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 [3–10]). 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 (∈, ∈ ∨ q_k)-fuzzy ideals such as (∈, ∈ ∨ q_k)-fuzzy left (right) ideals and (∈, ∈ ∨ q_k)-fuzzy (generalized) bi-ideals. Some classes of ordered semigroups are characterized by (∈, ∈ ∨ q_k)-fuzzy ideals. In addition, a minimum of two publications in 2012 characterized specific classes of ordered semigroups using (∈, ∈ ∨ q_k)-fuzzy ideals (see [21, 22]). Tang and Xie [23] studied the notion of (∈, ∈ ∨ q_k)-fuzzy left (and right) ideals, where k ∈ [0, 1). They demonstrated that when k = 0, a (∈, ∈ ∨ q_k)-fuzzy ideal is a (∈, ∈ ∨ q)-fuzzy ideal. Moreover, they characterized the prime properties of these fuzzy ideals.

The notions of (∈, ∈ ∨(k^*, q_k))-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, q_k))-fuzzy ideals, introducing the concept of (∈, ∈ ∨ (k, q_k))-fuzzy (m, n)-ideals. This novel extension is explained through the description of (∈, ∈ ∨ (k, q_k))-fuzzy (m, n)ideals using their level sets. Notably, the notion of (∈, ∈ ∨ (k, q_k))-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, C ⊆ S and x ∈ S, we define AB := {ab : a ∈ A and b ∈ B}, S_a := {(x, y) ∈ S × S : a ≤ xy} and

Lemma 2.1 [7]

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

• 1. A ⊆ (A];

• 2. A ⊆ B implies (A] ⊆ (B];

• 3. (A](B] = (AB];

• 4. (A ∪ B] ⊇ (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 SA ⊆ A;

• 2. a right ideal [7] of S if SA ⊆ A;

• 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 A ⊆ X is denoted by χ_A and is defined by

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

Let {f_i : i ∈ I} be a family of the fuzzy subsets of X. Fuzzy subsets ⋃_i∈If_i and ⋂_i∈If_i are defined as follows:

for all x ∈ X, where

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

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

for all x ∈ S.

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 x ≤ y. 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 x ∈ S;

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

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, y ∈ S,

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

• 2. x ≤ y implies f(x) ∨ α ≥ f(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, y ∈ S,

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

• 2. x ≤ y implies f(x) ∨ α ≥ f(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 (∈, ∈ ∨ q_k)-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 (∈, ∈ ∨ q_k)-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)} ∪ Δ_S, where Δ_S = {(x, x) : x ∈ S}. Thus, S ≔ ⟨S; ∘, ≤⟩ is an ordered semigroup. We define f : S → [0, 1] as

for any x ∈ S. 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 (∈, ∈ ∨ q_k)-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

for any x ∈ S. 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. a ≤ b 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 x ∈ S (see [32]). It is observed that fαβ(x)∈[α,β] for all x ∈ S.

Remark 3.6

Let f, g ∈ F(S) be We can observe that if f(x) ≤ g(y) for all x, y ∈ S, 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, g ∈ F(S). Then we have:

• 1. fαβ∪α=fαβ=fαβ∩β;

• 2. (f∩g)αβ=fαβ∩gαβ;

• 3. (fαβ)αβ=fαβ;

• 4. f(x) ∨ α ≥ f(y) ∧ β implies fαβ(x)≥fαβ(y) for all x, y ∈ S.

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 x ∈ S.

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 x ∈ S. The equality holds whenever S_x ≠ ∅︀.

• 4. fαβ∘αβgαβ=f∘αβg.

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, y ∈ S;

• 2. (f ∘ 1)(xy) ≥ f(x) for all x, y ∈ S;

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

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

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) ∨ α ≥ f(y) ∧ β whenever x ≤ y. 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.

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.

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. x ≤ y implies f(x) ∨ α ≥ f(y) ∧ β.

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. x ≤ y implies f(x) ∨ α ≥ f(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.

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].

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 a ∈ S, there exists x ∈ S such that a ≤ axa;

• 2. an intra-regular ordered semigroup if for any a ∈ S, there exists x, y ∈ S such that a ≤ xa²y.

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, B ⊆ S. Thus, the following statements hold:

• 1. (χA∩B)αβ=χA∩αβχB.

• 2. (χ(AB])αβ=χA∘αβχB.

Lemma 4.2

Let S be an ordered semigroup and A, B be a nonempty subset of S. Subsequently, we have A ⊆ B if and only if (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. R ∩ L = (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.

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.

Lemma 4.6 [34]

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

• 1. S is intra-regular.

• 2. L ∩ R ⊆ (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.

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.