EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3223-3241 ISSN 1307-5543 – ejpam.com Published by New York Business Global Characterizations Regular and Intra-regular Ordered Semigroups by using Generalized Interval Valued Bipolar Fuzzy Quasi-Ideals Thiti Gaketem1, Tanaphong Prommai2,∗ 1,2 Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand Abstract. In this article, we introduce the concept of a generalized interval-valued bipolar fuzzy quasi-ideal and investigate its properties. We explore the relationship between generalized interval- valued bipolar fuzzy quasi-ideals and generalized interval-valued bipolar fuzzy ideals. Furthermore, we characterize regular and intra-regular ordered semigroups by utilizing generalized interval valued bipolar fuzzy quasi-ideals. 2020 Mathematics Subject Classifications: 06F05, 06D72, 08A72 Key Words and Phrases: Ordered semigroup, interval valued bipolar fuzzy quasi-ideal, interval valued bipolar fuzzy ideals 1. Introduction The tool used phenomena of renowned vagueness and uncertainty of data scientists by L. A. Zadeh in 1965 [15]. The theory of fuzzy semigroups was contained by Kuroki in 1979 [10]. Later the theory of interval valued fuzzy sets was introduced by L. A. Zadeh in 1975 [16] as a generalization of the notion of fuzzy sets. Interval valued fuzzy sets have various applications in several areas like medical science [5], image processing [8], decision making [18], etc. In 2006, Narayanan and Manikantan [13] developed the theory of interval valued fuzzy subsemigroup and studied types interval valued fuzzy ideals in semigroups. In 1994 Zhang [17] introduced the notion of bipolar fuzzy sets with the extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [−1, 1], and used them for modeling and decision analysis. In 2000, Lee [11] used the term bipolar valued fuzzy sets and applied it to algebraic structures. In 2016, Mumtaz Ali et al. extended the concept of interval valued fuzzy set and bipolar fuzzy set to interval valued bipolar fuzzy set. In 2019, K. Arulmozhi et al. studied interval valued bipolar fuzzy set in ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5423 Email addresses: thiti.ga@up.ac.th (T. Gaketem), , tanaphong.pr@up.ac.th (T. Prommai) https://www.ejpam.com 3223 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3224 algebra structure. In 2019, A. Salm el at [4] characterize of regular ordered semigroups by (ε, ε,∨k, qk))-fuzzy quasi-ideals. In 2021, S. Lekkoksung [7] developed interval valued bipolar fuzzy ideal in ordered semigroup and characterized regular ordered semigroup in terms of generalized interval valued bipolar fuzzy ideal and bi-ideal. In 2024 P. Khamrot et. al [14] characterized weakly ordered semigroup in terms of generalized interval valued bipolar fuzzy ideal. There are also research studies related to ordered semigroup like fuzzy (m,n)-substructures [3], fuzzy (m,n)-ideal [1], fuzzy (m,n)-filters [2], fuzzy prime subset [12], etc. In this paper, we establish the concept of a generalized interval valued bipolar fuzzy quasi ideal. We prove properties of a generalized interval valued bipolar fuzzy quasi ideal in semigroups. Main results, we will characterize regular and intra-regular ordered semigroup in terms of generalized interval valued bipolar fuzzy quasi ideal. 2. Preliminaries In this section, we give some definitions and theory helpful in later sections. An ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. For a nonempty subset X and Y of ordered semigroup S, we write (X] := {a ∈ S | a ≤ b for some b ∈ X} and XY := {xy | x ∈ X and y ∈ Y }. A non-empty subset L of an ordered semigroup G is called (1) a subsemigroup of G if L2 ⊆ L, (2) a left (right) ideal of G if (GL] ⊆ L ((LG] ⊆ L) and x ∈ L and s ∈ G such that s ≤ x, then s ∈ L, that is (L] ⊆ L, (3) a bi-ideal of G if L is a subsemigorup and LGL ⊆ L, (4) an quasi-ideal of G if (LG] ∩ (GL] ⊆ L An ordered semigroup G is called a regular if, for each u ∈ G, there exists x ∈ G such that u ≤ uxu. An ordered semigroup G called an intra-regular if, for each u ∈ G, there exists a.b ∈ G such that u ≤ au2b. For any pi ∈ [0, 1], where i ∈ A, define ∨ i∈A pi := sup i∈A {pi} and ∧ i∈A pi := inf i∈A {pi}. We see that for any p, q ∈ [0, 1], we have p ∨ q = max{p, q} and p ∧ q = min{p, q}. A fuzzy set of a non-empty set T is a function ω : L → [0, 1]. Let Ω[0, 1] be the set of all closed subintervals of [0, 1], i.e., Ω[0, 1] = {ω = [ω−, ω+] | 0 ≤ ω− ≤ ω+ ≤ 1}. T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3225 We note that [ω, ω] = {ω} for all ω ∈ [0, 1]. For ω = 0 or 1 we shall denote [0, 0] by 0 and [1, 1] by 1. Let ω = [ω−, ω+] and ϖ = [ϖ−, ϖ+] ∈ Ω[0, 1]. Define the operations ⪯, =, ⋏ and ⋎ as follows: (1) ω ⪯ ϖ if and only if ω− ≤ ϖ− and ω+ ≤ ϖ+ (2) ω = ϖ if and only if ω− = ϖ− and ω+ = ϖ+ (3) ω ⋏ϖ = [(ω− ∧ϖ−), (ω+ ∧ϖ+)] (4) ω ⋎ϖ = [(ω− ∨ϖ−), (ω+ ∨ϖ+)]. If ω ⪰ ϖ, we mean ϖ ⪯ ω. For each interval ωi = [ω− i , ω + i ] ∈ Ω[0, 1], i ∈ A where A is an index set, we define ⋏ i∈A ωi = [ ∧ i∈A ω− i , ∧ i∈A ω+ i ] and ⋎ i∈A ωi = [ ∨ i∈A ω− i , ∨ i∈A ω+ i ]. Definition 1. [13] Let T be a non-empty set. Then the function f : T → Ω[0, 1] is called interval valued fuzzy set (shortly, IVF set) of T . Definition 2. [13] Let M be a subset of a non-empty set G. An interval valued charac- teristic function of M is defined to be a function χM : G → Ω[0, 1] by χM (e) = { 1 if e ∈ M, 0 if e /∈ M for all e ∈ G. Now, we review the definition of bipolar valued fuzzy set and the basic properties used in the next section. Definition 3. [11] Let T be a non-empty set. A bipolar fuzzy set (BF set) ω on T is an object having the form ω := {(k, ωp(k), ωn(k)) | k ∈ T}, where ωp : T → [0, 1] and ωn : T → [−1, 0]. Remark 1. For the sake of simplicity we shall use the symbol ω = (T ;ωp, ωn) for the BF set ω = {(k, ωp(k), ωn(k)) | k ∈ T}. The following example of a BF set. T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3226 Example 1. Let T = {21, 22, 23...}. Define ωp : T → [0, 1] is a function ωp(u) = { 0 if u is old number 1 if u is even number and ωn : T → [−1, 0] is a function ωn(u) = { −1 if u is old number 0 if u is even number. Then ω = (T ;ωp, ωn) is a BF set. For k ∈ T , define Fk = {(y, z) ∈ T × T | k = yz}. Definition 4. [6] Let M be a non-empty set of a semigroup T . A positive characteristic function and a negative characteristic function are respectively defined by χp M : T → [0, 1], k 7→ λp M (u) := { 1 k ∈ M , 0 k /∈ M , and χn M : T → [−1, 0], k 7→ λn M (k) := { −1 k ∈ M , 0 k /∈ M . Remark 2. For the sake of simplicity we shall use the symbol χM = (T ;χp M , χn M ) for the BF set χI := {(k, χp I(k), χ n I (k)) | k ∈ I}. Now, we review the definition of an interval valued bipolar fuzzy set and the basic properties used in the next section. Definition 5. [7] An interval valued bipolar fuzzy set (shortly, IVBF subset) T on an ordered semigroup G is form T := {e, ω p(e), ω n(e) | e ∈ G}, where ω p : G → Ω[0, 1] and ω n : G → Ω[−1, 0]. In this page we shall use the symbol T = (ω p, ω n) instead of the IVBF set T := {e, ω p(e), ω n(e) | e ∈ G}. For two IVBF sets T 1 = (ω p, ω n) and T 2 = (ϖ p, ϖ n) of an ordered semigroup G, define (1) T 1 ⊑ T 2 if and only if ω p(e) ≤ ϖ p(e) and ω n(e) ≤ ϖ n(e) for all e ∈ G, (2) T 1 = T 1 if and only if T 1 ⊑ T 2 and T 2 ⊑ T 1, (3) T 1⊔T 2 if and only if ω∪ϖ where (ω p∪ϖ p)(e) = ω p(e)∨ϖ p(e) and (ω n∪ϖ n)(e) = ω n(e) ∧ϖ n(e) for all e ∈ G, T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3227 (4) T 1⊓T 2 if and only if ω∩ϖ where (ω p∩ ϖ p)(e) = ω p(e)∧ϖ p(e) and (ω n∩ϖ n)(e) = ω n(e) ∨ϖ n(e)for all e ∈ G, (5) T 1◦T 2 if and on if ω ◦ϖ where (ω p ◦ϖ p)(e) =  ∨ (t,h)∈Fe {ω p(t) ∧ϖ p(h)} if Fe ̸= ∅, 0 if Fe = ∅, and (ω n ◦ϖ n)(e) =  ∧ (t,h)∈Fe {ω n(t) ∨ϖ n(h)} if Fe ̸= ∅, 0 if Fe = ∅, where Fe := {(t, h) ∈ G×G | e ≤ th} for all e ∈ G. Definition 6. [7] Let M be a non-empty set of an ordered semigroup G. An interval valued bipolar characteristic function are respectively defined by χ p M : G → Ω[0, 1], e 7→ χ p I(e) := { 1 e ∈ M , 0 e /∈ M , and χ n M : G → Ω[−1, 0], e 7→ χ n I (e) := { −1 e ∈ M , 0 e /∈ M . Remark 3. For the sake of simplicity we shall use the symbol χm = (G;χ p M , χ n M ) for the IVBF set χM := {(k, χ p M (k), χ n M (k)) | k ∈ M}. Now, we let λ p, δ p ∈ Ω[0, 1] be such that 0 ≤ λ p < δ p ≤ 1 and λ n, δ n ∈ Ω[−1, 0] be such that −1 ≤ δ n < λ n ≤ 1. Both λ, δ are arbitrary but fixed. Definition 7. [7] Let G be an ordered semigroup and T = (ω p, ω n) be an IVBF set of G is called an (λ, δ)-IVBF subsemigroup of G if (1) ω p(e1e2) ∨ λ p ≥ ω p(e1) ∧ ω p(e2) ∧ δ p. (2) ω n(e1e2) ∧ λ n ≤ ω n(e1) ∨ ω n(e2)⋏ δ n. for all e1, e2 ∈ G. Definition 8. [7] Let G be an ordered semigroup and T = (ω p, ω n) be an IVBF set of G is called an (λ, δ)-IVBF left ideal of G if (1) ω p(e1e2) ∨ λ p ≥ ω p(e2)⋏ δ p. (2) ω n(e1e2) ∧ λ n ≤ ω n(e2) ∨ δ n. (3) If e1 ≤ e2, then ω p(e1) ∨ λ p ≥ ω p(e2) ∧ δ p and ω n(e1) ∧ λ n ≤ ω n(e2) ∨ δ n. T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3228 for all e1, e2 ∈ G. Definition 9. [7] Let G be an ordered semigroup and T = (ω p, ω n) be an IVBF set of G is called an (λ, δ)-IVBF right ideal of G if (1) ω p(e1e2) ∨ λ p ≥ ω p(e1) ∧ δ p, (2) ω n(e1e2) ∧ λ n ≥ ω n(e1) ∨ δ n. (3) If e1 ≤ e2, then ω p(e1) ∨ λ p ≥ ω p(e2) ∧ δ p and ω n(e1) ∧ λ n ≤ ω n(e2) ∨ δ n, for all e1, e2 ∈ G. An IVBF set T = (ω p, ω n) of an ordered semigroup E is called (λ, δ)-IVBF ideal of G if it is both (λ, δ)-IVBF left ideal and (λ, δ)-IVBF right ideal of G. Definition 10. [7] Let G be an ordered semigroup and T = (ω p, ω n) be an IVBF set of G is called an (λ, δ)-IVBF bi-ideal of G if (1) T = (ω p, ω n) is an (λ, δ)-IVBF subsemigroup of G (2) ω p(e1e2e3) ∨ λ p ≥ ω p(e1) ∧ ω p(e3) ∧ δ p (3) ω n(e1e2e3) ∧ λ n ≤ ω n(e1) ∨ ω n(e3) ∨ δ n, for all e1, e2, e3 ∈ G. For two IVBF sets T 1 = (ω p, ω n) and T 2 = (ϖ p, ϖ n) of an ordered semigroup G, define (1) T λ δ (x) := ( (ω p)λ δ (x), (ω n)λ δ (x) ) = ( ((ω p)(x) ∧ λ p ) ∨ δ p , ((ω n)(x) ∨ λ n ) ∧ δ p ) , (2) (T 1 ⊓ T 2) λ δ (x) := ( (ω p ∩ϖ p)λ δ (x), (ω n ∩ϖ n)λ δ (x) ) = ( ((ω p(x) ∧ϖ p(x)) ∧ λ p ) ∨ δ p , ((ω n(x) ∨ϖ n(x)) ∨ λ n ) ∧ δ n ) , (3) (T 1 ◦ T 2) λ δ (x) := ( (ω p ◦ϖ p)λ δ (x), (ω n ◦ϖ n)λ δ (x) ) = ( ((ω p(x) ◦ϖ p(x)) ∧ λ p ) ∨ δ p , ((ω n(x) ◦ϖ n(x)) ∨ λ n ) ∧ δ n ) where (ω p ◦ϖ p)(e) =  ∨ (t,h)∈Fe {ω p(t) ∧ϖ p(h)} if Fe ̸= ∅, 0 if Fe = ∅, and (ω n ◦ϖ n)(e) =  ∧ (t,h)∈Fe {ω n(t) ∨ϖ n(h)} if Fe ̸= ∅, 0 if Fe = ∅, In the following theorem, we give a relationship between an ideal and the interval valued bipolar characteristic function which is proved easily. T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3229 Theorem 1. Let M be a non-empty subset of an ordered semigroup G. Then M is a left ideal (right ideal, ideal) of G with λ p < δ p and λ n > δ n if and only if χM = (G;χ p M , χ n M ) is an (λ, δ)-IVBF left ideal (right ideal, ideal) of G. 3. Generalized Interval Valued Bipolar Fuzzy Quasi-Ideals In this section, we give the concept of a generalized interval valued bipolar fuzzy quasi- ideal and investigate properties of generalized interval valued bipolar fuzzy quasi-ideal in ordered semigroups. Definition 11. Let G be an ordered semigroup and T = (ω p, ω n) be an IVBF set of G is called an (λ, δ)-IVBF quasi-ideal of G if (1) (G ◦ T )λ δ ⊓ (T ◦G)λ δ ⊑ T λ δ . (2) If e1 ≤ e2, then ω p(e1) ∨ λ p ≥ ω p(e2) ∧ δ p and ω n(e1) ∧ λ n ≤ ω n(e2) ∨ δ n, for all e1, e2 ∈ G. The following example is a (λ, δ)-IVBF quasi-ideal of a semigroup. Example 2. Let G be an ordered semigroup given by the following table. · α κ ρ α α α α κ α κ κ ρ α α κ Define IVBF set T = (ω p, ω n) in G as follows: µp(α) = [0.1, 0.8], µp(κ) = [0.1, 0.8], µp(ρ) = [0.3, 0.6] and µn(α) = [−0.1,−0.7], µn(κ) = [−0.1,−0.7], µn(ρ) = [−0.2,−0.5]. and de- fine a partial order relation ≤ on G as follows: ≤: {(α, κ), (α, ρ), (κ, ρ)}∪ △G, where △G is an equality relation on G. By routine calculation, T = (ω p, ω n) is an ([0.3, 0.3], [0.5, 0.5])- IVBF quasi-ideal of G. Theorem 2. Every (λ, δ)-IVBF left (right) ideal of an ordered semigroup G is an (λ, δ)- IVBF quasi ideal of G. Proof. Suppose that T = (ω p, ω n) is an (λ, δ)-IVBF left ideal of F and let e1, e2 ∈ G with e1 ≥ e2. Then ω p(e1)∨λ p ≥ ω p(e2)∧ δ p and ω n(e1)∧λ n ≤ ω n(e2)∨ δ n. Let e ∈ F. If Ae = ∅, then it is easy to verify that (G p ◦ ωp)δ λ (e) ∧ (ωp ◦ Gp )δ λ (e) ≤ (ω p)δ λ (e) and (G n ◦ ωn)λ δ (e) ∨ (ωn ◦Gn )λ δ (e) ≥ (ω n)λ δ (e). T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3230 If Ae ̸= ∅, then (G p ◦ ωp)δ λ (e) = ( ∨ (k,o)∈Ae {Gp (k) ∧ ωp(o)} ∧ δ p ) ∨ λ p = ( ∨ (k,o)∈Ae {1 ∧ ωp(o)} ∧ δ p ) ∨ λ p = ( ∨ (k,o)∈Ae {ωp(o)} ∧ δ p ) ∨ λ p = ( ∨ (k,o)∈Ae {ωp(o) ∧ δ p} ∧ δ p ) ∨ λ p ≤ ( ∨ (k,o)∈Ae {ωp(ko) ∨ λ p} ∧ δ p ) ∨ λ p = ((ωp(e) ∨ λ p ) ∧ δ p ) ∨ λ p = ((ωp(e) ∨ λ p ∨ λ p ) ∧ δ p ∨ λ p ) = ((ωp(e) ∨ λ p ) ∧ δ p ∨ λ p ) = (ωp(e) ∧ δ p ) ∨ λ p = (ω p)δ λ (e) and (G n ◦ ωn)λ δ (e) = ( ∧ (k,o)∈Ae {Gn (k) ∨ ωn(o)} ∧ λ n ) ∨ δ n = ( ∧ (k,o)∈Ae {−1 ∨ ωn(o)} ∧ λ n ) ∨ δ n = ( ∧ (k,o)∈Ae {ωn(o)} ∧ λ n ) ∨ δ n = ( ∧ (k,o)∈Ae {ωn(o) ∨ δ n} ∧ λ n ) ∨ δ n ≥ ( ∧ (k,o)∈Ae {ωn(ko) ∧ λ n} ∧ λ n ) ∨ δ n = ((ωp(e) ∧ λ n ) ∧ λ n ) ∨ δ n = (ωp(r) ∧ λ n ) ∨ δ n = (ω n)λ δ (e) Thus, (G p ◦ ωp)δ λ (e) ≤ (ω p)δ λ (e) and (G n ◦ ωn)λ δ (e) ≥ (ω n)λ δ (e) implies that, (G p ◦ ωp)δ λ (e) ∧ (ωp ◦ Gp )δ λ (r) ≤ (ω p)δ λ (e) and (G n ◦ ωn)λ δ (e) ∨ (ωn ◦ Gn )λ δ (e) ≥ (ω n)λ δ (e). Hence T = (ω p, ω n) is an (λ, δ)-IVBF quasi-ideal of G. The following theorem show that the (λ, δ)-IVBF quasi-ideal and (λ, δ)-IVBF subsemi- groups. Theorem 3. Every (λ, δ)-IVBF quasi-ideal of an ordered semigroup G is an (λ, δ)-IVBF subsemigroup of G. Proof. Assume that T = (ω p, ω n) is an (λ, δ)-IVBF quasi-ideal of G and let e1, e2 ∈ G with e1 ≥ e2. Then ω p(e1) ∨ λ p ≥ ω p(e2) ∧ δ p and ω n(e1) ∧ λ n ≤ ω n(e2) ∨ δ n. T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3231 Consider ωp(e1e2) ∨ λ p ≥ ωp(e1e2) ∧ δ p ∨ λ p ≥ (ωp ◦Gp )δ λ (e1e2) ∧ (G p ◦ ωp)δ λ (e1e2) = ( ∨ (i,j)∈Ae1e2 {ωp(i) ∧G p (j)} ∧ δ p ) ∨ λ p∧ ( ∨ (k,o)∈Ae1e2 {Gp (k) ∧ ωp(o)} ∧ δ p ) ∨ λ p ≥ (ωp(e1) ∧G p (e2) ∧ δ p ∨ λ p ) ∧ (G p (e1) ∧ ωp(e2) ∧ δ p ∨ λ p ) = (ωp(e1) ∧ 1 ∧ δ p ∨ λ p ) ∧ (1 ∧ ωp(e2) ∧ δ p ∨ λ p ) = (ωp(e1) ∧ δ p ∨ λ p ) ∧ (µp(e2) ∧ δ p ∨ λ p ) = (ωp(e1) ∧ ωp(e2)) ∧ δ p ∨ λ p = ωp(e1) ∧ ωp(e2) ∧ δ p ∨ λ p ≥ ωp(e1) ∧ ωp(e2) ∧ δ p and ωn(e1e2) ∧ λ n ≤ ωp(e1e2) ∧ λ n ∨ δ n ≤ (ωn ◦Gn )λ δ (e1e2) ∨ (G n ◦ ωn)λ δ (e1e2) = ( ∧ (i,j)∈Ae1e2 {ωn(i) ∨G n (j)} ∧ λ n ) ∨ δ n∨ ( ∧ (k,o)∈Ae1e2 {Gn (k) ∨ ωn(o)} ∧ λ n ) ∨ δ n ≤ (ωn(e1) ∨G n (e2) ∧ λ n ∨ δ n ) ∨ (G n (e1) ∨ ωn(e2) ∧ λ n ∨ δ n ) = (ωn(e1) ∨ −1 ∧ λ n ∨ δ n ) ∨ (−1 ∨ ωn(e2) ∧ λ n ∨ δ n ) = (ωn(e1) ∧ λ n ∨ δ n ) ∨ (ωn(e2) ∧ λ n ∨ δ n ) = (ωn(e1) ∨ ωn(e2)) ∧ λ n ∨ δ n = ωn(e1) ∨ ωnp(e2) ∧ λ n ∨ δ n ≤ ωn(e1) ∨ ωn(e2) ∨ δ n . Thus, ωp(e1e2) ∨ λ p ≥ ωp(e1) ∧ ωp(e2) ∧ δ p and ωn(e1e2) ∧ λ n ≤ ωn(e1) ∨ ωn(e2) ∨ δ n . Hence T = (ω p, ω n) is an (λ, δ)-IVBF subsemigroup of G. The following theorem show that the (λ, δ)-IVBF quasi-ideal and (λ, δ)-IVBF bi-ideal in semigroup. Theorem 4. Every (λ, δ)-IVBF quasi-ideal of an ordered semigroup G is a (λ, δ)-IVBF bi-ideal of G. Proof. Assume that T = (ω p, ω n) is an (λ, δ)-IVBF quasi-ideal of G and let e1, e2,∈ G. Then by Theorem 3, T = (ω p, ω n) is an (λ, δ)-IVBF subsemigroup of G. T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3232 Let e1, e2, e3 ∈ G. Then ωp(e1e2e3) ∨ λ p ≥ ωp(e1e2e3) ∧ δ p ∨ λ p ≥ (ωp ◦Gp )δ λ (e1e2e3) ∧ (G p ◦ ωp)δ λ (e1e2e3) = ( ∨ (i,j)∈Ae1e2e3 {ωp(i) ∧G p (j)} ∧ δ p ) ∨ λ p∧ ( ∨ (k,o)∈Ae1e2e3 {Gp (k) ∧ ωp(o)} ∧ δ p ) ∨ λ p ≥ (ωp(e1) ∧G p (e2e3) ∧ δ p ∨ λ p ) ∧ (G p (e1e2) ∧ ωp(e3) ∧ δ p ∨ λ p ) = (ωp(e1) ∧ 1 ∧ δ p ∨ λ p ) ∧ (1 ∧ ωp(e3) ∧ δ p ∨ λ p ) = (ωp(e1) ∧ δ p ∨ λ p ) ∧ (µp(e3) ∧ δ p ∨ λ p ) = (ωp(e1) ∧ ωp(e3)) ∧ δ p ∨ λ p = ωp(e1) ∧ ωp(e3) ∧ δ p ∨ λ p ≥ ωp(e1) ∧ ωp(e3) ∧ δ p and ωn(e1e2e3) ∧ λ n ≤ ωp(e1e2e3) ∧ λ n ∨ δ n ≤ (ωn ◦Gn )λ δ (e1e2e3) ∨ (G n ◦ ωn)λ δ (e1e2e3) = ( ∧ (i,j)∈Ae1e2e3 {ωn(i) ∨G n (j)} ∧ λ n ) ∨ δ n∨ ( ∧ (k,o)∈Ae1e2e3 {Gn (k) ∨ ωn(o)} ∧ λ n ) ∨ δ n ≤ (ωn(e1) ∨G n (e2e3) ∧ λ n ∨ δ n ) ∨ (G n (e1e2) ∨ ωn(e3) ∧ λ n ∨ δ n ) = (ωn(e1) ∨ −1 ∧ λ n ∨ δ n ) ∨ (−1 ∨ ωn(e3) ∧ λ n ∨ δ n ) = (ωn(e1) ∧ λ n ∨ δ n ) ∨ (ωn(e3) ∧ λ n ∨ δ n ) = (ωn(e1) ∨ ωn(e3)) ∧ λ n ∨ δ n = ωn(e1) ∨ ωnp(e3) ∧ λ n ∨ δ n ≤ ωn(e1) ∨ ωn(e3) ∨ δ n . Thus, ωp(e1e2e3) ∨ λ p ≥ ωp(e1) ∧ ωp(e3) ∧ δ p and ωn(e1e2e3) ∧ λ n ≤ ωn(e1) ∨ ωn(e3) ∨ δ n . Hence T = (ω p, ω n) is an (λ, δ)-IVBF bi-ideal of G. The following theorems are basic properties. Theorem 5. Let G be an ordered semigroup. Then the intersection of two (λ, δ)-IVBF quasi-ideal of G is an (λ, δ)-IVBF quasi-ideal of G. Proof. Assume that T 1 = (ω p, ω n) and T 2 = (ϖ p, ϖ n) are (λ, δ)-IVBF quasi-ideals T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3233 of G. Let e ∈ G. Then (ωp ⊓ϖp)(e) ∨ λ p ≥ (ωp ⊓ϖp)(e) ∨ λ p ∧ δ p = (ωp(e) ∧ϖp(e)) ∨ λ p ∧ δ p = (ωp(e) ∨ λ p ∧ δ p ) ∧ (ϖp(e) ∨ λ p ∧ δ p ) ≥ (ωp ◦Gp )δ λ (e) ∧ (G p ◦ ωp)δ λ (e)∧ (ϖp ◦Gp )δ λ (e) ∧ (G p ◦ϖp)δ p λ (e) = ( ∨ (i,j)∈Ae {ωp(i) ∧G p (j)} ∧ δ p ) ∨ λ p∧ ( ∨ (k,o)∈Ae {Gp (k) ∧ ωp(o)} ∧ δ p ) ∨ λ p∧ ( ∨ (i,j)∈Ae {ϖp(i) ∧G p (j)} ∧ δ) ∨ϖp∧ ( ∨ (k,o)∈Ae {Gp (k) ∧ϖp(o)} ∧ δ p ) ∨ λ p = ( ∨ (i,j)∈Ae {ωp(i) ∧ϖp(i) ∧G p (j) ∧ δ ∨ λ p}∧ ( ∨ (k,o)∈Ae {Gp (k) ∧ µp(o) ∧ϖp(o)} ∧ δ) ∨ λ p = ( ∨ (i,j)∈Ae {(ωp ⊓ϖp)(i) ∧G p (j)} ∧ δ) ∨ λ p∧ ( ∨ (k,o)∈Ae {Gp (k) ∧ (ωp ⊓ϖp)(o)} ∧ δ p ) ∨ λ p = (ωp ⊓ϖp ◦Gp )δ λ (e) ∧ (G p ◦ ωp ⊓ϖp)δ λ (e) and (ωn ⊔ϖn)(e) ∧ λ n ≤ (ωn ⊔ϖn)(e) ∧ λ n ∨ δ n = (ωn(e) ∨ϖn(e)) ∧ λ n ∨ δ n = (ωn(e) ∨ λ n ∨ δ n ) ∨ (ϖn(e) ∨ λ n ) ∨ δ n ≤ (ωn ◦Gn )λ δ (e) ∨ (G n ◦ ωn)λ δ (e)∨ (ϖn ◦Gn )λ δ (e) ∨ (G n ◦ϖn)λ δ (e) = ( ∧ (i,j)∈Ae {ωn(i) ∧G n (j)} ∧ λ n ) ∨ δ n∧ ( ∧ (k,o)∈Ae {Gn (k) ∧ ωn(o)} ∧ λ n ) ∨ δ n∨ ( ∧ (i,j)∈Ae {ϖn(i) ∧G n (j)} ∧ λ n ) ∨ δ n∨ ( ∧ (k,o)∈Ae {Gn (k) ∧ϖn(o)} ∧ λ n ) ∨ δ n = { ∧ (i,j)∈Ae (ωn(i) ∨ϖn(i) ∧G n (j) ∧ λ n ) ∨ δ n}∧ ( ∧ (k,o)∈Ae {Gp (k) ∧ µn(o) ∨ϖn(o)} ∧ λ n ) ∨ δ n = ( ∧ (i,j)∈Ae {(ωn ⊔ϖn)(i) ∧G n (j)} ∧ λ n ) ∨ δ n∧ ( ∧ (k,o)∈Ae {Gn (k) ∧ (ωn ⊔ϖp)(o)} ∧ λ n ) ∨ δ n = (ωn ⊔ϖn ◦Gn )λ δ (e) ∧ (G n ◦ ωn ⊔ϖn)λ δ (e). T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3234 Thus, (ωp ⊓ϖp)(e) ∨ λ p ≥ (ωp ⊓ϖp ◦Gp )δ λ (e) ∧ (G p ◦ ωp ⊓ϖp)δ λ (e) and (ωn ⊔ϖn)(e) ∧ λ n ≤ (ωn ⊔ϖn ◦Gn )λ δ (e) ∧ (G n ◦ ωn ⊔ϖn)λ δ (e) Hence T 1 ⊓ T 2 is an (λ, δ)-IVBF quasi-ideal of G. Theorem 6. If M is a quasi-ideal of an ordered semigroup G, then χM = (G;χ p M , χ n M ) is an (λ, δ)-IVBF quasi-ideal of G. Proof. Suppose that M is a quasi-ideal of G and e1, e2 ∈ G with e1 ≥ e2. Then ω p(e1) ∨ λ p ≥ ω p(e2) ∧ δ p and ω n(e1) ∧ λ n ≤ ω n(e2) ∨ δ n. Let e ∈ G. If e ∈ M or Ae = ∅, then (χp M )δ λ (e) ≥ (χp M ◦Gp )δ λ (e) ∧ (G p ◦ χp M )λ δ (e), (χn M )λ δ (e) ≤ (χn M ◦Gn )(e) ∨ (G n ◦ χn M )λ δ (e). Assume that e /∈ M and Ae ̸= ∅. Let K = {(i, j) | e = ij and (i /∈ M and j /∈ M)}. Thus K ⊆ Ae On the other hand if (i, j) ∈ Ae, then Ae = ij /∈ M which it implies that ij /∈ MG ∩ GM . Thus i /∈ M or j /∈ M e ∈ GM and e ∈ MG and so (i, j) ∈ M . Hence Ae ⊆ K. Therefore, Ae = K. That is, Ae = {(i, j) | i /∈ M or j /∈ M}. Thus, (χp M ◦G)δ λ ∧ (G ◦ χp M )δ λ = ( ∨ (i,j)∈Ae {χp M (i) ∧G p (j)} ∧ δ p ) ∨ λ p∧ ( ∨ (i,j)∈Ae {Gp (i) ∧ χp M (j)} ∧ δ p ) ∨ λ p = ( ∨ (i,j)∈Ae {χp M (i) ∧ χp M (j)} ∧ δ p ) ∨ λ p = λ p and (χn M ◦G)λ δ ∨ (G ◦ χn M )λ δ = ( ∧ (i,j)∈Ae {χn M (i) ∨G n (j)} ∧ λ n ) ∨ δ n∨ ( ∧ (i,j)∈Ae {Gn (i) ∨ χn M (j)} ∧ λ n ) ∨ δ n = ( ∧ (i,j)∈Ae {χn M (i) ∨ χn M (j)} ∧ λ n ) ∨ δ n = δ n . Hence χM = (G;χ p M , χ n M ) is an (λ, δ)-IVBF quasi-ideal of G. Lemma 1. If χM = (G;χ p M , χ n M ) is an (λ, δ)-IVBF quasi-ideal of an ordered semigroup G with λ p < δ p and λ n > δ n, then M is a quasi-ideal of G. Proof. Suppose that χM = (S;χ p M , χ n M ) is an (λ, δ)-IVBF quasi-ideal of G with λ p < δ p and λ n > δ n. Let e ∈ MG ∩ GM . Then there exist i, j ∈ G and y, z ∈ G such that e = iy and e = jz. Thus (χp M ◦ G)δ λ (e) = ( ∨ (i,n)∈Ae {χp M (i) ∧ G p (n)} ∧ δ p ) ∨ λ p ≥ ( ∨ (j,z)∈Ae {χp M (j) ∧ G p (z)} ∧ δ p ) ∨ λ p = δ p . Similarly (G ◦ χp M )δ λ = δ p . And (χn M ◦ G)λ δ = T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3235 ( ∧ (i,n)∈Ae {χn M (i)∨Gn (n)}∧λn )∨δn ≤ ( ∨ (j,z)∈Ae {χn M (j)∨Gn (z)}∧λn )∨δn = δ n . By assumption, (χp M )δ λ (e) ≥ (χp M ◦Gp )δ λ (e) ∧ (G p ◦ χp M )λ δ (e) and (χn M )λ δ (e) ≤ (χn M ◦Gn )(e)⋎ (G n ◦ χn M )λ δ (e). (1) If e /∈ M , then by (1) λ p ≥ δ pand λ n ≤ δ n. It is a contradiction. Hence e ∈ M . Therefore M is a quasi-ideal of G. 4. Characterizing ordered regular and intra-regular semigroups by using Generalized Interval Valued Bipolar Fuzzy Quasi-Ideals. In this topic, we will use knowledge of the characteristics of interval valued fuzzy set and bipolar fuzzy sets to characterize regular and intra-regular semigroups by using generalized interval valued bipolar fuzzy quasi-ideals in ordered semigroups. Theorem 7. [14] Let I and K be a non-empty subsets of G. Then (1) (χI ◦ χK)λ δ = (χIK)λ δ i.e. ⟨(χ p I ◦ χ p K)δ λ , (χ n I ◦ χ n K)λ δ ⟩ = ⟨(χ p IK)δ λ , (χ n IK)λ δ ⟩ (2) (χI ⊓χK)λ δ = (χ p I ⊓χK)λ δ i.e. ⟨(χ p I ∩χ p K)δ λ , (χ n I ∪χ n I ) λ δ ⟩ = ⟨(χ p I∩K)δ λ , (χ n I∪K)λ δ ⟩, where χI = (G;χ p I , χ n I ) and χK = (G;χ p K , χ n K). Remark 4. Since χI is an interval valued characteristic function we have (χ p I) λ δ (e) = { λ p if k ∈ I, δ p if k /∈ I and (χ n I ) λ δ (e) = { δ n if k ∈ I, λ n if k /∈ I Lemma 2. [4] For an ordered semigroup G, the following statements are equivalent. (1) G is a regular (2) Q ∩ L ⊆ (QL] for every quasi-ideal Q and every left ideal L of G. (3) R ∩Q ⊆ (RQ] for every right ideal R every quasi-ideal Q and of G. Theorem 8. For an ordered semigroup G, the following conditions are equivalent. (1) G is a regular, (2) (T ⊓J )λ δ ⊑ (T ◦ J )λ δ , for every (λ, δ)-IVBF quasi-ideal T = (ω p, ω n) of G and every (λ, δ)-IVBF left ideal J = (ϖ p, ϖ n) of G, T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3236 (3) (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ , for every (λ, δ)-IVBF right ideal T = (ω p, ω n) of G and every (λ, δ)-IVBF quasi-ideal J = (ϖ p, ϖ n) of G. Proof. (1) ⇒ (3) Let T = (ω p, ω n) and J = (ϖ p, ϖ n) be an (λ, δ)-IVBF right ideal and an (λ, δ)-IVBF quasi-ideal of G respectively and let e ∈ G. Since G is regular, there exists g ∈ G such that e ≤ ege. Thus (ωp ◦ϖp)δ λ (e) = ( ∨ (k,o)∈Ae {ωp(k) ∧ϖp(o)} ∧ δ p ) ∨ λ p = ( ∨ (k,o)∈Aege {ωp(k) ∧ϖp(o)} ∧ δ p ) ∨ λ p ≥ ((ωp(e) ∧ϖp(gr)) ∧ δ p ) ∨ λ p = (ωp(e) ∧ϖp(gr) ∨ λ p ) ∧ δ p ∨ λ p ≥ (ωp(e) ∧ϖp(e) ∧ δ p ) ∧ δ p ∨ λ p = ((ωp(e) ∧ϖp(e)) ∧ δ p ) ∨ λ p = (ωp ∩ϖp)δ λ (e) and (ωn ◦ϖn)λ δ (e) = ( ∧ (k,o)∈Ae {ωn(k) ∨ϖn(o)} ∧ λ n ) ∨ δ n = ( ∧ (k,o)∈Aege {ωn(k) ∨ϖn(o)} ∧ λ n ) ∨ δ n ≤ ((ωn(e) ∨ϖn(ge)) ∧ λ n ) ∨ δ n = (ωn(e) ∨ϖn(ge) ∧ λ n ) ∧ λ n ∨ δ n ≤ (ωn(e) ∨ϖn(e) ∨ δ n ) ∧ λ n ∨ δ n = ((ωp(n) ∨ϖn(e)) ∧ λ n ) ∨ δ n = (ωn ∪ϖn)λ δ (e), Thus, (ωp ◦ϖp)δ λ (e) ≥ (ωp ∩ϖp)δ λ (e) and (ωn ◦ϖn)λ δ (e) ≤ (ωn ∪ϖn)λ δ (e) Hence, (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ . (3) ⇒ (1) Let R and Q be a right ideal and quasi-ideal of F respectively. Then by Theorem 1 and 6, χR = (G;χ p R, χ n R) and χQ = (G;χ p Q, χ n Q) is an (λ, δ)-IVBF right ideal and an (λ, δ)-IVBF quasi-ideal of G respectively. By supposition and Thoerem7, we have (χ p (RQ]) δ λ (e) = (χ p R ◦ χ p Q) δ λ (e) ⊑ (χ p R ∩ χ p Q) δ λ (e) = (χ p R∩Q) δ λ (e) = δ p , and (χ n (RQ]) λ δ (e) = (χ n R ◦ χ p Q) λ δ (e) ⊑ (χ n R ∪ χ p Q) λ δ (e) = (χ n R∪Q) λ δ (e) = δ n . Thus, e ∈ (RQ]. Hence, R ∩Q ⊆ (RL]. Therefore by Lemma 2, G is regular. In a similar manner, it may be shown that (1) ⇔ (2). Corollary 1. For an ordered semigroup G and let T = (ω p, ω n) and J = (ϖ p, ϖ n), the following conditions are equivalent. (1) G is a regular, T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3237 (2) (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ , for every (λ, δ)-IVBF quasi-ideal T and every (λ, δ)-IVBF left ideal J of F, (3) (T ⊓J )λ δ ⊑ (T ◦J )λ δ , for every (λ, δ)-IVBF bi-ideal T and every (λ, δ)-IVBF left ideal J of F, (4) (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ , for every (λ, δ)-IVBF right ideal T and every (λ, δ)-IVBF quasi-ideal J of F, (5) (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ , for every (λ, δ)-IVBF right ideal T and every (λ, δ)-IVBF bi-ideal J of F. Some equivalent conditions are important properties for (λ, δ)-IVBF-subsemigroups of semigroups. Theorem 9. An (λ, δ)-IVBF set T = (ω p, ω n) is an (λ, δ)-IVBF subsemigroup of an ordered semigroup G if and only if (T ◦ T )λ δ ⊑ T λ δ . Proof. (⇒) Assume that T = (ω p, ω n) is an (λ, δ)-IVBF subsemigroup of an ordered semigroup G and let e ∈ G. If Ae = ∅, then it is easy to verify that, (ωp ◦ ωp)δ λ (e) ≤ (ωp)δ λ (e) and (µn ◦ ωn)λ δ (e) ≥ (µn)λ δ (e). If Ae ̸= ∅, then (ωp ◦ ωp)δ λ (e) = ( ∨ (k,o)∈Ae {ωp(k) ∧ ωp(o)} ∧ δ p ) ∨ λ p = ( ∨ (k,o)∈Ae {ωp(k) ∧ ωp(o) ∧ δ p} ∧ δ p ) ∨ λ p ≤ ( ∨ (k,o)∈Ae {ωp(ko) ∨ λ p} ∧ δ p ) ∨ λ p = (ωp(e) ∨ λ p ∧ δ p ) ∨ λ p = (ωp(e) ∧ δ p ) ∨ λ p = (ωp)δ λ (e) and (ωn ◦ ωn)λ δ (e) = ( ∧ (k,o)∈Ae {ωn(k) ∨ µn(o)} ∧ λ n ) ∨ δ n = ( ∧ (k,o)∈Ae {ωn(k) ∨ µn(o) ∨ δ n} ∧ λ n ) ∨ δ n ≤ ( ∧ (k,o)∈Ae {ωn(ko) ∧ λ n} ∧ λ n ) ∨ δ n = (ωn(e) ∧ λ n ∧ λ n ) ∨ δ n = (ωp(e) ∧ λ n ) ∨ δ n = (ωn)λ δ (e) Thus, (ωp ◦ ωp)δ λ (e) ≤ (ωp)δ λ (e) and (ωn ◦ ωn)λ δ (e) ≥ (µn)λ δ (e). Hence, (T ◦ T )λ δ ⊑ T λ δ . T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3238 (⇐) Suppose (T ◦ T )λ δ ⊑ T λ δ and let e1, e2 ∈ G. Then (ωp ◦ ωp)δ λ (e1e2) ≤ (ωp)δ λ (e1e2) and (ωn ◦ ωn)λ δ (e1e2) ≥ (µn)λ δ (r1r2). Thus ωp(e1e2) ∨ λ p ≥ (ωp(e1e2) ∧ δ p ) ∨ λ p = (ωp)δ λ (e1e2) ≥ (ωp ◦ ωp)δ λ (e1e2) = ( ∨ (k,o)∈Ae1e2 {µp(k) ∧ µp(o)} ∧ δ p ) ∨ λ p ≥ (µp(e1) ∧ µp(e2) ∧ δ p ) ∨ λ p ≥ µp(e1) ∧ µp(e2) ∧ δ p and ωn(e1e2) ∧ λ n ≤ (ωn(e1e2) ∧ λ n ) ∨ δ n = (ωn)λ δ (e1e2) ≥ (ωn ◦ ωn)λ δ (e1e2) = ( ∧ (k,o)∈Ae1e2 {µn(k) ∨ µn(o)} ∧ λ n ) ∨ δ p ≤ (µn(e1) ∨ µn(e2) ∧ λ n ) ∨ δ n ≤ µn(e1) ∨ µn(e2) ∨ δ n Hence, ωp(e1e2) ∨ λ p ≥ µp(e1) ∧ µp(e2) ∧ δ p and ωn(e1e2) ∧ λ n ≤ µn(e1) ∨ µn(e2) ∨ δ n . Therefore T = (ω p, ω n) is an (λ, δ)-IVBF subsemigroup of G. Lemma 3. [9] For an ordered semigroup G, the following conditions are equivalent. (1) G is regular and intra-regular. (2) Every quasi-ideal of F is idempotent. (3) Every bi-ideal of F is idempotent. Theorem 10. Let T = (ω p, ω n) and J = (ϖ p, ϖ n) be IVBF sets of an ordered semigroup G. Then the followings are equivalent. (1) G is both regular and intra-regular, (2) (T ◦ T )λ δ = T λ δ for every (λ, δ)-IVBF quasi-ideal T of F, (3) (T ◦ T )λ δ = T λ δ for every (λ, δ)-IVBF bi-ideal T of F, (4) (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ for every (λ, δ)-IVBF quasi-ideals T and J of F, (5) (T ⊓J )λ δ ⊑ (T ◦J )λ δ for every (λ, δ)-IVBF quasi-ideal T and every (λ, δ)-IVBF bi-ideal J of F, (6) (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ for every (λ, δ)-IVBF bi-ideals T and J of F. Proof. (1) ⇒ (6) Let T = (ω p, ω n) and J = (ϖ p, ϖ n) be (λ, δ)-IVBF bi-ideals of G and let e ∈ G. Since G is both regular and intra-regular, there exist k, l, e ∈ G such that T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 17 (4) (2024), 3223-3241 3239 e = rkr and e = km2e. Thus e = rkr = rkrkm = rk(lr2e)kr = (rklr)(relr). It follows that (ωp ◦ϖp)δ λ (e) = ∨ (i,j)∈Ae {ωp(i) ∧ λ p (j)} = ( ∨ (i,j)∈A(rklr)(relr) {ωp(i) ∧ϖp(j)} ∧ δ p ) ∨ λ p ≥ (ωp(rklr) ∧ϖp(relr) ∧ δ p ) ∨ λ p = (ωp(r(kl)r) ∨ λ p ∧ϖp(r(el)r) ∨ λ p ∧ δ p ) ∨ λ p ≥ (ωp(e) ∧ ωp(e) ∧ δ p ∧ϖp(e) ∧ϖp(e) ∧ δ p ∧ δ p ) ∨ λ p = (ωp(e) ∧ δ p ∧ϖp(e) ∧ δ p ∧ δ p ) ∨ λ p = ((ωp(e) ∧ϖp(e)) ∧ δ p ∧ δ p ) ∨ λ p = ((ωp(e) ∧ϖp(e)) ∧ δ p ) ∨ λ p = (ωp ⊓ϖp)δ λ (e), and (ωn ◦ϖn)λ δ (e) = ∧ (i,j)∈Ae {ωn(i) ∨ λ n (j)} = ( ∧ (i,j)∈A(rklr)(relr {ωn(i) ∨ϖn(j)} ∧ λ n ) ∨ δ n ≤ (ωn(rklr) ∨ϖn(relr) ∧ λ n ) ∨ δ n = (ωn(r(kl)r) ∧ λ n ∨ϖn(r(el)r) ∧ λ n ∧ λ n ) ∨ δ n ≤ (ωn(e) ∨ ωn(e) ∨ δ n ∨ϖn(e) ∨ϖn(e) ∨ δ n ∧ λ n ) ∨ δ n = (ωn(e) ∨ δ n ∨ϖn(e) ∨ δ n ∧ λ n ) ∨ δ n = ((ωn(e) ∨ϖn(e)) ∨ δ n ∧ λ n ) ∨ δ n = ((ωn(e) ∨ϖn(e)) ∧ λ n ) ∨ δ n = (ωn ⊓ϖn)λ δ (e). Hence, (ωp ◦ϖp)δ λ (e) ≥ (ωp ⊓ϖp)δ λ (e) and (ωn ◦ϖn)λ δ (e) ≤ (ωn ⊓ϖn)λ δ (e). Therefore, (T ⊓ J )λ δ ⊑ (T ◦ J )λ δ . (6) ⇒ (5) ⇒ (4) and (3) ⇒ (2) This is obvious because every (λ, δ)-IVBF quasi-ideal is an (λ, δ)-IVBF bi-ideal of G. (4) ⇒ (2) Take T = J in (4), we get T λ δ = (T ⊓ T )λ δ ⊑ (T ◦ T )λ δ . Since every (λ, δ)- IVBF quasi-ideal of G is an (λ, δ)-IVBF subsemigroup of G and by Theorem 9, we have (T ◦ T )λ δ ⊑ T λ δ . Thus, (T ◦ T )λ δ = T λ δ . (2) ⇒ (1) Let Q be a quasi-ideal of G. Then by Theorem 6, χQ = (G;χ p Q, χ n Q) is an (λ, δ)-IVBF quasi-ideal of G. By supposition and Thoerem7, we have (χ p (Q2] )δ λ (e) = (χ p Q ◦ χ p Q) δ λ (e) = (χ p Q) δ λ (e) = δ p , and (χ n (Q2]) λ δ (e) = (χ n Q ◦ χ n Q) λ δ (e) = (χ n Q) λ δ (e) = δ n . Thus, (QQ] = Q. An application of Lemma 3 shows us that G is both regular and intra-regular. REFERENCES 3240 5. Conclusion The theory of fuzzy sets, initially introduced by L. A. Zadeh, was later extended to interval-valued fuzzy sets. Building upon this foundation, K. Arulmozhi et al. explored interval-valued bipolar fuzzy sets in algebraic structures. In 2021, S. Lekkoksung advanced this area by developing the concept of interval-valued bipolar fuzzy ideals in ordered semigroups and characterized regular ordered semigroups in terms of generalized interval- valued bipolar fuzzy ideals and bi-ideals. In this paper, we introduce new definitions of generalized interval-valued bipolar fuzzy quasi-ideals and establish their properties. Using intra-regular ordered semigroups, we prove several properties of these generalized quasi- ideals. Furthermore, we provide a characterization of intra-regular ordered semigroups through the framework of generalized interval-valued bipolar fuzzy quasi-ideals. We hope that the study of intra-regular ordered semigroups in terms of generalized interval valued bipolar fuzzy quasi-ideal are useful mathematical tools. In the future, we study character- ized semisimple ordered semigroups in terms of generalized interval valued bipolar fuzzy interior ideals. 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