INTRODUCTION An Abel-Grassmann’s groupoid (abbreviated as AG-groupoid) or Left Almost Semigroup (briefly LA-semigroup) is a groupoidS satisfying the left invertive law, defined as, .ab/cD.cb/afor all a

Vol. 19 (2018), No. 2, pp. 931–944 DOI: 10.18514/MMN.2018.2206

CONGRUENCES AND DECOMPOSITIONS OF AG-GROUPOIDS

WAQAR KHAN, FAISAL YOUSAFZAI, AND KOSTAQ HILA Received 12 January, 2017

Abstract. We introduce the concept of completely left inverse AG-groupoids and study some basic congruences and a congruence pair by means of the kernel and trace approach of completely left inverse AG-groupoids. Also, we provide separative and anti-separative decomposition of locally associative AG-groupoids.

2010Mathematics Subject Classification: 20N02; 08A30

Keywords: AG-groupoid, completely left inverse AG-groupoid, trace of congruence, kernel of congruence, decomposition, locally associative AG-groupoid

1. INTRODUCTION

An Abel-Grassmann’s groupoid (abbreviated as AG-groupoid) or Left Almost Semigroup (briefly LA-semigroup) is a groupoidS satisfying the left invertive law, defined as, .ab/cD.cb/afor all a; b; c2S:Inverse AG-groupoids, their different characterisations and congruences on inverse AG-groupoids using the kernel-normal system and kernel-trace approaches have been studied by many authors which can be found in the literature (see [1–4,6,7,11]).

In this paper, we introducecompletely left inverseAG-groupoids and investigate a congruence pair consisting a kernel and trace of a congruence of a completely left inverse AG-groupoid. In the second section, some preliminaries and basic results on completely inverse AG-groupoids are mentioned. In Section 3, we introduce completely left inverse AG-groupoids and investigate some basic congruences us- ing the congruence pair. We show that ifis a congruence on a completely left in- verse AG-groupoid, then.ker;tr/is a congruence. In Section 4, we discuss sep- arative and anti-separative decompositions of a locally associative AG-groupoid.

Before the proofs of the main results, it is important to recall the basic knowledge and necessary terminology.

2. PRELIMINARIES

An AG-groupoidS is regular ifa2.aS /afor alla2S:If fora2S;there exists an elementa⁰ such thataD.aa⁰/aanda⁰D.a⁰a/a⁰;then we say thata⁰ is inverse

c 2018 Miskolc University Press

ofa:In addition, if inverses commute, that isa⁰aDaa⁰, thenS is called completely regular. Ifa2S, then

V .a/D fa⁰2SWaD.aa⁰/aanda⁰D.a⁰a/a⁰g

is called the set of all inverses ofa2S:Note that ifa⁰ 2V .a/andb⁰ 2V .b/, then a2V .a⁰/anda⁰b⁰2V .ab/:

An AG-groupoidS in which every element has a unique inverse is called inverse AG-groupoid. If a ¹ is the unique inverse of a 2S; then a groupoid satisfying the following identities is called a completely inverse AG-groupoid, that is for alla; b; c2S

.ab/cD.cb/a; a.bc/Db.ac/

aD.aa ¹/a; a ¹D.a ¹a/a ¹andaa ¹Da ¹a:

IfSis a completely inverse AG-groupoid, thena ¹a2E_S, whereE_S is the set of idempotents ofS. IfS is a completely inverse AG-groupoid, thenES is either empty or a semilattice. For any idempotente inE_S,e ¹De:Moreover, the setE_S of an AG-groupoidS is a rectangular AG-band, that is for alle; f 2ES,eD.ef /e.

For futher concepts and results, the reader is referred to [3]. The set of idempotents ES of an AG-groupoid S is called left (respectively; right) regular AG-band if it satisfies

.ef /eDef .respectively;.ef /eDf e/for alle; f 2ES: Note that ifSis an AG-groupoid, then fore; f 2ES

ef D.ee/.ff /D.ff /.ee/Df e which shows left and right AG-bands serve the same purpose.

Lemma 1([3]). LetS be a completely inverseAG-groupoid and leta; b2S such thatab2E_S:ThenabDba:

Lemma 2([3]). Completely inverseAG-groupoids are idempotent-surjective.

If is a congruence on a completely inverse AG-groupoid, then S= is also completely inverse AG-groupoid. The natural morphism mapsS ontoS=by the rulex!.x/and by the uniqueness of inverses.x ¹/D.x/¹:If.a; b/2, then .a ¹; b ¹/2and.aa ¹; bb ¹/2:

3. CONGRUENCES IN COMPLETELY LEFT INVERSEAG-GROUPOID

In this section, we introduce the notion of completely left inverse AG-groupoids and study certain congruences by means of their kernel and trace for this class of groupoids. The essential part is to describe such congruence in terms of a congruence pair which comprises of a normal subgroupoid and a congruence of a completely left inverse AG-groupoids.

Definition 1. A completely inverse AG-groupoid is called completely left in- verse AG-groupoid if the setES of idempotents ofS is a left regular AG-band.

Proposition 1. Letbe a congruence on a completely inverseAG-groupoidS. If.a; e/2fore2ES;then.a; a ¹/2and.a; a ¹a/2.

Lemma 3. LetSbe a completely left inverseAG-groupoid. Ifis a congruence onS;thenS=is a completely left inverseAG-groupoid.

Proof. It is straightforward, and so it is omitted.

Definition 2. A nonempty subsetN of a completely left inverse AG-groupoidSis called normal if

(1)ES N;

(2) for everyx2S; xN x ¹N;

(3) for everya2N,a ¹2N:

Letbe a congruence on a completely left inverse AG-groupoidS andES be the set of idempotents ofS. The restriction ofonES, that isjES is the trace of denoted by tr. The subset

kerD fa2SW.9e2ES/.a; e/2g is the kernel of.

Lemma 4. Letbe a congruence on a completely left inverseAG-groupoidS:

.1/keris a normalAG-subgroupoid ofS:

.2/For anya2S,e2ES;ifea2kersuch that.e; aa ¹/2tr;thena2ker:

.3/For anya2S, ifa2ker, then.a ¹a; aa ¹/2tr:

Proof. .1/ Letbe a congruence. If a; b2ker, then .a; e/2; .b; f /2so that.ab; ef /2for somee; f 2ES:Henceab2kerand keris a subgroupoid of S:Obviously, all the idempotents ofS lie in ker. Leta2ker;then.a; e/2for e2E_S:Therefore for allx2S; .x ¹ax; x ¹ex/2:Sincex ¹exDex ¹x2 ES, thusx ¹ax2ker:Now ifa2ker, then forg2ES; .a; g/2. SinceS=;

is left inverse, it is clear that.a/¹2V ..a//:Moreover, ifh2ES, thena ¹2V .h/

so that.a ¹; h/2. That isa ¹2kerfor everya2ker:

.2/ If for a 2S; e 2ES and ea 2ker, then there exists f 2ES such that .ea; f /2:Since.e; aa ¹/2tr;thenaDaa ¹aeaf:Hence.a; f /2 anda2ker:

.3/ Leta2ker. Then.a; e/2 for somee 2ES: By Proposition 1, we have .a ¹a; a ¹a/2:Since trDjES;it follows that.a ¹a; aa ¹/2tr:

Lemma 5. Letbe a congruence on a completely left inverseAG-groupoidS:

Ifa ¹b2ker, thenab ¹2kerand for alla; b2S; ..a ¹bab ¹/; a ¹b/2. Proof. Letbe a congruence onS:Ifa ¹b2ker;then.a ¹b; e/2for somee2 E_S:Then it is clear that.ab ¹/is inverse of.a ¹b/inS=and it follows immedi- ately from the preliminaries and Lemma3that.a ¹b/2ES=:Hence.ab ¹; f /2

for somef 2ES:Thusab ¹2ker:Moreover, sinceS=is inverse andE=is a left regular AG-band, we have

.a ¹bab ¹/D.e/.f /D..e/.f //.e/D.e/D.a ¹b/:

Lemma 6. Letbe a congruence on a completely left inverseAG-groupoidS and leta; b2S ande2E_S:If.aa ¹; bb ¹/2trandab ¹2ker, then

.aea ¹; beb ¹/2tr:

Proof. Letbe a congruence onS:Leta; b2Ssuch that.aa ¹; bb ¹/2trand ab ¹2ker:Then for alle2E_S;we have

aea ¹ a.e.a ¹aa ¹/

a.e.b ¹ba ¹// .since.aa ¹; bb ¹/2tr/

a.e.a ¹bb ¹//

a.e..a ¹b.a ¹b/ ¹/b ¹// .since.a ¹b/2ES=/ a..a ¹abb ¹/.eb ¹//

a.b ¹beb ¹/ .since.aa ¹; bb ¹/2tr/

eab ¹

e.ab ¹a ¹b/ .by Lemma5/

e.aa ¹bb ¹/

beb ¹ .since.aa ¹; bb ¹/2tr/:

Thus.aea ¹; beb ¹/2tr:

Definition 3. LetN be normal subgroupoid of a completely left inverse AG- groupoid S and be congruence on a left regular AG-band ES: Then .N; / is a congruence pair ofSif for alla; b2S ande2ES the following conditions hold.

(1) Ifea2N and.e; a ¹a/2;thena2N;

(2) If.aa ¹; bb ¹/2 anda ¹b2N;then.aea ¹; beb ¹/2:

Theorem 1. Let S be a completely left inverse AG-groupoid and .N; / is congruence pair onS:Then the relation

_{.N; /}D f.a; b/2SSW .aa ¹; bb ¹/2 anda ¹b2Ng is a congruence relation.

Proof. Clearly,.N; /is reflexive. _{.N; /} is symmetric. In fact: is symmetric and by Definition 2(3), a ¹2N for any a2N: Also, if.a; b/; .b; c/2, then .aa ¹; bb ¹/2; .bb ¹; cc ¹/2 and a ¹b; b ¹c 2N: Then by Lemma 1,

cb ¹2N: Hence.aa ¹; cc ¹/2: Since .c ¹f.aa ¹bb ¹/cg/2ES andN is normal AG-subgroupoid, then

.c ¹f.aa ¹bb ¹/cg/.a ¹c/D fc ¹.a ¹ac/g.a ¹c/ .since.a ¹a; bb ¹/2 / D fa ¹ac ¹cg.a ¹c/

D fa ¹ab ¹bg.a ¹c/ .since.bb ¹; cc ¹/2 / D.bb ¹/.a ¹c/ .since.a ¹a; bb ¹/2 / D.cb ¹/.a ¹b/2N:

Moreover,

.a ¹c/ ¹.a ¹c/Dc ¹.aa ¹c/

Dc ¹..aa ¹cc ¹/c/ c ¹..aa ¹bb ¹/c/

.sincebb ¹ cc ¹/:

Hence by Definition3(1),a ¹c2N which implies that.a; c/2_{.N; /}:Thus_{.N; /} is equivalence relation.

Let.a; b/2_{.N; /};then.ac; bc/2_{.N; /}:In fact: if.aa ¹; bb ¹/2anda ¹b2 ker;then by Definition2(2), .ac/ ¹.bc/2N: Further, using Definition3(2), we have

.ac/.ac/ ¹D.aa ¹/.cc ¹/D.c ¹c/.aa ¹/Da.c ¹ca ¹/ b.c ¹cb ¹/:

Hence by definition of the relation.N; /; .ac; bc/2.N; /:

Similarly, since is a congruence, then.aa ¹; bb ¹/2 implies that

.ca/.ca/ ¹Dcac ¹a ¹Da ¹acc ¹Dc.aa ¹c ¹/ c.bb ¹c ¹/:

It remains to show that.ca/ ¹.cb/2N:Therefore

.b ¹..c ¹caa ¹/b//..ca/ ¹.cb//D..c ¹caa ¹/.b ¹b//.c ¹a ¹cb/

D..c ¹caa ¹/.b ¹b//.c ¹ca ¹b/

D..c ¹c/.aa ¹b ¹b/.c ¹c//.a ¹b/

D..c ¹c/.aa ¹b ¹b//.a ¹b/

(sinceES is left regular)

D.b ¹..aa ¹cc ¹/b//.a ¹b/2N:

Moreover,

b ¹..c ¹caa ¹/b/D..c ¹caa ¹/.c ¹c//.b ¹b/ .sinceES is left regular/

D..a ¹cac ¹/.c ¹c//.b ¹b/

D..acc ¹/.a ¹c ¹c//.b ¹b/

D.b.cc ¹a ¹//.b ¹.c ¹ca//:

Thus.b ¹..c ¹caa ¹/b// .b.cc ¹a ¹//.b ¹.c ¹ca//:Hence by Definition3 (1) it follows thata ¹.c ¹cb/2N:Thus.ca; cb/2_{.N; /}:

Corollary 1. Let S be a completely left inverse AG-groupoid and .N; / is congruence pair onS:Then the relation

.N; /D f.a; b/2SSW .aa ¹; bb ¹/2 andba ¹2Ng is a congruence relation.

Theorem 2. LetS be a completely left inverseAG-groupoid. Ifis a congru- ence onS;then (ker,tr) is a congruence ofS:Conversely, if.N; /is congruence pair onS, then the relation

_{.N; /}D f.a; b/2SSW .aa ¹; bb ¹/2 anda ¹b2Ng is a congruence relation onS:Furthermore,

ker_{.N; /}DN;tr_{.N; /}D and_.ker;tr/D:

Proof. The proof of the first part can be followed from Lemma4,6and Theorem 1. We show that ker.N; /DN and tr.N; /D:Leta2ker.N; /, then for somee2 ES; .a; e/2.N; /:It follows that.ee ¹; aa ¹/2andea2N:Thus by Definition 3(1),a2N;that is ker.N; /N:Conversely, suppose thata2N. Thena ¹2N:

Leta ¹aDe;it is clear that.ee ¹; aa ¹/2 ande ¹aDeaDa ¹aaDa2N:

Thus.e; a/2_{.N; /}:Hencea2.e/.N; /ker_{.N; /}:Thus ker_{.N; /}DN:

Similarly, we show that tr_{.N; /} and tr_{.N; /}: Lete; f 2E_S such that .e; f /2tr_{.N; /}: Then sinceES is left regular, thereforee D.ee ¹/eDee ¹ ff ¹D.ff ¹/f Df and hence tr_{.N; /}:Conversely, ifef, thenee ¹D e f Dff ¹ and e ¹f 2 ES N: Thus by definition of _{.N; /}; it follows .e; f /2.N; /\ESESDtr:Thus tr.N; /D:

Finally, suppose that .a; b/2. Then .a ¹a; a ¹b/2 so that a ¹b 2ker:

If a ¹ is the inverse ofa and sinceS= is completely left inverse, then .a ¹/ 2 V ..a//DV ..b//:Also,.b ¹/2V ..b//DV ..a//:It is clear that.a/.a ¹/D .a/.b ¹/D.b/.b ¹/which further implies that

.aa ¹; bb ¹/2tr: Thus .a; b/2.ker;tr/ and .ker;tr/: Conversely, let .a; b/2_.ker;tr/. Then.aa ¹; bb ¹/2tranda ¹b2ker:By Lemma5,ab ¹2 kerand.ab ¹/2E_S=:Then there existse2E_S such that.ab ¹/D.e/, where .e/2Ej:SinceE_S=is left regular, thus by Lemma5, we have

.ab ¹/D.ab ¹/..ab ¹/ ¹/: Then

aaa ¹aDbb ¹a .since.aa ¹; bb ¹/2tr/

ab ¹b

..ab ¹/.ab ¹/ ¹/b .aa ¹b ¹b/b

b ¹bb .since.aa ¹; bb ¹/2tr/

b:

Hence.ker;tr/D:This completes the proof.

4. DECOMPOSITIONS OF LOCALLY ASSOCIATIVEAG-GROUPOIDS

An AG-groupoid has many characteristics similar to that of a commutative semi- group. Let us consider x²y² Dy²x²; which holds for all x; y in a commutative semigroup:On the other hand one can easily see that it holds in an AG^**-groupoid:

This simply gives that how an AG^**-groupoid has closed connections with commut- ative algebra:In this section, we generalize the results of Hewitt and Zuckerman for commutative semigroups [5]:

An AG-groupoidS is called a locally associative AG-groupoid ifaaaDaaa for alla2S [8]:

Note that a locally associative AG-groupoid does not necessarily have associative powers:For example;in a locally associative AG-groupoidSD fa; b; cg;defined by the following table [8]W

a b c a c c b b b b b c b b b

.aaa/aDb¤cDa.aaa/:

Definition 4. A locally associative AG^**-groupoid is an AG^**-groupoidS satis- fying an identityaaaDaaafor alla2S:

Example1. Let us consider an AG^**-groupoidSD fa; b; c; d; egin the following multiplication table:

a b c d e

a a a a a a

b a e e c e c a e e b e d a b c d e e a e e e e

It is easy to verify thatS is a locally associativeAG-groupoid.

Proposition 2. The following statements hold:

(1) Every locally associative AG-groupoid has associative powers, that is aaⁿDaⁿafor alla2S and positive integern[8].

(2) In anAG^**-groupoidS; a^maⁿDa^mCn for alla2S and positive integers m; n[8].

(3) In a locally associativeAG^**-groupoidS; .a^m/ⁿDa^mn for alla2S and positive integersm; n[10].

(4) IfSis a locally associativeAG^**-groupoid anda; b2S;then.ab/ⁿDaⁿbⁿ for anyn1and.ab/ⁿDbⁿaⁿfor anyn2[9].

(5) LetS be a locally associative AG^**-groupoid: ThenaⁿDa^{n 1}aDaa^{n 1} for alla2Sandn > 1[10].

(6) IfSis a locally associativeAG^**-groupoid anda; b2S;thenaⁿb^mDb^maⁿ form; n > 1[8].

Note thata^{n 1}aD....aa/a/a/:::a/aandaa^{n 1}Da....aa/a/a/:::a/:

4.1. Separative decomposition

IfS is a locally associative AG^**-groupoid;thenabⁿcDabⁿc is not generally true for alla; b; c2S;that is.S xⁿ/S¤S.xⁿS /for somex2S:

Let us define the relationsandin a locally associative AG-groupoidS as follows:

for all a; b2S, ab”there exists n2N;such that aⁿ2S.bⁿS /andbⁿ2 S.aⁿS /.

for alla; b2S, ab”there exists n2N;such thataⁿ2.S bⁿ/S andbⁿ2 .Saⁿ/S.

Theorem 3. is equivalent toon a locally associativeAG-groupoidS:

Proof. Letaⁿ2S.bⁿS /:Then by using Proposition2(3), we get

a²ⁿD.aⁿ/²2.SbⁿS /²D.SbⁿS /.SbⁿS /D.S S /.bⁿSbⁿS / D.S S /.bⁿbⁿS S /

D.bⁿbⁿ/.S SS S /D.S SS S /.bⁿbⁿ/D.bⁿbⁿS S /.S S / D.S Sbⁿbⁿ/.S S /

.S b²ⁿ/S:

Similarly, we can show thatbⁿ2S.aⁿS /impliesb²ⁿ2.Sa²ⁿ/S:

Conversely, assume thataⁿ2.S bⁿ/S:Then by using Proposition2(3), we get a²ⁿD.aⁿ/²2.S bⁿS /²D.S bⁿS /.S bⁿS /D.S bⁿS bⁿ/.S S /

D.S Sbⁿbⁿ/.S S /

D.S S /.bⁿbⁿS S /S.b²ⁿS /:

Similarly, we can show thatbⁿ2.Saⁿ/S impliesb²ⁿ2S.a²ⁿS /:Thusis equi-

valent to:

Theorem 4. The relationon a locally associativeAG-groupoidS is a con- gruence relation:

Proof. Clearlyis reflexive and symmetric: For transitivity;let us suppose that ab andbc; such thataⁿ2S.bⁿS /andbⁿ2ScⁿS for alla; b; c 2S with as- sumption thatn > 1:By using Proposition2(3), we get

aⁿ2S.bⁿS /Dbⁿ.S S /.ScⁿS /S D.cⁿS S /S.cⁿS /SD.S S /cⁿ DS Sc^{n 1}cDcc^{n 1}S S Dcⁿ.S S /DS.cⁿS /:

Similarly, we can show thatcⁿ2S.aⁿS /:Henceis an equivalence relation:To show thatis compatible;assume thatab such that forn > 1; aⁿ2S.bⁿS /and bⁿ2S.aⁿS /for alla; b2S:Letc2S;then

.ac/ⁿDaⁿcⁿ2.SbⁿS /cⁿD.bⁿS S /cⁿD.b^{n 1}bS S /cⁿD.S Sbb^{n 1}/cⁿ D.S Sbⁿ/cⁿDcⁿbⁿS S DbⁿcⁿS S DS.bⁿcⁿS /DS.bc/ⁿS:

Similarly, we can show that.ca/ⁿ2S..cb/ⁿS /:Henceis a congruence relation

onS:

Definition 5. A congruenceis said to be separative congruence inS;ifaba² andab b²implies thata b:

Theorem 5. The relationon a locally associativeAG^**-groupoidSis separative:

Proof. Leta; b2Ssuch thataba²andabb²:Then for a positive integer n;

.ab/ⁿ2S.a²/ⁿS; .a²/ⁿ2S.ab/ⁿS and

.ab/ⁿ2S.b²/ⁿS; .b²/ⁿ2S.ab/ⁿS:

Now

a²ⁿD.a²/ⁿ2S.ab/ⁿS2S.S.b²/ⁿS /S D.S.b²/ⁿS /.S S / D..b²/ⁿS S /.S S /

D.S SS S /.bⁿbⁿ/D.bⁿbⁿ/.S SS S /D.S S /.bⁿbⁿS S /S.b²ⁿS /:

Similarly we can show thatb²ⁿ2S.a²ⁿS /:Henceis separative:

Proposition 3. IfSis a locally associativeAG-groupoid;thenabbafor all a; b2S;that isis commutative:

Proof. Leta; b2S such thatab andn be a positive integer:Then by using Proposition2(4), we get

.ab/ⁿDaⁿbⁿ2.SbⁿS /.SaⁿS /D.S S /.bⁿSaⁿS / D.S S /.bⁿaⁿS S /S.ba/ⁿS:

Similarly, we can show that.ba/ⁿ2S.ab/ⁿS:Henceabba:

Corollary 2. LetS be a locally associativeAG-groupoid: ThenS=is a sep- arative commutative image ofS:

Let us define a relation on a locally associative AG-groupoidS as follows:

for all x; y 2S, xy ” there exists n2N; such that .xa/ⁿ2 .ya/ⁿS and .ya/ⁿ2.xa/ⁿS, for somea2S.

Theorem 6. The relation is a congruence relation on a locally associative AG-groupoidS:

Proof. Clearly is reflexive and symmetric: For transitivity let us suppose that xy and y´; then there exist positive integers m; n such that .xa/ⁿ2 .ya/ⁿS;

.ya/ⁿ2.xa/ⁿSand.ya/^m2.´a/^mS and.´a/^m2.ya/^mS, for somea2S. More specifically;there existst12S such that.xa/ⁿD.ya/ⁿt1: Assume thatm; n > 1:

Now by using Proposition2(3) and Proposition2(4), we get .xa/^mnD .xa/ⁿm

D .ya/ⁿt1m

D .ya/^mn

t₁^m .´a/^mSn

S D.´a/^mnSⁿS

D.S Sⁿ/.´a/^mnD.S Sⁿ/.´a/^{mn 1}.´a/D.´a/.´a/^{mn 1}.SⁿS / .´a/^mnS:

Similarly we can show that.´a/^mn2.xa/^mnS:Henceis an equivalence relation onS:

To show compatibility; let xy; then there exists a positive integer n such that .xa/ⁿ2.ya/ⁿS and.ya/ⁿ2.xa/ⁿS:Hence there existst32S such that.xa/ⁿD .ya/ⁿt3: Now using Proposition 2(3), Proposition 2(4) and Proposition 2(6) with assumption thatn > 1;we get

.x´a/²ⁿD .x´a/ⁿ2

D .x´/ⁿaⁿ2

D.xⁿ´ⁿaⁿ/²D.´ⁿxⁿaⁿ/² D.aⁿxⁿ´ⁿ/²D.xⁿaⁿ´ⁿ/²D .xa/ⁿ´ⁿ2

D .ya/ⁿt3´ⁿ2

D .´ⁿt3/.ya/ⁿ2

D.´²ⁿt₃²/.ya/²ⁿD.´ⁿ´ⁿt3t3/.ya/²ⁿ D.t3t3´ⁿ´ⁿ/.ya/²ⁿ

D.t₃²´²ⁿ/.ya/²ⁿD .t3´ⁿ/.ya/ⁿ2

D .ya/ⁿ´ⁿt3

2

D .yⁿaⁿ/´ⁿt32

D .aⁿyⁿ/´ⁿt32

D .´ⁿyⁿ/aⁿt32

D .yⁿ´ⁿ/aⁿt32

D .y´a/ⁿt3

2

D.y´a/²ⁿt₃²2.y´a/²ⁿS:

Similarly, we can show that.y´a/²ⁿ2.x´a/²ⁿS:Thereforex´y´:Similarly we can show that is left compatible:Hence is a congruence relation onS:

4.2. Anti-separative decomposition

In this section, we show thatS= is a maximal anti-separative commutative image of a locally associative AG-groupoidS;where is defined as followsW

a bif and only ifabⁿDb^nC1andbaⁿDa^nC1for alla; b2Sand a positive integern:

Lemma 7. LetS be a locally associative AG-groupoid: If ab^mDb^mC1 and baⁿDa^nC1fora; b2S and positive integersm; n;thena b:

Proof. Without loss of generality let us suppose thatn > m:Thus by using Pro- position2(2), we get

b^{n m}b^mC1Db^{n m}ab^mDab^{n m}b^mDabⁿ:

Hencea b:

Theorem 7. The relation on a locally associativeAG-groupoidS is a con- gruence relation:

Proof. Clearly is reflexive and symmetric:For transitivity;leta b andb c;so there exist positive integers m; n such that abⁿDb^nC1; baⁿDa^nC1 and bc^mD c^mC1; cb^mDb^mC1:LetkD.nC1/.mC1/ 1;that iskDn.mC1/Cm:Now by using Proposition2(3) and Proposition2(6), we get

ac^kDac^n.mC1/CmDac^n.mC1/c^mDa.c^mC1/ⁿc^mDa.bc^m/ⁿc^m Da.bⁿc^mn/c^m

Dbⁿc^mnac^mDbⁿac^mnc^mDc^mc^mnabⁿDc^mc^mnb^nC1 Dc^m.nC1/b^nC1

Dc^{m.nC1/ 1}cbⁿbDbbⁿcc^{m.nC1/ 1}Db^nC1c^m.nC1/D.bc^m/^nC1 D.c^mC1/^nC1

Dc^.mC1/.nC1/Dc^kC1:

Similarly, we can show thatca^k Da^kC1: Thus is an equivalence relation: To show that is compatible;assume thata bsuch that for some integern;

abⁿDb^nC1andbaⁿDa^nC1: Letc2S:By using Proposition2(4), we get

.ac/.bc/ⁿDacbⁿcⁿDabⁿccⁿDb^nC1c^nC1D.bc/^nC1:

Similarly, we can show that.bc/.ac/ⁿD.ac/^nC1:Hence is a congruence rela-

tion onS:

Definition 6. A congruence is said to be anti-separative congruence inS; if ab a²andba b²implies thata b:

Theorem 8. The relation is anti-separative:

Proof. Leta; b2S such that ab a² andba b²: Then by definition of there exist positive integersmandnsuch that

.ab/.a²/^mD.a²/^mC1; a².ab/^mD.ab/^mC1; and

.ba/.b²/ⁿD.b²/^nC1; b².ba/ⁿD.ba/^nC1: Now by using Proposition2(2) and Proposition2(3), we get

ba^2mC1Dba^2maDa^2mbaDa^ma^mbaDaba^ma^mDaba^2m D.ab/.a²/^mD.a²/^mC1Da^2mC2;

and

ab^2nC1Dab²ⁿbDb²ⁿabDbⁿbⁿaDbabⁿbⁿDbab²ⁿ D.ba/.b²/ⁿD.b²/^nC1Db^2nC2:

Thus by using Lemma7; ab:Hence is anti-separative:

Proposition 4. IfS is a locallyAG-groupoid;thenab bafor alla; b2S;

that isis commutative:

Proof. Leta; b2S andn be a positive integer: Then by using Proposition2(6), Proposition2(2) and Proposition2(4) with assumption thatn > 1;we get

.ab/.ba/ⁿDabbⁿaⁿDabaⁿbⁿDaaⁿbbⁿDbⁿbaⁿaDb^nC1a^nC1D.ba/^nC1: Similarly we can show that.ba/.ab/ⁿD.ab/^nC1:Henceabba:

Theorem 9. LetS be a locally associativeAG-groupoid:ThenS= is a max- imal anti-separative commutative image ofS:

Proof. By Theorem8; is anti-separative;and henceS= is anti-separative:We now show thatis contained in every anti-separative congruence relationonS:Let a bso that there exists a positive integernsuch that

abⁿDb^nC1andbaⁿDa^nC1:

We need to show thatab;where is an anti-separative congruence onS:Let kbe a positive integer such that

ab^kb^kC1andba^ka^kC1: Suppose thatk > 2:Now by using Proposition2(2), we get

.ab^{k 1}/²Dab^{k 1}ab^{k 1}Daab^{k 1}b^{k 1}Da²b^{2k 2}; a²b^{2k 2}Daab^{k 2}b^kDab^{k 2}ab^kab^{k 2}b^kC1

Dab^{k 2}b^kbDab^kb^{k 2}bDab^kb^{k 1};

and from above;we have

a²b^{2k 2}ab^kb^{k 1}Db^{k 1}b^kaDb^kb^{k 1}aDab^{k 1}b^k:

Thus .ab^{k 1}/² ab^kb^{k 1}: Since ab^k b^kC1 implies that ab^kb^{k 1}

b^kC1b^{k 1}:Hence.ab^{k 1}/².b^k/²:It further implies that .ab^{k 1}/²a²b^{2k 2}Db^{2k 2}a².b^k/²: Thusab^{k 1}b^k:Similarly we can show thatba^{k 1}a^k:

By induction down fromk; it follows that for kD1; ab b² andba a²: Hence by using anti-separativity and Proposition4;it follows thatS= is a maximal

anti-separative commutative image ofS:

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Authors’ addresses

Waqar Khan

School of Mathematics & Statistics, Southwest University, Beibei, Chongqing 400715, P. R. China, Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

E-mail address:waqarmaths@gmail.com

Faisal Yousafzai

Military College of Engineering, National University of Sciences & Technology (NUST), Islamabad, Pakistan.

E-mail address:yousafzaimath@gmail.com

Kostaq Hila

Department of Mathematics & Computer Science, University of Gjirokastra, Gjirokastra 6001, Al- bania, or: Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana, Albania

E-mail address:kostaq hila@yahoo.com