𝒩-Fuzzy Ideals in Ordered Semigroups

Abstract

We introduce the concept of 𝒩-fuzzy left (right) ideals in ordered semigroups and characterize ordered semigroups in terms of 𝒩-fuzzy left (right) ideals. We characterize left regular (right regular) and left simple (right simple) ordered semigroups in terms of 𝒩-fuzzy left (𝒩-fuzzy right) ideals. The semilattice of left (right) simple semigroups in terms of 𝒩-fuzzy left (right) ideals is discussed.

1. Introduction

A fuzzy subset 𝑓 of a given set 𝑆 is described as an arbitrary function 𝑓∶𝑆→[0,1], where [0,1] is the usual closed interval of real numbers. This fundamental concept was first introduced by Zadeh in his pioneering paper [1] of 1965, which provides a natural framework for the generalizations of some basic notions of algebra, for example, set theory, group theory, ring theory, groupoids, real analysis, measure theory, topology, and differential equations, and so forth. Rosenfeld (see [2]) was the first who introduced the concept of a fuzzy set in a group. The concept of a fuzzy ideal in semigroups was first developed by Kuroki (see [3–8]). He studied fuzzy ideals, fuzzy bi-ideals, fuzzy quasi-ideals, and fuzzy semiprime ideals of semigroups. Fuzzy ideals and Green's relations in semigroups were studied by McLean and Kummer in [9]. Dib and Galham in [10] introduced the definitions of a fuzzy groupoid and a fuzzy semigroup and studied fuzzy ideals and fuzzy bi-ideals of a fuzzy semigroup. Ahsan et al. in [11] characterized semisimple semigroups in terms of fuzzy ideals. A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in [12], where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines, and fuzzy languages. The monograph by Mordeson and Malik (see [13]) deals with the applications of fuzzy approach to the concept of automata and formal languages. Fuzzy sets in ordered semigroups/ordered groupoids were first introduced by Kehayopulu and Tsingelis in [14]. They also introduced the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in ordered semigroups in [14, 15].

In [16], Shabir and Khan introduced the concept of a fuzzy generalized bi-ideal of ordered semigroups and characterized different classes of ordered semigroups by using fuzzy generalized bi-ideals. They also gave the concept of fuzzy left (resp., bi-) filters in ordered semigroups and gave the relations of fuzzy bi-filters and fuzzy bi-ideal subsets of ordered semigroups in [17].

In this paper, we introduce the concept of 𝒩-fuzzy left (resp., right) ideals and characterize regular, left, and right simple ordered semigroups and completely regular ordered semigroups in terms of 𝒩-fuzzy left (resp., right) ideals. In this respect, we prove that a regular ordered semigroup 𝑆 is left simple if and only if every 𝒩-fuzzy left ideal 𝑓 of 𝑆 is a constant function. We also prove that 𝑆 is left regular if and only if for every 𝒩-fuzzy left ideal 𝑓 of 𝑆 we have 𝑓(𝑎)=𝑓(𝑎²) for every 𝑎∈𝑆. Next we characterize semilattice of left simple ordered semigroups in terms of 𝒩-fuzzy left ideals. We prove that an ordered semigroup 𝑆 is a semilattice of left simple semigroups if and only if for every 𝒩-fuzzy left ideal 𝑓 of 𝑆, 𝑓(𝑎)=𝑓(𝑎²) and 𝑓(𝑎𝑏)=𝑓(𝑏𝑎) for all 𝑎,𝑏∈𝑆.

2. Preliminaries

By an ordered semigroup (or po-semigroup) we mean a structure (𝑆,⋅,≤) in which

(OS1)(𝑆,⋅) is a semigroup,(OS2)(𝑆,≤) is a poset,(OS3) (∀𝑎,𝑏,𝑥∈𝑆) (𝑎≤𝑏⇒𝑎𝑥≤𝑏𝑥 and 𝑥𝑎≤𝑥𝑏).

Let (𝑆,⋅,≤) be an ordered semigroup. A nonempty subset 𝐴 of 𝑆 is called a left (resp., right) ideal of 𝑆 (see [14]) if

(i)𝑆𝐴⊆𝐴 (resp., 𝐴𝑆⊆𝐴),(ii)(∀𝑎∈𝐴) (∀𝑏∈𝑆) (𝑏≤𝑎⇒𝑏∈𝐴).

𝐴 is called a two-sided ideal or simply an ideal if 𝐴 is both left and right ideal of 𝑆.

For 𝐴⊆𝑆, denote (𝐴]∶={𝑡∈𝑆∣𝑡≤ℎ for some ℎ∈𝐴}. If 𝐴={𝑎}, then we write (𝑎] instead of {(𝑎]}. Let 𝐴,𝐵⊆𝑆, then 𝐴⊆(𝐴], (𝐴](𝐵]⊆(𝐴𝐵], and ((𝐴]]=(𝐴].

Let 𝑆 be an ordered semigroup and 𝑓 a fuzzy subset of 𝑆. Then 𝑓 is called a fuzzy left (resp., right) ideal of 𝑆 if

(i)(∀𝑥,𝑦∈𝑆) (𝑥≤𝑦⇒𝑓(𝑥)≥𝑓(𝑦)),(ii)(𝑥,𝑦∈𝑆) (𝑓(𝑥𝑦)≥𝑓(𝑦) (resp., 𝑓(𝑥𝑦)≥𝑓(𝑥))).

A fuzzy left and right ideal of 𝑆 is called a fuzzy two-sided ideal or simply a fuzzy ideal of 𝑆.

3. 𝓝-Fuzzy Left (Resp., Right) Ideals

Let 𝑆 be an ordered semigroup. By a negative fuzzy subset (briefly 𝒩-fuzzy subset) 𝑓 of 𝑆 we mean a mapping 𝑓∶𝑆→[−1,0].

Definition 3.1. Let (𝑆,⋅,≤) be an ordered semigroups and 𝑓 an 𝒩-fuzzy subset of 𝑆. Then 𝑓 is called an 𝒩-fuzzy left (resp., right) ideal of 𝑆 if(1)(∀𝑥,𝑦∈𝑆) (𝑥≤𝑦⇒𝑓(𝑥)≤𝑓(𝑦)),(2)(∀𝑥,𝑦∈𝑆) (𝑓(𝑥𝑦)≤𝑓(𝑦) (resp., 𝑓(𝑥𝑦)≤𝑓(𝑥))).
An 𝒩-fuzzy left and right ideal 𝑓 of 𝑆 is called an 𝒩-fuzzy two-sided ideal of 𝑆.
For any 𝒩-fuzzy subset 𝑓 of 𝑆 and 𝑡∈[−1,0) the set 𝐿(𝑓;𝑡)∶={𝑥∈𝑆∣𝑓(𝑥)≤𝑡} (3.1) is called the 𝒩-level subset of 𝑓.

Theorem 3.2. Let (𝑆,⋅,≤) be an ordered semigroup. An 𝒩-fuzzy subset 𝑓 of 𝑆 is an 𝒩-fuzzy left (resp., right) ideal of 𝑆 if and only if it satisfies (∀𝑡∈[−1,0)𝐿(𝑓;𝑡)(≠∅)isaleftidealiff𝑓isan𝒩-fuzzyleftideal). (3.2)

Proof. Suppose that 𝑓 is an 𝒩-fuzzy left ideal of 𝑆. Let 𝑥,𝑦∈𝑆 be such that 𝑥≤𝑦. If 𝑦∈𝐿(𝑓;𝑡), then 𝑓(𝑦)≤𝑡. Since 𝑥≤𝑦, we have 𝑓(𝑥)≤𝑓(𝑦) and 𝑓(𝑥)≤𝑡 and we have 𝑥∈𝐿(𝑓;𝑡). Let 𝑥,𝑦∈𝑆 be such that 𝑦∈𝐿(𝑓;𝑡). Then 𝑓(𝑦)≤𝑡, since 𝑓(𝑥𝑦)≤𝑓(𝑦). Then one has 𝑓(𝑥𝑦)≤𝑡 implies that 𝑥𝑦∈𝐿(𝑓;𝑡). Thus 𝑆𝐿(𝑓;𝑡)⊆𝐿(𝑓;𝑡).
Conversely, assume that for all 𝑡∈[−1,0) such that 𝐿(𝑓;𝑡)≠∅, the set 𝐿(𝑓;𝑡) is a left ideal of 𝑆. Let 𝑥,𝑦∈𝑆 be such that 𝑥≤𝑦. If 𝑓(𝑦)=0, then since 𝑓(𝑥)≤0 for all 𝑥∈𝑆, we have 𝑓(𝑦)≤𝑓(𝑥). If 𝑓(𝑦)=𝑡, then 𝑦∈𝐿(𝑓;𝑡) and since 𝑥≤𝑦∈𝐿(𝑓;𝑡), and 𝐿(𝑓;𝑡) is a left ideal of 𝑆, we have 𝑥∈𝐿(𝑓;𝑡) and so 𝑓(𝑥)≤𝑡=𝑓(𝑦). Let 𝑥,𝑦∈𝑆. If 𝑓(𝑦)=0, then since 𝑓(𝑥𝑦)≤0 for all 𝑥,𝑦∈𝑆 we have 𝑓(𝑥𝑦)≤𝑓(𝑦). If 𝑓(𝑦)=𝑡, then 𝑦∈𝐿(𝑓;𝑡) and since 𝐿(𝑓;𝑡) is a left ideal of 𝑆, we have 𝑥𝑦∈𝐿(𝑓;𝑡). Then 𝑓(𝑥𝑦)≤𝑡=𝑓(𝑦).

By left-right dual of the above theorem, we have the following theorem.

Theorem 3.3. Let (𝑆,⋅,≤) be an ordered semigroup. An 𝒩-fuzzy subset 𝑓 of 𝑆 is an 𝒩-fuzzy left (resp., right) ideal of 𝑆 if and only if it satisfies (∀𝑡∈[−1,0)𝐿(𝑓;𝑡)(≠∅)isarightidealiff𝑓isan𝒩-fuzzyrightideal). (3.3)

Example 3.4. Let 𝑆={𝑎,𝑏,𝑐,𝑑,𝑒,𝑓} be the ordered semigroup defined by the multiplication and the order as follows: ⋅𝑎𝑏𝑐𝑑𝑒𝑓𝑎𝑎𝑎𝑎𝑑𝑎𝑎𝑏𝑎𝑏𝑏𝑑𝑏𝑏𝑐𝑎𝑏𝑐𝑑𝑒𝑒𝑑𝑎𝑎𝑑𝑑𝑑𝑑𝑒𝑎𝑏𝑐𝑑𝑒𝑒𝑓𝑎𝑏𝑐𝑑𝑒𝑓≤∶={(𝑎,𝑎),(𝑏,𝑏),(𝑐,𝑐),(𝑑,𝑑),(𝑒,𝑒),(𝑓,𝑒),(𝑓,𝑓)}. (3.4)
Then left ideals of 𝑆 are {𝑎},{𝑑},{𝑎,𝑏},{𝑎,𝑑},{𝑎,𝑏,𝑑},{𝑎,𝑏,𝑐,𝑑},{𝑎,𝑏,𝑑,𝑒,𝑓}, and 𝑆 (see [18]). Define 𝑓∶𝑆→−1,0] by 𝑓(𝑎)=−0.8,𝑓(𝑏)=−0.6,𝑓(𝑑)=−0.5,𝑓(𝑐)=−0.4,𝑓(𝑒)=𝑓(𝑓)=−0.2. (3.5)
Then 𝑆𝐿(𝑓;𝑡)∶=if𝑡∈[−0.2,0),{𝑎,𝑏,𝑐,𝑑}if𝑡∈[−0.4,−0.2),{𝑎,𝑏,𝑑}if𝑡∈[−0.5,−0.4),{𝑎,𝑏}if𝑡∈[−0.6,−0.5),{𝑎}if∅𝑡∈[−0.8,−0.6),if𝑡∈[−1,−0.8). (3.6)
Then 𝐿(𝑓;𝑡) is a left ideal of 𝑆, and by Theorem 3.2, 𝑓 is an 𝒩-fuzzy left ideal of 𝑆.
Let 𝑆 be an ordered semigroup and ∅≠𝐴⊆𝑆. The characteristic 𝒩-function 𝜅_𝐴∶𝑆→{−1,0} of 𝐴 is defined by 𝜅_𝐴∶𝑆→[−1,0],𝑥↦𝜅_𝐴(𝑥)∶=−1if0𝑥∈𝐴,if𝑥∉𝐴. (3.7)

Theorem 3.5. Let (𝑆,⋅,≤) be an ordered semigroup and ∅≠𝐴⊆𝑆. Then the followings are equivalent. (i)𝐴 is a left (resp., right) ideal of 𝑆.(ii)The characteristic 𝒩-function 𝜅_𝐴(𝑥)=−1if0𝑥∈𝐴,if𝑥∉𝐴 (3.8) of 𝐴 is an 𝒩-fuzzy left (resp., right) ideal of 𝑆.

Proof. (i)⇒(ii) Suppose that 𝐴 is a left ideal of 𝑆. Let 𝑥,𝑦∈𝑆, 𝑥≤𝑦. If 𝑦∈𝐴, then 𝜅_𝐴(𝑦)=−1. Since 𝑥≤𝑦∈𝐴 and 𝐴 is a left ideal of 𝑆, we have 𝑥∈𝐴. Then 𝜅_𝐴(𝑥)=−1 and hence 𝜅_𝐴(𝑥)≤𝜅_𝐴(𝑦). If 𝑦∉𝐴, then 𝜅_𝐴(𝑦)=0. Since 𝜅_𝐴 is an 𝒩-fuzzy subset of 𝑆, we have 𝜅_𝐴(𝑥)≤0 for all 𝑥∈𝑆. Hence 𝜅_𝐴(𝑥)≤𝜅_𝐴(𝑦).
Let 𝑥∈𝑆 and 𝑦∈𝐴. Then 𝜅_𝐴(𝑦)=−1. Since 𝐴 is a left ideal of 𝑆 and 𝑦∈𝐴, we have 𝑥𝑦∈𝐴. Then 𝜅_𝐴(𝑥𝑦)=−1, and we have 𝜅_𝐴(𝑥𝑦)≤𝜅_𝐴(𝑦). If 𝑦∉𝐴. Then 𝜅_𝐴(𝑦)=0. Since 𝜅_𝐴(𝑥𝑦)≤0 for all 𝑥,𝑦∈𝑆. Hence 𝜅_𝐴(𝑥𝑦)≤𝜅_𝐴(𝑦). Thus 𝜅_𝐴 is an 𝒩-fuzzy left ideal of 𝑆.
(ii)⇒(i) Assume that 𝜅_𝐴(𝑥)=−1if0𝑥∈𝐴,if𝑥∉𝐴 (3.9) is an 𝒩-fuzzy left ideal of 𝑆. Let 𝑥,𝑦∈𝑆, 𝑥≤𝑦. If 𝑦∈𝐴, then 𝜅_𝐴(𝑦)=−1. Since 𝑥≤𝑦, we have 𝜅_𝐴(𝑥)≤𝜅_𝐴(𝑦). Then 𝜅_𝐴(𝑥)=−1 and we have 𝑥∈𝐴.
Let 𝑥∈𝑆 and 𝑦∈𝐴. Then 𝜅_𝐴(𝑦)=−1. Since 𝜅_𝐴 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝜅_𝐴(𝑥𝑦)≤𝜅_𝐴(𝑦). Hence 𝜅_𝐴(𝑥𝑦)=−1 and 𝑥𝑦∈𝐴. Thus 𝐴 is a left ideal of 𝑆.

A subset 𝑇 of an ordered semigroup 𝑆 is called semiprime (see [15]) if for every 𝑎∈𝑆 such that 𝑎²∈𝑇, we have 𝑎∈𝑇. Equivalent definition: for each subset 𝐴 of 𝑆 such that 𝐴²⊆𝑇, we have 𝐴⊆𝑇.

4. Characterization of Left Simple and Left Regular Ordered Semigroups

Lemma 4.1 (cf. [15]). Let (𝑆,⋅,≤) be an ordered semigroup. Then the followings are equivalent.(i)(𝑥)_𝒩 is a left simple subsemigroup of 𝑆, for every 𝑥∈𝑆.(ii)Every left ideal of 𝑆 is a right ideal of 𝑆 and semiprime.
An ordered semigroup 𝑆 is regular (see [19]) if for every 𝑎∈𝑆, there exists 𝑥∈𝑆 such that 𝑎≤𝑎𝑥𝑎.
Equivalent definitions are(1)(∀𝑎∈𝑆) (𝑎∈(𝑎𝑆𝑎]),(2)(∀𝐴⊆𝑆) (𝐴⊆(𝐴𝑆𝐴]).
An ordered semigroup 𝑆 is left (resp., right ) simple (see [15]) if for every left (resp., right) ideal 𝐴 of 𝑆, one has 𝐴=𝑆, and 𝑆 is called simple if it is left simple and right simple.

Lemma 4.2 (cf. [15]). An ordered semigroup (𝑆,⋅,≤) is left (resp., right) simple if and only if for every 𝑎∈𝑆, (𝑆𝑎]=𝑆 (resp., (𝑎𝑆]=𝑆).

Theorem 4.3. For a regular ordered semigroup 𝑆, the following conditions are equivalent.(i)𝑆 is left simple.(ii)Every 𝒩-fuzzy left ideal of 𝑆 is a constant 𝒩-function.

Proof. (i)⇒(ii) Let 𝑆 be a left simple ordered semigroup, 𝑓 an 𝒩-fuzzy left ideal of 𝑆, and 𝑎∈𝑆. We consider the set 𝐸_𝑆∶={𝑒∈𝑆∣𝑒²≥𝑒}. (4.1)
Then 𝐸_𝑆≠∅. In fact, since 𝑆 is regular and 𝑎∈𝑆, there exists 𝑥∈𝑆 such that 𝑎≤𝑎𝑥𝑎. It follows from (OS3) that (𝑎𝑥)²=(𝑎𝑥𝑎)𝑥≥𝑎𝑥, (4.2) and so 𝑎𝑥∈𝐸_𝑆 and hence 𝐸_𝑆≠∅.
(1) 𝑓 is a constant 𝒩-function on 𝐸_𝑆. Let 𝑡∈𝐸_𝑆. Since 𝑆 is left simple and 𝑡∈𝑆 we have (𝑆𝑡]=𝑆. Since 𝑒∈𝑆, 𝑒∈(𝑆𝑡], so there exists 𝑧∈𝑆 such that 𝑒≤𝑧𝑡. Hence 𝑒²≤(𝑧𝑡)(𝑧𝑡)=(𝑧𝑡𝑧)𝑡. Since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑒²)≤𝑓((𝑧𝑡𝑧)𝑡)≤𝑓(𝑡). (4.3)
Since 𝑒∈𝐸_𝑆, we have 𝑒²≥𝑒. Since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑒)≤𝑓(𝑒²). Thus 𝑓(𝑒)≤𝑓(𝑡). Besides, since 𝑆 is left simple and 𝑒∈𝑆, we have (𝑆𝑒]=𝑆. Since 𝑡∈𝑆=(𝑆𝑒], so there exists 𝑦∈𝑆 such that 𝑡≤𝑦𝑒. Hence 𝑡²≤(𝑦𝑒)(𝑦𝑒)=(𝑦𝑒𝑦)𝑒. Since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑡²)≤𝑓((𝑦𝑒𝑦)𝑒)≤𝑓(𝑒). (4.4) On the other hand, 𝑡∈𝐸_𝑆, we have 𝑡²≥𝑡 and so 𝑓(𝑡)≤𝑓(𝑡²)≤𝑓((𝑦𝑒𝑦)𝑒)≤𝑓(𝑒). Hence 𝑓(𝑡)=𝑓(𝑒).
(2) 𝑓 is a constant 𝒩-function on 𝑆. Let 𝑎∈𝑆. Since 𝑆 is regular, there exists 𝑥∈𝑆 such that 𝑎≤𝑎𝑥𝑎. We consider the element 𝑥𝑎∈𝑆. Then it follows by (OS3) that (𝑥𝑎)²=𝑥(𝑎𝑥𝑎)≥𝑥𝑎. (4.5)
Hence 𝑥𝑎∈𝐸_𝑆 and by (1), we have 𝑓(𝑥𝑎)=𝑓(𝑡). Besides, since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑥𝑎)≤𝑓(𝑎). Thus 𝑓(𝑡)≤𝑓(𝑎). On the other hand, since 𝑆 is left simple and 𝑡∈𝑆, 𝑆=(𝑆𝑡]. Since 𝑎∈𝑆, we have 𝑎≤𝑠𝑡 for some 𝑠∈𝑆. Since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑎)≤𝑓(𝑠𝑡)≤𝑓(𝑡). Thus 𝑓(𝑡)=𝑓(𝑎).
(ii)⇒(i) Let 𝑎∈𝑆. Then the set (𝑆𝑎] is a left ideal of 𝑆. In fact, 𝑆(𝑆𝑎]=(𝑆](𝑆𝑎]⊆(𝑆𝑆𝑎]⊆(𝑆𝑎]. If 𝑥∈(𝑆𝑎] and 𝑆∋𝑦≤𝑥, then 𝑦∈((𝑆𝑎]]=(𝑆𝑎]. Since (𝑆𝑎] is a left ideal of 𝑆, by Theorem 3.5, the characteristic 𝒩-function 𝜅_(𝑆𝑎] of (𝑆𝑎], 𝜅_(𝑆𝑎]∶𝑆→[−1,0],𝑥↦𝜅_(𝑆𝑎](𝑥)∶=−1if0𝑥∈(𝑆𝑎],if𝑥∉(𝑆𝑎], (4.6) is a fuzzy left ideal of 𝑆. By hypothesis, 𝜅_(𝑆𝑎] is a constant 𝒩-function; that is, there exists 𝑐∈{−1,0} such that 𝜅_(𝑆𝑎](𝑥)=𝑐,forevery𝑥∈𝑆. (4.7)
Let (𝑆𝑎]⊂𝑆 and 𝑡∈𝑆 be such that 𝑡∉(𝑆𝑎]. Then 𝜅_(𝑆𝑎](𝑡)=0. On the other hand, since 𝑎²∈(𝑆𝑎], we have 𝜅_(𝑆𝑎](𝑎²)=0, a contradiction to the fact that 𝜅_(𝑆𝑎] is a constant 𝒩-mapping. Hence 𝑆=(𝑆𝑎].

From left-right dual of Theorem 4.3, we have the following.

Theorem 4.4. For a regular ordered semigroup. The following statements are equivalent.(1)𝑆 is right simple.(2)Every 𝒩-fuzzy right ideal of 𝑆 is a constant 𝒩-function.
An ordered semigroup (𝑆,⋅,≤) is left (resp., right ) regular [20] if for every 𝑎∈𝑆 there exists 𝑥∈𝑆 such that 𝑎≤𝑥𝑎² (resp., 𝑎≤𝑎²𝑥).
Equivalent definitions are(1)(∀𝑎∈𝑆) (𝑎∈(𝑆𝑎²] (resp., 𝑎∈(𝑎²𝑆])),(2)(∀𝐴⊆𝑆) (𝐴⊆(𝑆𝐴²] (resp., 𝐴⊆(𝐴²𝑆])).
An ordered semigroup (𝑆,⋅,≤) is intraregular (see [16]) if for every 𝑎∈𝑆, there exist 𝑥,𝑦∈𝑆 such that 𝑎≤𝑥𝑎²𝑦.
Equivalent definitions are(1)(∀𝑎∈𝑆) (𝑎∈(𝑆𝑎²𝑆]),(2)(∀𝐴⊆𝑆) (𝐴⊆(𝑆𝐴²𝑆]).
An ordered semigroup 𝑆 is called completely regular (see [21]) if it is regular, left regular, and right regular.

Lemma 4.5 (cf. [21]). An ordered semigroup 𝑆 is completely regular if and only if 𝐴⊆(𝐴²𝑆𝐴²] for every 𝐴⊆𝑆. Equivalently, 𝑎∈(𝑎²𝑆𝑎²] for every 𝑎∈𝑆.

Theorem 4.6. An ordered semigroup (𝑆,⋅,≤) is left regular if and only if for each 𝒩-fuzzy left ideal 𝑓 of 𝑆, one has 𝑓(𝑎)=𝑓(𝑎²)∀𝑎∈𝑆. (4.8)

Proof. Suppose that 𝑓 is an 𝒩-fuzzy left ideal of 𝑆 and let 𝑎∈𝑆. Since 𝑆 is left regular, there exists 𝑥∈𝑆 such that 𝑎≤𝑥𝑎². Since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑎)≤𝑓(𝑥𝑎²)≤𝑓(𝑎²)≤𝑓(𝑎). (4.9)
Conversely, let 𝑎∈𝑆. We consider the left ideal 𝐿(𝑎²)=(𝑎²∪𝑆𝑎²] of 𝑆, generated by 𝑎². Then by Theorem 3.5, the characteristic 𝒩-function 𝜅_𝐿(𝑎2)∶𝑆→[−1,0]∣𝑥↦𝜅_𝐿(𝑎2)(𝑥)∶=−1if𝑥∈𝐿(𝑎²0),if𝑥∉𝐿(𝑎²) (4.10) is an 𝒩-fuzzy left ideal of 𝑆. By hypothesis we have 𝜅_𝐿(𝑎2)(𝑎)=𝜅_𝐿(𝑎2)(𝑎²). (4.11)
Since 𝑎²∈𝐿(𝑎²), we have 𝜅_𝐿(𝑎2)(𝑎²)=−1 and 𝜅_𝐿(𝑎2)(𝑎)=−1. Then 𝑎∈𝐿(𝑎²)=(𝑎²∪𝑆𝑎²] and 𝑎≤𝑦 for some 𝑦∈𝑎²∪𝑆𝑎². If 𝑦=𝑎², then 𝑎≤𝑦=𝑎²=𝑎𝑎=𝑎𝑎²∈𝑆𝑎² and 𝑎∈(𝑆𝑎²]. If 𝑦=𝑥𝑎² for some 𝑥∈𝑆, then 𝑎≤𝑦=𝑥𝑎²∈𝑆𝑎², and 𝑎∈(𝑆𝑎²].

From left-right dual of Theorem 4.6, we have the following theorem.

Theorem 4.7. An ordered semigroup (𝑆,⋅,≤) is right regular if and only if for each 𝒩-fuzzy right ideal 𝑓 of 𝑆, one has 𝑓(𝑎)=𝑓(𝑎²)∀𝑎∈𝑆. (4.12)

From [22] and by Theorems 4.6, 4.7, and Lemma 4.5, we have the following characterization theorem for completely regular ordered semigroups.

Theorem 4.8. Let (𝑆,⋅,≤) be an ordered semigroup. Then the following statements are equivalent.(i)𝑆 is completely regular.(ii)For each 𝒩-fuzzy bi-ideal 𝑓 of 𝑆 one has 𝑓(𝑎)=𝑓(𝑎²)∀𝑎∈𝑆. (4.13)(iii)For each 𝒩-fuzzy left ideal 𝑔 and each 𝒩-fuzzy right ideal ℎ of 𝑆 we have 𝑔(𝑎)=𝑔(𝑎²),ℎ(𝑎)=ℎ(𝑎²)∀𝑎∈𝑆. (4.14)
An ordered semigroup (𝑆,⋅,≤) is called left (resp., right ) duo if every left (resp., right) ideal of 𝑆 is a two-sided ideal of 𝑆. An ordered semigroup 𝑆 is called duo if it is both left and right duo.

Definition 4.9. An ordered semigroup (𝑆,⋅,≤) is called 𝒩-fuzzy left (resp., right) duo if every 𝒩-fuzzy left (resp., right) ideal of 𝑆 is an 𝒩-fuzzy two-sided ideal of 𝑆. An ordered semigroup 𝑆 is called 𝒩-fuzzy duo if it is both 𝒩-fuzzy left and 𝒩-fuzzy right duo.

Theorem 4.10. Let (𝑆,⋅,≤) be a regular ordered semigroup. Then the followings are equivalent.(i)𝑆 is left duo.(ii)𝑆 is 𝒩-fuzzy left duo.

Proof. (i)⇒(ii) Let 𝑆 be left duo and 𝑓 an 𝒩-fuzzy left ideal of 𝑆. Let 𝑎,𝑏∈𝑆. Then the set (𝑆𝑎] is a left ideal of 𝑆. In fact, 𝑆(𝑆𝑎]=(𝑆](𝑆𝑎]⊆(𝑆𝑆𝑎]⊆(𝑆𝑎] and if 𝑥∈(𝑆𝑎] and 𝑆∋𝑦≤𝑥, then 𝑦∈((𝑆𝑎]]=(𝑆𝑎]. Since 𝑆 is left duo, then (𝑆𝑎] is a two-sided ideal of 𝑆. Since 𝑆 is regular, there exists 𝑥∈𝑆 such that 𝑎≤𝑎𝑥𝑎, 𝑎𝑏≤(𝑎𝑥𝑎)𝑏∈(𝑎𝑆𝑎)𝑏⊆(𝑆𝑎)𝑆⊆(𝑆𝑎]𝑆⊆(𝑆𝑎]. (4.15)
Thus 𝑎𝑏∈((𝑆𝑎]]=(𝑆𝑎] and 𝑎𝑏≤𝑥𝑎 for some 𝑥∈𝑆. Since 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑎𝑏)≤𝑓(𝑥𝑎)≤𝑓(𝑎). (4.16)
Let 𝑥,𝑦∈𝑆 be such that 𝑥≤𝑦. Then 𝑓(𝑥)≤𝑓(𝑦), because 𝑓 is an 𝒩-fuzzy left ideal of 𝑆. Thus 𝑓 is an 𝒩-fuzzy right deal of 𝑆 and 𝑆 is 𝒩-fuzzy left duo.
(ii)⇒(i) Let 𝑆 be 𝒩-fuzzy left duo and 𝐴 a left ideal of 𝑆. Then the characteristic 𝒩-function 𝜅_𝐴 of 𝐴 is an 𝒩-fuzzy left ideal of 𝑆. By hypothesis, 𝜅_𝐴 is an 𝒩-fuzzy right ideal of 𝑆, and by Theorem 3.5, 𝐴 is a right ideal of 𝑆. Thus 𝑆 is left duo.

By the left-right dual of Theorem 4.10, we have the following.

Theorem 4.11. Let (𝑆,⋅,≤) be a regular ordered semigroup. Then the followings are equivalent.(i)𝑆 is right duo.(ii)𝑆 is 𝒩-fuzzy right duo.

Theorem 4.12. Let (𝑆,⋅,≤) be a regular ordered semigroup. Then the followings are equivalent.(i)Every bi-ideal of 𝑆 is a right ideal of 𝑆.(ii)Every 𝒩-fuzzy bi-ideal of 𝑆 is an 𝒩-fuzzy right ideal of 𝑆.

Proof. (i)⇒(ii) Let 𝑎,𝑏∈𝑆 and 𝑓 an 𝒩-fuzzy bi-ideal of 𝑆. Then (𝑎𝑆𝑎] is a bi-ideal of 𝑆. In fact, (𝑎𝑆𝑎]²⊆(𝑎𝑆𝑎](𝑎𝑆𝑎]⊆(𝑎𝑆𝑎], (𝑎𝑆𝑎]𝑆(𝑎𝑆𝑎]=(𝑎𝑆𝑎](𝑆](𝑎𝑆𝑎]⊆(𝑎𝑆𝑎] and if 𝑥∈(𝑎𝑆𝑎] and 𝑆∋𝑦≤𝑥∈(𝑎𝑆𝑎], then 𝑦∈((𝑎𝑆𝑎]]=(𝑎𝑆𝑎]. Since (𝑎𝑆𝑎] is a bi-ideal of 𝑆, by hypothesis (𝑎𝑆𝑎] is right ideal of 𝑆. Since 𝑎∈𝑆 and 𝑆 is regular, there exists 𝑥∈𝑆 such that 𝑎≤𝑎𝑥𝑎, then 𝑎𝑏≤(𝑎𝑥𝑎)𝑏∈(𝑎𝑆𝑎)𝑆⊆(𝑎𝑆𝑎]𝑆⊆(𝑎𝑆𝑎]. (4.17)
Then 𝑎𝑏≤𝑎𝑧𝑎 for some 𝑧∈𝑆. Since 𝑓 is an 𝒩-fuzzy bi-ideal of 𝑆, we have 𝑓(𝑎𝑏)≤𝑓(𝑎𝑧𝑎)≤max{𝑓(𝑎),𝑓(𝑎)}=𝑓(𝑎). (4.18)
Let 𝑥,𝑦∈𝑆 be such that 𝑥≤𝑦. Then 𝑓(𝑥)≤𝑓(𝑦) because 𝑓 is an 𝒩-fuzzy bi-ideal of 𝑆. Thus 𝑓 is an 𝒩-fuzzy right ideal of 𝑆.
(ii)⇒(i) Let 𝐴 be a bi-ideal of 𝑆. Then by Theorem 3.5, 𝜅_𝐴 is an 𝒩-fuzzy bi-ideal of 𝑆. By hypothesis 𝜅_𝐴 is an 𝒩-fuzzy right ideal of 𝑆. By Theorem 3.5, 𝐴 is a right ideal of 𝑆.

By left-right dual of Theorem 4.12, we have the following.

Theorem 4.13. Let (𝑆,⋅,≤) be a regular ordered semigroup. Then the followings are equivalent.(i)Every bi-ideal of 𝑆 is a left ideal of 𝑆.(ii)Every 𝒩-fuzzy bi-ideal of 𝑆 is an 𝒩-fuzzy left ideal of 𝑆.

5. Characterization of Intraregular Ordered Semigroups in Terms of 𝒩-Fuzzy Ideals

Definition 5.1 (cf. [22]). Let (𝑆,⋅,≤) be an ordered semigroup and 𝑓 an 𝒩-fuzzy subset of 𝑆. Then 𝑓 is called an semiprime 𝒩-fuzzy subset of 𝑆 if 𝑓(𝑎)≤𝑓(𝑎²)∀𝑎∈𝑆. (5.1)

Theorem 5.2. Let (𝑆,⋅,≤) be an ordered semigroup and ∅≠𝐴⊆𝑆. Then the followings are equivalent.(i)𝐴 is semiprime.(ii)The characteristic 𝒩-function 𝜅_𝐴 of 𝐴 is an 𝒩-fuzzy semiprime subset.

Proof. (i)⇒(ii) Suppose that 𝐴 is semiprime subset. Let 𝜅_𝐴(𝑎²)=0. Since 𝜅_𝐴(𝑎)≤0 for all 𝑎∈𝑆, thus 𝜅_𝐴(𝑎)≤𝜅_𝐴(𝑎²). If 𝜅_𝐴(𝑎²)=−1, then 𝑎²∈𝐴. Since 𝐴 is semiprime, we have 𝑎∈𝐴. Then 𝜅_𝐴(𝑎)=−1 and; hence 𝜅_𝐴(𝑎)≤𝜅_𝐴(𝑎²).
(ii)⇒(i) Assume that 𝜅_𝐴 is 𝒩-fuzzy semiprime subset. Let 𝑎∈𝑆 be such that 𝑎²∈𝐴. Then 𝜅_𝐴(𝑎²)=−1. Since 𝜅_𝐴(𝑎)≤𝜅_𝐴(𝑎²), we have 𝜅_𝐴(𝑎)=−1, hence 𝑎∈𝐴.

Theorem 5.3. Let (𝑆,⋅,≤) be an ordered semigroup and let 𝑓 be an 𝒩-fuzzy subsemigroup of 𝑆. Then 𝑓 is an 𝒩-fuzzy semiprime if and only if for every 𝑎∈𝑆, one has 𝑓(𝑎)=𝑓(𝑎²). (5.2)

Proof. Suppose that 𝑓 is an 𝒩-fuzzy subsemigroup of 𝑆 such that 𝑓 is semiprime. Let 𝑎∈𝑆. Then 𝑓(𝑎)≤𝑓(𝑎²)=𝑓(𝑎𝑎)≤max{𝑓(𝑎),𝑓(𝑎)}=𝑓(𝑎). (5.3)
The converse is obvious.

Theorem 5.4. An ordered semigroup 𝑆 is intraregular if and only if every 𝒩-fuzzy ideal of 𝑆 is semiprime.

Proof. Suppose that 𝑆 is intraregular and 𝑓 an 𝒩-fuzzy ideal of 𝑆. Let 𝑎∈𝑆. Then 𝑓(𝑎)≤𝑓(𝑎²). In fact, since 𝑆 is intraregular, there exist 𝑥,𝑦∈𝑆 such that 𝑎≤𝑥𝑎²𝑦=𝑥(𝑎²𝑦). Then 𝑓(𝑎)≤𝑓(𝑥(𝑎²𝑦))≤𝑓(𝑎²𝑦)≤𝑓(𝑎²). (5.4)
Assume that 𝑓 is an 𝒩-fuzzy ideal of 𝑆 such that 𝑓(𝑎)≤𝑓(𝑎²) for all 𝑎∈𝑆. Consider the ideal 𝐼(𝑎²)=(𝑎²∪𝑆𝑎²∪𝑎²𝑆∪𝑆𝑎²𝑆] of 𝑆 generated by 𝑎²(𝑎∈𝑆). Then by Theorem 3.5, the characteristic 𝒩-function 𝜅_𝐼(𝑎2) is an 𝒩-fuzzy ideal of 𝑆, and by hypothesis, we have 𝜅_𝐼(𝑎2)(𝑎)≤𝜅_𝐼(𝑎2)(𝑎²). Since 𝑎²∈𝐼(𝑎²), then 𝜅_𝐼(𝑎2)(𝑎²)=−1 and 𝜅_𝐼(𝑎2)(𝑎)=−1⇒𝑎∈𝐼(𝑎²)=(𝑎²∪𝑆𝑎²∪𝑎²𝑆∪𝑆𝑎²𝑆]. Then 𝑎≤𝑥 for some 𝑥∈𝑎²∪𝑆𝑎²∪𝑎²𝑆∪𝑆𝑎²𝑆. If 𝑥=𝑎², then 𝑎≤𝑎²=𝑎𝑎≤𝑎²𝑎²=𝑎𝑎²𝑎∈𝑆𝑎²𝑆 and 𝑎∈(𝑆𝑎²𝑆]. If 𝑥=𝑦𝑎² for some 𝑦∈𝑆, then 𝑎≤𝑦𝑎²=𝑦𝑎𝑎≤𝑦(𝑦𝑎²)𝑎=𝑦𝑦𝑎²𝑎∈𝑆𝑎²𝑆 and 𝑎∈(𝑆𝑎²𝑆]. If 𝑥=𝑎²𝑧, then 𝑎≤𝑎²𝑧=𝑎𝑎𝑧≤𝑎(𝑎²𝑧)𝑧=𝑎𝑎²𝑧𝑧∈𝑆𝑎²𝑆 and 𝑎∈(𝑆𝑎²𝑆].

6. Some Semilattices of Left Simple Ordered Semigroups in Terms of 𝒩-Fuzzy Left Ideals

Let (𝑆,⋅,≤) be an ordered semigroup. A subsemigroup 𝐹 of 𝑆 is called filter (see [15]) of 𝑆 if

(1)(∀𝑎,𝑏∈𝑆) (𝑎𝑏∈𝐹⇒𝑎∈𝐹 and 𝑏∈𝐹),(2)(∀𝑐∈𝑆) (𝑐≥𝑎∈𝐹⇒𝑐∈𝐹).

For 𝑥∈𝑆, we denote by 𝑁(𝑥) the filter of 𝑆 generated by 𝑥 (𝑥∈𝑆) (i.e., the least filter with respect to inclusion relation containing 𝑥). 𝒩 denotes the equivalence relation on 𝑆 defined by 𝒩∶={(𝑥,𝑦)∈𝑆×𝑆∣𝑁(𝑥)=𝑁(𝑦)} (see [15]).

Definition 6.1 (cf. [15]). Let 𝑆 be an ordered semigroup. An equivalence relation 𝜎 on 𝑆 is called congruence if (𝑎,𝑏)∈𝜎 implies (𝑎𝑐,𝑏𝑐)∈𝜎 and (𝑐𝑎,𝑐𝑏)∈𝜎 for every 𝑐∈𝑆. A congruence 𝜎 on 𝑆 is called semilattice congruence if (𝑎²,𝑎)∈𝜎 and (𝑎𝑏,𝑏𝑎)∈𝜎 for each 𝑎,𝑏∈𝑆. If 𝜎 is a semilattice congruence on 𝑆, then the 𝜎-class (𝑥)_𝜎 of 𝑆 containing 𝑥 is a subsemigroup of 𝑆 for every 𝑥∈𝑆.
An ordered semigroup 𝑆 is called a semilattice of left simple semigroups if there exists a semilattice congruence 𝜎 on 𝑆 such that the 𝜎-class (𝑥)_𝜎 of 𝑆 containing 𝑥 is a left simple subsemigroup of 𝑆 for every 𝑥∈𝑆.
Equivalent definition: there exists a semilattice 𝑌 and a family {𝑆_𝛼}_𝛼∈𝑌 of left simple subsemigroups of 𝑆 such that(1)𝑆_𝛼∩𝑆_𝛽=∅ for all 𝛼,𝛽∈𝑌, 𝛼≠𝛽,(2)𝑆=⋃_𝛼∈𝑌𝑆_𝛼,(3)𝑆_𝛼𝑆_𝛽⊆𝑆_𝛼𝛽 for all 𝛼,𝛽∈𝑌.
In ordered semigroups the semilattice congruences are defined exactly same as in the case of semigroups—without order—so the two definitions are equivalent (see [15]).

Lemma 6.2 (cf. [22]). An ordered semigroup (𝑆,⋅,≤) is a semilattice of left simple semigroups if and only if for all left ideals 𝐴, 𝐵 of 𝑆 one has (𝐴²]=𝐴,(𝐴𝐵]=(𝐵𝐴]. (6.1)

Theorem 6.3. An ordered semigroup (𝑆,⋅,≤) is a semilattice of left simple semigroups if and only if for every 𝒩-fuzzy left ideal 𝑓 of 𝑆, one has 𝑓(𝑎²)=𝑓(𝑎),𝑓(𝑎𝑏)=𝑓(𝑏𝑎)∀𝑎,𝑏∈𝑆. (6.2)

Proof. ⇒ (A) Let 𝑆 be a semilattice of left simple semigroups. By hypothesis, there exists a semilattice 𝑌 and a family {𝑆_𝛼}_𝛼∈𝑌 of left simple subsemigroups of 𝑆 such that(i)𝑆_𝛼∩𝑆_𝛽=∅ for all 𝛼,𝛽∈𝑌, 𝛼≠𝛽,(ii)𝑆=⋃_𝛼∈𝑌𝑆_𝛼,(iii)𝑆_𝛼𝑆_𝛽⊆𝑆_𝛼𝛽 for all 𝛼,𝛽∈𝑌.
Let 𝑓 be an 𝒩-fuzzy left ideal of 𝑆 and 𝑎∈𝑆. Then 𝑓(𝑎)=𝑓(𝑎²). In fact, by Theorem 4.6, it is enough to prove that 𝑎∈(𝑆𝑎²] for every 𝑎∈𝑆. Let 𝑎∈𝑆, then there exists 𝛼∈𝑌 such that 𝑎∈𝑆_𝛼. Since 𝑆_𝛼 is left simple, we have 𝑆_𝛼=(𝑆_𝛼𝑎] and 𝑎≤𝑥𝑎forsome𝑥∈𝑆_𝛼. (6.3)
Since 𝑥∈𝑆_𝛼, we have 𝑥∈(𝑆_𝛼𝑎] and 𝑥≤𝑦𝑎 for some 𝑦∈𝑆_𝛼. Thus we have 𝑎≤𝑥𝑎≤(𝑦𝑎)𝑎=𝑦𝑎² (6.4) since 𝑦∈𝑆, we have 𝑎∈(𝑆𝑎²].
(B) Let 𝑎,𝑏∈𝑆. Then by (A), we have 𝑓(𝑎𝑏)=𝑓((𝑎𝑏)²)=𝑓(𝑎(𝑏𝑎)𝑏)≥𝑓(𝑏𝑎). (6.5)
By symmetry we can prove that 𝑓(𝑏𝑎)≥𝑓(𝑎𝑏). Hence 𝑓(𝑎𝑏)=𝑓(𝑏𝑎).
⇐ Assume that for every fuzzy left ideal 𝑓 of 𝑆, we have 𝑓(𝑎²)=𝑓(𝑎),𝑓(𝑎𝑏)=𝑓(𝑏𝑎)∀𝑎,𝑏∈𝑆, (6.6) by condition (1) and Theorem 4.6, we have that 𝑆 is left regular. Let 𝐴 be a left ideal of 𝑆 and let 𝑎∈𝐴. Then 𝑎∈𝑆, since 𝑆 is left regular there exists 𝑥∈𝑆 such that 𝑎≤𝑥𝑎²=(𝑥𝑎)𝑎∈(𝑆𝐴)𝐴⊆𝐴𝐴=𝐴², (6.7) then 𝑎∈(𝐴²] and 𝐴⊆(𝐴]. On the other hand, since 𝐴 is a left ideal of 𝑆, we have 𝐴²⊆𝑆𝐴⊆𝐴, then (𝐴²]⊆(𝐴]=𝐴. Let 𝐴 and 𝐵 be left ideals of 𝑆 and let 𝑥∈(𝐵𝐴] then 𝑥≤𝑏𝑎 for some 𝑎∈𝐴 and 𝑏∈𝐵. We consider the left ideal 𝐿(𝑎𝑏) generated by 𝑎𝑏, that is, the set 𝐿(𝑎𝑏)=(𝑎𝑏∪𝑆𝑎𝑏]. Then by Theorem 3.5, the characteristic 𝒩-function 𝑓_{𝐿(𝑎𝑏)} of 𝐿(𝑎𝑏) defined by 𝑓_{𝐿(𝑎𝑏)}∶𝑆→[0,1]∣𝑥↦𝑓_{𝐿(𝑎𝑏)}1(𝑥)∶=if0𝑥∈𝐿(𝑎𝑏),if𝑥∉𝐿(𝑎𝑏) (6.8) is a fuzzy left ideal of 𝑆. By hypothesis, we have 𝑓_{𝐿(𝑎𝑏)}(𝑎𝑏)=𝑓_{𝐿(𝑎𝑏)}(𝑏𝑎). Since 𝑎𝑏∈𝐿(𝑎𝑏), we have 𝑓_{𝐿(𝑎𝑏)}(𝑎𝑏)=1 and 𝑓_{𝐿(𝑎𝑏)}(𝑏𝑎)=1 and hence 𝑏𝑎∈𝐿(𝑎𝑏)=(𝑎𝑏∪𝑆𝑎𝑏]. Then 𝑏𝑎≤𝑎𝑏 or 𝑏𝑎≤𝑦𝑎𝑏 for some 𝑦∈𝑆. If 𝑏𝑎≤𝑎𝑏, then 𝑥≤𝑎𝑏∈𝐴𝐵 and 𝑥∈(𝐴𝐵]. If 𝑏𝑎≤𝑦𝑎𝑏, then 𝑥≤𝑦𝑎𝑏∈(𝑆𝐴)𝐵⊆𝐴𝐵 and 𝑥∈(𝐴𝐵]. Thus (𝐵𝐴]⊆(𝐴𝐵]. By symmetry we can prove that (𝐴𝐵]⊆(𝐵𝐴]. Therefore (𝐴𝐵]=(𝐵𝐴], and by Lemma 6.2, it follows that 𝑆 is a semilattice of left simple semigroups.

From left-right dual of Theorem 6.3, we have the following.

Theorem 6.4. An ordered semigroup (𝑆,⋅,≤) is a semilattice of right simple semigroups if and only if for every 𝒩-fuzzy right ideal 𝑓 of 𝑆, one has 𝑓(𝑎²)=𝑓(𝑎),𝑓(𝑎𝑏)=𝑓(𝑏𝑎)∀𝑎,𝑏∈𝑆. (6.9)

Lemma 6.5. Let (𝑆,⋅,≤) be an ordered semigroup and 𝑓 an 𝒩-fuzzy left (resp., right) ideal of 𝑆, and 𝑎∈𝑆 such that 𝑎≤𝑎². Then 𝑓(𝑎)=𝑓(𝑎²).

Proof. Since 𝑎≤𝑎² and 𝑓 is an 𝒩-fuzzy left ideal of 𝑆, we have 𝑓(𝑎)≤𝑓(𝑎²)=𝑓(𝑎𝑎)≤𝑓(𝑎), (6.10) and so 𝑓(𝑎)=𝑓(𝑎²).

7. Conclusion

Here we provided the concept of an 𝒩-fuzzy ideal in ordered semigroups and characterized some classes in terms of 𝒩-fuzzy left (resp., right) ideals of ordered semigroups. In this regard, we provided the characterizations of left (resp., right) regular, left (resp., right) simple, and completely regular ordered semigroups in terms of 𝒩-fuzzy left (resp., right) ideals.

In our future work we will consider 𝒩-fuzzy prime ideals and 𝒩-fuzzy filters in ordered semigroups and will establish the relations between them. We will also try to discuss the quotient structure of ordered semigroup in terms of 𝒩-fuzzy ideals.

Acknowledgments

The authors would like to thank the learned referees for their valuable comments and suggestions which improved the presentation of the paper and for their interest in their work. We would also like to thank Professor Dost Muhammad for his excellent guidance and valuable suggestions during this work.