Representations of ordered semigroups and their interconnection

1 Introduction

The concept of fuzzy sets, firstly introduced by Zadeh [33] in 1965, is an important mathematical tool in dealing with uncertain data. Following his introduction, fuzzy sets were applied in various fields, such as engineering, control system, decision-making, and biology (see [1, 30]). In mathematics, fuzzy sets also involve many branches in pure and applied mathematics, for example, optimization, stability, number theory, and algebra (see [3, 35]). Their research in a wide range of disciplines highlights the benefits of fuzzy sets.

In algebra, Rosenfeld [28] was the first to apply fuzzy sets to groups. Kuroki [25] studied semigroups, a generalization of groups, employing fuzzy sets in 1980. The idea of ordered semigroups is one of the many generalizations of semigroups. Kehayopulu and Tsingelis [17] firstly investigated ordered semigroups in terms of fuzzy sets. They proposed the notions of fuzzy left and fuzzy right ideals. Moreover, the relationships between left (resp., right) ideals and fuzzy left (resp., right) ideals were provided. One year later, in 2003, the same authors examined the powerful results in fuzzy ordered semigroups. They showed that an algebraic system consisting of the set of all fuzzy sets of an ordered semigroup together with a binary operation and a partial order forms an ordered semigroup. They also discovered that this algebraic system is a representation of ordered semigroups (see [18]). These results demonstrate the significance of the conclusion made by Kehayopulu and Tsinglis in [18]. The representation result is analogous to semigroup theory in that any semigroup embeds into a transformation semigroup, and any group embeds into a symmetric group.

Numerous scholars have been interested in investigating fuzzy ordered semigroups for the past two decades due to the aforementioned noteworthy result. The research on fuzzy ideals in ordered semigroups has attracted much attention. Although it is impossible to list all such research here, a few are highlighted as follows. In 2005, Kehayopulu and Tsingelis [19] introduced the concepts of fuzzy bi-ideals in ordered semigroups. Left and right simple ordered semigroups were characterized by fuzzy bi-ideals. The prime properties of fuzzy ideals were studied in [32]. The notion of fuzzy interior ideals was introduced by Kehayopulu and Tsingelis in [20]. They demonstrated the coincidence of fuzzy ideals and fuzzy interior ideals in regular and intra-regular ordered semigroups. The semiprime attribute of fuzzy sets was applied to characterize left regular and intra-regular ordered semigroups in [21]. Regular ordered semigroups were described in terms of fuzzy left (resp., right, quasi-) ideals by Kehayopulu and Tsingelis (see [22]). In [23], Kehayopulu and Tsingelis gave a representation of fuzzy quasi-ideals by fuzzy left and fuzzy right ideals. In addition to the use mentioned of fuzzy ideals in the examination of ordered semigroups, several generalizations of fuzzy ideals are also applied to the study of ordered semigroups. Jun et al. [9] introduced the notion of (∈ , ∈ ∨ q)-fuzzy bi-ideals in ordered semigroups. The concepts of many kinds of (∈ , ∈ ∨ q)-fuzzy ideals were extend to (∈ , ∈ ∨ q _k )-fuzzy ideals by Khan et al. [13] in 2012. Later, Khan et al. [11] and Ali Khan et al. [2] defined the concepts of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy ideals and ( ∈ , ∈ ∨ q k δ ) -fuzzy ideals, respectively.

After the proof that any ordered semigroup can be embedded into a fuzzy ordered semigroup by Kehayopulu and Tsingelis in 2003, there was a growing number of studies of ordered semigroups through fuzzy ideals of ordered semigroups. However, there is no evidence to provide a relation between the algebraic systems corresponding to generalized fuzzy ideals and ordered semigroups. Therefore, the present paper aims to provide the readers with a relation between algebraic systems corresponding to various notions of generalized fuzzy ideals and ordered semigroups.

The structure of the current paper is as follows. Section 2 provides an introduction to ordered semigroups and fuzzy ordered semigroups. We are reminded of the embedding theorem, a significant result in the study of ordered semigroups by fuzzy ideals. A binary operation defined on the set of all fuzzy sets of ordered semigroups, a concept Khan et al. [15] introduced, is also recalled. According to some literature, this binary operation generalizes the operations that characterize ordered semigroups into classes based on linear inequalities. In Section 3, we examine that the binary operation defined by Khan et al. [15] is associative. As a result, two ordered semigroups are obtained and illustrated as representations of any ordered semigroup. Additionally, we discover a connection between each representation.

2 Preliminaries

All necessary concepts are offered. We are reminded of the ideas of fuzzy sets, ordered semigroups, and fuzzy ordered semigroups. Some auxiliary results are also stated in this section.

An ordered semigroup 〈S ; · , ≤ 〉 is an algebraic system consisting of a semigroup 〈S ; · 〉 and a partially ordered set 〈S ; ≤ 〉 such that for any a, b ∈ S, a ≤ b implies c · a ≤ c · b and a · c ≤ b · c for all c ∈ S. We often denote an ordered semigroup 〈S ; · , ≤ 〉 by S the boldface of its universe set. Furthermore, the product of any two elements x and y of S is usually written by xy instead of x · y. The readers are referred to [4] for more information about ordered semigroups.

Let us say we simultaneously mention two or more ordered semigroups. In that situation, we can conduct the operations and relations of such ordered semigroups more precisely for clarity. For example, let S and T be ordered semigroups. In this case, S and T is referred to an ordered semigroup 〈S ; ·  _S , ≤  _S 〉 and an ordered semigroup 〈T ; ·  _T , ≤  _T 〉, respectively.

Let X be a set. A mapping fcolonX → [0, 1] is called a fuzzy set [33] of X. We denote the set of all fuzzy sets of X by F (X). Any α ∈ [0, 1] can be regarded as a fuzzy set of X designated by α (x) : = α for all x ∈ X. A relation ⊆ on the set F (X) is defined by

f ⊆ g if f ( x ) ≤ g ( x )(1) for all x ∈ X. We say that f = g if f ⊆ g and g ⊆ f. Let {f _i  : i ∈ I} be a nonempty family of fuzzy sets of X. Then the fuzzy sets ⋃_i∈If _I and ⋂_i∈If _I of X are assigned by (⋃ _i∈If _i ) (x) : = ⋁ _i∈If _i  (x) and (⋂ _i∈If _i ) (x) : = ⋀ _i∈If _i  (x), where ⋁ i ∈ I f i ( x ) = sup i ∈ I { f i ( x ) } and ⋀ i ∈ I f i ( x ) = inf i ∈ I { f i ( x ) } .

Kehayopulu and Tsingelis [17] applied the concept of fuzzy sets to study ordered semigroups by the following settings. Let S be an ordered semigroup. The set of all fuzzy sets of the universe set of S is denoted by F (S). For any x ∈ S, we let S _x  : = {(u, v) ∈ S × S : x ≤ uv}. The binary operation ∘ on the set F (S) is defined by ( f ∘ g ) ( x ) : = { ⋁ ( u , v ) ∈ S x [ f ( u ) ∧ g ( u ) ] , S x ¬ = ∅ , 0 , S x = ∅ , for all x ∈ S.

By this defining, the following result was obtained.

Theorem 2.1. ([18]). Let S be an ordered semigroup. Then F (S) : = 〈F (S) ; ∘ , ⊆ 〉 is an ordered semigroup. Moreover, S can be embedded into F (S).

Theorem 2.1 illustrates that F (S) is a representation of S. The structural properties of F (S) can be used to characterize S into classes by linear inequations (see [20, 31]).

In 2012, Khan et al. [15] considered fuzzy sets in ordered semigroups more generally. The following configuration was used to define a new binary operation on the set F (S). Let α, β ∈ [0, 1] such that 0 ≤ α < β ≤ 1. Then we can consider I : = [α, β] as a closed subinterval of the closed unit interval [0, 1]. Define a binary operation ∘ _I colonF (S) × F (S) → F (S) by ( f ∘ I g ) ( x ) : = [ ( f ∘ g ) ( x ) ∧ β ] ∨ α for all x ∈ S. For any fuzzy set f of S, the fuzzy set f _I is assigned by f _I  (x) : = [f (x) ∧ β] ∨ α for all x ∈ S. We note that if f (x) ∈ I for all x ∈ S, then f _I  (x) = f (x).

The operation ∘ _I was used to study ordered semigroups through various fuzzy ideals.

If I = [0, 0.5], then ∘ _I is a binary operation considered in [7, 12].

Remark 2.2.

Let f, g ∈ F (S) and I a closed subinterval of [0, 1]. We have that f ⊆ g implies f _I  ⊆ g _I . The converse does not necessary hold. Indeed, let I = [0.4.0.6] , f (x) =0.3 and g (x) =0.2. We see that f _I  (x) = g _I  (x) but f (x) > g (x). The converse of the statement is true whenever f (x) and g (x) belong to I for all x ∈ S.

The following result can be easily obtained by the definition of ∘ _I .

Lemma 2.3. ([10, 15]). Let f, g ∈ F (S) and I a closed subinterval of [0, 1]. Then:

if S _x ¬ = ∅ for any x ∈ S, f ∘  _I g = (f ∘ g)  _I  = f _I  ∘ g _I ;

In the next section, we construct new algebraic systems using the set of all fuzzy sets of an ordered semigroup. We demonstrate that these systems are ordered semigroups. Additionally, they are identified as representing ordered semigroups. Finally, there is a connection between these two models and the one that Kehayopulu and Tsingelis [18] proposed.

In this section, we construct two classes of algebraic systems through the binary operation defined by Khan et al. [15]. The structural properties of these systems are also investigated. From now on, we suppose that I : = [α, β] is a closed subinterval of the closed unit interval [0, 1] such that α < β.

As works of literature considering the operation defined by Khan et al. [15], we can see that the operation ∘ _I is significant in studying ordered semigroups in detail (see [9–11, 27]). However, there is no evidence that this operation is associative. Our first result demonstrates the associativity of ∘ _I shown as follows.

Theorem 3.1. The algebraic system 〈F (S) ; ∘  _I 〉 is a semigroup.

Proof. Let f, g, h ∈ F (S). Let x ∈ S. We put t : = [f ∘  _I  (g ∘  _I h)] (x) and s : = [(f ∘  _I g) ∘  _I h] (x). First of all, we show that t ≥ s. If S _x  =∅, then t ≥ α = s. Suppose that S _x ¬ = ∅. We show that t ≥ f _I  (c) ∧ (g ∘  _I h) (d) for all (c, d) ∈ S _x . Let (c, d) ∈ S _x . Then x ≤ cd.

Case 1. If t ≥ f _I  (c), then t ≥ f _I  (c) ∧ (g ∘  _I h) (d).

Case 2. Suppose that t < f _I  (c). If S _d  =∅, then t ≥ α = (g ∘  _I h) (d) ≥ f _I  (c) ∧ (g ∘  _I h) (d). Assume that S _d ¬ = ∅. We show that t ≥ g _I  (d₁) ∧ h _I  (d₂) for all (d₁, d₂) ∈ S _d . Let (d₁, d₂) ∈ S _d . Then d ≤ d₁d₂. This implies that x ≤ (cd₁) d₂.

Case 2.1. If t ≥ g _I  (d₁), then t ≥ g _I  (d₁) ∧ h _I  (d₂).

Case 2.2. Suppose that t < g _I  (d₁). Now, t ≥ [(f ∘  _I g) (cd₁) ∧ h (d₂)]  _I  = (f ∘  _I g) (cd₁) ∧ h _I  (d₂). But, (f ∘  _I g) (cd₁) = (f _I  ∘ g _I ) (cd₁) ≥ f _I  (c) ∧ g _I  (d₁) > t by the facts that t < f _I  (c) and t < g _I  (d₁). Thus, t ≥ h _I  (d₂). This implies that t ≥ h _I  (d₂) ≥ g _I  (d₁) ∧ h _I  (d₂). By Case 2.1 and 2.2, we have that t ≥ (g _I  ∘ h _I ) (d) = (g ∘  _I h) (d) ≥ f _I  (c) ∧ (g ∘  _I h) (d). Therefore, by Case 1 and 2, we have t ≥ [ f I ∘ ( g ∘ I h ) ] ( x ) = [ f I ∘ ( g ∘ I h ) I ] ( x ) since (g ∘I h)(y) ∈ I forall y ∈ S = [ f ∘ I ( g ∘ I h ) ] ( x ) = s . For illustrating that s ≥ t can be done similarly. Altogether, we have f ∘  _I  (g ∘  _I h) = (f ∘  _I g) ∘  _I h. □

We obtain the following consequence by the definition of ⊆ given by (1) in Section 2.

Proposition 3.2. The algebraic system F ^I  (S) : = 〈F (S) ; ∘  _I , ⊆ 〉 is an ordered semigroup.

Proof. By Theorem 3.1, 〈F (S) ; ∘  _I 〉 is a semigroup. It is not difficult to verify that 〈F (S) ; ⊆ 〉 is a partially ordered set. Thus, it is necessary to show that ⊆ is compatible with ∘ _I . Let f, g ∈ F (S) such that f ⊆ g. Given h ∈ F (S). By Theorem 2.1, we have that h ∘ f ⊆ h ∘ g and f ∘ h ⊆ g ∘ h. Applying Remark 2.2, we obtain that h ∘  _I f = (h ∘ f)  _I  ⊆ (h ∘ g)  _I  = h ∘  _I g and f ∘  _I h = (f ∘ h)  _I  ⊆ (g ∘ h)  _I  = g ∘  _I h. That is, ⊆ is compatible with ∘ _I . Hence, F ^I  (S) is an ordered semigroup. □

It is easily observed that if I is the closed unit interval, then F ^I  (S) and F (S) are the same structure. In fact, F^[0,1] (S) = F (S).

To provide a connection between S and F ^I  (S), we need the concept presented in [18]. Let S and T be ordered semigroups. A function φcolonS → T is said to be:

isotone if for any x, y ∈ S, we have φ (x) ≤  _T φ (y) whenever x ≤  _S y;

Definition 3.3. Let S and T be ordered semigroups, and φ is a mapping from S into T. Then φ is called:

a homomorphism from S into T if φ is isotone and φ (x ·  _S y) = φ (x) ·  _T φ (y) for all x, y ∈ S;

A connection between S and F ^I  (S) is presented as follows.

Theorem 3.4. Let S be an ordered semigroup. Then S embeds into F ^I  (S).

Proof. Define φcolonS → F (S) by φ (a) : = f _a for all a ∈ S, where f a ( x ) : = { α if x ≰ a , β if x ≤ a , for all x ∈ S. Suppose that a, b ∈ S such that a = b. Let x ∈ S. If x ≰ a, then x ≰ b and f _a  (x) = α = f _b  (x). If x ≤ a, then x ≤ b and f _a  (x) = β = f _b  (x). This illustrates that φ is well-defined.

Let a, b ∈ S. Given x ∈ S. If S _x  =∅, then f _ab  (x) = α = (f _a  ∘  _I f _b ) (x). Suppose that S _x ¬ = ∅. We let s : = (f _a  ∘ f _b ) (x). If x ≤ ab, then f _ab  (x) = β and s ≥ f _a  (a) ∧ f _b  (b) = β. This implies that f _ab  (x) = β = s _I  = (f _a  ∘  _I f _b ) (x). Assume that x ≰ ab. We show that f _a  (u) ∧ f _b  (v) = α for all (u, v) ∈ S _x . Let (u, v) ∈ S _x . Then x ≤ uv. It evidences that f _a  (u) = α or f _b  (v) = α, otherwise it contradicts to our assumption that x ≰ ab. Without loss of generality, we suppose that f _a  (u) = α. Then f _a  (u) ∧ f _b  (v) = α. This means that s = α. Therefore, f _ab  (x) = α = (f _a  ∘  _I f _b ) (x). This verify that φ (ab) = φ (a) ∘  _I φ (b).

Finally, let a, b ∈ S. Suppose that a ≤ b. Let x ∈ S. If x ≰ a, then it is clear that f _a  (x) ≤ f _b  (x). If x ≤ a, then x ≤ b and f _a  (x) = f _b  (x). This illustrates that φ is isotone. Conversely, suppose that f _a  ⊆ f _b . We observe that β = f _a  (a) ≤ f _b  (a) implies f _b  (a) = β. This means that a ≤ b. Therefore, φ is reverse isotone.

Altogether, we obtain that S embeds into F ^I  (S). □

By the above result, we know that F ^I  (S) is another representation of S. Now, we construct a new ordered semigroup such that any ordered semigroup can be embedded into it. We define a binary relation ⊆ _I on F (S) by f ⊆  _I g if f _I  ⊆ g _I for all f, g ∈ F (S). We observe that ⊆ _I is a quasiorder relation. We can see by Remark 2.2 that ⊆ _I is not anti-reflexive.

Since ⊆ _I is a quasiorder relation, it induces an equivalence relation ≡ _I defined on F (S) by f ≡  _I g if and only if f ⊆  _I g and g ⊆  _I f for all f, g ∈ F (S). On the set F (S)/≡  _I , we define a binary operation ◊ _I and a binary relation ⊑ _I by f / ≡ I ◊ I g / ≡ I : = ( f ∘ I g ) / ≡ I and f / ≡ I ⊑ I g / ≡ I if and only if f ⊆ I g for all f/≡  _I , g/≡  _I  ∈ F (S)/≡  _I .

By the above construction, we obtain the following result.

Theorem 3.5. The algebraic system F I ( S ) : = 〈 F ( S ) / ≡ I ; ◊ I , ⊑ I 〉 dered semigroup.

Proof. Let f/≡  _I , g/≡  _I , h/≡  _I  ∈ F (S)/≡  _I . Then ( f / ≡ I ◊ I g / ≡ I ) ◊ I h / ≡ I = ( f ∘ I g ) / ≡ I ◊ I h / ≡ I = [ ( f ∘ I g ) ∘ I h ] / ≡ I = [ f ∘ I ( g ∘ I h ) ] / ≡ I byTheorem 3.1 = f / ≡ I ◊ I ( g ∘ I h ) / ≡ I = f / ≡ I ◊ I ( g / ≡ I ◊ I h / ≡ I ) . This shows that 〈F (S)/≡  _I  ; ◊  _I 〉 is a semigroup. It is not difficult to verify that 〈F (S)/≡  _I  ; ⊑  _I 〉 is a partially ordered set. Indeed, for the anti-reflexivity, we let f/≡  _I , g/≡  _I  ∈ F (S)/≡  _I such that f/≡  _I  ⊑  _I g/≡  _I and g/≡  _I  ⊑  _I f/≡  _I . Then, we have f ⊆  _I g and g ⊆  _I f. That is, f ≡  _I g. We conclude that f/≡  _I  = g/≡  _I . Finally, we show that ◊ _I is compatible with ⊑ _I . Let f/≡  _I , g/≡  _I  ∈ F (S)/≡  _I such that f/≡  _I  ⊑  _I g/≡  _I . Then f ⊆  _I g. This means that f _I  ⊆ g _I . Given h/≡  _I  ∈ F (S)/≡  _I . By Theorem 2.1, h _I  ∘ f _I  ⊆ h _I  ∘ g _I . This implies, by Remark 2.2, that (h _I  ∘ f _I )  _I  ⊆ (h _I  ∘ g _I )  _I . That is, by Lemma 2.3(2), we have (h ∘  _I f)  _I  = h _I  ∘  _I f _I  ⊆ h _I  ∘  _I g _I  = (h ∘  _I g)  _I . Therefore, we have h/≡  _I  ◊  _I f/≡  _I  ⊑  _I h/≡  _I  ◊  _I g/≡  _I . By similar arguments, we can illustrate that f/≡  _I  ◊  _I h/≡  _I  ⊑  _I g/≡  _I  ◊  _I h/≡  _I . Altogether, F _I  (S) is an ordered semigroup. □

We observe that ≡ [ 0 , 1 ] = { ( f , g ) ∈ F ( S ) 2 : f / ≡ [ 0 , 1 ] = g / ≡ [ 0 , 1 ] } = { ( f , g ) ∈ F ( S ) 2 : f ⊆ [ 0 , 1 ] g and g ⊆ [ 0 , 1 ] f } = { ( f , g ) ∈ F ( S ) 2 : f = g } = Δ F ( S ) , where Δ_F(S) is a diagonal relation on F (S). Hence, F (S) is isomorphic to F_[0,1] (S). This means that F^[0,1] (S) also isomorphic to F_[0,1] (S).

The relationship between S and F _I  (S) is illustrated by the following result.

Proposition 3.6. Let S be an ordered semigroup, and I a closed subinterval of [0, 1]. Then S embeds into F _I  (S).

Proof. Define φcolonS → F (S) by φ (a) : = f _a /≡  _I for all a ∈ S, where f a ( x ) : = { α if x ≰ a , β if x ≤ a , for all x ∈ S. It is not difficult to verify that for any a, b ∈ S such that a = b, we have (f _a )  _I  (x) = (f _b )  _I  (x) for all x ∈ S. This shows that φ is well-defined.

Let a, b ∈ S. Next, we show that (f _ab )  _I  = (f _a  ∘ f _b )  _I . Let x ∈ S. If S _x  =∅, then (f _ab )  _I  (x) = α = (f _a  ∘ f _b )  _I  (x). Suppose that S _x ¬ = ∅. We put s : = ( f a ∘ f b ) ( x ) = ⋁ ( u , v ) ∈ S x [ f a ( u ) ∧ f b ( v ) ] .

Case 1. If x ≤ ab, then (f _ab )  _I  (x) = β. Moreover, s ≥ f _a  (a) ∧ f _b  (a) = β. Thus, (f _ab )  _I  (x) = β = s _I  = (f _a  ∘ f _b )  _I  (x).

Case 2. Suppose that x ≰ ab. Then (f _ab )  _I  (x) = α. We illustrate that f _a  (u) ∧ f _b  (v) = α for all (u, v) ∈ S _x . Let (u, v) ∈ S _x . Then x ≤ uv. It is impossible that f _a  (u) ¬ = α and f _b  (v) ¬ = α since it contradicts to our presumption that x ≰ ab. Thus, we have that f _a  (u) = α or f _b  (v) = α. Without loss of generality, we assume that f _a  (u) = α. Since f _b  (v) ≥ α, we have that f _a  (u) ∧ f _b  (v) = α. This means that s = α. Therefore, (f _ab )  _I  (x) = α = s _I  = (f _a  ∘ f _b )  _I  (x). Altogether, φ (ab) = φ (a) ◊  _I φ (b).

Finally, we verify that φ is isotone and reverse isotone. Let a, b ∈ S such that a ≤ b. Since f _a  (x) , f _b  (x) ∈ I for all x ∈ S, it suffices to show that f _a  ⊆ f _b . Let x ∈ S. If x ≰ a, then f _a  (x) = α ≤ f _b  (x). If x ≤ a, then a ≤ b. This implies that f _a  (x) = f _b  (x). Thus, f _a  ⊆ f _b . On the other hand, we suppose that f _a  ⊆ f _b . Then β = f _a  (a) ≤ f _b  (a). By the definition of f _b , we have a ≤ b. This demonstrates evidence that φ is isotone and reverse isotone.

Altogether, we have that S can be embedded into F _I  (S). □

Let f₁, f₂, g₁, g₂ ∈ F (S). Suppose that 0 ≤ f₁ (x) + g₁ (x) ≤1 and 0 ≤ f₂ (x) - g₂ (x) ≤1 for all x ∈ S. Then we define fuzzy sets f₁ + g₁ and f₂ - g₂ by ( f 1 + g 1 ) ( x ) : = f 1 ( x ) + g 1 ( x ) and ( f 2 - g 2 ) ( x ) : = f 2 ( x ) - g 2 ( x ) for all x ∈ S.

Lemma 3.7. Let f, g ∈ F (S) and ɛ ∈ [0, 1]. Suppose that 0 ≤ f (x) + ɛ ≤ 1 and 0 ≤ g (x) + ɛ ≤ 1 for all x ∈ S. Then ( f + ɛ ) ∘ ( g + ɛ ) ⊆ ( f ∘ g ) + ɛ . The equality holds if S _x ¬ = ∅ for all x ∈ S.

Proof. We observe that (f ∘ g) + ɛ is a fuzzy set of S since f + ɛ and g + ɛ are fuzzy sets of S. Let x ∈ S. If S _x  =∅, then [ ( f + ɛ ) ∘ ( g + ɛ ) ] ( x ) = 0 ≤ [ ( f ∘ g ) + ɛ ] ( x ) . Suppose that S _x ¬ = ∅. Then [ ( f + ɛ ) ∘ ( g + ɛ ) ] ( x ) = ⋁ ( u , v ) ∈ S x [ ( f + ɛ ) ( u ) ∧ ( g + ɛ ) ( v ) ] = ⋁ ( u , v ) ∈ S x [ ( f ( u ) + ɛ ) ∧ ( g ( v ) + ɛ ) ] = ⋁ ( u , v ) ∈ S x [ f ( u ) ∧ g ( v ) ] + ɛ = ( f ∘ g ) ( x ) + ɛ = [ ( f ∘ g ) + ɛ ] ( x ) . By both cases, we obtain our claim. □

Similarly, we obtain the following lemma.

Lemma 3.8. Let f, g ∈ F (S) and ɛ ∈ [0, 1]. Suppose that 0 ≤ f (x) - ɛ ≤ 1 and 0 ≤ g (x) - ɛ ≤ 1 for all x ∈ S. Then ( f - ɛ ) ∘ ( g - ɛ ) ⊆ ( f ∘ g ) - ɛ . The equality holds if S _x ¬ = ∅ for all x ∈ S.

The following theorem provides a connection between two ordered semigroups F _{I 1}  (S) and F _{I 2}  (S), where I₁ and I₂ are closed subintervals of [0, 1].

Theorem 3.9. Let I₁ : = [α₁, β₁] and I₂ : = [α₂, β₂] be closed subintervals of [0, 1] such that |I₁| ≤ |I₂|. Then F _{I 1}  (S) can be embedded into F _{I 2}  (S).

Proof. We separate our considerations into two cases: α₁ ≤ α₂ and α₂ ≤ α₁.

Let us suppose first that α₁ ≤ α₂. Put ɛ : = α₂ - α₁. Define φcolonF (S)/≡  _{I 1}  → F (S)/≡  _{I 2} by φ ( f / ≡ I 1 ) : = ( f I 1 + ɛ ) / ≡ I 2 for all f/≡  _{I 1}  ∈ F (S)/≡  _{I 1} . First of all, we verify that φ is well-defined. Let f/≡  _{I 1} , g/≡  _{I 1}  ∈ F (S)/≡  _{I 1} such that f/≡  _{I 1}  = g/≡  _{I 1} . Then f ≡  _{I 1} g. That is, f ⊆  _{I 1} g and g ⊆  _{I 1} f. This implies that f _{I 1}  + ɛ ⊆ g _{I 1}  + ɛ and g _{I 1}  + ɛ ⊆ f _{I 1}  + ɛ. By Remark 2.2, we have that f _{I 1}  + ɛ ⊆  _{I 2} g _{I 1}  + ɛ and g _{I 1}  + ɛ ⊆  _{I 2} f _{I 1}  + ɛ. Thus, (f _{I 1}  + ɛ)/≡  _{I 2}  = (g _{I 1}  + ɛ)/≡  _{I 2} . This means that φ is well-defined.

Next, we prove that φ ( f / ≡ I 1 ◊ I 1 g / ≡ I 1 ) = φ ( f / ≡ I 1 ) ◊ I 2 φ ( g / ≡ I 1 ) . Let f/≡  _{I 1} , g/≡  _{I 1}  ∈ F (S)/≡  _{I 1} . It is sufficient to show that (f ∘  _{I 1} g)  _{I 1}  + ɛ = (f _{I 1}  + ɛ) ∘  _{I 2}  (g _{I 1}  + ɛ). Let x ∈ S. If S _x  =∅, then [ ( f ∘ I 1 g ) I 1 + ɛ ] ( x ) = ( f ∘ I 1 g ) I 1 ( x ) + ɛ = α 1 + ɛ = α 2 = [ ( f I 1 + ɛ ) ∘ I 2 ( g I 1 + ɛ ) ] ( x ) . Suppose that S _x ¬ = ∅. Thus, [ ( f ∘ I 1 g ) I 1 + ɛ ] ( x ) = ( f ∘ I 1 g ) I 1 ( x ) + ɛ = ( f ∘ I 1 g ) ( x ) + ɛ = [ ( f ∘ I 1 g ) + ɛ ] ( x ) = [ ( f ∘ I 1 g ) + ɛ ] I 2 ( x ) = [ ( f I 1 ∘ g I 1 ) + ɛ ] I 2 ( x ) byLemma 2.3 ( 1 ) = [ ( f I 1 + ɛ ) ∘ ( g I 1 + ɛ ) ] I 2 ( x ) byLemma 3.7 = [ ( f I 1 + ɛ ) ∘ I 2 ( g I 1 + ɛ ) ] ( x ) byLemma 2.3 ( 1 ) This shows that φ (f/≡  _{I 1}  ◊  _{I 1} g/≡  _{I 1} ) = φ (f/≡  _{I 1} ) ◊  _{I 2} φ (g/≡  _{I 1} ).

Finally, we demonstrate that φ is isotone and reverse isotone. Let f/≡  _{I 1} , g/≡  _{I 1}  ∈ F (S)/≡  _{I 1} . Then, we have φ ( f / ≡ I 1 ) ⊑ I 2 φ ( g / ≡ I 1 ) ⇔ ( f I 1 + ɛ ) / ≡ I 2 ⊑ I 2 ( g I 1 + ɛ ) / ≡ I 2 ⇔ f I 1 + ɛ ⊆ I 2 g I 1 + ɛ ⇔ ( f I 1 + ɛ ) I 2 ⊆ ( g I 1 + ɛ ) I 2 ⇔ f I 1 + ɛ ⊆ g I 1 + ɛ ⇔ f I 1 ⊆ g I 1 ⇔ f ⊆ I 1 g ⇔ f / ≡ I 1 ⊑ I 1 g / ≡ I 1 . Therefore, φ is isotone and reverse isotone. This examines that φ is an embedding.

Now, we suppose that α₂ ≤ α₁. Put ɛ : = α₁ - α₂. Similarly, we can prove that φcolonF (S)/≡  _{I 1}  → F (S)/≡  _{I 2} defined by φ ( f / ≡ I 1 ) : = ( f I 1 - ɛ ) / ≡ I 2 for all f/≡  _{I 1}  ∈ F (S)/≡  _{I 1} , is an embedding.

Altogether, by both two cases, F _{I 1}  (S) is embedded in F _{I 2}  (S). □

By Theorem 3.9, we obtain the following result.

Corollary 3.10. Let I₁ : = [α₁, β₁] and I₂ : = [α₂, β₂] be closed subintervals of [0, 1] such that |I₁| = |I₂|. Then F _{I 1}  (S) and F _{I 2}  (S) are embedded in each other.

4 Conclusion

In this paper, we construct ordered semigroups F ^I  (S) and F _I  (S) from fuzzy sets defined on ordered semigroup S. It turns out that these algebraic systems are representations of an ordered semigroup S. We describe a relation between these ordered semigroups and ordered semigroup F (S). We obtain that F (S) = F^[0,1] (S) and F (S) ≅ F_[0,1] (S). We also know that there is a relationship between F _{I 1}  (S) and F _{I 2}  (S) for any closed subinverals I₁ and I₂ of [0, 1]. It is still an open problem in finding the relevance of F ^{I ₁}  (S) and F ^{I ₂}  (S).