New Generalizations of sup -Hesitant Fuzzy Ideals of Semigroups

. 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.

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 intervalvalued 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.

Preliminaries
In this section we first give some basic definitions and results which will be used in this paper.
In what follows, unless otherwise specified, let A be a semigroup, B be a nonempty set, ℘(B) be the power set of B and , ∈ ℘([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 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 numberȃ we mean an interval [a − , a + ], where a − , a + ∈ [0, 1] and a − ≤ a + . We are defined by:  [33,40] of A ifλ(p) λ (pq) for all p, q ∈ A, that is,λ L andλ U are FRIs of A, (2) an interval-valued fuzzy left ideal (IvFLI) [33,40] of A ifλ(q) λ (pq) for all p, q ∈ A, that is,λ L andλ U 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,λ L andλ U are FIs of A, and (4) an interval-valued fuzzy interior ideal (IvFII) [40] of A ifλ(w ) λ (pw q) for all p, q, w ∈ A, that is,λ L andλ U are FIIs of A.
A cubic set [19] in B is defined to be a function λ , η : (1) a cubic right ideal (CuRI) [20] of A ifλ is an IvFRI and η is an AFRI of A, (2) a cubic left ideal (CuLI) [20] of A ifλ 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 ifλ is an IvFII and η is an AFII of A.
A hybrid set in A over a set B is defined to be a function ( ε, η) : where ε : A → ℘(B) and η : A → [0, 1]. Note that every cubic set in A is a hybrid set in A over otherwise, for all p ∈ B, Let SC( ε) be the set of all supremum complements of ε. Then, we obtain that (2) f ω (p) = 1 − SUP ε(p) for all ω ∈ SC( ε) and p ∈ B, for all ω ∈ SC( ε) and p ∈ 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. (1) ε is a sup-HFI of A, (2) f ε is a FI of A, (1) ε 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.
3.1. sup + α -hesitant fuzzy ideals. In this part, we introduce a sup + α -hesitant fuzzy right ideal, a sup + αhesitant 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.
Similarly, we can prove the other results.
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 − βhesitant 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.
Similarly, we can prove the other results.
Similarly, we can prove the other results.
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 ε be a HFS on A. Then the followings are true: (1) ε is a sup-HFRI of A if and only if ε(p) ε(pq) for all p, q ∈ A, (2) ε is a sup-HFLI of A if and only if ε(q) ε(pq) for all p, q ∈ A, (3) ε is a sup-HFII of A if and only if ε(w ) ε(pw q) for all p, q, w ∈ A.
Proof. It follows from Proposition 3.6.
. It is directly obtained from taking k = α.
for all p, q ∈ A. Hence ε(p) + α ε(pq) for all p, q ∈ A, which implies that ε is a sup + α -HFRI of A.
. It is directly obtained from taking k = β.
Proof. It follows from Theorems 3.1 and 3.3.
Proof. It follows from Theorems 3.2 and 3.4.
Proof. It follows from Theorems 3.3 and 3.7.
Proof. It follows from Theorems 3.3 and 3.11.
Proof. It follows from Theorems 3.4 and 3.12.
Proof. It follows from Theorems 4.2 and 4.3.
Then it can be easily seen the following conditions: (1) if ε is constant, then ε is sup-constant, (2) if ε is sup-constant, then ε is both sup + α -constant and sup − β -constant.
(1) ⇒ (2). Assume that (1) holds, k ∈ [0, 1] and ε 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 = w 1 q and q = w 2 p for some p, q ∈ A.
Therefore, A is left simple.
The following theorem can be seen in a similar way as in the proof of Theorem 4.5. (1) A is simple, (2) ε is sup + k -constant for every k ∈ [0, 1] and sup + k -HFI ε of A, (3) ε is sup − k -constant for every k ∈ [0, 1] and sup − k -HFI ε of A, (4) ε is sup-constant for every sup-HFI ε of A.