̂(p) v− β ε̂(pq). On the other hand, suppose that SUPε̂(p) > β. Let H = (β, 1]. Then SUP− β ε̂(p) + β = max{SUPε̂(p),β} = SUPε̂(p) ∈H(ε̂,H) SUP (p) ⊆H(ε̂,H) SUP (pq). Thus SUP− β ε̂(pq) + β ≥ (SUPε̂(pq) −β) + β = SUPε̂(pq) ≥ SUP− β ε̂(p) + β. Hence SUP− β ε̂(p) ≤ SUP−αε̂(pq) and so ε̂(p) v − β ε̂(pq). Therefore, ε̂ is a sup− β -HFRI of A. � Theorem 3.9. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, (2) (H(ε̂,H) SUP , (fε̂ ∗ )−α ) is a HyRI (resp., HyLI, HyII, HyI) of A over [0, 1] for all H∈ ℘([0, 1 −α]), (3) (H(ε̂,H) SUP , (fω̂)−α ) is a HyRI (resp., HyLI, HyII, HyI) of A over [0, 1] for all ω̂ ∈ SC(ε̂) and H∈ ℘([0, 1 −α]), and (4) (H(ε̂,H) SUP , (fω̂)− k ) is a HyRI (resp., HyLI, HyII, HyI) of A over [0, 1] for all ω̂ ∈SC(ε̂), k ∈ [α, 1] and H∈ ℘([0, 1 −α]). Proof. It follows from Theorems 3.3 and 3.7. � Theorem 3.10. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, (2) (H(ε̂,H) SUP , (fε̂ ∗ )+ β ) is a HyRI (resp., HyLI, HyII, HyI) of A over [0, 1] for all H∈ ℘((β, 1]), (3) (H(ε̂,H) SUP , (fω̂)+ β ) is a HyRI (resp., HyLI, HyII, HyI) of A over [0, 1] for all ω̂ ∈ SC(ε̂) and H∈ ℘((β, 1]), and Int. J. Anal. Appl. (2022), 20:58 19 (4) (H(ε̂,H) SUP , (fω̂)+ k ) is a HyRI (resp., HyLI, HyII, HyI) of A over [0, 1] for all ω̂ ∈SC(ε̂), k ∈ [β, 1] and H∈ ℘((β, 1]). Proof. It follows from Theorems 3.4 and 3.8. � 3.5. Interval-valued fuzzy sets and cubic sets. In this part, we characterize sup+α-HFRIs, sup + α- HFLIs, sup+α-HFIIs, sup + α-HFIs, sup − β -HFRIs, sup− β -HFLIs, sup− β -HFIIs, and sup− β -HFIs of semigroups in terms of IvFSs and cubic sets. Let ξ and η be FSs of A such that ξ ≤ η, the followings are true. (1) [ξ,ξ], [ξ− β ,η], [ξ,η+α ] and [ξ − β ,η+α ] are IvFSs on A. (2) If α ≥ β, then [ξ−α,η − β ] is an IvFS on A. (3) If α ≤ β, then [ξ+α,η + β ] is an IvFS on A. (4) If λ̆ is an IvFS on A and λ̆ = [ξ,η], then λ̆L = ξ and λ̆U = η. Theorem 3.11. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, (2) [(fε̂)+α, (f ε̂)+ k ] is an IvFRI (resp., IvFLI, IvFII, IvFI) of A for all k ∈ [α, 1], (3) [(fε̂)+ k , (fε̂)+ β ] is an IvFRI (resp., IvFLI, IvFII, IvFI) of A for all k,β ∈ [α, 1] with k ≤ β, (4) H(ε̂,[0,1−α]) SUP is an IvFRI (resp., IvFLI, IvFII, IvFI) of A. Proof. (1) ⇔ (2) and (1) ⇔ (3). It follows from Theorem 3.1. (1) ⇒ (4). Assume that ε̂ is a sup+α-HFRI of A and p,q ∈ A. Then ε̂(p) v+α ε̂(pq) and so SUP+αε̂(p) ≤ SUP + αε̂(pq). Thus H(ε̂,[0,1−α]) SUP (p) = [0, SUP+αε̂(p) −α] - [0, SUP + αε̂(pq) −α] = H (ε̂,[0,1−α]) SUP (pq). Therefore, H(ε̂,[0,1−α]) SUP is an IvFRI of A. (4) ⇒ (1). Assume that H(ε̂,[0,1−α]) SUP is an IvFRI of A and p,q ∈A. Then [0, SUP+αε̂(p) −α] = H (ε̂,[0,1−α]) SUP (p) -H(ε̂,[0,1−α]) SUP (pq) = [0, SUP+αε̂(pq) −α]. Thus SUP+αε̂(p) = (SUP + αε̂(p) −α) + α ≤ (SUP + αε̂(pq) −α) + α = SUP + αε̂(pq). Hence ε̂(p) v+α ε̂(pq). Therefore, ε̂ is a sup+α-HFRI of A. � Theorem 3.12. For a HFS ε̂ on A, the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, (2) [(fε̂)− k , (fε̂)− β ] is an IvFRI (resp., IvFLI, IvFII, IvFI) of A for all k ∈ [β, 1], and (3) [(fε̂)− k , (fε̂)−α ] is an IvFRI (resp., IvFLI, IvFII, IvFI) of A for all k,α ∈ [β, 1] with α ≤ k. Proof. It follows from Theorem 3.2. � 20 Int. J. Anal. Appl. (2022), 20:58 Theorem 3.13. Let ε̂ be a HFS on A and λ̆ = H(ε̂,[0,1−α]) SUP . Then the followings are equivalent: (1) ε̂ is a sup+α-HFRI (resp., sup + α-HFLI, sup + α-HFII, sup + α-HFI) of A, (2) 〈λ̆, (fε̂ ∗ )−α〉 is a CuRI (resp., CuLI, CuII, CuI) of A, (3) 〈λ̆, (fω̂)−α〉 is a CuRI (resp., CuLI, CuII, CuI) of A for all ω̂ ∈SC(ε̂), (4) 〈λ̆, (fω̂)− k 〉 is a CuRI (resp., CuLI, CuII, CuI) of A for all ω̂ ∈SC(ε̂) and k ∈ [α, 1]. Proof. It follows from Theorems 3.3 and 3.11. � Theorem 3.14. Let ε̂ be a HFS on A, k ∈ [β, 1] and λ̆ = [(fε̂)− k , (fε̂)− β ]. Then the followings are equivalent: (1) ε̂ is a sup− β -HFRI (resp., sup− β -HFLI, sup− β -HFII, sup− β -HFI) of A, (2) 〈λ̆, (fε̂ ∗ )+ β 〉 is a CuRI (resp., CuLI, CuII, CuI) of A, (3) 〈λ̆, (fω̂)+ β 〉 is a CuRI (resp., CuLI, CuII, CuI) of A for all ω̂ ∈SC(ε̂). Proof. It follows from Theorems 3.4 and 3.12. � 4. Characterizing semigroups by sup+α-type and sup − β -type of HFSs In this section, we characterize intra-regular, completely regular, left (right) regular, left (right) simple and simple semigroups and groups in terms of sup+α-type and sup − β -type of HFSs. A semigroup A is called (1) intra-regular if for each w ∈A, there exist p,q ∈A such that w = pw2q, (2) completely regular if for each p ∈A there exists q ∈A such that p = pqp and pq = qp, (3) left regular if for each p ∈A there exists q ∈A such that p = qp2, (4) right regular if for each p ∈A there exists q ∈A such that p = p2q, (5) left simple if A = B for each left ideal B of A, (6) right simple if A = B for each right ideal B of A, (7) simple if A = B for each ideal B of A, (8) group if it is both left simple and right simple. It is well-known that A is completely regular if and only if it is both left and right regular. Proposition 4.1. Let ε̂ be a HFS on an intra-regular semigroup A. Then ε̂ is a sup+α-HFII (resp., sup− β -HFII) of A if and only if ε̂ is a sup+α-HFI (resp., sup − β -HFI) of A. Proof. (⇒). Assume that ε̂ is a sup+α-HFII of A and p,q ∈ A. Then there exist w1,w2,w3,w4 ∈ A such that p = w1p2w2 and q = w3q2w4. Thus ε̂(p) v+α ε̂(w1p2w2q) = ε̂(pq) and ε̂(q) v+α ε̂(pw3q 2w4) = ε̂(pq). Therefore, ε̂ is a sup+α-HFI of A. (⇐). It follows from Proposition 3.4. � Theorem 4.1. Let A be a semigroup. The followings are equivalent: Int. J. Anal. Appl. (2022), 20:58 21 (1) A is intra-regular, (2) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -HFI ε̂ of A and p ∈A, (3) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -HFII ε̂ of A and p ∈A, (4) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -HFI ε̂ of A and p ∈A, (5) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -HFII ε̂ of A and p ∈A, (6) ε̂(p) ∼= ε̂(p2) for each sup-HFI ε̂ of A and p ∈A, (7) ε̂(p) ∼= ε̂(p2) for each sup-HFII ε̂ of A and p ∈A. Proof. (7) ⇒ (6). It follows from Proposition 3.4. (2) ⇔ (4) ⇔ (6) and (3) ⇔ (5) ⇔ (7). They follow from Proposition 3.10. (1) ⇒ (2). Assume that (1) holds, k ∈ [0, 1], ε̂ is a sup+ k -HFI of A and p ∈ A. There exist q,w ∈A such that p = qp2w. Thus ε̂(p) v+ k ε̂(p2) v+ k ε̂(p2w) v+ k ε̂(qp2w) = ε̂(p). Hence ε̂(p) ∼=+k ε̂(p 2). Therefore, ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -HFI ε̂ of A and p ∈A. (1) ⇒ (3). It is similar to prove (1) ⇒ (2). (7) ⇒ (1). Assume that (7) holds and p ∈ A. Then J[p2] = {p2} ∪ Ap2 ∪ p2A ∪ Ap2A is an interior ideal of A and by Theorem 2.9, χ̂J[p2] is a sup-HFII of A. Since p2 ∈ J[p2], we get χ̂J[p2](p) ∼= χ̂J[p2](p2) = [0, 1] which implies that χ̂J[p2](p) = [0, 1]. Thus p ∈ J[p2] and so p ∈Ap2A. Hence A is intra-regular. (6) ⇒ (1). It is similar to prove (7) ⇒ (1). � Proposition 4.2. Let ε̂ be a HFS of an intra-regular semigroup A. Then the followings are true: (1) ε̂(pq) ∼=+α ε̂(qp) for each sup+α-HFII of A and p,q ∈A, (2) ε̂(pq) ∼=+α ε̂(qp) for each sup+α-HFI of A and p,q ∈A. Proof. (1). Let ε̂ be a sup+α-HFII of A and p,q ∈A. By Theorem 4.1, we have ε̂(qp) v+α ε̂(p(qp)q) = ε̂((pq) 2) ∼=+α ε̂(pq) v + α ε̂(q(pq)p) = ε̂((qp) 2) ∼=+α ε̂(qp). Thus ε̂(pq) ∼=+α ε̂(qp). (2). It follows from (1) and Proposition 4.1. � Similarly we can prove the following theorem. Proposition 4.3. Let ε̂ be a HFS of an intra-regular semigroup A. Then the followings are true: (1) ε̂(pq) ∼=−β ε̂(qp) for each sup − β -HFII of A and p,q ∈A, (2) ε̂(pq) ∼=−β ε̂(qp) for each sup − β -HFI of A and p,q ∈A. Theorem 4.2. Let A be a semigroup. The followings are equivalent: (1) A is left regular, 22 Int. J. Anal. Appl. (2022), 20:58 (2) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -HFLI of A and p ∈A, (3) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -HFLI of A and p ∈A, (4) ε̂(p) ∼= ε̂(p2) for each sup-HFLI of A and p ∈A. Proof. (2) ⇔ (3) ⇔ (4). It follows from Proposition 3.10. (1) ⇒ (2). Assume that (1) holds, k ∈ [0, 1], ε̂ is a sup+ k -HFLI of A and p ∈ A. Since A is left regular, there exists q ∈A such that p = qp2. Thus, by ε̂ is a sup+ k -HFLI of A, we obtain ε̂(p) v+ k ε̂(p2) v+ k ε̂(qp2) = ε̂(p). Hence ε̂(p) ∼=+k ε̂(p 2). Therefore ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -HFLI of A and p ∈A. (4) ⇒ (1). Assume that (4) holds and p ∈ A. Then L[p2] = {p2}∪Ap2 is a left ideal of A and by Theorem 2.11, we get that χ̂L[p2] is a sup-HFLI of A. Since p2 ∈ L[p2], we have χ̂L[p2](p) ∼= χ̂L[p2](p 2) = [0, 1] and so χ̂L[p2](p) = [0, 1]. Thus p ∈ L[p2] which implies that p ∈Ap2. Therefore, A is left regular. � The left-right dual of Theorem 4.2 reads as follows: Theorem 4.3. Let A be a semigroup. The followings are equivalent: (1) A is right regular, (2) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -HFRI ε̂ of A and p ∈A, (3) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -HFRI ε̂ of A and p ∈A, (4) ε̂(p) ∼= ε̂(p2) for each sup-HFRI ε̂ of A and p ∈A. Theorem 4.4. Let A be a semigroup. The followings are equivalent: (1) A is completely regular, (2) ε̂(p) ∼=+k ε̂(p 2) and ω̂(p) ∼=+k ω̂(p 2) for each k ∈ [0, 1], sup+ k -HFRI ε̂, sup+ k -HFLI ω̂ of A and p ∈A, (3) ε̂(p) ∼=−k ε̂(p 2) and ω̂(p) ∼=−k ω̂(p 2) for each k ∈ [0, 1], sup− k -HFRI ε̂, sup− k -HFLI ω̂ of A and p ∈A, (4) ε̂(p) ∼= ε̂(p2) and ω̂(p) ∼= ω̂(p2) for each sup-HFRI ε̂, sup-HFLI ω̂ of A and p ∈A. Proof. It follows from Theorems 4.2 and 4.3. � A HFS ε̂ on A is called (1) constant if ε̂(p) = ε̂(q) for all p,q ∈A, (2) sup+α-constant if ε̂(p) ∼=+α ε̂(q) for all p,q ∈A, (3) sup− β -constant if ε̂(p) ∼=−β ε̂(q) for all p,q ∈A, and (4) sup-constant if ε̂(p) ∼= ε̂(q) for all p,q ∈A. Then it can be easily seen the following conditions: (1) if ε̂ is constant, then ε̂ is sup-constant, Int. J. Anal. Appl. (2022), 20:58 23 (2) if ε̂ is sup-constant, then ε̂ is both sup+α-constant and sup − β -constant. Theorem 4.5. Let A be a semigroup. The followings are equivalent: (1) A is left simple, (2) ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -HFLI ε̂ of A, (3) ε̂ is sup− k -constant for every k ∈ [0, 1] and sup− k -HFLI ε̂ of A, (4) ε̂ is sup-constant for every sup-HFLI ε̂ of A. Proof. (2) ⇔ (3) ⇔ (4). It follows from Proposition 3.10. (1) ⇒ (2). Assume that (1) holds, k ∈ [0, 1] and ε̂ is a sup+ k -HFLI of A. Let p,q ∈A. Since A is left simple, we have p ∈ A = Aq and q ∈ A = Ap. Thus p = w1q and q = w2p for some p,q ∈ A. Since ε̂ is a sup+ k -HFLI of A, we get ε̂(p) v+ k ε̂(w2p) = ε̂(q) v+k ε̂(w1q) = ε̂(p). Then ε̂(p) ∼=+k ε̂(q). Hence ε̂ is sup + k -constant. Therefore, we obtain that ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -HFLI ε̂ of A. (4) ⇒ (1). Assume that (4) holds. Let L be a left ideal of A and w ∈ L. Then, by Theorem 2.11, we have χ̂L is sup-HFLI of A. By assumption (4), we get that χ̂L is sup-constant. Thus χ̂L(p) ∼= χ̂L(w) = [0, 1] foll p ∈ A which implies that χ̂L(p) = [0, 1] for all p ∈ A. Hence A = L. Therefore, A is left simple. � The left-right dual of Theorem 4.5 reads as follows: Theorem 4.6. Let A be a semigroup. The followings are equivalent: (1) A is right simple, (2) ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -HFRI ε̂ of A, (3) ε̂ is sup− k -constant for every k ∈ [0, 1] and sup− k -HFRI ε̂ of A, (4) ε̂ is sup-constant for every sup-HFRI ε̂ of A. The following theorem can be seen in a similar way as in the proof of Theorem 4.5. Theorem 4.7. Let A be a semigroup. The followings are equivalent: (1) A is simple, (2) ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -HFI ε̂ of A, (3) ε̂ is sup− k -constant for every k ∈ [0, 1] and sup− k -HFI ε̂ of A, (4) ε̂ is sup-constant for every sup-HFI ε̂ of A. From Theorems 4.5 and 4.6, we have the following theorem. Theorem 4.8. Let A be a semigroup. The followings are equivalent: (1) A is group, 24 Int. J. Anal. Appl. (2022), 20:58 (2) ε̂ and ω̂ are sup+ k -constant for every k ∈ [0, 1], sup+ k -HFLI ε̂ and sup+ k -HFRI ω̂ of A, (3) ε̂ and ω̂ are sup− k -constant for every k ∈ [0, 1], sup− k -HFLI ε̂ and sup− k -HFRI ω̂ of A, (4) ε̂ and ω̂ are sup-constant for every sup-HFLI ε̂ and sup-HFRI ω̂ of A. 5. Conclusions and Future Works In present paper, we have introduced the concepts of sup+α-HFRIs (resp., sup + α-HFLIs, sup + α-HFIIs, sup+α-HFIs) and sup − β -HFRIs (resp., sup− β -HFLIs, sup− β -HFIIs, sup− β -HFIs) which are generalizations of the concepts of sup-HFRIs (resp., sup-HFLIs, sup-HFIIs, sup-HFIs) of semigroups, and discussed their some properties. Furthermore, the concepts have been established by FSs, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets, PFSs, HFSs, hybrid sets, IvFSs and cubic sets. Finally, we have charac- terized intra-regular, left (right) regular, completely regular, left (right) simple and simple semigroups in terms of sup+α-type and sup − β -type of hesitant fuzzy sets. The following are objectives for study and research in semigroups and other algebras: • to introduce and study sup+α-type and sup − β -type of HFSs based on bi-ideals of semigroups, • to introduce and study sup+α-type and sup − β -type of HFSs based on ideal theory in BCK/BCI- algebras, ternary semigroups, Γ-semigroups and LA-semigroups, • to introduce and study sup+α-type and sup − β -type of HFSs based on substructures of GE- algebras, BRK-algebras, BE-algebras and IUP-algebras [5,6,11,26], • to apply this study to the concept of rough sets according to Ansari’s study [2,3]. Acknowledgment: This work was supported by Research Development and Research Management Fund of Pibulsongkram Rajabhat University for 2022 [Grant number RDI-2-65-19]. 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The Industrial Fuzzy Control and Intelligent Systems Conference, and the NASA Joint Technology Wo, IEEE, San Antonio, TX, USA, 1994: pp. 305–309. https://doi.org/10.1109/IJCF.1994.375115. https://digitalcommons.pvamu.edu/aam/vol14/iss1/32 https://digitalcommons.pvamu.edu/aam/vol14/iss1/32 https://doi.org/10.1007/BF02831952 https://doi.org/10.1007/BF01001956 https://doi.org/10.24507/ijicic.18.01.121 https://doi.org/10.24507/ijicic.18.01.121 https://doi.org/10.28924/2291-8639-20-2022-34 https://doi.org/10.1080/16168658.2021.1993668 https://doi.org/10.1002/int.20418 https://doi.org/10.1109/FUZZY.2009.5276884 https://doi.org/10.1002/int.21584 https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375 https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375 https://doi.org/10.1016/S0019-9958(65)90241-X https://doi.org/10.1016/S0019-9958(65)90241-X https://doi.org/10.1016/0020-0255(75)90036-5 https://doi.org/10.1109/IJCF.1994.375115 1. Introduction 2. Preliminaries 3. Generalized sup-hesitant fuzzy ideals 3.1. sup+-hesitant fuzzy ideals 3.2. sup--hesitant fuzzy ideals 3.3. Fuzzy sets, Łukasiewicz fuzzy sets, Łukasiewicz anti-fuzzy sets and Pythagorean fuzzy sets 3.4. Hesitant fuzzy sets and hybrid sets 3.5. Interval-valued fuzzy sets and cubic sets 4. Characterizing semigroups by sup+-type and sup--type of HFSs 5. Conclusions and Future Works Acknowledgment: References