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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 141-152

Published online June 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.2.141

© The Korean Institute of Intelligent Systems

## Exploring Regularities of Ordered Semigroups Through Generalized Fuzzy Ideals

Young Bae Jun1, Kittisak Tinpun2, and Nareupanat Lekkoksung3

1Department of Mathematics Education, Gyeongsang National University, Jinju, Korea
2Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani, Thailand
3Division of Mathematics, Faculty of Engineering, Rajamangala University of Technology Isan, Khon Kaen, Thailand

Correspondence to :
Nareupanat Lekkoksung (nareupanat.le@rmuti.ac.th)

Received: February 17, 2022; Revised: August 18, 2023; Accepted: May 3, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider (α, β)-fuzzy ideals, which are a generalized version of fuzzy ideals, in ordered semigroups. A connection between (α, β)-fuzzy quasi-ideals and (α, β)-fuzzy left (right) ideals is provided. The notion of (α, β)-fuzzy ideals is characterized in terms of a particular operation. We describe regular and intra-regular ordered semigroups using the concept of (α, β)fuzzy ideals.

Keywords: (α, β)-fuzzy left ideal, (α, β)-fuzzy right ideal, Ordered semigroup, Regular, Intra-regular

The concept of ordered semigroups is a generalization of semigroups. It is an algebraic structure of type (2; 2) comprising a semigroup and a partially ordered set defined on the same set, such that the order relation is compatible with the associative operation (see [1, 2]). This concept has been extensively studied. Ideals are crucial in studying ordered semigroups from numerous perspectives (refer to [310]). Kehayopulu [7] introduced the concepts of left and right ideals in ordered semigroups. Furthermore, the author explored a more nuanced interpretation of the prime properties inherent in these ideals, called weakly prime ideals. Kehayopulu pioneered the concept of quasi-ideals in ordered semi-groups. Using quasi-, left, and right ideals helped describe regularly ordered semigroups and intra-regular ordered semigroups across diverse analytical avenues (see [11]). Given that ideals comprise a fundamental aspect when examining the algebraic attributes of ordered semigroups, numerous mathematical tools have been employed to expand the scope of the ideal theory. These tools were subsequently utilized to further explore and analyze the ordered semigroups. Fuzzy sets are important tools used to study the structural properties of ordered semigroups.

The concept of fuzzy sets was proposed by Zadeh [12] in 1965. Serving as an extension of crisp sets, this conceptual framework has broad applications in diverse mathematical disciplines, including algebra. Rosenfeld [13] pioneered the exploration of group properties within the field of fuzzy sets, known as fuzzy groups. Kuroki introduced the concept of fuzzy semigroups. Fuzzy sets have been applied to examine various properties of semigroups, as demonstrated in [14].

In 2003, Kehayopulu and Tsingelis [15] implemented the concept of fuzzy sets to ordered semigroups. A fuzzy ordered semigroup is an ordered semigroup whose universe set is the set of all the fuzzy subsets on an ordered semigroup with a particular binary operation and order relation. They demonstrated that any ordered semigroup can be embedded into a fuzzy-ordered semigroup. An extensive exploration of ordered semigroups from multiple perspectives was conducted using fuzzy ideals. For example, Kehayopulu and Tsingelis [9] introduced fuzzy quasi-ideals that allowed for the characterization of regularly ordered semigroups through the application of fuzzy left (resp., right, or quasi-) ideals. In a parallel vein, in their subsequent study [16], the concept of fuzzy bi-ideals in ordered semigroups was introduced. The authors delved into the complicated relationships among these specific types of fuzzy ideals, revealing the broader scope of fuzzy bi-ideals as an extension of fuzzy left (resp., right, or quasi) ideals in ordered semigroups. The coincidence between fuzzy quasi-ideals and fuzzy bi-ideals was established, further underlining their inherent interconnections. Furthermore, fuzzy left and fuzzy right ideals have attained an insightful characterization of fuzzy quasi-ideals.

Numerous researchers have attempted to expand the concepts of fuzzy ideals and their applications to examine specific properties of ordered semigroups. Drawing inspiration from the “belongs to” relation (∈) and the “quasi-coincident” relation (q), Khan and Shabir [17] introduced the concept of (α, β)-fuzzy interior ideals in ordered semigroups, where α, β ∈ {∈, q, ∈ ∨ q, ∈ ∧ q} with the condition that α ≠ ∈ ∧ q. They primarily focused on the scenario in which α = ∈ and β = ∈ ∨ q. In this context, the authors established definitions of (∈, ∈ ∨ q)-fuzzy left (right, two-sided, interior) ideals in ordered semigroups and explored the relationships between these (∈, ∈ ∨ q)-fuzzy ideals. Jun et al. [18] extended this notion to (α, β)-fuzzy bi-ideals in ordered semigroups with a specific focus on (∈, ∈ ∨ q)-fuzzy bi-ideals. They characterized distinct classes of ordered semigroups using this concept. Furthermore, Khan et al. [19] conducted an in-depth investigation of the properties of (∈, ∈ ∨ q)-fuzzy left-hand ideals and (∈, ∈ ∨ q)-fuzzy right-hand ideals in ordered semigroups. Their research culminated in the characterization of regularly ordered semigroups using (∈, ∈ ∨ q)-fuzzy left ideals and (∈, ∈ ∨ q)-fuzzy right ideals.

The concept of fuzzy ideals in ordered semigroups, defined as belonging to the and quasi-coincident relations, was first extended by Khan et al. [20]. They introduced various types of (∈, ∈ ∨ qk)-fuzzy ideals such as (∈, ∈ ∨ qk)-fuzzy left (right) ideals and (∈, ∈ ∨ qk)-fuzzy (generalized) bi-ideals. Some classes of ordered semigroups are characterized by (∈, ∈ ∨ qk)-fuzzy ideals. In addition, a minimum of two publications in 2012 characterized specific classes of ordered semigroups using (∈, ∈ ∨ qk)-fuzzy ideals (see [21, 22]). Tang and Xie [23] studied the notion of (∈, ∈ ∨ qk)-fuzzy left (and right) ideals, where k ∈ [0, 1). They demonstrated that when k = 0, a (∈, ∈ ∨ qk)-fuzzy ideal is a (∈, ∈ ∨ q)-fuzzy ideal. Moreover, they characterized the prime properties of these fuzzy ideals.

The notions of (∈, ∈ ∨(k*, qk))-fuzzy ideals and (∈, qkδ)-fuzzy ideals represent the same concept in ordered semigroups. These concepts, which are extensions of the independently introduced generalized (∈, ∈ ∨ qk)-fuzzy ideals by Khan et al. [24] and Ali Khan et al. [25], converge to underline the specific aspects of fuzzy ideals. Their investigations illuminated ordered semi-groups through a new perspective of mathematical tools, employing novel fuzzy ideals with distinctive properties. Recently, Muhiuddin et al. [26] have extended the scope of the (∈, ∈ ∨ (k, qk))-fuzzy ideals, introducing the concept of (∈, ∈ ∨ (k, qk))-fuzzy (m, n)-ideals. This novel extension is explained through the description of (∈, ∈ ∨ (k, qk))-fuzzy (m, n)ideals using their level sets. Notably, the notion of (∈, ∈ ∨ (k, qk))-fuzzy (m, n)ideals characterizes the (m, n)regular ordered semigroups.

The generalization of fuzzy ideals in ordered semigroups is based on the concept of (α, β)-fuzzy ideals, where 0 ≤ α < β ≤ 1. Feng and Corsini [27] defined the concept of (α, β)-fuzzy left (right, interior, quasi-, bi-) ideals in ordered semigroups. The authors studied the connections between these concepts. Independently defined the notion of (α, β)-fuzzy bi-ideals by Khan et al. [28]. They completely characterized regular-ordered semigroups based on the properties of (α, β)-fuzzy bi-ideals. Feng and Corsini [29] investigated the relationship between (α, β)-fuzzy ideals and (α, β)-fuzzy interior ideals in ordered semigroups. They demonstrated that these notions coincided with regular and intra-regular ordered semigroups.

In this study, we center our investigation on the concepts proposed by Feng and Corsini [27] and Khan et al. [28], particularly focusing on the notions of (α, β)-fuzzy ideals in ordered semigroups. Feng and Corsini [29] demonstrated that any (α, β)-fuzzy left (right) ideal is an (α, β)-fuzzy quasi-ideal. For example, we illustrate that the opposite of this statement does not generally hold. A relation between (α, β)-fuzzy left (right) ideals and (α, β)-fuzzy quasi-ideals is presented. To provide more connections between (α, β)-fuzzy left (right) ideals and left (right) ideals in ordered semigroups, we attempt to describe both concepts using their level sets and characteristic functions. We also describe (α, β)-fuzzy left (right) ideals of their products. Moreover, we characterize regular and intra-regular ordered semigroups using (α, β)-fuzzy left and (α, β)-fuzzy right ideals.

In this section, we introduce some basic terminologies for ordered semigroups and fuzzy subsets, which will be used in the subsequent section.

An algebraic structure ⟨S; ·, ≤⟩ of type (2; 2) is called an ordered semigroup if:

• 1. ⟨S; ·⟩ is a semigroup;

• 2. ⟨S; ≤⟩ is partially ordered set;

• 3. ≤ compatible with · the associative binary operation.

For convenience, we write an ordered semigroup ⟨S; ·, ≤⟩ by S the bold letter of its universe set. We denote the product x · y as xy.

Let S be an ordered semigroup. For any A, B, CS and xS, we define AB := {ab : aA and bB}, Sa := {(x, y) ∈ S × S : axy} and

(C]:={xS:xysuch that yC}.

### Lemma 2.1 [7]

Let S be an ordered semigroup, and A, B, CS. Then:

• 1. A ⊆ (A];

• 2. AB implies (A] ⊆ (B];

• 3. (A](B] = (AB];

• 4. (AB] ⊇ (A] ∪ (B];

• 5. ((A]] = (A].

Let S be an ordered semigroup. A nonempty subset A of S is said to be a subsemigroup of S if ⟨A; ·|A×A, ≤|A×A⟩ is an ordered semigroup. A nonempty subset A of S such that (A] ⊆ A is called

• 1. a left ideal [7] of S if SAA;

• 2. a right ideal [7] of S if SAA;

• 3. a two-sided ideal (ideal) [7] of S if it is both a left and a right ideal of S;

• 4. a quasi-ideal [30] of S if (SA] ∩ (AS] ⊆ A.

Let X be a nonempty set. A fuzzy subset [12] of X (or a fuzzy set in X) is a mapping from X to [0, 1] the closed unit interval. Denoted by F(S) the set of fuzzy subsets of X.

The characteristic function of AX is denoted by χA and is defined by

χA(x)={1if xA,0otherwise,

for all xX. For any α ∈ [0, 1], we consider α as a constant fuzzy subset of X with α(x) = α for all xX.

Let {fi : iI} be a family of the fuzzy subsets of X. Fuzzy subsets ⋃iIfi and ⋂iIfi are defined as follows:

(iIfi)(x):=iI{fi(x)}   and   (iIfi)(x):=iI{fi(x)},

for all xX, where

iIfi(x)=supiI{fi(x)}   and   iIfi(x)=infiI{fi(x)}.

For fuzzy subset f and g of X, we define fg as follows: if f(x) ≤ g(x) for all xX.

We denote the set F(S), where S is an ordered semigroup, by F(S). For any f, gF(S), we define the product fg of f and g as follows:

(fg)(x):={(u,v)Sx{f(u)g(v)}if Sx,0if Sx=,

for all xS.

The results were as follows:

### Theorem 2.2 [15]

Let S be an ordered semigroup. Subsequently, F(S) := ⟨F(S); ∘, ⊆⟩ is an ordered semigroup.

### Lemma 2.3 [16]

Let S be an ordered semigroup. Subsequently, ⟨F(S);∪, ∩⟩ is the distributive lattice.

Some important fuzzy ideals in ordered semigroups are as follows:

Let S be an ordered semigroup and f be a fuzzy subset of S such that f(x) ≥ f(y) whenever xy. Subsequently, f is called:

• 1. a fuzzy left (resp., right) ideal [31] of S if f(xy) ≥ f(y) (resp., f(xy) ≥ f(x)) for all xS;

• 2. a fuzzy quasi-ideal [31] of S if f(x) ≥ (f ∘ 1)(x) ∧ (1 ∘ f)(x) for all xS.

In the following, we assume 0 ≤ α < β ≤ 1. In this section, we reintroduce the concepts of (α, β)-fuzzy left ideals, (α, β)-fuzzy right ideals, and (α, β)-fuzzy quasi-ideals for ordered semigroup S. We show that an (α, β)-fuzzy quasi-ideal is the intersection of an (α, β)-fuzzy left ideal and an (α, β)-fuzzy right ideal of S, and vice versa. The notions of left (right) ideals are characterized in terms of (α, β)-fuzzy left (right) ideals. Moreover, we characterize (α, β)-fuzzy left (right) ideals.

### Definition 3.1 [27]

Let S be an ordered semigroup. A fuzzy subset f of S is called an (α, β)-fuzzy left ((α, β)-fuzzy right) ideal of S if for any x, yS,

• 1. f(xy) ∨ αf(y) ∧ β (resp., f(xy) ∨ αf(x) ∧ β);

• 2. xy implies f(x) ∨ αf(y) ∧ β.

We occassionally call a (α, β)-fuzzy left ((α, β)-fuzzy right) ideal an (α, β)-fuzzy one-sided ideal. A fuzzy subset f of S is said to be a (α, β)-fuzzy two-sided ideal if it is both the (α, β)-fuzzy left ideal and (α, β)-fuzzy right ideal of S.

### Definition 3.2 [27]

Let S be an ordered semigroup. A fuzzy subset f of S is called an (α, β)-fuzzy quasi-ideal of S if, for any x, yS,

• 1. f(x) ∨ α ≥ (f ∘ 1)(x) ∧ (1 ∘ f)(x) ∧ β;

• 2. xy implies f(x) ∨ αf(y) ∧ β.

### Remark 3.3

Let S be an ordered semigroup. We observe that:

• 1. every fuzzy left (resp., right, quasi-) ideal of S is a (0, 1)-fuzzy (resp., right, quasi-) ideal of S;

• 2. every (α, β)-fuzzy left (resp., right, quasi-) ideal of S is a fuzzy left (resp., right, quasi-) ideal of S whenever the image of such (α, β)-fuzzy left (resp., right, quasi-) ideal lies between of α and β.

Furthermore, the notion of (α, β)-fuzzy one-sided ideals can be regarded as a generalization of fuzzy one-sided ideals with the following settings:

• 1. an (∈, ∈ ∨ q)-fuzzy one-sided ideal is a (0, 0.5)-fuzzy one-sided ideal (see [19]);

• 2. an (∈, ∈ ∨ qk)-fuzzy one-sided ideal is a (0,1-k2)-fuzzy one-sided ideal, where k ∈ [0, 1) (see [23]).

Hence, the (α, β)-fuzzy one-sided ideal is a generalization of an (∈, ∈ ∨ q)-fuzzy one-sided ideal and a (∈, ∈ ∨ qk)-fuzzy one-sided ideal.

### Example 3.4

Let S = {a, b, c, d}. We define a binary operation ∘ and a binary relation ≤ on S as follows:

≤ ≔ {(a, b)} ∪ ΔS, where ΔS = {(x, x) : xS}. Thus, S ≔ ⟨S; ∘, ≤⟩ is an ordered semigroup. We define f : S → [0, 1] as

f(x):={0.8if x=a,0.9if x=b,0if x=c,0.7if x=d,0.4if x=e,

for any xS. Subsequently, by routine calculation, we obtain f as the (0.4, 0.6)-fuzzy right ideal of S. Since f(ea) ≱ f(e) ∧ 0.5 and f(ea)f(e)1-k2 for any k ∈ [0, 1), f is not an (∈, ∈ ∨ q)-fuzzy right ideal and is not an (∈, ∈ ∨ qk)-fuzzy right ideal of S, respectively.

The following example shows that an (α, β)-fuzzy quasi-ideal does not require an (α, β)-fuzzy one-sided ideal:

### Example 3.5

Consider the ordered semigroup S defined in Example 3.4. We define f : S → [0, 1] as

f(x):={0.7if x=a,0.8if x=b,0.3if x=c,0.4if x=d,e,

for any xS. Then, by routine calculations, we have f as a (0.4, 0.6)-fuzzy quasi-ideal of S. We observe that f is not a fuzzy quasi-ideal of S because

• 1. ab but f(a) < f(b);

• 2. f(a) < (1 ∘ f)(a) ∧ (f ∘ 1)(a).

Furthermore, f is not a (0.4, 0.6)-fuzzy left ideal of S because f(ca) ∨ 0.4 < f(a) ∧ 0.8 and f is not a (0.4, 0.6)-fuzzy right ideal of S because f(ad) ∨ 0.4 < f(a) ∧ 0.8.

For any fuzzy subset f of S, we define a new fuzzy subset fαβ of S as fαβ(x):=(f(x)β)α for all xS (see [32]). It is observed that fαβ(x)[α,β] for all xS.

### Remark 3.6

Let f, gF(S) be We can observe that if f(x) ≤ g(y) for all x, yS, then fαβ(x)gαβ(y). In general, the opposite of this statement did not hold. Indeed, suppose that α = 0.3, β = 0.6, f(x) = 0.2 and f(y) = 0.1. We can see that fαβ(x)=fαβ(x) but f(x) > f(y).

The following lemma is straightforward; thus, we omit it.

### Lemma 3.7

Let S be an ordered semigroup, and f, gF(S). Then we have:

• 1. fαβα=fαβ=fαββ;

• 2. (fg)αβ=fαβgαβ;

• 3. (fαβ)αβ=fαβ;

• 4. f(x) ∨ αf(y) ∧ β implies fαβ(x)fαβ(y) for all x, yS.

Khan et al. [28] defined two new binary relations and a new binary operation for the set of all fuzzy subsets of S as follows: For any fuzzy subset f and g of S, we define

• 1. (fαβg)(x):=[(fg)(x)β]α;

• 2. (fαβg)(x):=[(fg)(x)β]α;

• 3. (fαβg)(x):=[(fg)(x)β]α;

for all xS.

In 2023, Lekkoksung et al. [33] emphasized the significance of the operation, denoted as αβ. Their investigation led to highlight that F(S) can be embedded into an ordered semigroup Fαβ(S):=F(S);αβ,. This realization prompted the exploration of (α, β)-fuzzy left-(right-) ideals and (α, β)-fuzzy quasi-ideals in ordered semigroups.

### Lemma 3.8

Let S be an ordered semigroup, and f, g be fuzzy subsets of S. Thus, the following statements hold:

• 1. fαβg=fαβgαβ=fαβαβgαβ.

• 2. fαβg=fαβgαβ=fαβαβgαβ.

• 3. (fαβg)(x)(fαβgαβ)(x) for all xS. The equality holds whenever Sx ≠ ∅︀.

• 4. fαβαβgαβ=fαβg.

Proof

The proof can be readily completed by applying Lemma 2.3.

### Lemma 3.9 [16]

Let S be an ordered semigroup, and f be a fuzzy subset of S. Then we have:

• 1. (1 ∘ f)(xy) ≥ f(y) for all x, yS;

• 2. (f ∘ 1)(xy) ≥ f(x) for all x, yS;

• 3. (f ∘ 1)(x) ≥ (f ∘ 1)(y) whenever xy;

• 4. (1 ∘ f)(x) ≥ (1 ∘ f)(y) whenever xy.

The following lemma is an essential tool for providing the relation between (α, β)-fuzzy left (right) ideals and (α, β)-fuzzy quasi-ideals.

### Lemma 3.10

Let S be an ordered semigroup, and f be a fuzzy subset of S such that f(x) ∨ αf(y) ∧ β whenever xy. Thus, we obtain the following statement.

• 1. The fuzzy subset [f(1f)]αβ is an (α, β)-fuzzy left ideal of S.

• 2. The fuzzy subset [f(f1)]αβ is an (α, β)-fuzzy right ideal of S.

Proof

1. Let x, yS. As [f ∪ (1 ∘ f)](xy) ≥ (1 ∘ f)(xy) ≥ f(y), from Remark 3.6, we obtain

[f(1f)]αβ(xy)(1f)αβ(xy)fαβ(y).

Therefore, from Lemmas 3.7(1), [f(1f)]αβ(xy)αfαβ(y)β: Suppose that xy, From Lemmas 3.9(4), we have (1 ∘ f)(x) ≥ (1 ∘ f)(y). Subsequently, by Remark 3.6, (1f)αβ(x)(1f)]αβ(y) and fαβ(x)fαβ(y). By applying lemmas 3.8(2), we find that [f(1f)]αβ(x)[f(1f)]αβ(y). Thus, by Lemma 3.7(1), [f(1f)]αβ(x)α[f(1f)]αβ(y)β.

2. This is similar to the results above.

The following result provides the connection between (α, β)-fuzzy quasi-ideals, (α, β)-fuzzy right ideals, and (α, β)-fuzzy right ideals of an ordered semigroup:

### Theorem 3.11

Let S be an ordered semigroup, and f be a fuzzy subset of S. Subsequently, f is an (α, β)-quasi-ideal of S if and only if fαβ=(gh)αβ for some (α, β)-fuzzy right ideal g and (α, β)-fuzzy left ideal h of S.

Proof

We assume that f is (α, β)-quasi-ideal S. From Lemma 3.10, we set g=[f(f1)]αβ and h=[f(1f)]αβ. Evidently, f ⊆ [f ∪(f ∘ 1)]∩[f ∪(1 ∘ f)]. From Remark 3.6 and Lemma 3.7, we have

fαβ[([f(f1)][f(1f)])αβ]αβ([f(f1)]αβ[f(1f)]αβ)αβ.

However, as [f ∪(f ∘1)]∩[f ∪(1∘f)] = f ∪[(f ∘1)∩(1∘f)], we have

[f(f1)]αβ[f(1f)]αβ=(f[(f1)(1f)])αβ=fαβ([(f1)(1f)]αββ)=fαβ[([(f1)(1f)β]α)β]fαβ[(fα)β](since fis an (α,β)-quasi-ideal of S)=fαβ[(fβ)α]=fαβfαβ=fαβ.

This implies that ([f(f1)]αβ[f(1f)]αβ)αβ(fαβ)αβ=fαβ.

Conversely, suppose that there exists an (α, β)-fuzzy right ideal g and an (α, β)-fuzzy left ideal h of S such that fαβ=(gh)αβ. We demonstrate that f is a (α, β)-fuzzy quasi-ideal of S. Let xS. If Sx = ∅︀ and (f ∘1)(x)∧(1 ∘f)(x)∧β = 0 ≤ f(x) ∨ α, We assume that Sx ≠ ∅︀. Then, for any (u, v) ∈ Sx, we have xuv. This implies that g(x) ∨ αg(uv) ∧ β. By the (α, β)-fuzzy right ideality of g, and Lemma 3.7(1), we obtain that gαβ(x)gαβ(u)fαβ(u). Similarly, hαβ(x)hαβ(v)fαβ(v). Thus,

(f1)αβ(x)=[((u,v)Sx{f(u)1(v)})β]α=(u,v)Sx{fαβ(u)}gαβ(x),

and

(1f)αβ(x)=[((u,v)Sx{1(u)f(v)})β]α=(u,v)Sx{fαβ(v)}hαβ(x).

Therefore,

f(x)αfαβ(x)=(gh)αβ(x)=gαβ(x)hαβ(x)(f1)αβ(x)(1f)αβ(x)(f1)(x)(1f)(x)β.

Let x, yS such that xy. Then, g(x) ∨ αg(y) ∧ β and h(x) ∨ αh(y) ∧ β. Hence, f(x)αfαβ(x)=(gh)αβ(x)[g(y)β][h(y)β]=(gh)(y)β. This follows that

f(x)α(gh)αβ(y)=fαβ(y)f(y)β.

Altogether, this proof is complete.

By setting α = 0 and β = 1 in Proposition 3.11, we obtain Proposition 4, as presented in [16]. Next, we characterize (α, β)-fuzzy left ideals in terms of operation αβ.

### Theorem 3.12

Let S be an ordered semigroup, and f be a fuzzy subset of S. Subsequently, f is an (α, β)-fuzzy left ideal of S if and only if the following statements hold.

• 1. βαβffαβ.

• 2. xy implies f(x) ∨ αf(y) ∧ β.

Proof

Let xS. If Sx = ∅, (βαβf)(x)=αfαβ(x). We assume that Sx ≠ 0. Subsequently,

(βαβf)(x)=[((u,v)Sx{β(u)f(v)})β]α=[((u,v)Sx{βf(v)})β]α[((u,v)Sx{f(uv)α})β]α=[((u,v)Sx{f(uv)β}α)β]α[((u,v)Sx{f(x)α}α)β]α=(f(x)β)α=fαβ(x).

This implies that βαβffαβ. Because f is an (α, β)-fuzzy left ideal of S, we obtain (2) as

Conversely, let x, yS. Thus,

f(xy)α[f(xy)β]α=fαβ(xy)(βαβf)(xy)=[((u,v)Sxy{β(u)f(v)})β]α(f(y)β)αf(y)β.

Thus, f(xy) ∨ αf(y) ∧ β. Because (2) holds true, we have f as the (α, β)-fuzzy left ideal of S.

Similarly, we obtain the characterization of (α, β)-fuzzy right ideals as follows:

### Theorem 3.13

Let S be an ordered semigroup, and f be a fuzzy subset of S. Then, f is an (α, β)-fuzzy right ideal of S if and only if the following statements hold.

• 1. fαββfαβ.

• 2. xy implies f(x) ∨ αf(y) ∧ β.

We observe that if we set α = 0 and β = 1, Theorems 3.12 and 3.13 becomes Lemmas 2.8 and 2.9, respectively, as presented in [34].

The notions of left(right) ideals, along with their (α, β)-fuzzifications in ordered semigroups, exhibit a close relationship. The following proposition encapsulates the characterization of left (right) ideals through the utilization of (α, β)-fuzzy left (right) ideals.

### Proposition 3.14

Let S be an ordered semigroup, and A be a nonempty subset of S. Thus, the following statements are equivalent:

• 1. (χA)αβis an (α, β)-fuzzy left (resp., right) ideal of S.

• 2. A is a left (resp., right) ideal of S.

Proof

(1) ⇒ (2). Let xS and yA. Subsequently, (χA)αβ(xy)α(χA)αβ(y)β=β. This implies that xyA. Let x, yS such that yA and xy. Subsequently, (χA)αβ(x)α(χA)αβ(y)β=β. This implies that xA. Therefore, A is a left-handed ideal of S.

(2) ⇒ (1). Let x, yS. If yA, two possibilities must be considered: xyA and xyA. If xyA, (χA)αβ(xy)α=βα=(χA)αβ(y)β. If xyA, (χA)αβ(xy)α=α=(χA)αβ(y)β. We assume that yA. Thus, xyA. This implies that (χA)αβ(xy)α=β=(χA)αβ(y)β. Let x, yS such that xy. If yA, then xA or xA. If xA, (χA)αβ(x)α=βα=(χA)αβ(y)β. If xA, (χA)αβ(x)α=α=(χA)αβ(y)β. We assume that yA. As A is the left ideal of S, xA. Therefore, (χA)αβ(x)α=β=(χA)αβ(y)β. Hence, (χA)αβ is the (α, β)-fuzzy left ideal of S.

To illustrate, (χA)αβ is an (α, β)-fuzzy right ideal of S if and only if A is the right ideal of S can be performed similarly.

By substituting α = 0 and β = 1, the theorem above becomes Proposition 2, as given by Kehayopulu and Tsingelis [31].

In contrast to Proposition 3.14, the description of (α, β)-fuzzy left (right) ideals in ordered semigroups can be obtained using the left (right) ideals. To accomplish this, it is necessary to introduce the following set: For any fF(S) and t ∈ [0, 1], an (α, β)-level set of f is defined as

levαβ(f;t):={xS:f(x)αtβ}.

### Proposition 3.15

Let S be an ordered semigroup, and f be a fuzzy subset of S. Thus, the following statements are equivalent:

• 1. f is an (α, β)-fuzzy left (resp., right) ideal of S.

• 2. The nonempty (α, β)-level set levαβ(f;t) of f is a left (resp., right) ideal of S for any t ∈ (α, β].

Proof

(1) ⇒ (2). We assume that fαβ is an (α, β)-fuzzy left ideal of S. Let t ∈ (α, β]. Given xS and ylevαβ(f;t). We then obtain that f(y) ∨ αtβ. From our assumptions, we obtain

f(xy)a[f(y)β]α=[f(y)α]β(tβ)β=tβ.

This implies that Slevαβ(f;t)levαβ(f;t). Next, we let x, yS such that xy and ylevαβ(f;t). Subsequently, xy implies f(x) ∨ αf(y) ∧ β. It follows that

f(x)α[f(y)β]α=[f(y)α]β(tβ)         (since ylevαβ(f;t))=tβ.

Thus, xlevαβ(f;t). Therefore, (levαβ(f;t)]levαβ(f;t). Overall, levαβ(f;t) is the ideal on the left for S.

(2) ⇒ (1). Let x, yS. If f(y) ≤ α, then f(xy) ∨ αf(y) ∧ β. We assume that f(y) > α. We set t = f(y) ∧ β, Then t ∈ (α, β]. Additionally, f(y) ∨ α = f(y) ≥ f(y) ∧ β = tβ, This implies that ylevαβ(f;t). Because levαβ(f;t) is the left ideal of S, we have xylevαβ(f;t). In other words, f(xy) ∨ αtβ = f(y) ∧ β. Hence, for any x, yS, f(xy) ∨ αf(y) ∧ β, Let x, yS such that xy. If f(y) ≤ α, f(x) ∨ αf(y) = f(y) ∧ β. We assume that f(y) > α. We set t = f(y) ∧ β, Subsequently, t ∈ (α, β] and f(y) ∨ α = f(y) ≥ tβ. This implies that ylevαβ(f;t). As levαβ(f;t) is the left-hand ideal of S, we obtain xlevαβ(f;t). Thus, f(x) ∨ αtβ = f(y) ∧ β. Hence, xy implies f(x) ∨ αf(y) ∧ β for any x, yS. Altogether, we have f as an (α, β)-fuzzy left ideal of S.

To illustrate, f is an (α, β)-fuzzy right ideal of S if and only if the nonempty (α, β)-level set levαβ(f;t) of f is the right ideal of S for any t ∈ (α, β]. This can be

Propositions 3.14 and 3.15 establish a direct relationship between (α, β)-fuzzy left (right) ideals and left (right) ideals in ordered semigroups. These results effectively connect set-theoretical ideals with the process of ideals’ fuzzification.

### 4. Characterizing Some Regularities of Ordered Semigroups

In this section, certain classes of ordered semigroups, regular and intra-regular ordered semigroups, use (α, β)-fuzzy left and (α, β)-fuzzy right ideals, respectively.

An ordered semigroup S is said to be:

• 1. a regular ordered semigroup if for any aS, there exists xS such that aaxa;

• 2. an intra-regular ordered semigroup if for any aS, there exists x, yS such that axa2y.

We need the following results before characterizing the regular and intra-regular ordered semigroups.

### Lemma 4.1

Let S be an ordered semigroup, and A, BS. Thus, the following statements hold:

• 1. (χAB)αβ=χAαβχB.

• 2. (χ(AB])αβ=χAαβχB.

Proof

From Proposition 7 of [9], we obtain χAB = χAχB. Applying Remark 3.6 and Lemma 3.8(1), we obtain that (χAB)αβ=(χAχB)αβ=χAαβχB. Thus, (1) holds as a claim.

By Lemma 2.5 in [35], we have χ(AB] = χAχB. It follows, by Remark 3.6 and the definition of αβ, that (χ(AB])αβ=(χAχB)αβ=χAαβχB. Hence, (2) holds.

### Lemma 4.2

Let S be an ordered semigroup and A, B be a nonempty subset of S. Subsequently, we have AB if and only if (χA)αβ(χB)αβ.

Proof

Let xA. We assume β=(χA)αβ(x)(χB)αβ(x). It follows that (χB)αβ(x)=β because α(χB)αβ(x)β. That is, xB.

Conversely, let xS. If xA, then (χA)αβ(x)=α(χB)αβ(x). If xA, then xB. Thus, (χA)αβ(x)=β=(χB)αβ(x). Hence, (χA)αβ(χB)αβ.

### Lemma 4.3 [36]

Let S be an ordered semigroup. Thus, the following statements are equivalent:

• 1. S is regular.

• 2. RL = (RL] for any right ideal R and left ideal L of S.

### Lemma 4.4

Let S be an ordered semigroup, f an (α, β)-fuzzy right ideal, and g the (α, β)-fuzzy left ideal of S. Subsequently, fαβgfαβg.

Proof

Let xS. If S = ∅, (fαβg)(x)=α(fαβg)(x). We assume that S ≠ ∅. Then

(fαβg)(x)=[((u,v)Sx{f(u)g(v)})β]α=[((u,v)Sx{(f(u)β)(g(v)β)})β]α[((u,v)Sx{(f(uv)α)(g(uv)α)})β]α=[((u,v)Sx{(f(uv)β}(g(uv)β)})β]α[((u,v)Sx{(f(x)α}(g(x)α)})β]α=[(fg)(x)β]α=(fααg)(x).

Hence, the proof is complete.

We characterize the regular-ordered semigroups as follows:

### Theorem 4.5

Let S be an ordered semigroup. Thus, the following statement is equivalent:

• 1. S is regular.

• 2. fαβg=fαβg for any (α, β)-fuzzy right ideal f and (α, β)-fuzzy left ideal g of S.

Proof

(1) ⇒ (2). Let f and g be the (α, β)-fuzzy right ideals and (α, β)-fuzzy right ideals of S. From Lemma 4.4, it is sufficient to show that fαβgfαβg. Let aS. Because S is regular, there exists xS such that aaxa. It follows that Sa ≠ ∅ because (ax, a) ∈ Sa. Thus,

(fαβg)(a)=[((u,v)Sa{f(u)g(v)})β]α(f(ax)g(a)β)α=[(f(ax)α)(g(a)β)β]α[(f(a)α)(g(a)β)β]α=(f(a)g(a)β)α=[(fg)(x)β]α=(fααg)(a).

This indicates that fαβgfαβg.

(2) ⇒ (1). Let R and L be the right and left ideals of S, respectively. From Lemma 4.3, we show that RL = (RL]. By Theorem 3.14, we have that (χR)αβ and (χL)αβ is an (α, β)-fuzzy right ideal and an (α, β)-fuzzy left ideal of S, respectively. Subsequently, based on our assumption and Lemma 4.1, we have

(χRL)αβ=χRαβχL=χRαβχL=(χ(RL])αβ.

From Lemma 4.2, we have RL = (RL]. Therefore, by Lemma 4.3, S is regular.

### Lemma 4.6 [34]

Let S be an ordered semigroup. Thus, the following statements are equivalent:

• 1. S is intra-regular.

• 2. LR ⊆ (LR] for any left ideal L and right ideal R of S.

### Theorem 4.7

Let S be an ordered semigroup. Thus, the following statement is equivalent:

• 1. S is intra-regular.

• 2. fαβgfαβg for any (α, β)-fuzzy left ideal f and (α, β)-fuzzy right ideal g of S.

Proof

(1) ⇒ (2). Let f and g be the (α, β)-fuzzy left ideal, and let (α, β) the fuzzy right ideal of S. Let aS. Because S is intra-regular, there exist x, yS such that axa2y. Sa ≠ ∅ because (xa, ay) ∈ Sa. Thus,

(fαβg)(a)=[((u,v)Sa{f(u)g(v)})β]α(f(xa)g(ay)β)α=[(f(xa)α)(g(ay)α)β]α[(f(a)β)(g(a)β)β]α=(f(a)g(a))β]α=(fαβg)(a).

This indicates that fαβgfαβg.

(2) ⇒ (1). Let L and R be the left and right ideals of S, respectively. From Lemma 4.6, we show that LR ⊆ (LR]. By Theorem 3.14, we have that (χL)αβ and (χR)αβ is an (α, β)-fuzzy left ideal and an (α, β)-fuzzy right ideal of S, respectively. Subsequently, based on our assumption and Lemma 4.1, we have

(χLR)αβ=χLαβχR=χLαβχR=(χ(LR])αβ.

From Lemma 4.2, we have LR ⊆ (LR]. Therefore, by Lemma 4.6, S is intra-regular.

In this study, we explored (α, β)-fuzzy left (right) and (α, β)-fuzzy quasi-ideals within the context of ordered semigroups. These concepts generalize their corresponding fuzzy left (resp., right) ideals and fuzzy quasi-ideals, respectively. The initial segment of our work was devoted to establishing foundational results concerning these concepts. We now present the intersection property of (α, β)-fuzzy quasi-ideals. In fact, the connection among (α, β)-fuzzy left-hand ideals, (α, β)-fuzzy right-hand ideals, and (α, β)-fuzzy quasi-ideals in ordered semigroups is provided. Subsequently, we characterize the (α, β)-fuzzy left ideals and (α, β)-fuzzy right ideals through a specific product. This product is crucial in effectively representing fuzzy-ordered semigroups. Moreover, we establish meaningful links between (α, β)-fuzzy left (resp., right) ideals and their conventional counterparts, namely, left (resp., right) ideals. Collectively, these findings underscore the inherent similarities between ideals and their fuzzy counterparts. In the final section of our study, we focus on classifying ordered semigroups as precise, regular, and intra-regular ordered semigroups. These classes are characterized by applying (α, β)-fuzzy left ideals and (α, β)-fuzzy right ideals. As a parting note, we encourage our readers to explore the utility of (α, β)-fuzzy ideals in characterizing other classes of ordered semigroups and investigate their prime, semiprime, and weakly semiprime properties.

The author would like to thank Rajamangala University of Technology Isan, Khon Kaen Campus.

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Young Bae Jun is an Emeritus Professor of Gyeongsang National University, Korea. He is currently the Editor-in-Chief of two international journals and the editorial board of six international journals. He has published over 800 papers in international journals. His primary scientific interests include BCK/BCI-algebras and fuzzy (soft, rough) logical algebras. He was on the list of 2016, 2017, 2018 and 2019 Highly Cited Researchers which has been published in Clarivate Analytics.

Email: skywine@gmail.com

Kittisak Tinpun received Ph.D from University of Potsdam, Germany. He is currently working as an assistant professor at the Prince of Songkla University, Pattani Campus, Thailand. His research interests are semigroup theory, transformation semigroup theories.

Email: kittisak.ti@psu.ac.th

Nareupanat Lekkoksung received M.Sc. and Ph.D. from Khon Kaen University. He is currently an assistant professor at Rajamangala University of Technology Isan, Khon Kaen Campus, since 2019. His research interests are hypersubstitution theory, semigroup theory, and fuzzy set theory.

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 141-152

Published online June 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.2.141

## Exploring Regularities of Ordered Semigroups Through Generalized Fuzzy Ideals

Young Bae Jun1, Kittisak Tinpun2, and Nareupanat Lekkoksung3

1Department of Mathematics Education, Gyeongsang National University, Jinju, Korea
2Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani, Thailand
3Division of Mathematics, Faculty of Engineering, Rajamangala University of Technology Isan, Khon Kaen, Thailand

Correspondence to:Nareupanat Lekkoksung (nareupanat.le@rmuti.ac.th)

Received: February 17, 2022; Revised: August 18, 2023; Accepted: May 3, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We consider (α, β)-fuzzy ideals, which are a generalized version of fuzzy ideals, in ordered semigroups. A connection between (α, β)-fuzzy quasi-ideals and (α, β)-fuzzy left (right) ideals is provided. The notion of (α, β)-fuzzy ideals is characterized in terms of a particular operation. We describe regular and intra-regular ordered semigroups using the concept of (α, β)fuzzy ideals.

Keywords: (&alpha,, &beta,)-fuzzy left ideal, (&alpha,, &beta,)-fuzzy right ideal, Ordered semigroup, Regular, Intra-regular

### 1. Introduction

The concept of ordered semigroups is a generalization of semigroups. It is an algebraic structure of type (2; 2) comprising a semigroup and a partially ordered set defined on the same set, such that the order relation is compatible with the associative operation (see [1, 2]). This concept has been extensively studied. Ideals are crucial in studying ordered semigroups from numerous perspectives (refer to [310]). Kehayopulu [7] introduced the concepts of left and right ideals in ordered semigroups. Furthermore, the author explored a more nuanced interpretation of the prime properties inherent in these ideals, called weakly prime ideals. Kehayopulu pioneered the concept of quasi-ideals in ordered semi-groups. Using quasi-, left, and right ideals helped describe regularly ordered semigroups and intra-regular ordered semigroups across diverse analytical avenues (see [11]). Given that ideals comprise a fundamental aspect when examining the algebraic attributes of ordered semigroups, numerous mathematical tools have been employed to expand the scope of the ideal theory. These tools were subsequently utilized to further explore and analyze the ordered semigroups. Fuzzy sets are important tools used to study the structural properties of ordered semigroups.

The concept of fuzzy sets was proposed by Zadeh [12] in 1965. Serving as an extension of crisp sets, this conceptual framework has broad applications in diverse mathematical disciplines, including algebra. Rosenfeld [13] pioneered the exploration of group properties within the field of fuzzy sets, known as fuzzy groups. Kuroki introduced the concept of fuzzy semigroups. Fuzzy sets have been applied to examine various properties of semigroups, as demonstrated in [14].

In 2003, Kehayopulu and Tsingelis [15] implemented the concept of fuzzy sets to ordered semigroups. A fuzzy ordered semigroup is an ordered semigroup whose universe set is the set of all the fuzzy subsets on an ordered semigroup with a particular binary operation and order relation. They demonstrated that any ordered semigroup can be embedded into a fuzzy-ordered semigroup. An extensive exploration of ordered semigroups from multiple perspectives was conducted using fuzzy ideals. For example, Kehayopulu and Tsingelis [9] introduced fuzzy quasi-ideals that allowed for the characterization of regularly ordered semigroups through the application of fuzzy left (resp., right, or quasi-) ideals. In a parallel vein, in their subsequent study [16], the concept of fuzzy bi-ideals in ordered semigroups was introduced. The authors delved into the complicated relationships among these specific types of fuzzy ideals, revealing the broader scope of fuzzy bi-ideals as an extension of fuzzy left (resp., right, or quasi) ideals in ordered semigroups. The coincidence between fuzzy quasi-ideals and fuzzy bi-ideals was established, further underlining their inherent interconnections. Furthermore, fuzzy left and fuzzy right ideals have attained an insightful characterization of fuzzy quasi-ideals.

Numerous researchers have attempted to expand the concepts of fuzzy ideals and their applications to examine specific properties of ordered semigroups. Drawing inspiration from the “belongs to” relation (∈) and the “quasi-coincident” relation (q), Khan and Shabir [17] introduced the concept of (α, β)-fuzzy interior ideals in ordered semigroups, where α, β ∈ {∈, q, ∈ ∨ q, ∈ ∧ q} with the condition that α ≠ ∈ ∧ q. They primarily focused on the scenario in which α = ∈ and β = ∈ ∨ q. In this context, the authors established definitions of (∈, ∈ ∨ q)-fuzzy left (right, two-sided, interior) ideals in ordered semigroups and explored the relationships between these (∈, ∈ ∨ q)-fuzzy ideals. Jun et al. [18] extended this notion to (α, β)-fuzzy bi-ideals in ordered semigroups with a specific focus on (∈, ∈ ∨ q)-fuzzy bi-ideals. They characterized distinct classes of ordered semigroups using this concept. Furthermore, Khan et al. [19] conducted an in-depth investigation of the properties of (∈, ∈ ∨ q)-fuzzy left-hand ideals and (∈, ∈ ∨ q)-fuzzy right-hand ideals in ordered semigroups. Their research culminated in the characterization of regularly ordered semigroups using (∈, ∈ ∨ q)-fuzzy left ideals and (∈, ∈ ∨ q)-fuzzy right ideals.

The concept of fuzzy ideals in ordered semigroups, defined as belonging to the and quasi-coincident relations, was first extended by Khan et al. [20]. They introduced various types of (∈, ∈ ∨ qk)-fuzzy ideals such as (∈, ∈ ∨ qk)-fuzzy left (right) ideals and (∈, ∈ ∨ qk)-fuzzy (generalized) bi-ideals. Some classes of ordered semigroups are characterized by (∈, ∈ ∨ qk)-fuzzy ideals. In addition, a minimum of two publications in 2012 characterized specific classes of ordered semigroups using (∈, ∈ ∨ qk)-fuzzy ideals (see [21, 22]). Tang and Xie [23] studied the notion of (∈, ∈ ∨ qk)-fuzzy left (and right) ideals, where k ∈ [0, 1). They demonstrated that when k = 0, a (∈, ∈ ∨ qk)-fuzzy ideal is a (∈, ∈ ∨ q)-fuzzy ideal. Moreover, they characterized the prime properties of these fuzzy ideals.

The notions of (∈, ∈ ∨(k*, qk))-fuzzy ideals and (∈, $∈∨qkδ$)-fuzzy ideals represent the same concept in ordered semigroups. These concepts, which are extensions of the independently introduced generalized (∈, ∈ ∨ qk)-fuzzy ideals by Khan et al. [24] and Ali Khan et al. [25], converge to underline the specific aspects of fuzzy ideals. Their investigations illuminated ordered semi-groups through a new perspective of mathematical tools, employing novel fuzzy ideals with distinctive properties. Recently, Muhiuddin et al. [26] have extended the scope of the (∈, ∈ ∨ (k, qk))-fuzzy ideals, introducing the concept of (∈, ∈ ∨ (k, qk))-fuzzy (m, n)-ideals. This novel extension is explained through the description of (∈, ∈ ∨ (k, qk))-fuzzy (m, n)ideals using their level sets. Notably, the notion of (∈, ∈ ∨ (k, qk))-fuzzy (m, n)ideals characterizes the (m, n)regular ordered semigroups.

The generalization of fuzzy ideals in ordered semigroups is based on the concept of (α, β)-fuzzy ideals, where 0 ≤ α < β ≤ 1. Feng and Corsini [27] defined the concept of (α, β)-fuzzy left (right, interior, quasi-, bi-) ideals in ordered semigroups. The authors studied the connections between these concepts. Independently defined the notion of (α, β)-fuzzy bi-ideals by Khan et al. [28]. They completely characterized regular-ordered semigroups based on the properties of (α, β)-fuzzy bi-ideals. Feng and Corsini [29] investigated the relationship between (α, β)-fuzzy ideals and (α, β)-fuzzy interior ideals in ordered semigroups. They demonstrated that these notions coincided with regular and intra-regular ordered semigroups.

In this study, we center our investigation on the concepts proposed by Feng and Corsini [27] and Khan et al. [28], particularly focusing on the notions of (α, β)-fuzzy ideals in ordered semigroups. Feng and Corsini [29] demonstrated that any (α, β)-fuzzy left (right) ideal is an (α, β)-fuzzy quasi-ideal. For example, we illustrate that the opposite of this statement does not generally hold. A relation between (α, β)-fuzzy left (right) ideals and (α, β)-fuzzy quasi-ideals is presented. To provide more connections between (α, β)-fuzzy left (right) ideals and left (right) ideals in ordered semigroups, we attempt to describe both concepts using their level sets and characteristic functions. We also describe (α, β)-fuzzy left (right) ideals of their products. Moreover, we characterize regular and intra-regular ordered semigroups using (α, β)-fuzzy left and (α, β)-fuzzy right ideals.

### 2. Basic Concepts

In this section, we introduce some basic terminologies for ordered semigroups and fuzzy subsets, which will be used in the subsequent section.

An algebraic structure ⟨S; ·, ≤⟩ of type (2; 2) is called an ordered semigroup if:

• 1. ⟨S; ·⟩ is a semigroup;

• 2. ⟨S; ≤⟩ is partially ordered set;

• 3. ≤ compatible with · the associative binary operation.

For convenience, we write an ordered semigroup ⟨S; ·, ≤⟩ by S the bold letter of its universe set. We denote the product x · y as xy.

Let S be an ordered semigroup. For any A, B, CS and xS, we define AB := {ab : aA and bB}, Sa := {(x, y) ∈ S × S : axy} and

$(C]:={x∈S:x≤y such that y∈C}.$

### Lemma 2.1 [7]

Let S be an ordered semigroup, and A, B, CS. Then:

• 1. A ⊆ (A];

• 2. AB implies (A] ⊆ (B];

• 3. (A](B] = (AB];

• 4. (AB] ⊇ (A] ∪ (B];

• 5. ((A]] = (A].

Let S be an ordered semigroup. A nonempty subset A of S is said to be a subsemigroup of S if ⟨A; ·|A×A, ≤|A×A⟩ is an ordered semigroup. A nonempty subset A of S such that (A] ⊆ A is called

• 1. a left ideal [7] of S if SAA;

• 2. a right ideal [7] of S if SAA;

• 3. a two-sided ideal (ideal) [7] of S if it is both a left and a right ideal of S;

• 4. a quasi-ideal [30] of S if (SA] ∩ (AS] ⊆ A.

Let X be a nonempty set. A fuzzy subset [12] of X (or a fuzzy set in X) is a mapping from X to [0, 1] the closed unit interval. Denoted by F(S) the set of fuzzy subsets of X.

The characteristic function of AX is denoted by χA and is defined by

$χA(x)={1if x A,0otherwise,$

for all xX. For any α ∈ [0, 1], we consider α as a constant fuzzy subset of X with α(x) = α for all xX.

Let {fi : iI} be a family of the fuzzy subsets of X. Fuzzy subsets ⋃iIfi and ⋂iIfi are defined as follows:

$(∪i∈Ifi)(x):=⋁i∈I{fi(x)} and (∩i∈Ifi)(x):=⋀i∈I{fi(x)},$

for all xX, where

$⋁i∈Ifi(x)=supi∈I{fi(x)} and ⋀i∈Ifi(x)=infi∈I{fi(x)}.$

For fuzzy subset f and g of X, we define fg as follows: if f(x) ≤ g(x) for all xX.

We denote the set F(S), where S is an ordered semigroup, by F(S). For any f, gF(S), we define the product fg of f and g as follows:

$(f∘g)(x):={∨(u,v)∈Sx{f(u)∧g(v)}if Sx≠∅,0if Sx=∅,$

for all xS.

The results were as follows:

### Theorem 2.2 [15]

Let S be an ordered semigroup. Subsequently, F(S) := ⟨F(S); ∘, ⊆⟩ is an ordered semigroup.

### Lemma 2.3 [16]

Let S be an ordered semigroup. Subsequently, ⟨F(S);∪, ∩⟩ is the distributive lattice.

Some important fuzzy ideals in ordered semigroups are as follows:

Let S be an ordered semigroup and f be a fuzzy subset of S such that f(x) ≥ f(y) whenever xy. Subsequently, f is called:

• 1. a fuzzy left (resp., right) ideal [31] of S if f(xy) ≥ f(y) (resp., f(xy) ≥ f(x)) for all xS;

• 2. a fuzzy quasi-ideal [31] of S if f(x) ≥ (f ∘ 1)(x) ∧ (1 ∘ f)(x) for all xS.

### 3. On (α, β)-Fuzzy Ideals

In the following, we assume 0 ≤ α < β ≤ 1. In this section, we reintroduce the concepts of (α, β)-fuzzy left ideals, (α, β)-fuzzy right ideals, and (α, β)-fuzzy quasi-ideals for ordered semigroup S. We show that an (α, β)-fuzzy quasi-ideal is the intersection of an (α, β)-fuzzy left ideal and an (α, β)-fuzzy right ideal of S, and vice versa. The notions of left (right) ideals are characterized in terms of (α, β)-fuzzy left (right) ideals. Moreover, we characterize (α, β)-fuzzy left (right) ideals.

### Definition 3.1 [27]

Let S be an ordered semigroup. A fuzzy subset f of S is called an (α, β)-fuzzy left ((α, β)-fuzzy right) ideal of S if for any x, yS,

• 1. f(xy) ∨ αf(y) ∧ β (resp., f(xy) ∨ αf(x) ∧ β);

• 2. xy implies f(x) ∨ αf(y) ∧ β.

We occassionally call a (α, β)-fuzzy left ((α, β)-fuzzy right) ideal an (α, β)-fuzzy one-sided ideal. A fuzzy subset f of S is said to be a (α, β)-fuzzy two-sided ideal if it is both the (α, β)-fuzzy left ideal and (α, β)-fuzzy right ideal of S.

### Definition 3.2 [27]

Let S be an ordered semigroup. A fuzzy subset f of S is called an (α, β)-fuzzy quasi-ideal of S if, for any x, yS,

• 1. f(x) ∨ α ≥ (f ∘ 1)(x) ∧ (1 ∘ f)(x) ∧ β;

• 2. xy implies f(x) ∨ αf(y) ∧ β.

### Remark 3.3

Let S be an ordered semigroup. We observe that:

• 1. every fuzzy left (resp., right, quasi-) ideal of S is a (0, 1)-fuzzy (resp., right, quasi-) ideal of S;

• 2. every (α, β)-fuzzy left (resp., right, quasi-) ideal of S is a fuzzy left (resp., right, quasi-) ideal of S whenever the image of such (α, β)-fuzzy left (resp., right, quasi-) ideal lies between of α and β.

Furthermore, the notion of (α, β)-fuzzy one-sided ideals can be regarded as a generalization of fuzzy one-sided ideals with the following settings:

• 1. an (∈, ∈ ∨ q)-fuzzy one-sided ideal is a (0, 0.5)-fuzzy one-sided ideal (see [19]);

• 2. an (∈, ∈ ∨ qk)-fuzzy one-sided ideal is a $(0,1-k2)$-fuzzy one-sided ideal, where k ∈ [0, 1) (see [23]).

Hence, the (α, β)-fuzzy one-sided ideal is a generalization of an (∈, ∈ ∨ q)-fuzzy one-sided ideal and a (∈, ∈ ∨ qk)-fuzzy one-sided ideal.

### Example 3.4

Let S = {a, b, c, d}. We define a binary operation ∘ and a binary relation ≤ on S as follows:

$oabcdeaaaaddbabaddcccceedaaaddecccee$

≤ ≔ {(a, b)} ∪ ΔS, where ΔS = {(x, x) : xS}. Thus, S ≔ ⟨S; ∘, ≤⟩ is an ordered semigroup. We define f : S → [0, 1] as

$f(x):={0.8if x=a,0.9if x=b,0if x=c,0.7if x=d,0.4if x=e,$

for any xS. Subsequently, by routine calculation, we obtain f as the (0.4, 0.6)-fuzzy right ideal of S. Since f(ea) ≱ f(e) ∧ 0.5 and $f(ea)≱f(e)∧1-k2$ for any k ∈ [0, 1), f is not an (∈, ∈ ∨ q)-fuzzy right ideal and is not an (∈, ∈ ∨ qk)-fuzzy right ideal of S, respectively.

The following example shows that an (α, β)-fuzzy quasi-ideal does not require an (α, β)-fuzzy one-sided ideal:

### Example 3.5

Consider the ordered semigroup S defined in Example 3.4. We define f : S → [0, 1] as

$f(x):={0.7if x=a,0.8if x=b,0.3if x=c,0.4if x=d,e,$

for any xS. Then, by routine calculations, we have f as a (0.4, 0.6)-fuzzy quasi-ideal of S. We observe that f is not a fuzzy quasi-ideal of S because

• 1. ab but f(a) < f(b);

• 2. f(a) < (1 ∘ f)(a) ∧ (f ∘ 1)(a).

Furthermore, f is not a (0.4, 0.6)-fuzzy left ideal of S because f(ca) ∨ 0.4 < f(a) ∧ 0.8 and f is not a (0.4, 0.6)-fuzzy right ideal of S because f(ad) ∨ 0.4 < f(a) ∧ 0.8.

For any fuzzy subset f of S, we define a new fuzzy subset $fαβ$ of S as $fαβ(x):=(f(x)∧β)∨α$ for all xS (see [32]). It is observed that $fαβ(x)∈[α,β]$ for all xS.

### Remark 3.6

Let f, gF(S) be We can observe that if f(x) ≤ g(y) for all x, yS, then $fαβ(x)≤gαβ(y)$. In general, the opposite of this statement did not hold. Indeed, suppose that α = 0.3, β = 0.6, f(x) = 0.2 and f(y) = 0.1. We can see that $fαβ(x)=fαβ(x)$ but f(x) > f(y).

The following lemma is straightforward; thus, we omit it.

### Lemma 3.7

Let S be an ordered semigroup, and f, gF(S). Then we have:

• 1. $fαβ∪α=fαβ=fαβ∩β$;

• 2. $(f∩g)αβ=fαβ∩gαβ$;

• 3. $(fαβ)αβ=fαβ$;

• 4. f(x) ∨ αf(y) ∧ β implies $fαβ(x)≥fαβ(y)$ for all x, yS.

Khan et al. [28] defined two new binary relations and a new binary operation for the set of all fuzzy subsets of S as follows: For any fuzzy subset f and g of S, we define

• 1. $(f∩αβg) (x):=[(f∩g) (x)∧β]∨α$;

• 2. $(f∪αβg) (x):=[(f∪g) (x)∧β]∨α$;

• 3. $(f∘αβg) (x):=[(f∘g) (x)∧β]∨α$;

for all xS.

In 2023, Lekkoksung et al. [33] emphasized the significance of the operation, denoted as $∘αβ$. Their investigation led to highlight that F(S) can be embedded into an ordered semigroup $Fαβ(S):=⟨F(S);∘αβ,⊆⟩$. This realization prompted the exploration of (α, β)-fuzzy left-(right-) ideals and (α, β)-fuzzy quasi-ideals in ordered semigroups.

### Lemma 3.8

Let S be an ordered semigroup, and f, g be fuzzy subsets of S. Thus, the following statements hold:

• 1. $f∩αβg=fαβ∩gαβ=fαβ∩αβgαβ$.

• 2. $f∪αβg=fαβ∪gαβ=fαβ∪αβgαβ$.

• 3. $(f∘αβg) (x)≥(fαβ∘gαβ) (x)$ for all xS. The equality holds whenever Sx ≠ ∅︀.

• 4. $fαβ∘αβgαβ=f∘αβg$.

Proof

The proof can be readily completed by applying Lemma 2.3.

### Lemma 3.9 [16]

Let S be an ordered semigroup, and f be a fuzzy subset of S. Then we have:

• 1. (1 ∘ f)(xy) ≥ f(y) for all x, yS;

• 2. (f ∘ 1)(xy) ≥ f(x) for all x, yS;

• 3. (f ∘ 1)(x) ≥ (f ∘ 1)(y) whenever xy;

• 4. (1 ∘ f)(x) ≥ (1 ∘ f)(y) whenever xy.

The following lemma is an essential tool for providing the relation between (α, β)-fuzzy left (right) ideals and (α, β)-fuzzy quasi-ideals.

### Lemma 3.10

Let S be an ordered semigroup, and f be a fuzzy subset of S such that f(x) ∨ αf(y) ∧ β whenever xy. Thus, we obtain the following statement.

• 1. The fuzzy subset $[f∪(1∘f)]αβ$ is an (α, β)-fuzzy left ideal of S.

• 2. The fuzzy subset $[f∪(f∘1)]αβ$ is an (α, β)-fuzzy right ideal of S.

Proof

1. Let x, yS. As [f ∪ (1 ∘ f)](xy) ≥ (1 ∘ f)(xy) ≥ f(y), from Remark 3.6, we obtain

$[f∪(1∘f)]αβ(xy)≥(1∘f)αβ(xy)≥fαβ(y).$

Therefore, from Lemmas 3.7(1), $[f∪(1∘f)]αβ(xy)∨α≥fαβ(y)∧β$: Suppose that xy, From Lemmas 3.9(4), we have (1 ∘ f)(x) ≥ (1 ∘ f)(y). Subsequently, by Remark 3.6, $(1∘f)αβ(x)≥(1∘f)]αβ(y)$ and $fαβ(x)≥fαβ(y)$. By applying lemmas 3.8(2), we find that $[f∪(1∘f)]αβ(x)≥[f∪(1∘f)]αβ(y)$. Thus, by Lemma 3.7(1), $[f∪(1∘f)]αβ(x)∨α≥[f∪(1∘f)]αβ(y)∧β$.

2. This is similar to the results above.

The following result provides the connection between (α, β)-fuzzy quasi-ideals, (α, β)-fuzzy right ideals, and (α, β)-fuzzy right ideals of an ordered semigroup:

### Theorem 3.11

Let S be an ordered semigroup, and f be a fuzzy subset of S. Subsequently, f is an (α, β)-quasi-ideal of S if and only if $fαβ=(g∩h)αβ$ for some (α, β)-fuzzy right ideal g and (α, β)-fuzzy left ideal h of S.

Proof

We assume that f is (α, β)-quasi-ideal S. From Lemma 3.10, we set $g=[f∪(f∘1)]αβ$ and $h=[f∪(1∘f)]αβ$. Evidently, f ⊆ [f ∪(f ∘ 1)]∩[f ∪(1 ∘ f)]. From Remark 3.6 and Lemma 3.7, we have

$fαβ⊆[([f∪(f∘1)]∩[f∪(1∘f)])αβ]αβ⊆([f∪(f∘1)]αβ∩[f∪(1∘f)]αβ)αβ.$

However, as [f ∪(f ∘1)]∩[f ∪(1∘f)] = f ∪[(f ∘1)∩(1∘f)], we have

$[f∪(f∘1)]αβ∩[f∪(1∘f)]αβ=(f∪[(f∘1)∩(1∘f)])αβ=fαβ∪([(f∘1)∩(1∘f)]αβ∩β)=fαβ∪[([(f∘1)∩(1∘f)∩β]∪α)∩β]⊆fαβ∪[(f∪α)∩β] (since f is an (α,β)-quasi-ideal of S)=fαβ∪[(f∩β)∪α]=fαβ∪fαβ=fαβ.$

This implies that $([f∪(f∘1)]αβ∩[f∪(1∘f)]αβ)αβ⊆(fαβ)αβ=fαβ$.

Conversely, suppose that there exists an (α, β)-fuzzy right ideal g and an (α, β)-fuzzy left ideal h of S such that $fαβ=(g∩h)αβ$. We demonstrate that f is a (α, β)-fuzzy quasi-ideal of S. Let xS. If Sx = ∅︀ and (f ∘1)(x)∧(1 ∘f)(x)∧β = 0 ≤ f(x) ∨ α, We assume that Sx ≠ ∅︀. Then, for any (u, v) ∈ Sx, we have xuv. This implies that g(x) ∨ αg(uv) ∧ β. By the (α, β)-fuzzy right ideality of g, and Lemma 3.7(1), we obtain that $gαβ(x)≥gαβ(u)≥fαβ(u)$. Similarly, $hαβ(x)≥hαβ(v)≥fαβ(v)$. Thus,

$(f∘1)αβ(x)=[(⋁(u,v)∈Sx{f(u)∧1(v)})∧β]∨α=⋁(u,v)∈Sx{fαβ(u)}≤gαβ(x),$

and

$(1∘f)αβ(x)=[(⋁(u,v)∈Sx{1(u)∧f(v)})∧β]∨α=⋁(u,v)∈Sx{fαβ(v)}≤hαβ(x).$

Therefore,

$f(x)∨α≥fαβ(x)=(g∩h)αβ(x)=gαβ(x)∧hαβ(x)≥(f∘1)αβ(x)∧(1∘f)αβ(x)≥(f∘1)(x)∧(1∘f)(x)∧β.$

Let x, yS such that xy. Then, g(x) ∨ αg(y) ∧ β and h(x) ∨ αh(y) ∧ β. Hence, $f(x)∨α≥fαβ(x)=(g∩h)αβ(x)≥[g(y)∧β]∧[h(y)∧β]=(g∩h)(y)∧β$. This follows that

$f(x)∨α≥(g∩h)αβ(y)=fαβ(y)≥f(y)∧β.$

Altogether, this proof is complete.

By setting α = 0 and β = 1 in Proposition 3.11, we obtain Proposition 4, as presented in [16]. Next, we characterize (α, β)-fuzzy left ideals in terms of operation $∘αβ$.

### Theorem 3.12

Let S be an ordered semigroup, and f be a fuzzy subset of S. Subsequently, f is an (α, β)-fuzzy left ideal of S if and only if the following statements hold.

• 1. $β∘αβf⊆fαβ$.

• 2. xy implies f(x) ∨ αf(y) ∧ β.

Proof

Let xS. If Sx = ∅, $(β∘αβf)(x)=α≤fαβ(x)$. We assume that Sx ≠ 0. Subsequently,

$(β∘αβf)(x)=[(⋁(u,v)∈Sx{β(u)∧f(v)})∧β]∨α=[(⋁(u,v)∈Sx{β∧f(v)})∧β]∨α≤[(⋁(u,v)∈Sx{f(uv)∧α})∧β]∨α=[(⋁(u,v)∈Sx{f(uv)∧β}∨α)∧β]∨α≤[(⋁(u,v)∈Sx{f(x)∨α}∨α)∧β]∨α=(f(x)∧β)∨α=fαβ(x).$

This implies that $β∘αβf⊆fαβ$. Because f is an (α, β)-fuzzy left ideal of S, we obtain (2) as

Conversely, let x, yS. Thus,

$f(xy)∨α≥[f(xy)∧β]∨α=fαβ(xy)≥(β∘αβf) (xy)=[(⋁(u,v)∈Sxy{β(u)∧f(v)})∧β]∨α≥(f(y)∧β)∨α≥f(y)∧β.$

Thus, f(xy) ∨ αf(y) ∧ β. Because (2) holds true, we have f as the (α, β)-fuzzy left ideal of S.

Similarly, we obtain the characterization of (α, β)-fuzzy right ideals as follows:

### Theorem 3.13

Let S be an ordered semigroup, and f be a fuzzy subset of S. Then, f is an (α, β)-fuzzy right ideal of S if and only if the following statements hold.

• 1. $f∘αββ⊆fαβ$.

• 2. xy implies f(x) ∨ αf(y) ∧ β.

We observe that if we set α = 0 and β = 1, Theorems 3.12 and 3.13 becomes Lemmas 2.8 and 2.9, respectively, as presented in [34].

The notions of left(right) ideals, along with their (α, β)-fuzzifications in ordered semigroups, exhibit a close relationship. The following proposition encapsulates the characterization of left (right) ideals through the utilization of (α, β)-fuzzy left (right) ideals.

### Proposition 3.14

Let S be an ordered semigroup, and A be a nonempty subset of S. Thus, the following statements are equivalent:

• 1. $(χA)αβ$is an (α, β)-fuzzy left (resp., right) ideal of S.

• 2. A is a left (resp., right) ideal of S.

Proof

(1) ⇒ (2). Let xS and yA. Subsequently, $(χA)αβ(xy)∨α≥(χA)αβ(y)∧β=β$. This implies that xyA. Let x, yS such that yA and xy. Subsequently, $(χA)αβ(x)∨α≥(χA)αβ(y)∧β=β$. This implies that xA. Therefore, A is a left-handed ideal of S.

(2) ⇒ (1). Let x, yS. If yA, two possibilities must be considered: xyA and xyA. If xyA, $(χA)αβ(xy)∨α=β≥α=(χA)αβ(y)∧β$. If xyA, $(χA)αβ(xy)∨α=α=(χA)αβ(y)∧β$. We assume that yA. Thus, xyA. This implies that $(χA)αβ(xy)∨α=β=(χA)αβ(y)∧β$. Let x, yS such that xy. If yA, then xA or xA. If xA, $(χA)αβ(x)∨α=β≥α=(χA)αβ(y)∧β$. If xA, $(χA)αβ(x)∨α=α=(χA)αβ(y)∧β$. We assume that yA. As A is the left ideal of S, xA. Therefore, $(χA)αβ(x)∨α=β=(χA)αβ(y)∧β$. Hence, $(χA)αβ$ is the (α, β)-fuzzy left ideal of S.

To illustrate, $(χA)αβ$ is an (α, β)-fuzzy right ideal of S if and only if A is the right ideal of S can be performed similarly.

By substituting α = 0 and β = 1, the theorem above becomes Proposition 2, as given by Kehayopulu and Tsingelis [31].

In contrast to Proposition 3.14, the description of (α, β)-fuzzy left (right) ideals in ordered semigroups can be obtained using the left (right) ideals. To accomplish this, it is necessary to introduce the following set: For any fF(S) and t ∈ [0, 1], an (α, β)-level set of f is defined as

$levαβ(f;t):={x∈S:f(x)∨α≥t∧β}.$

### Proposition 3.15

Let S be an ordered semigroup, and f be a fuzzy subset of S. Thus, the following statements are equivalent:

• 1. f is an (α, β)-fuzzy left (resp., right) ideal of S.

• 2. The nonempty (α, β)-level set $levαβ(f;t)$ of f is a left (resp., right) ideal of S for any t ∈ (α, β].

Proof

(1) ⇒ (2). We assume that $fαβ$ is an (α, β)-fuzzy left ideal of S. Let t ∈ (α, β]. Given xS and $y∈levαβ(f;t)$. We then obtain that f(y) ∨ αtβ. From our assumptions, we obtain

$f(xy)∨a≥[f(y)∧β]∨α=[f(y)∨α]∧β≥(t∧β)∧β=t∧β.$

This implies that $S levαβ(f;t)⊆levαβ(f;t)$. Next, we let x, yS such that xy and $y∈levαβ(f;t)$. Subsequently, xy implies f(x) ∨ αf(y) ∧ β. It follows that

$f(x)∨α≥[f(y)∧β]∨α=[f(y)∨α]∧β≥(t∧β)∧ (since y∈levαβ(f;t))=t∧β.$

Thus, $x∈levαβ(f;t)$. Therefore, $(levαβ(f;t)]⊆levαβ(f;t)$. Overall, $levαβ(f;t)$ is the ideal on the left for S.

(2) ⇒ (1). Let x, yS. If f(y) ≤ α, then f(xy) ∨ αf(y) ∧ β. We assume that f(y) > α. We set t = f(y) ∧ β, Then t ∈ (α, β]. Additionally, f(y) ∨ α = f(y) ≥ f(y) ∧ β = tβ, This implies that $y∈levαβ(f;t)$. Because $levαβ(f;t)$ is the left ideal of S, we have $xy∈levαβ(f;t)$. In other words, f(xy) ∨ αtβ = f(y) ∧ β. Hence, for any x, yS, f(xy) ∨ αf(y) ∧ β, Let x, yS such that xy. If f(y) ≤ α, f(x) ∨ αf(y) = f(y) ∧ β. We assume that f(y) > α. We set t = f(y) ∧ β, Subsequently, t ∈ (α, β] and f(y) ∨ α = f(y) ≥ tβ. This implies that $y∈levαβ(f;t)$. As $levαβ(f;t)$ is the left-hand ideal of S, we obtain $x∈levαβ(f;t)$. Thus, f(x) ∨ αtβ = f(y) ∧ β. Hence, xy implies f(x) ∨ αf(y) ∧ β for any x, yS. Altogether, we have f as an (α, β)-fuzzy left ideal of S.

To illustrate, f is an (α, β)-fuzzy right ideal of S if and only if the nonempty (α, β)-level set $levαβ(f;t)$ of f is the right ideal of S for any t ∈ (α, β]. This can be

Propositions 3.14 and 3.15 establish a direct relationship between (α, β)-fuzzy left (right) ideals and left (right) ideals in ordered semigroups. These results effectively connect set-theoretical ideals with the process of ideals’ fuzzification.

### 4. Characterizing Some Regularities of Ordered Semigroups

In this section, certain classes of ordered semigroups, regular and intra-regular ordered semigroups, use (α, β)-fuzzy left and (α, β)-fuzzy right ideals, respectively.

An ordered semigroup S is said to be:

• 1. a regular ordered semigroup if for any aS, there exists xS such that aaxa;

• 2. an intra-regular ordered semigroup if for any aS, there exists x, yS such that axa2y.

We need the following results before characterizing the regular and intra-regular ordered semigroups.

### Lemma 4.1

Let S be an ordered semigroup, and A, BS. Thus, the following statements hold:

• 1. $(χA∩B)αβ=χA∩αβχB$.

• 2. $(χ(AB])αβ=χA∘αβχB$.

Proof

From Proposition 7 of [9], we obtain χAB = χAχB. Applying Remark 3.6 and Lemma 3.8(1), we obtain that $(χA∩B)αβ=(χA∩χB)αβ=χA∩αβχB$. Thus, (1) holds as a claim.

By Lemma 2.5 in [35], we have χ(AB] = χAχB. It follows, by Remark 3.6 and the definition of $∘αβ$, that $(χ(AB])αβ=(χA∘χB)αβ=χA∘αβχB$. Hence, (2) holds.

### Lemma 4.2

Let S be an ordered semigroup and A, B be a nonempty subset of S. Subsequently, we have AB if and only if $(χA)αβ⊆(χB)αβ$.

Proof

Let xA. We assume $β=(χA)αβ(x)≤(χB)αβ(x)$. It follows that $(χB)αβ(x)=β$ because $α≤(χB)αβ(x)≤β$. That is, xB.

Conversely, let xS. If xA, then $(χA)αβ(x)=α≤(χB)αβ(x)$. If xA, then xB. Thus, $(χA)αβ(x)=β=(χB)αβ(x)$. Hence, $(χA)αβ⊆(χB)αβ$.

### Lemma 4.3 [36]

Let S be an ordered semigroup. Thus, the following statements are equivalent:

• 1. S is regular.

• 2. RL = (RL] for any right ideal R and left ideal L of S.

### Lemma 4.4

Let S be an ordered semigroup, f an (α, β)-fuzzy right ideal, and g the (α, β)-fuzzy left ideal of S. Subsequently, $f∘αβg⊆f∩αβg$.

Proof

Let xS. If S = ∅, $(f∘αβg)(x)=α≤(f∩αβg) (x)$. We assume that S ≠ ∅. Then

$(f∘αβg) (x)=[(⋁(u,v)∈Sx{f(u)∧g(v)})∧β]∨α=[(⋁(u,v)∈Sx{(f(u)∧β)∧(g(v)∧β)})∧β]∨α≤[(⋁(u,v)∈Sx{(f(uv)∨α)∧(g(uv)∨α)})∧β]∨α=[(⋁(u,v)∈Sx{(f(uv)∧β}∧(g(uv)∧β)})∧β]∨α≤[(⋁(u,v)∈Sx{(f(x)∨α}∧(g(x)∨α)})∧β]∨α=[(f∩g) (x)∧β]∨α=(f∩ααg) (x).$

Hence, the proof is complete.

We characterize the regular-ordered semigroups as follows:

### Theorem 4.5

Let S be an ordered semigroup. Thus, the following statement is equivalent:

• 1. S is regular.

• 2. $f∩αβg=f∘αβg$ for any (α, β)-fuzzy right ideal f and (α, β)-fuzzy left ideal g of S.

Proof

(1) ⇒ (2). Let f and g be the (α, β)-fuzzy right ideals and (α, β)-fuzzy right ideals of S. From Lemma 4.4, it is sufficient to show that $f∩αβg⊆f∘αβg$. Let aS. Because S is regular, there exists xS such that aaxa. It follows that Sa ≠ ∅ because (ax, a) ∈ Sa. Thus,

$(f∘αβg) (a)=[(⋁(u,v)∈Sa{f(u)∧g(v)})∧β]∨α≥(f(ax)∧g(a)∧β)∨α=[(f(ax)∨α)∧(g(a)∧β)∧β]∨α≥[(f(a)∨α)∧(g(a)∧β)∧β]∨α=(f(a)∧g(a)∧β)∨α=[(f∩g) (x)∧β]∨α=(f∩ααg) (a).$

This indicates that $f∩αβg⊆f∘αβg$.

(2) ⇒ (1). Let R and L be the right and left ideals of S, respectively. From Lemma 4.3, we show that RL = (RL]. By Theorem 3.14, we have that $(χR)αβ$ and $(χL)αβ$ is an (α, β)-fuzzy right ideal and an (α, β)-fuzzy left ideal of S, respectively. Subsequently, based on our assumption and Lemma 4.1, we have

$(χR∩L)αβ=χR∩αβχL=χR∘αβχL=(χ(RL])αβ.$

From Lemma 4.2, we have RL = (RL]. Therefore, by Lemma 4.3, S is regular.

### Lemma 4.6 [34]

Let S be an ordered semigroup. Thus, the following statements are equivalent:

• 1. S is intra-regular.

• 2. LR ⊆ (LR] for any left ideal L and right ideal R of S.

### Theorem 4.7

Let S be an ordered semigroup. Thus, the following statement is equivalent:

• 1. S is intra-regular.

• 2. $f∩αβg⊆f∘αβg$ for any (α, β)-fuzzy left ideal f and (α, β)-fuzzy right ideal g of S.

Proof

(1) ⇒ (2). Let f and g be the (α, β)-fuzzy left ideal, and let (α, β) the fuzzy right ideal of S. Let aS. Because S is intra-regular, there exist x, yS such that axa2y. Sa ≠ ∅ because (xa, ay) ∈ Sa. Thus,

$(f∘αβg) (a)=[(⋁(u,v)∈Sa{f(u)∧g(v)})∧β]∨α≥(f(xa)∧g(ay)∧β)∨α=[(f(xa)∨α)∧(g(ay)∨α)∧β]∨α≥[(f(a)∧β)∧(g(a)∧β)∧β]∨α=(f(a)∧g(a))∧β]∨α=(f∩αβg) (a).$

This indicates that $f∩αβg⊆f∘αβg$.

(2) ⇒ (1). Let L and R be the left and right ideals of S, respectively. From Lemma 4.6, we show that LR ⊆ (LR]. By Theorem 3.14, we have that $(χL)αβ$ and $(χR)αβ$ is an (α, β)-fuzzy left ideal and an (α, β)-fuzzy right ideal of S, respectively. Subsequently, based on our assumption and Lemma 4.1, we have

$(χL∩R)αβ=χL∩αβχR=χL∘αβχR=(χ(LR])αβ.$

From Lemma 4.2, we have LR ⊆ (LR]. Therefore, by Lemma 4.6, S is intra-regular.

### 5. Conclusion

In this study, we explored (α, β)-fuzzy left (right) and (α, β)-fuzzy quasi-ideals within the context of ordered semigroups. These concepts generalize their corresponding fuzzy left (resp., right) ideals and fuzzy quasi-ideals, respectively. The initial segment of our work was devoted to establishing foundational results concerning these concepts. We now present the intersection property of (α, β)-fuzzy quasi-ideals. In fact, the connection among (α, β)-fuzzy left-hand ideals, (α, β)-fuzzy right-hand ideals, and (α, β)-fuzzy quasi-ideals in ordered semigroups is provided. Subsequently, we characterize the (α, β)-fuzzy left ideals and (α, β)-fuzzy right ideals through a specific product. This product is crucial in effectively representing fuzzy-ordered semigroups. Moreover, we establish meaningful links between (α, β)-fuzzy left (resp., right) ideals and their conventional counterparts, namely, left (resp., right) ideals. Collectively, these findings underscore the inherent similarities between ideals and their fuzzy counterparts. In the final section of our study, we focus on classifying ordered semigroups as precise, regular, and intra-regular ordered semigroups. These classes are characterized by applying (α, β)-fuzzy left ideals and (α, β)-fuzzy right ideals. As a parting note, we encourage our readers to explore the utility of (α, β)-fuzzy ideals in characterizing other classes of ordered semigroups and investigate their prime, semiprime, and weakly semiprime properties.

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