# 𝒩-Fuzzy Ideals in Ordered Semigroups

Authors: Asghar Khan; Young Bae Jun; Muhammad Shabir
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We introduce the concept of𝒩-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.


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## Body

## 1. Introduction

A fuzzy subsetf 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 of𝒩-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 anordered 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.ForA⊆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].LetS 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 ofS is called a fuzzy two-sided ideal or simply a fuzzy ideal of S.


## 3.N-Fuzzy Left (Resp., Right) Ideals

LetS be an ordered semigroup. By a negative  fuzzy subset (briefly 𝒩-fuzzy subset) f of S we mean a mappingf: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))).
An𝒩-fuzzy left and right ideal f of S is called an 𝒩-fuzzy two-sided ideal of S.
For any𝒩-fuzzy subset f of S and t∈[-1,0) the set
(3.1)L(f;t):={x∈S∣f(x)≤t}
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
(3.2)(∀t∈[-1,0)L(f;t)(≠∅)isaleftidealifffisan𝒩-fuzzyleftideal).Proof.
Suppose thatf 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 allt∈[-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
(3.3)(∀t∈[-1,0)L(f;t)(≠∅)isarightidealifffisan𝒩-fuzzyrightideal).Example 3.4.
LetS={a,b,c,d,e,f} be the ordered semigroup defined by the multiplication and the order as follows:
(3.4)·abcdefaaaadaababbdbbcabcdeedaaddddeabcdeefabcdef≤:={(a,a),(b,b),(c,c),(d,d),(e,e),(f,e),(f,f)}.
Then left ideals ofS 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
(3.5)f(a)=-0.8,f(b)=-0.6,f(d)=-0.5,f(c)=-0.4,f(e)=f(f)=-0.2.
Then(3.6)L(f;t):={Sift∈[-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}ift∈[-0.8,-0.6),∅ift∈[-1,-0.8).
ThenL(f;t) is a left ideal of S, and by Theorem 3.2, f is an 𝒩-fuzzy left ideal of S.
LetS be an ordered semigroup and ∅≠A⊆S. The characteristic𝒩-functionκA:S→{-1,0} of A is defined by
(3.7)κA:S→[-1,0],x↦κA(x):={-1ifx∈A,0ifx∉A.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 characteristic𝒩-function
(3.8)κA(x)={-1ifx∈A,0ifx∉A
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 κA(y)=-1. Since x≤y∈A and A is a left ideal of S, we have x∈A. Then κA(x)=-1 and hence κA(x)≤κA(y). If y∉A, then κA(y)=0. Since κA is an 𝒩-fuzzy subset of S, we have κA(x)≤0 for all x∈S. Hence κA(x)≤κA(y).
Letx∈S and y∈A. Then κA(y)=-1. Since A is a left ideal of S and y∈A, we have xy∈A. Then κA(xy)=-1, and we have κA(xy)≤κA(y). If y∉A. Then κA(y)=0. Since κA(xy)≤0 for all x,y∈S. Hence κA(xy)≤κA(y). Thus κA is an 𝒩-fuzzy left ideal of S.
(ii)⇒(i) Assume that
(3.9)κA(x)={-1ifx∈A,0ifx∉A
is an 𝒩-fuzzy left ideal of S. Let x,y∈S, x≤y. If y∈A, then κA(y)=-1. Since x≤y, we have κA(x)≤κA(y). Then κA(x)=-1 and we have x∈A.
Letx∈S and y∈A. Then κA(y)=-1. Since κA is an 𝒩-fuzzy left ideal of S, we have κA(xy)≤κA(y). Hence κA(xy)=-1 and xy∈A. Thus A is a left ideal of S.A subsetT 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 ofS is a right ideal of S and semiprime.
An ordered semigroupS 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 semigroupS 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 semigroupS, the following conditions are equivalent.(i)
S is left simple.(ii)
Every𝒩-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
(4.1)ES:={e∈S∣e2≥e}.
ThenES≠∅. In fact, since S is regular and a∈S, there exists x∈S such that a≤axa. It follows from (OS3) that
(4.2)(ax)2=(axa)x≥ax,
and so ax∈ES and hence ES≠∅.
(1)f is a constant 𝒩-function on ES. Let t∈ES. 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
(4.3)f(e2)≤f((ztz)t)≤f(t).
Sincee∈ES, 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
(4.4)f(t2)≤f((yey)e)≤f(e).
On the other hand, t∈ES, 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
(4.5)(xa)2=x(axa)≥xa.
Hencexa∈ES 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],
(4.6)κ(Sa]:S→[-1,0],x↦κ(Sa](x):={-1ifx∈(Sa],0ifx∉(Sa],
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
(4.7)κ(Sa](x)=c,foreveryx∈S.
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 Theorem4.3, we have the following.Theorem 4.4.
For a regular ordered semigroup. The following statements are equivalent.(1)
S is right simple.(2)
Every𝒩-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 semigroupS is called completely regular (see  [21]) if it is regular, left regular, and right regular.Lemma 4.5 (cf. [21]).
An ordered semigroupS 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
(4.8)f(a)=f(a2)∀a∈S.Proof.
Suppose thatf 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
(4.9)f(a)≤f(xa2)≤f(a2)≤f(a).
Conversely, leta∈S. We consider the left ideal L(a2)=(a2∪Sa2] of S, generated by a2. Then by Theorem 3.5, the characteristic 𝒩-function
(4.10)κL(a2):S→[-1,0]∣x↦κL(a2)(x):={-1ifx∈L(a2),0ifx∉L(a2)
is an 𝒩-fuzzy left ideal of S. By hypothesis we have
(4.11)κL(a2)(a)=κL(a2)(a2).
Sincea2∈L(a2), we have κL(a2)(a2)=-1 and κL(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 Theorem4.6, 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
(4.12)f(a)=f(a2)∀a∈S.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 each𝒩-fuzzy bi-ideal f of S one has
(4.13)f(a)=f(a2)∀a∈S.(iii)
For each𝒩-fuzzy left ideal g and each 𝒩-fuzzy right ideal h of S we have
(4.14)g(a)=g(a2),h(a)=h(a2)∀a∈S.
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,
(4.15)ab≤(axa)b∈(aSa)b⊆(Sa)S⊆(Sa]S⊆(Sa].
Thusab∈((Sa]]=(Sa] and ab≤xa for some x∈S. Since f is an 𝒩-fuzzy left ideal of S, we have
(4.16)f(ab)≤f(xa)≤f(a).
Letx,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 κA of A is an 𝒩-fuzzy left ideal of S. By hypothesis, κA 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 Theorem4.10, 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 ofS is a right ideal of S.(ii)
Every𝒩-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
(4.17)ab≤(axa)b∈(aSa)S⊆(aSa]S⊆(aSa].
Thenab≤aza for some z∈S. Since f is an 𝒩-fuzzy bi-ideal of S, we have
(4.18)f(ab)≤f(aza)≤max{f(a),f(a)}=f(a).
Letx,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, κA is an 𝒩-fuzzy bi-ideal of S. By hypothesis κA is an 𝒩-fuzzy right ideal of S. By Theorem 3.5, A is a right ideal of S.By left-right dual of Theorem4.12, we have the following.Theorem 4.13.
Let(S,·,≤) be a regular ordered semigroup. Then the followings are equivalent.(i)
Every bi-ideal ofS is a left ideal of S.(ii)
Every𝒩-fuzzy bi-ideal of S is an 𝒩-fuzzy left ideal of S.


## 5. Characterization of Intraregular Ordered Semigroups in Terms  of𝒩-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
(5.1)f(a)≤f(a2)∀a∈S.Theorem 5.2.
Let(S,·,≤) be an ordered semigroup and ∅≠A⊆S. Then the followings are equivalent.(i)
A is semiprime.(ii)
The characteristic𝒩-function κA of A is an 𝒩-fuzzy semiprime subset.Proof.
(i)⇒(ii) Suppose that A is semiprime subset. Let κA(a2)=0. Since κA(a)≤0 for all a∈S, thus κA(a)≤κA(a2). If κA(a2)=-1, then a2∈A. Since A is semiprime, we have a∈A. Then κA(a)=-1 and; hence κA(a)≤κA(a2).
(ii)⇒(i) Assume that κA is 𝒩-fuzzy semiprime subset. Let a∈S be such that a2∈A. Then κA(a2)=-1. Since κA(a)≤κA(a2), we have κA(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
(5.2)f(a)=f(a2).Proof.
Suppose thatf is an 𝒩-fuzzy subsemigroup of S such that f is semiprime. Let a∈S. Then
(5.3)f(a)≤f(a2)=f(aa)≤max{f(a),f(a)}=f(a).
The converse is obvious.Theorem 5.4.
An ordered semigroupS is intraregular if and only if every 𝒩-fuzzy ideal of S is semiprime.Proof.
Suppose thatS 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
(5.4)f(a)≤f(x(a2y))≤f(a2y)≤f(a2).
Assume thatf 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 κI(a2) is an 𝒩-fuzzy ideal of S, and by hypothesis, we have κI(a2)(a)≤κI(a2)(a2). Since a2∈I(a2), then κI(a2)(a2)=-1 and κI(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  of𝒩-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).Forx∈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]).
LetS 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 semigroupS 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 semilatticeY and a family {Sα}α∈Y of left simple subsemigroups of S such that(1)
Sα∩Sβ=∅ for all α,β∈Y, α≠β,(2)
S=⋃α∈YSα,(3)
SαSβ⊆Sαβ 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
(6.1)(A2]=A,(AB]=(BA].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
(6.2)f(a2)=f(a),f(ab)=f(ba)∀a,b∈S.Proof.
⇒ (A) Let S be a semilattice of left simple semigroups. By hypothesis, there exists a semilattice Y and a family {Sα}α∈Y of left simple subsemigroups of S such that(i)
Sα∩Sβ=∅ for all α,β∈Y, α≠β,(ii)
S=⋃α∈YSα,(iii)
SαSβ⊆Sαβ for all α,β∈Y.
Letf 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∈Sα. Since Sα is left simple, we have Sα=(Sαa] and
(6.3)a≤xaforsomex∈Sα.
Sincex∈Sα, we have x∈(Sαa] and x≤ya for some y∈Sα. Thus we have
(6.4)a≤xa≤(ya)a=ya2
since y∈S, we have a∈(Sa2].
(B) Leta,b∈S. Then by (A), we have
(6.5)f(ab)=f((ab)2)=f(a(ba)b)≥f(ba).
By symmetry we can prove thatf(ba)≥f(ab). Hence f(ab)=f(ba).
⇐ Assume that for every fuzzy left ideal f of S, we have
(6.6)f(a2)=f(a),f(ab)=f(ba)∀a,b∈S,
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
(6.7)a≤xa2=(xa)a∈(SA)A⊆AA=A2,
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 fL(ab) of L(ab) defined by
(6.8)fL(ab):S→[0,1]∣x↦fL(ab)(x):={1ifx∈L(ab),0ifx∉L(ab)
is a fuzzy left ideal of S. By hypothesis, we have fL(ab)(ab)=fL(ab)(ba). Since ab∈L(ab), we have fL(ab)(ab)=1 and fL(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 Theorem6.3, 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
(6.9)f(a2)=f(a),f(ab)=f(ba)∀a,b∈S.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.
Sincea≤a2 and f is an 𝒩-fuzzy left ideal of S, we have
(6.10)f(a)≤f(a2)=f(aa)≤f(a),
and so f(a)=f(a2).


## 7. Conclusion

Here we provided the concept of an𝒩-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 consider𝒩-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.



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