# A Study of Ordered Ag-Groupoids in terms of Semilattices via Smallest (Fuzzy) Ideals

Authors: Venus Amjid; Faisal Yousafzai; Kostaq Hila
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

An ordered AG-groupoid can be referred to as an ordered left almost semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest one-sided ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and fuzzy one-sided ideals. We also introduce the concept of an orderedAG⁎⁎⁎-groupoid and investigate its structural properties by using the generated ideals and fuzzy one-sided ideals. These concepts will verify the existing characterizations and will help in achieving more generalized results in future works.


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## Body

## 1. Introduction

The concept of fuzzy set was first proposed by Zadeh [1] in 1965. Fuzzy algebra is becoming popular day by day due to wide applications of fuzzification in almost every field. It gives us a tool to model the uncertainty present in phenomena that do not have sharp boundaries. Many papers on fuzzy sets have been published which show its importance and applications to set theory, algebra, real analysis, measure theory, topology, etc. Several algebraists extended the concepts and results of algebra to the broader frame work of fuzzy set theory. Rosenfeld [2] was the first to consider the case when S is a groupoid. Kuroki and Mordeson have widely explored fuzzy semigroups in [3, 4]. F. Yousafzai et. al. have also studied the structural properties of certain classes of an ordered LA-semigroup and characterized them in terms of different fuzzy ideals as well [5–8].In the present paper, we introduce and investigate the notions of smallest left (right) ideals and study the relationship between these ideals along with fuzzy left (right) ideals. As an application of our results we get characterizations of a strongly regular class of a unitary ordered AG-groupoid (an orderedAG∗∗-groupoid) in terms of its semilattices, one-sided (two-sided) ideals based on fuzzy sets, and generated commutative monoids.


## 2. Preliminaries

An Abelian-Grassmann-groupoid (AG-groupoid) is a nonassociative and a noncommutative algebraic structure lying in a grey area between a groupoid and a commutative semigroup. Commutative law is given byabc=cba in ternary operations. By putting brackets on the left of this equation, i.e., (ab)c=(cb)a, in 1972, M. A. Kazim and M. Naseeruddin introduced a new algebraic structure called a left almost semigroup abbreviated as an LA-semigroup [9]. This identity is called the left invertive law. P. V. Protic and N. Stevanovic called the same structure an Abel-Grassmann’s groupoid abbreviated as an AG-groupoid [10].This structure is closely related to a commutative semigroup because a commutative AG-groupoid is a semigroup [11]. It was proved in [9] that an AG-groupoid S is medial; that is, ab·cd=ac·bd holds for all a,b,c,d∈S. An AG-groupoid may or may not contain a left identity. The left identity of an AG-groupoid permits the inverses of elements in the structure. If an AG-groupoid contains a left identity, then this left identity is unique [11]. In an AG-groupoid S with left identity (unitary AG-groupoid), the paramedial law ab·cd=dc·ba holds for all a,b,c,d∈S. By using medial law with left identity, we get a·bc=b·ac for all a,b,c∈S. An AG-groupoid satisfying the identities a(bc)=b(ac) and (ab)(cd)=(dc)(ba) for all a,b,c,d∈S is called an AG∗∗-groupoid (cf. [9, 12, 13]). We should genuinely acknowledge that much of the ground work has been done by M. A. Kazim, M. Naseeruddin, Q. Mushtaq, M.S. Kamran, P.V. Protic, N. Stevanovic, M. Khan, W.A. Dudek, and R.S. Gigon. One can be referred to [10, 11, 14–19] in this regard.An AG-groupoid(S,·) together with a partial order ≤ on S that is compatible with an AG-groupoid operation, meaning that for x,y,z∈S, x≤y⇒zx≤zy and xz≤yz, is called an ordered AG-groupoid [7].Let us define a binary operation “ ∘ e ” ( e -sandwich operation ) on an ordered AG-groupoid ( S , · , ≤ ) with left identity e as follows: (1) a  ∘ e  b = a e · b ,  ∀ a , b ∈ S .Then ( S , ∘ e , ≤ ) becomes an ordered semigroup[7].Note that an ordered AG-groupoid is the generalization of an ordered semigroup because if an ordered AG-groupoid has a right identity then it becomes an ordered semigroup.Let∅≠A⊆S; we denote (A] by (A]≔{x∈S/x≤a for some a∈A}. If A={a}, then we write ({a}]. For ∅≠A,B⊆S, we denote AB≕{ab/a∈A,b∈B}.(1) A nonempty subsetA of S is called an AG-subgroupoid of S if A2⊆A.(2) A nonempty subsetA of an ordered AG-groupoid S is called a left (right) ideal of S if we have the following:(i) S A ⊆ A   ( A S ⊆ A );(ii) if a∈A and b∈S such that b≤a, then b∈A.Equivalently, a nonempty subsetA of an ordered AG-groupoid S is called left (right) ideal of S if (SA]⊆A((AS]⊆A).(3) By two-sided ideal or, simply, ideal, we mean a nonempty subset of an ordered AG-groupoidS which is both left and right ideal of S.Lemma 1 (see [7]).
LetS be an ordered AG-groupoid and ∅≠A,B⊆S. Then the following hold:(i)
A⊆A;(ii)
IfA⊆B, then A⊆B;(iii)
AB⊆AB;(iv)
A=A;(v)
AB=AB;(vi)
T=T, for every ideal T of S;(vii)
(SS]=S=SS, if S has a left identity.


## 3. Fuzzy Set Theory

A fuzzy subset or a fuzzy set of a nonempty setS is an arbitrary mapping f:S→[0,1], where [0,1] is the unit segment of real line. A fuzzy subset f is a class of objects with grades of membership having the form f={(s, f(s))/s∈S}.The product of any fuzzy subsetsf and g of S is defined by(2)f∘gx=⋁y,z∈Axfy∧gzifAx≠∅0ifAx=∅,whereAx=y,z∈S×S∖x≤yz.The order relation⊆ between any two fuzzy subsets f and g of S is defined by f⊆g if and only if f(x)≤g(x), ∀x∈S.The symbolsf∩g and f∪g will mean the following fuzzy subsets of S: (3)f∩gx=min⁡fx,gx=fx∧gx,∀x∈S,and (4)f∪gx=max⁡fx,gx=fx∨gx,∀x∈S.For∅≠A⊆S, we define A=t∈S∣t≤a,forsomea∈A. For A={a}, we usually write a.(1)
A fuzzy setf of an ordered AG-groupoid is called
(i)
afuzzy left ideal of S if it satisfies the following:
(a)
f(xy)≥f(y);(b)
x≤y⇒fx≥fy,∀x,y∈S;(ii)
afuzzy right ideal of S if it satisfies
(a)
f(xy)≥f(x);(b)
x≤y⇒f(x)≥f(y),∀x, y∈S:(iii)
afuzzy ideal ofS, if it is both fuzzy left ideal and fuzzy right ideal of S.(2)
LetA be a nonempty subset of S. We denote by CA thecharacteristic functionof A and define it as follows: (5)CA=1ifx∈A,0ifx∉A.(3)
A fuzzy subsetf of S is called f-semiprime if f(x)≥f(x2), ∀x∈A.Lemma 2.
LetA be any right (left) ideal of an ordered AG-groupoid S. Then Ais semiprime if and only ifCAisf-semiprime.Proof.
LetA be a right (left) ideal of S; then, by Lemma 3, CA is a fuzzy left ideal of S. Let a2∈A, and then CA(a)≥CA(a2), and therefore CA(a2)=U⊆CA(a), and this implies CA(a)=1. Thus a∈A and therefore A is semiprime. Converse is simple.The following results are available in [7, 8].Lemma 3.
For a nonempty subsetA of an ordered AG-groupoid S, the following conditions are equivalent:(i)
A is a left (right) ideal of S;(ii)
the characteristic functionCA of A is a fuzzy left (right) ideal of S.Lemma 4.
Letf be any fuzzy subset of an ordered AG-groupoid S. Then fis a fuzzy left(right)ideal ofSif and only iff∘S⊆fS∘f⊆f.Remark 5.
The set(S,∘,⊆) forms an ordered AG-groupoid and satisfies all the basic laws.Remark 6.
IfS is an ordered AG-groupoid, then S∘S=S.Lemma 7.
LetS be an ordered AG-groupoid. For ∅≠A,B⊆S, the following assertions hold:(i)
A⊆B⇔CA⊆CB.(ii)
CA∩CB=CA∩B.(iii)
CA∪CB=CA∪B.(iv)
CA∘CB=C(AB].


## 4. On Strongly Regular Ordered AG-Groupoids

By a unitary ordered AG-groupoid, we shall mean an ordered AG-groupoid with left identity unless otherwise specified.


### 4.1. Basic Results

This section contains some basic results which will be essential for upcoming section.Lemma 8.
LetR be a right ideal and L be a left ideal of a unitary ordered AG-groupoid S. Then (RL] is a left ideal of S.Proof.
LetR be a right ideal and L be a left ideal of S. Then, by using Lemma 1, we get S(RL]=(SS](RL]⊆(SS·RL]=(SR·SL]⊆(SR·(SL]]=(SR·L]=((SS]R·L]⊆((SS)R·L]=((RS)S·L]⊆((RS]S·L]⊆(RL], which shows that (RL] is a left ideal of S.The converse inclusion of above lemma is not true in general. Let us consider a unitary ordered AG-groupoid(S,·,≤) with the following multiplication table and order below.(6)
·
1
2
3
4
5
1
2
2
4
4
5
2
2
2
2
2
5
3
1
2
3
4
5
4
1
2
1
2
5
5
1
5
5
5
5
≤
=
1,1
,
1,2
,
3,3
,
1,3
,
4,4
,
1,5
,
5,5
,
2,2
.SetR={1,2} and L={4,5}. Then it is easy to verify that (RL] is a left ideal of S but R(resp., L) is not a right (resp., left) ideal of S.Lemma 9.
LetS be a unitary ordered AG-groupoid. Then Ra=(Sa∪Sa2] is the smallest right ideal of S containing a, for all a∈S.Proof.
Leta∈S. Then, by using Lemma 1, we have (7)Sa∪Sa2S=Sa∪Sa2S⊆Sa∪Sa2S=Sa·S∪Sa2·S=Sa·SS∪Sa2·SS=S·aS∪S·a2S=a·SS∪a2·SS=a·SS∪a2·SS=SS·a∪SS·a2=Sa∪Sa2,
which shows that(Sa∪Sa2] is a right ideal of S. It is easy to see that a∈(Sa∪Sa2]. Let R be another right ideal of S containing a. Since (8)Sa∪Sa2=SS·a∪a·Sa=aS·S∪a·Sa⊆RS·S∪RS⊆R,
hence(Sa∪Sa2] is the smallest right ideal of S containing a.Lemma 10.
LetS be a unitary ordered AG-groupoid and a=a2 for all a∈S. Then S becomes a commutative monoid.Proof.
It is simple.Corollary 11.
R a = ( S a ∪ S a 2 ] is the smallest right ideal of an ordered commutative monoid S containing a, for all a∈S.Lemma 12.
LetS be a unitary ordered AG-groupoid and a∈S. Then La=(Sa] is the smallest left ideal of S containing a.Proof.
It is simple.Theorem 13.
LetS be a unitary ordered AG-groupoid and ∅≠E={x∈S:x=x2}⊆S. Then the following assertions hold:(i)
E forms a semilattice.(ii)
E is a singleton set, if a=ax·a,∀a,x∈S.Proof.
( i ) . It is simple.
( i i ) . Let y,z∈E. Then, by using (i), we get (9)y=yz·y=zy·y=yy·z=yz=zy=zz·y=yz·z=zy·z=z.(i) Recall that an orderedAG∗∗-groupoid is an ordered AG-groupoid in which a·bc=b·ac,∀a,b,c∈S. Note that an ordered AG∗∗-groupoid also satisfies the paramedial law as well.Now let us introduce the concept of an orderedAG∗∗∗-groupoid as follows:(ii) An orderedAG∗∗-groupoid S is called an ordered AG∗∗∗-groupoid if S=S2.Note that every unitary ordered AG-groupoid is an orderedAG∗∗∗-groupoid but the converse is not true in general.Corollary 14.
LetS be an ordered AG∗∗∗-groupoid and ∅≠E⊆S. Then the following assertions hold:(i)
E forms a semilattice, where E={x∈S:x=x2};(ii)
E is a singleton set if a=ax·a, ∀a,x∈S.Lemma 15.
LetS be an ordered AG∗∗∗-groupoid and a∈S. Then Ra2=(Sa2∪a2](La=(Sa∪a]) is the right (resp. left) ideal of S.Proof.
Lettinga∈S, then by using Lemma 1, we get (10)Sa2∪a2S=Sa2∪a2S=Sa2∪a2S=Sa2·S∪a2S=SS·a2S∪SS·aa=S·a2S∪Sa2=a2·SS∪Sa2=Sa2⊆Sa2∪a2,
which is what we set out to prove. Similarly we can prove thatS(Sa∪a]⊆(Sa∪a].(i) An elementa of an ordered AG-groupoid S is called a strongly regular element of S, if there exists some x in S such that a≤ax·a and ax=xa, where x is called a pseudoinverse of a. S is called strongly regular ordered AG-groupoid if all elements of S are strongly regular.Theorem 16.
LetS be a unitary ordered AG-groupoid (an ordered AG∗∗∗-groupoid). An element a of S is strongly regular if and only if a≤ax·ayfor somex,y∈S.Proof.
(Necessity). Let a∈S be strongly regular; then a≤ax·a≤(ax)·(xa)(ax·a)=(ax)·(a·ax)(ax)=(ax)·a((a·ax)x)=ax·ay, where (a·ax)x=y∈S. Thus a≤ax·ay for some x,y∈S.
(Sufficiency). Let a∈S such that a≤ax·ay for some x,y∈S; then a≤ax·ay=(ay·x)a=(xy·a)a=ua·a, where xy=u∈S. Thus au≤(ua·a)u=ua·ua=u(ua·a)≤ua, and a≤ua·a=au·a. Thus S is strongly regular.Remark 17 (see [7]).
Every fuzzy right ideal of a unitary orderedAG-groupoid S is a fuzzy left ideal of S but the converse inclusion is not true in general.Lemma 18.
Letf be any fuzzyleft (right) ideal of astrongly regular unitary ordered AG-groupoid S. Then the following assertions hold:(i)
f=f∘S(f=S∘f, f=(f∘S)∘f).(ii)
f is f-semiprime.Proof.
It is simple.


### 4.2. Characterization Problems

In this section, we generalize the results of an ordered semigroup and get some interesting characterizations which we usually do not find in an ordered semigroup.From now onward,R(resp.L) will denote any right (resp. left) ideal of an ordered AG-groupoid S; Ra(resp.La) will denote any smallest right (resp. smallest left) ideal of S containing a. Any fuzzy left ideal (resp., fuzzy right ideal) of an ordered AG-groupoid S will be denoted by f(resp.g) unless otherwise specified.Theorem 19.
Let f , g be any fuzzy left ideals of aunitary orderedAG-groupoid S. Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
(RaLa]∩La=((Ra·RaLa)La·La], ∀a∈S.(iii)
RL∩L=R·RLL·L.(iv)
f∩g=f∘g∘f.(v)
S is strongly regular and E=1, (a=ax·a, ∀a,x∈E).(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Theorem 13(i).
( v i ) ⇒ ( v ) : It can follow from Theorem 13(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy left ideals of a strongly regular S. Now for a∈S, there exist some x,y∈S such that a≤ax·ay=ya·xa≤y(ax·ay)·xa=(ax)(y·ay)·xa=(ay·y)(xa)·xa=(y2a·xa)(xa). Thus (y2a·xa,xa)∈Aa. Therefore (11)f∘g∘fa=⋁y2a·xa,xa∈Aaf∘gy2a·xa∧fxa≥⋁y2a·xa≤y2a·xafy2a∧gxa∧fxa≥fy2a∧gxa∧fxa≥fa∧ga,
which shows that(f∘g)∘f⊇f∩g. By using Lemmas 4 and 18, it is easy to show that (f∘g)∘f⊆f∩~g. Thus f∩g=(f∘g)∘f.
( i v ) ⇒ ( i i i ) : Let R and L be any right and left ideals of S, respectively. Then, by using Lemmas 3 and 8, we have the following.
C ( R L ] and CL are the fuzzy left ideals of S. Now by using Lemma 7, we get C(RL]∩L=C(RL]∩CL=(C(RL]∘CL)∘C(RL]=C((RL]L·(RL]], which give us (RL]∩L=((RL]L·(RL]]. Now, by using Lemma 1, we get (12)RLL·RL=RLL·RL=L2R·RL=LR·RL2=RLR·L2=RL2·RL=RR·L2L=R·RL3=RR·L2L=RL2·RL=R·RLL·L.
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : Since (Sa∪Sa2] is the smallest right ideal of S containing a and (Sa] is the smallest left ideal of S containing a, ∀a∈S, thus, by using given assumption and Lemma 1, we get (13)a∈Sa∪Sa2Sa∩Sa=Sa∪Sa2·Sa∪Sa2SaSa·Sa=Sa∪Sa2·Sa∪Sa2SaSa·Sa⊆SSa·Sa=S2a·Sa=Sa·Sa=aS·aS.Hence, by using Theorem 16, S is strongly regular.Theorem 20.
Let S be a unitary ordered A G-groupoid. Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
Ra∩La=(Ra(LaRa·Ra)], ∀a∈S).(iii)
R∩L=RLR·R.(iv)
f∩g=f3∘g.(v)
S is strongly regular and E=1, (a=ax·a, ∀a,x∈E).(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Theorem 13(i).
( v i ) ⇒ ( v ) : It can follow from Theorem 13(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy right ideal and fuzzy left ideal of a strongly regular S, respectively. From Lemma 4, it is easy to show that f3∘g⊆f∩g. Now, for a∈S, there exist some x,y∈S such that (14)a≤ax·ay≤ax·ayx·ax·ayy=yax·ay·xax·ay=axy·ay·axx·ay=axay2·axa·xy=y2axa·axa·xy=axa·xyxa·y2a=axa·xyex·a·y2a=axa·xyax·e·y2a=bc·y2a=d·y2a,whered=bc=axa·xyax·e.
Thus(15)f∘f∘fd=⋁d≤bcf∘fb∧fc≥f∘fb∧fc=⋁b≤axa·xyfax∧fa·xy∧fax·e≥fax∧fa·xy∧fax·e≥fa.
Therefore(16)f3∘ga=⋁a≤d·y2af∘f∘fd∧gy2a≥fa∧ga,
which shows thatf∩g⊆f3∘g. Thus f∩g=f3∘g.
( i v ) ⇒ ( i i i ): Let R and L be any right and left ideals of S, respectively. Then, by using Lemma 3, CR and CL are the fuzzy right ideal and fuzzy left ideal of S, respectively. Now, by using Lemma 7, we get (17)CR∩L=CR∩CL=CR∘CR∘CR∘CL=CR3∘CL=CR3L,
which implies thatR∩L=((R3]L]. Now by using Lemma 1, we get R∩L=((R3]L]=(R3L]=(R2R·L]=(LR·R2]=(R2·RL]=(R·R2L]=(R(LR·R)].
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : ( S a ∪ S a 2 ] is the smallest right ideal of S containing a and (Sa] is the smallest left ideal of S containing a. Thus, by using given assumption and Lemma 1, we get (18)a∈Sa∪Sa2∩Sa=Sa∪Sa2SaSa∪Sa2·Sa∪Sa2=Sa∪Sa2SaSa∪Sa2·Sa∪Sa2⊆SSSa∪Sa2·Sa∪Sa2=SS2a∪S2a2Sa∪Sa2=S2a∪S2a2SSa∪Sa2=S2a∪S2a2S2a∪S2a2=Sa∪a2S2Sa∪a2S2=Sa∪S2a·aSa∪S2a·a⊆Sa∪SaSa∪Sa=Sa·Sa=aS·aS.
Hence, by using Theorem16, S is strongly regular.LetS be an ordered AG∗∗∗-groupoid. From now onward, R(resp.L) will denote any right (resp.left) ideal of S; Ra2 will denote a right ideal (Sa2∪a2] of S containing a2 and La will denote a left ideal (Sa∪a] of S containing a; f(resp.g) will denote any fuzzy left ideal (resp., fuzzy right ideal) of S unless otherwise specified.Theorem 21.
Let S be an ordered A G ∗ ∗ ∗-groupoid.ThenSis strongly regular if and only if Ra2∩La=(Ra22La2] and Ra2 is semiprime.Proof.
(Necessity). Let S be strongly regular. It is easy to see that (Ra22La2]⊆Ra2∩La. Let a∈Ra2∩La. Then there exist some x,y∈S such that (19)a≤ax·ay≤ax·ayx·ax·ayy=x·ayax·y·ayax=a·xyax·ay2ax=a·xyax·xay2a∈Ra2S·Ra2SSLa·SLa⊆Ra22La2,
which shows thatRa2∩La=(Ra22La2]. It is easy to see that Ra2 is semiprime.
(Sufficiency). Since (Sa2∪a2] and (Sa∪a] are the right and left ideals of S containing a2 and a, respectively, thus, by using given assumption and Lemma 1, we get (20)a∈Sa2∪a2∩Sa∪a=Sa2∪a22Sa∪a2=Sa2∪a2Sa2∪a·Sa∪aSa∪a⊆SSa2∪a·SSa∪a=S·Sa2∪SaS·Sa∪Sa=a2S·S∪SaaS·S∪Sa=a2S·S∪SaaS·S∪Sa=Sa2∪SaSa∪Sa=a2S∪SaSa∪Sa=Sa·a∪SaSa∪Sa⊆Sa∪SaSa∪Sa=Sa·Sa=aS·aS.This implies that S is strongly regular.Corollary 22.
Let S be an ordered A G ∗ ∗ ∗-groupoid. ThenSis strongly regular if and only if Ra2∩La=(La2Ra22] and Ra2 is semiprime.Theorem 23.
Let S be an ordered A G ∗ ∗ ∗-groupoid.Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
Ra2∩La=(La2Ra22]andRa2is semiprime.(iii)
R∩L=(L2R2] and R semiprime.(iv)
f∩g=(f∘g)∘(f∘g) and f is f-semiprime.(v)
S is strongly regular and E=1, (a=ax·a, ∀a,x∈E).(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Corollary 14(i).
( v i ) ⇒ ( v ) : It can follow from Corollary 14(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy right ideal and fuzzy left ideal of a strongly regular S, respectively. From Lemma 4, it is easy to show that (f∘g)∘(f∘g)⊆f∩g. Now, for a∈S, there exist some x,y∈S such that (21)a≤ax·ay≤ax·ayx·ax·ayy=ax·ay·ax·ayxy=ax·ay·yxax·ay=ax·ay·axyx·ay=ax·ay·ay·yxxa=ax·ay·yx·yaxa=axyx·ya·ayxa=axba·ayxa,whereyx·y=b.
Thus(ax·ba,ay·xa)∈Aa. Therefore (22)f∘g∘f∘ga=⋁ax·ba,ay·xa∈Aaf∘gax·ba∧f∘gay·xa≥⋁ax·ba≤ax·bafax∧gba∧⋁ay·xa=ay·xafay∧gxa≥fax∧gba∧fay∧gxa≥fa∧ga,
which shows that(f∘g)∘(f∘g)⊇f∩g. Hence f∩g=(f∘g)∘(f∘g). Also, by using Lemma 18, f is f-semiprime.
( i v ) ⇒ ( i i i ) : Let R and L be any left and right ideals of S. Then, by using Lemma 3, CR and CL are the fuzzy right ideal and fuzzy left ideal of S. Now, by using Lemma 7, we get CR∩L=CR∩CL=(CR∘CL)∘(CR∘CL)=(CR∘CR)∘(CL∘CL)=C(R2]∘C(L2]=C(R2L2]=C(L2R2], which implies that R∩L=(L2R2].
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : It can follow from Corollary 22.Theorem 24.
Let S be an ordered A G ∗ ∗ ∗-groupoid.Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
Ra2∩La=(Ra2La·Ra2]andRa2is semiprime.(iii)
R∩L=(RL·R] and R is semiprime.(iv)
f∩g=(f∘g)∘fand f is f-semiprime.(v)
S is strongly regular and E=1, a=ax·a,∀a,x∈E.(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Corollary 14(i).
( v i ) ⇒ ( v ) : It can follow from Corollary 14(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy left ideals of a strongly regular S. Now, for a∈S, there exist some x,y∈S such that a≤ax·ay≤ax·(ax·ay)y=((ax·ay)y·x)a=(xy·(ax·ay))a=(ax·(xy·ay))a=(ax·(a·(xy)y))a.
Thus(ax·(a·(xy)y),a)∈Aa. Therefore (23)f∘g∘fa=⋁ax·a·xyy,a∈Aaf∘gax·a·xyy∧ga≥⋁ax·a·xyy≤ax·a·xyyfax∧ga·xyy∧ga≥fax∧ga·xyy∧ga≥fa∧ga,
which shows that(f∘g)∘f⊇f∩g. By using Lemmas 4 and 18, it is easy to show that (f∘g)∘f⊆f∩g. Thus f∩g=(f∘g)∘f. Also, by using Lemma 18, f is f-semiprime.
( i v ) ⇒ ( i i i ) : Let R and L be any left and right ideals of S. Then, by Lemma 3, CR and CL are the fuzzy right ideal and fuzzy left ideal of S, respectively. Now by using Lemmas 7, 8, and 1, we get (24)CR∩L=CR∩CL=CR∘CL∘CL=CRL·R=CRL·R,which shows that R∩L=(RL·R]. Also, by using Lemma 2, R is semiprime.
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : Since (Sa2∪a2] and (Sa∪a] are the right and left ideals of S containing a2 and a, respectively, thus by using given assumption and Lemma 1, we get (25)a∈Sa2∪a2∩Sa∪a=Sa2∪a2Sa∪a·Sa2∪a2=Sa2∪a2Sa∪a·Sa2∪a2⊆SSa∪a·Sa2∪a2=S2a∪SaSa2∪a2=S2a·Sa2∪S2a·a2∪Sa·Sa2∪S2a·a2⊆Sa·a2S∪Sa·Sa∪Sa·a2S∪Sa·Sa⊆Sa·Sa∪Sa·Sa∪Sa·Sa∪Sa·Sa=Sa·Sa=aS·aS. Hence S is strongly regular.


## 4.1. Basic Results

This section contains some basic results which will be essential for upcoming section.Lemma 8.
LetR be a right ideal and L be a left ideal of a unitary ordered AG-groupoid S. Then (RL] is a left ideal of S.Proof.
LetR be a right ideal and L be a left ideal of S. Then, by using Lemma 1, we get S(RL]=(SS](RL]⊆(SS·RL]=(SR·SL]⊆(SR·(SL]]=(SR·L]=((SS]R·L]⊆((SS)R·L]=((RS)S·L]⊆((RS]S·L]⊆(RL], which shows that (RL] is a left ideal of S.The converse inclusion of above lemma is not true in general. Let us consider a unitary ordered AG-groupoid(S,·,≤) with the following multiplication table and order below.(6)
·
1
2
3
4
5
1
2
2
4
4
5
2
2
2
2
2
5
3
1
2
3
4
5
4
1
2
1
2
5
5
1
5
5
5
5
≤
=
1,1
,
1,2
,
3,3
,
1,3
,
4,4
,
1,5
,
5,5
,
2,2
.SetR={1,2} and L={4,5}. Then it is easy to verify that (RL] is a left ideal of S but R(resp., L) is not a right (resp., left) ideal of S.Lemma 9.
LetS be a unitary ordered AG-groupoid. Then Ra=(Sa∪Sa2] is the smallest right ideal of S containing a, for all a∈S.Proof.
Leta∈S. Then, by using Lemma 1, we have (7)Sa∪Sa2S=Sa∪Sa2S⊆Sa∪Sa2S=Sa·S∪Sa2·S=Sa·SS∪Sa2·SS=S·aS∪S·a2S=a·SS∪a2·SS=a·SS∪a2·SS=SS·a∪SS·a2=Sa∪Sa2,
which shows that(Sa∪Sa2] is a right ideal of S. It is easy to see that a∈(Sa∪Sa2]. Let R be another right ideal of S containing a. Since (8)Sa∪Sa2=SS·a∪a·Sa=aS·S∪a·Sa⊆RS·S∪RS⊆R,
hence(Sa∪Sa2] is the smallest right ideal of S containing a.Lemma 10.
LetS be a unitary ordered AG-groupoid and a=a2 for all a∈S. Then S becomes a commutative monoid.Proof.
It is simple.Corollary 11.
R a = ( S a ∪ S a 2 ] is the smallest right ideal of an ordered commutative monoid S containing a, for all a∈S.Lemma 12.
LetS be a unitary ordered AG-groupoid and a∈S. Then La=(Sa] is the smallest left ideal of S containing a.Proof.
It is simple.Theorem 13.
LetS be a unitary ordered AG-groupoid and ∅≠E={x∈S:x=x2}⊆S. Then the following assertions hold:(i)
E forms a semilattice.(ii)
E is a singleton set, if a=ax·a,∀a,x∈S.Proof.
( i ) . It is simple.
( i i ) . Let y,z∈E. Then, by using (i), we get (9)y=yz·y=zy·y=yy·z=yz=zy=zz·y=yz·z=zy·z=z.(i) Recall that an orderedAG∗∗-groupoid is an ordered AG-groupoid in which a·bc=b·ac,∀a,b,c∈S. Note that an ordered AG∗∗-groupoid also satisfies the paramedial law as well.Now let us introduce the concept of an orderedAG∗∗∗-groupoid as follows:(ii) An orderedAG∗∗-groupoid S is called an ordered AG∗∗∗-groupoid if S=S2.Note that every unitary ordered AG-groupoid is an orderedAG∗∗∗-groupoid but the converse is not true in general.Corollary 14.
LetS be an ordered AG∗∗∗-groupoid and ∅≠E⊆S. Then the following assertions hold:(i)
E forms a semilattice, where E={x∈S:x=x2};(ii)
E is a singleton set if a=ax·a, ∀a,x∈S.Lemma 15.
LetS be an ordered AG∗∗∗-groupoid and a∈S. Then Ra2=(Sa2∪a2](La=(Sa∪a]) is the right (resp. left) ideal of S.Proof.
Lettinga∈S, then by using Lemma 1, we get (10)Sa2∪a2S=Sa2∪a2S=Sa2∪a2S=Sa2·S∪a2S=SS·a2S∪SS·aa=S·a2S∪Sa2=a2·SS∪Sa2=Sa2⊆Sa2∪a2,
which is what we set out to prove. Similarly we can prove thatS(Sa∪a]⊆(Sa∪a].(i) An elementa of an ordered AG-groupoid S is called a strongly regular element of S, if there exists some x in S such that a≤ax·a and ax=xa, where x is called a pseudoinverse of a. S is called strongly regular ordered AG-groupoid if all elements of S are strongly regular.Theorem 16.
LetS be a unitary ordered AG-groupoid (an ordered AG∗∗∗-groupoid). An element a of S is strongly regular if and only if a≤ax·ayfor somex,y∈S.Proof.
(Necessity). Let a∈S be strongly regular; then a≤ax·a≤(ax)·(xa)(ax·a)=(ax)·(a·ax)(ax)=(ax)·a((a·ax)x)=ax·ay, where (a·ax)x=y∈S. Thus a≤ax·ay for some x,y∈S.
(Sufficiency). Let a∈S such that a≤ax·ay for some x,y∈S; then a≤ax·ay=(ay·x)a=(xy·a)a=ua·a, where xy=u∈S. Thus au≤(ua·a)u=ua·ua=u(ua·a)≤ua, and a≤ua·a=au·a. Thus S is strongly regular.Remark 17 (see [7]).
Every fuzzy right ideal of a unitary orderedAG-groupoid S is a fuzzy left ideal of S but the converse inclusion is not true in general.Lemma 18.
Letf be any fuzzyleft (right) ideal of astrongly regular unitary ordered AG-groupoid S. Then the following assertions hold:(i)
f=f∘S(f=S∘f, f=(f∘S)∘f).(ii)
f is f-semiprime.Proof.
It is simple.


## 4.2. Characterization Problems

In this section, we generalize the results of an ordered semigroup and get some interesting characterizations which we usually do not find in an ordered semigroup.From now onward,R(resp.L) will denote any right (resp. left) ideal of an ordered AG-groupoid S; Ra(resp.La) will denote any smallest right (resp. smallest left) ideal of S containing a. Any fuzzy left ideal (resp., fuzzy right ideal) of an ordered AG-groupoid S will be denoted by f(resp.g) unless otherwise specified.Theorem 19.
Let f , g be any fuzzy left ideals of aunitary orderedAG-groupoid S. Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
(RaLa]∩La=((Ra·RaLa)La·La], ∀a∈S.(iii)
RL∩L=R·RLL·L.(iv)
f∩g=f∘g∘f.(v)
S is strongly regular and E=1, (a=ax·a, ∀a,x∈E).(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Theorem 13(i).
( v i ) ⇒ ( v ) : It can follow from Theorem 13(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy left ideals of a strongly regular S. Now for a∈S, there exist some x,y∈S such that a≤ax·ay=ya·xa≤y(ax·ay)·xa=(ax)(y·ay)·xa=(ay·y)(xa)·xa=(y2a·xa)(xa). Thus (y2a·xa,xa)∈Aa. Therefore (11)f∘g∘fa=⋁y2a·xa,xa∈Aaf∘gy2a·xa∧fxa≥⋁y2a·xa≤y2a·xafy2a∧gxa∧fxa≥fy2a∧gxa∧fxa≥fa∧ga,
which shows that(f∘g)∘f⊇f∩g. By using Lemmas 4 and 18, it is easy to show that (f∘g)∘f⊆f∩~g. Thus f∩g=(f∘g)∘f.
( i v ) ⇒ ( i i i ) : Let R and L be any right and left ideals of S, respectively. Then, by using Lemmas 3 and 8, we have the following.
C ( R L ] and CL are the fuzzy left ideals of S. Now by using Lemma 7, we get C(RL]∩L=C(RL]∩CL=(C(RL]∘CL)∘C(RL]=C((RL]L·(RL]], which give us (RL]∩L=((RL]L·(RL]]. Now, by using Lemma 1, we get (12)RLL·RL=RLL·RL=L2R·RL=LR·RL2=RLR·L2=RL2·RL=RR·L2L=R·RL3=RR·L2L=RL2·RL=R·RLL·L.
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : Since (Sa∪Sa2] is the smallest right ideal of S containing a and (Sa] is the smallest left ideal of S containing a, ∀a∈S, thus, by using given assumption and Lemma 1, we get (13)a∈Sa∪Sa2Sa∩Sa=Sa∪Sa2·Sa∪Sa2SaSa·Sa=Sa∪Sa2·Sa∪Sa2SaSa·Sa⊆SSa·Sa=S2a·Sa=Sa·Sa=aS·aS.Hence, by using Theorem 16, S is strongly regular.Theorem 20.
Let S be a unitary ordered A G-groupoid. Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
Ra∩La=(Ra(LaRa·Ra)], ∀a∈S).(iii)
R∩L=RLR·R.(iv)
f∩g=f3∘g.(v)
S is strongly regular and E=1, (a=ax·a, ∀a,x∈E).(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Theorem 13(i).
( v i ) ⇒ ( v ) : It can follow from Theorem 13(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy right ideal and fuzzy left ideal of a strongly regular S, respectively. From Lemma 4, it is easy to show that f3∘g⊆f∩g. Now, for a∈S, there exist some x,y∈S such that (14)a≤ax·ay≤ax·ayx·ax·ayy=yax·ay·xax·ay=axy·ay·axx·ay=axay2·axa·xy=y2axa·axa·xy=axa·xyxa·y2a=axa·xyex·a·y2a=axa·xyax·e·y2a=bc·y2a=d·y2a,whered=bc=axa·xyax·e.
Thus(15)f∘f∘fd=⋁d≤bcf∘fb∧fc≥f∘fb∧fc=⋁b≤axa·xyfax∧fa·xy∧fax·e≥fax∧fa·xy∧fax·e≥fa.
Therefore(16)f3∘ga=⋁a≤d·y2af∘f∘fd∧gy2a≥fa∧ga,
which shows thatf∩g⊆f3∘g. Thus f∩g=f3∘g.
( i v ) ⇒ ( i i i ): Let R and L be any right and left ideals of S, respectively. Then, by using Lemma 3, CR and CL are the fuzzy right ideal and fuzzy left ideal of S, respectively. Now, by using Lemma 7, we get (17)CR∩L=CR∩CL=CR∘CR∘CR∘CL=CR3∘CL=CR3L,
which implies thatR∩L=((R3]L]. Now by using Lemma 1, we get R∩L=((R3]L]=(R3L]=(R2R·L]=(LR·R2]=(R2·RL]=(R·R2L]=(R(LR·R)].
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : ( S a ∪ S a 2 ] is the smallest right ideal of S containing a and (Sa] is the smallest left ideal of S containing a. Thus, by using given assumption and Lemma 1, we get (18)a∈Sa∪Sa2∩Sa=Sa∪Sa2SaSa∪Sa2·Sa∪Sa2=Sa∪Sa2SaSa∪Sa2·Sa∪Sa2⊆SSSa∪Sa2·Sa∪Sa2=SS2a∪S2a2Sa∪Sa2=S2a∪S2a2SSa∪Sa2=S2a∪S2a2S2a∪S2a2=Sa∪a2S2Sa∪a2S2=Sa∪S2a·aSa∪S2a·a⊆Sa∪SaSa∪Sa=Sa·Sa=aS·aS.
Hence, by using Theorem16, S is strongly regular.LetS be an ordered AG∗∗∗-groupoid. From now onward, R(resp.L) will denote any right (resp.left) ideal of S; Ra2 will denote a right ideal (Sa2∪a2] of S containing a2 and La will denote a left ideal (Sa∪a] of S containing a; f(resp.g) will denote any fuzzy left ideal (resp., fuzzy right ideal) of S unless otherwise specified.Theorem 21.
Let S be an ordered A G ∗ ∗ ∗-groupoid.ThenSis strongly regular if and only if Ra2∩La=(Ra22La2] and Ra2 is semiprime.Proof.
(Necessity). Let S be strongly regular. It is easy to see that (Ra22La2]⊆Ra2∩La. Let a∈Ra2∩La. Then there exist some x,y∈S such that (19)a≤ax·ay≤ax·ayx·ax·ayy=x·ayax·y·ayax=a·xyax·ay2ax=a·xyax·xay2a∈Ra2S·Ra2SSLa·SLa⊆Ra22La2,
which shows thatRa2∩La=(Ra22La2]. It is easy to see that Ra2 is semiprime.
(Sufficiency). Since (Sa2∪a2] and (Sa∪a] are the right and left ideals of S containing a2 and a, respectively, thus, by using given assumption and Lemma 1, we get (20)a∈Sa2∪a2∩Sa∪a=Sa2∪a22Sa∪a2=Sa2∪a2Sa2∪a·Sa∪aSa∪a⊆SSa2∪a·SSa∪a=S·Sa2∪SaS·Sa∪Sa=a2S·S∪SaaS·S∪Sa=a2S·S∪SaaS·S∪Sa=Sa2∪SaSa∪Sa=a2S∪SaSa∪Sa=Sa·a∪SaSa∪Sa⊆Sa∪SaSa∪Sa=Sa·Sa=aS·aS.This implies that S is strongly regular.Corollary 22.
Let S be an ordered A G ∗ ∗ ∗-groupoid. ThenSis strongly regular if and only if Ra2∩La=(La2Ra22] and Ra2 is semiprime.Theorem 23.
Let S be an ordered A G ∗ ∗ ∗-groupoid.Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
Ra2∩La=(La2Ra22]andRa2is semiprime.(iii)
R∩L=(L2R2] and R semiprime.(iv)
f∩g=(f∘g)∘(f∘g) and f is f-semiprime.(v)
S is strongly regular and E=1, (a=ax·a, ∀a,x∈E).(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Corollary 14(i).
( v i ) ⇒ ( v ) : It can follow from Corollary 14(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy right ideal and fuzzy left ideal of a strongly regular S, respectively. From Lemma 4, it is easy to show that (f∘g)∘(f∘g)⊆f∩g. Now, for a∈S, there exist some x,y∈S such that (21)a≤ax·ay≤ax·ayx·ax·ayy=ax·ay·ax·ayxy=ax·ay·yxax·ay=ax·ay·axyx·ay=ax·ay·ay·yxxa=ax·ay·yx·yaxa=axyx·ya·ayxa=axba·ayxa,whereyx·y=b.
Thus(ax·ba,ay·xa)∈Aa. Therefore (22)f∘g∘f∘ga=⋁ax·ba,ay·xa∈Aaf∘gax·ba∧f∘gay·xa≥⋁ax·ba≤ax·bafax∧gba∧⋁ay·xa=ay·xafay∧gxa≥fax∧gba∧fay∧gxa≥fa∧ga,
which shows that(f∘g)∘(f∘g)⊇f∩g. Hence f∩g=(f∘g)∘(f∘g). Also, by using Lemma 18, f is f-semiprime.
( i v ) ⇒ ( i i i ) : Let R and L be any left and right ideals of S. Then, by using Lemma 3, CR and CL are the fuzzy right ideal and fuzzy left ideal of S. Now, by using Lemma 7, we get CR∩L=CR∩CL=(CR∘CL)∘(CR∘CL)=(CR∘CR)∘(CL∘CL)=C(R2]∘C(L2]=C(R2L2]=C(L2R2], which implies that R∩L=(L2R2].
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : It can follow from Corollary 22.Theorem 24.
Let S be an ordered A G ∗ ∗ ∗-groupoid.Then the following conditions are equivalent:(i)
S is strongly regular.(ii)
Ra2∩La=(Ra2La·Ra2]andRa2is semiprime.(iii)
R∩L=(RL·R] and R is semiprime.(iv)
f∩g=(f∘g)∘fand f is f-semiprime.(v)
S is strongly regular and E=1, a=ax·a,∀a,x∈E.(vi)
S is strongly regular and ∅≠E⊆S is semilattice.Proof.
( i ) ⇒ ( v i ) : It can follow from Corollary 14(i).
( v i ) ⇒ ( v ) : It can follow from Corollary 14(ii).
( v ) ⇒ ( i v ) : Let f and g be any fuzzy left ideals of a strongly regular S. Now, for a∈S, there exist some x,y∈S such that a≤ax·ay≤ax·(ax·ay)y=((ax·ay)y·x)a=(xy·(ax·ay))a=(ax·(xy·ay))a=(ax·(a·(xy)y))a.
Thus(ax·(a·(xy)y),a)∈Aa. Therefore (23)f∘g∘fa=⋁ax·a·xyy,a∈Aaf∘gax·a·xyy∧ga≥⋁ax·a·xyy≤ax·a·xyyfax∧ga·xyy∧ga≥fax∧ga·xyy∧ga≥fa∧ga,
which shows that(f∘g)∘f⊇f∩g. By using Lemmas 4 and 18, it is easy to show that (f∘g)∘f⊆f∩g. Thus f∩g=(f∘g)∘f. Also, by using Lemma 18, f is f-semiprime.
( i v ) ⇒ ( i i i ) : Let R and L be any left and right ideals of S. Then, by Lemma 3, CR and CL are the fuzzy right ideal and fuzzy left ideal of S, respectively. Now by using Lemmas 7, 8, and 1, we get (24)CR∩L=CR∩CL=CR∘CL∘CL=CRL·R=CRL·R,which shows that R∩L=(RL·R]. Also, by using Lemma 2, R is semiprime.
( i i i ) ⇒ ( i i ) : It is simple.
( i i ) ⇒ ( i ) : Since (Sa2∪a2] and (Sa∪a] are the right and left ideals of S containing a2 and a, respectively, thus by using given assumption and Lemma 1, we get (25)a∈Sa2∪a2∩Sa∪a=Sa2∪a2Sa∪a·Sa2∪a2=Sa2∪a2Sa∪a·Sa2∪a2⊆SSa∪a·Sa2∪a2=S2a∪SaSa2∪a2=S2a·Sa2∪S2a·a2∪Sa·Sa2∪S2a·a2⊆Sa·a2S∪Sa·Sa∪Sa·a2S∪Sa·Sa⊆Sa·Sa∪Sa·Sa∪Sa·Sa∪Sa·Sa=Sa·Sa=aS·aS. Hence S is strongly regular.


## 5. Conclusions

We have got some interesting and new characterizations which we usually do not find in other algebraic structures. This paper is the continuation of the work carried out in [7, 8]. We introduce the concept of an ordered AG∗∗∗-groupoid and characterize its strongly regular class in terms of fuzzy left/right ideals via its semilattices and generated commutative monoids.Some important issues for future work are as follows:(i)
To develop strategies for obtaining more valuable results in related areas(ii)
To apply these notions and results for studying fuzzy expert sets and applications in decision making problems.



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