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### SEMIGROUP CONGRUENCES

M. EL-GHALI M. ABDALLAH, L. N. GAB-ALLA, AND SAYED K. M. ELAGAN

Received 6 April 2006; Accepted 6 April 2006

A known result in groups concerning the inheritance of minimal conditions on normal subgroups by subgroups with finite indexes is extended to semilattices of groups [E(S), Se,ϕe,f] with identities in which allϕe,f are epimorphisms (calledqpartial groups).

For-mulation of this result in terms ofqcongruences is also obtained.

1. Introduction and preliminaries

A partial group as defined in [3] is a semigroupSwhich satisfies the following axioms. (i) For every x∈S, there exists a (necessarily unique) element ex∈S, called the

partial identity ofxsuch thatexx=xex=xand ifyx=xy=xthenexy=yex=ex. (ii) For every x∈S, there exists a (necessarily unique) elementx−1S, called the

partial inverse ofxsuch thatxx−1=x1x=exandexx1=x1ex=x1.

(iii) The operationx→exis a homomorphism fromSintoS, that is,exy=exeyfor all x,y∈S, and the operationx→x−1is an antihomomorphism, that is, (xy)1=

y−1x1for allx,yS.

Consequently, a partial group is precisely a Cliﬀord semigroup, that is, a regular semi-group with central idempotents, and this is characterized by Cliﬀord structure theorem (see [4, Chapter IV, Theorem 2.1] or [5, Chapter II, Theorem 2]) as a (strong) semilattice of groups.

Thus, in particular, a partial groupSmay be viewed as a strong semilattice of groups S=[E(S);Se,ϕe,f], whereSeis the maximal subgroup ofSwith identitye(e∈E(S)) and fore≥f inE(S),ϕe,f is the homomorphism of groupsSe→Sf,x→x f. HereE(S) is the

semilattice (e≥f if and only ife f = f) of idempotents (partial identities) inS.

LetSbe a partial group. A subpartial group ofSis a subsemigroup ofSclosed under the unary operations ofS. A subpartial group ofSis wide (or full) if it containsE(S).

A normal subpartial group of Sis a wide subpartial groupKofSsuch thatx−1KxK

for allx∈S. This notion is standard in the literature, and we refer in particular to [2] for the following consequences.

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 76951, Pages1–9

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LetKbe normal inS,Keis a normal subgroup ofSefor everye∈E(S).

ρK= {(x,y)∈S×S:ex=eyandxy−1K}is an idempotent-separating congruence

onSwith kerρK=K.

Conversely, ifρis an idempotent-separating congruence onS, thenK=kerρis a nor-mal subpartial group ofSandρK=ρ.

Let N(S) be the set of all normal subpartial groups ofS, and let Ci(S) be the set of all

idempotent-separating congruences onS. ForN,M∈N(S),N∨Mis the joinNM , that is, the smallest normal subpartial group ofScontainingNM, and we haveN∨ M=NM=MN. Forρ,σ∈Ci(S),ρσ is defined similarly, and we haveρσ=σρ=

ρ◦σ. AlsoρNM=ρN◦ρMfor allN,M∈N(S). Moreover, we have the following theorem.

Theorem 1.1 [2]. (N(S),⊂,,∨) and (Ci(S),,,) are complete modular lattices and

N(S)Ci(S),NρNis a lattice isomorphism.

For sake of reference, cite from [1] the material required for the present work.

Aq partial group is a partial groupSwith identity 1 such thatϕ1,e:S1→Se is an

epi-morphism (i.e.,S1e=Se) for everye∈E(S). This implies clearly thatϕe,f :Se→Sf is an epimorphism for everye,f ∈E(S) withe≥ f.

Theorem 1.2 [1]. LetSbe a partial group with identity 1. Every wide subpartial groupKof Scontains a unique maximalqsubpartial group, denoted by Q(K), ofSgiven by (Q(K))e= Im,ϕ1,e|K1=K1efor alle∈E(S). That is, Q(K) is a strong semilatticeE(S) of groupsK1e.

Also Q(K) is non trivial (i.e., Q(K)=E(S)) if and only ifK1is a nontrivial group.

On the class of partial groups with identities, we have idempotent unary operation S→Q(S), where Q(S), defind as above, is the unique maximalqsubpartial group ofS.

Lemma 1.3. LetSbe a partial group with identity and let{Si,i∈I}be a family of wide sub-partial groups ofS. Then Q(i∈ISi )= i∈IQ(Si) . In particular, the join of any family ofqsubpartial groups ofSis aqsubpartial group ofS.

A normal subpartial group of a partial groupSwith identity need not be aqsubpartial group, even ifSis aqpartial group (e.g., [1, Example 4.4 ]). On the other hand, ifSis aq partial group andNis normal inS, then Q(N) is normal inS.

LetSbe aqpartial group and let QN(S) be the set of allqnormal subpartial groups ofS.

Lemma 1.4. QN(S) is a complete modular lattice with meet and join defined by

M∧N=Q(M∩N),

M∨N=MN= M∪N . (1.1)

An idempotent-separating congruenceρon a partial groupSwith identity 1 is called aqcongruence if for allx,y∈S,xρyimplies thatx=syfor somes∈(kerρ)1, that is, for

somes∈S1with1.

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LetSbe a partial group with identity 1. The Q operation is defined on idempotent-separating congruences as follows:

Q(ρ)=(x,y) :xρy,x=sy(orx=ys) for somes∈Swith1. (1.2)

Precisely, Q(ρ) is the unique maximalqcongruence onScontained inρ.

Lemma 1.6. LetSbe aqpartial group and letρ∈Ci(S). Then

Q(ρ)Q(N), whereN=kerρ. (1.3)

LetSbe aqpartial group and let QCi(S) be the set of allqcongruences onS.

Theorem 1.7. With join and meet given byρ∨σ=ρ◦σ=σ◦ρandρ∧σ=Q(ρσ), QCi(S) is a complete modular lattice and the mapping

QN(S)−→QCi(S), N−→ρN, (1.4)

is a lattice isomorphism.

As observed in [1, Section 1],qpartial groups exist naturally as partial mappings from sets to groups. In the present paper, we consider minimal conditions on normal subpartial groups of partial groups with identities and we discuss the situations with which such conditions could be inherited by subpartial groups with finite indexes. Our principal goal is to extend the following known result in groups to appropriate classes of partial groups.

Theorem 1.8 [6, Theorem 3.1.8]. If a groupGsatisfies min−nandHis a subgroup ofG with finite index, thenHsatisfies min−n.

In this theorem, min−n is the minimal condition on normal subgroups. In other words a groupGis said to satisfy min−nif any nonempty family of normal subgroups has a minimal member, or equivalently there does not exist a proper descending chain N1⊃N2⊃ ···of normal subgroups ofG. The proof of the above result depends on the

notion of normal closures and cores in groups. Here we give the definitions and some properties (see [6, Section I.3]).

IfXis a nonempty subset of a groupG, the normal closure ofXinG, denoted byXG, is the intersection of all normal subgroups ofGwhich containsX. Dually, the core ofX inG, denoted byXG, is the join of all normal subgroups ofGthat contains byX. In other words,XGis the smallest normal subgroup ofGcontainingX, whereasXGis the largest one contained inX.

Fact 1.9. XG= g−1Xg:gG .

Fact 1.10. For any subgroupH ofG,HG=g∈Gg−1Hg.

For our work, we introduce inSection 2normal closures and cores in partial groups, showing that ifSis a partial group and∅ =X⊂S, thenXSis characterized as inFact 1.9. If eitherXSe= ∅for alle∈E(S) orShas an identity 1 and 1∈X, thatSis aqpartial group, andK is a normal subgroup ofS1(the maximal subgroup ofSwith identity 1),

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We also characterize the (normal) coreHSof a wide subpartial groupHof aqpartial groupSin terms of the group-theoretic cores ofHeinSe(e∈E(S)).HSneed not be aq subpartial group ofS, whence we introduce the notion of aq (normal) core showing that theqcore of a wide subpartial groupHof aqpartial groupSis precisely Q(HS).

InSection 3, we show that aqpartial groupSsatisfies min−qn(the minimal condi-tion onqnormal subpartial groups) if and only ifS1 satisfies min−n. Introducing the

notions of finite index and local finite index in partial groups (with identities), we show that min−qninqpartial groups is inherited byqsubpartial groups with local finite in-dexes extendingTheorem 1.8.

Whereas for partial groups with identities, we show that min−nimplies min−qnfor wide subpartial groups with finite indexes.

These two results can be formulated in terms of (q) congruences. In particular, the former result may have the version.

Inqpartial groups, the minimal condition onqcongruences is inherited byq subpar-tial groups with local finite indexes.

2. Normal closures and cores

Given a partial groupSand a nonempty subsetX ofS, we define the normal closure of X, denoted byXS, to be the intersection of all normal subpartial groups ofScontaining X. Evidently,XS is the smallest normal subpartial group ofScontainingX. Analogous to the known characterization of normal closures in groups (seeSection 1), we have the following lemma.

Lemma 2.1. LetSbe a partial group and let∅ =X⊂S. (i) IfX∩Se= ∅for alle∈E(S), then

XS=s1Xs:sS. (2.1)

In particular, ifHis a wide subpartial group ofS, then

HS=s1Hs:sS. (2.2)

(ii) IfShas an identity 1 and 1∈X, then

XS=s1Xs:sS. (2.3)

Proof. (i) Clearly, s−1Xs:sS is a subpartial group of S containingX, and by the

hypothesis, it is also wide. Lets∈Sand let y∈ s−1Xs:sS be a generator, say y=

s−1

1 xs1, for somes1∈S,x∈X. Thens−1ys=s−1s−11xs1s=(s1s)1x(s1s)∈ s−1Xs:s∈S .

Ify1,...,ynare generators ins−1Xs:s∈S , then

s−1y

1y2···yn s=s−1y1ss−1y2ss−1···ss−1yns

=s−1y

1s s−1y2s ···

s−1yns s1Xs:sS. (2.4)

Hence,s−1Xs:sS is a normal subpartial group ofScontainingX. Suppose thatK

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(sincex∈K), and sos−1Xs:sS ⊂K. Therefore,s1Xs:sS is the normal closure

ofXand (i) is proved.

(ii) For anye∈E(S),e=e−11e∈ s1Xs:sS . Thuss1Xs:sS is a wide

sub-partial group ofScontainingX. The result follows as in (i).

Lemma 2.2. LetSbe aqpartial group and letKbe a normal subgroup ofS1. ThenKSis aq

normal subpartial group ofS.

Proof. LetH=KS. ByLemma 2.1(ii),H= s1Ks:sS . Thus, it is suﬃcient to show

thatHe⊂H1efor alle∈E(S) (which implies thatϕ1,e:H1→Heis an epimorphism for

alle∈E(S)). Lete∈E(S) be fixed but arbitrary, and letx∈He. SinceKis a subgroup of S1, thenxmay be written as a product of generatorsx=x11k1x1x−21k2x2···x−n1knxnfor somexi∈Sandki∈K, andi=1,...,n. We haveeki=1 for alli∈I, and so

e=ex=ex1ex2···exn. (2.5)

SinceSis aqpartial group, we have

xi=siexi (2.6)

for somesi∈S1,i=1, 2,...,n. Thus

x=s−1

1 k1s1s−21k2s2···s−n1knsnex1ex2···exn. (2.7)

SinceKis normal inS1,ki=si−1kisi∈K,i=1, 2,...,n, and we have x=k

1k2···kne=ke, wherek=k1k2···kn∈K, (2.8)

and sox∈Ke. Clearly,H1=(KS)1= s−1Ks:s∈S1 =K. Thusx∈H1e. ThereforeHe⊂

H1e.

The notion of a core (or normal interior) in groups can be also extended to partial groups. LetS be a partial group and letH be a wide subpartial group of S. The core (normal interior) ofHinS, denoted byHS, is the join of all normal subpartial groups ofS contained inH. In other words,HSis the largest normal subpartial group ofScontained inH. Cores inqpartial groups can be characterized in terms of cores in groups. We have the following theorem.

Theorem 2.3. LetSbe aqpartial group and letH be a wide subpartial group ofS. Then (HS)eis the group-theoretic core ofHeinSefor everye∈E(S), that is,HSis a semilattice of groups

HS=E(S),Ke,ϕe,f, (2.9)

whereKeis the core of the subgroupHeinSe,e∈E(S).

Proof. By hypothesis,K=e∈E(S)Keis a union of disjoint groups indexed by the

semi-latticeE(S). ByFact 1.10, for everye∈E(S),Ke=s∈Ses−1Hes. Fore≥f inE(S), define

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Thus, letx∈Ke=s∈Ses−1Hes. We must show thatx f ∈Kf =

s∈Sfs−1Hfs. For this, let

sbe an arbitrary element ofSf. Since (by hypothesis)ϕe,f :Se→Sf is an epimorphism, there existss1∈Se such thats=s1f. Now,x∈

s∈Ses−1Hesimplies thatx=s−

1 1 ys1, for

some y∈He. Whence,x f =(s1f)1(y f)(s1f)∈s−1Hfs. Since sis arbitrary inSf, we

obtainx f ∈s∈Sfs−1Hfs=Kf. HenceK is a (strong) semilattice of groups. It follows

thatKis a subpartial group ofSwhich is clearly wide. To complete the proof, we have to show thatHS=K. Thus it is suﬃcient to show thatK is normal inS, thatK⊂H, and thatKis the largest with respect to this property. ThatK⊂H follows trivially from the definition ofK. To show thatKis normal inS, letx∈Sand lety∈K, sayx∈Se,y∈Kf, for somee,f ∈E(S). Thus,xy=xye f, and we havex−1yx=(x f)1(ye f)(x f). Clearly,

x f ∈Se f and ye f ∈Ke f. By the construction ofK,Ke f is normal inSe f. It follows that x−1yxKe f K. ThusK is a normal subpartial group ofS. Finally, letN be a normal

subpartial group ofSsuch thatN⊂H. We have for everye∈E(S),Neis normal inSeand Ne⊂He. SinceKeis the core ofHeinSe, we obtainNe⊂Ke. ThereforeN⊂K. LetSbe a partial group with 1 and letHbe a wide subpartial group ofS. We define the qcore of Hin Sto be the join of allqnormal subpartial groups ofScontained inH.

Theorem 2.4. IfSis aqpartial group andHis a wide subpartial group ofS, then theqcore ofH inSis Q(HS), whereHSis the core ofHinS.

Proof. We haveHS= N:NS,N⊂H , let NH be the family of all normal subpartial groupsN ofScontained inH. ThenHS= N:N∈NH . LetqNH be the family of allq normal subpartial groups ofScontained inH, we haveK∈qNHif and only ifK=Q(N), for someN∈NH, that is,qNH⊂ {Q(N) :N∈NH}. On the other hand, ifN∈NH, then, since Q(N) is normal inS[1], we have Q(N)∈qNH. Thus,{Q(N) :N∈NH} ⊂qNH. Then,qNH= {Q(N) :N∈NH}. By definition and [1, Lemma 4.2], we have theqcore of HinSwhich is

K:K∈qNH=Q(N) :N∈NH

=QN:N∈NH =QHS . (2.10)

3. Minimal conditions onqcongruences

We use the machinary that has been developed so far to extend a result in groups con-cerning the inheritance of minimal condition on normal subgroups of a given group to subgroups with finite indexes, and obtain analogous results for partial groups andq par-tial groups.

Recall that a partially ordered set (P,≤) is said to satisfy the minimal condition if any nonempty subset ofPcontains a minimal element. This is equivalent to saying thatP sat-isfies the descending chain condition: there does not exist an infinite properly descending chainx1> x2>···inP.

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condition, or equivalently if there does not exist an infinite properly descending chain K1⊃K2⊃ ···in QN(S). Here N(S) and QN(S) are defined as inSection 1.

Lemma 3.1. LetSbe a partial group with identity 1. (i) IfSsatisfies min−n, thenS1satisfies min−n.

(ii) IfS1satisfies min−n, thenSsatisfies min−qn.

Proof. (i) Suppose thatSsatisfies min−nbutS1does not. There exists an infinite properly

descending chain

N1⊃N2⊃ ··· in N

S1 . (3.1)

For eachi=1, 2,...,NiS(the normal closure ofNiinS) is a normal subpartial group ofS and we have a descending chain

NS

1⊃N2S⊃ ··· in N(S). (3.2)

By min−nofS, we obtain

NS

j =NSj+1 for some j. (3.3)

SinceNj⊃Nj+1andNj=Nj+1, there exists some elements∈Njwiths /∈Nj+1. We have

s∈NS

j =NSj+1. Thus,smay be written as an expansion,

s=y−1

1 x1y1y−21x2y2···yn−1xnyn, (3.4) withyi∈S,xi∈Nj+1,i=1, 2,...,n. Sincees=1, we must haveeyi=1 for alli=1, 2,...,n,

that is,yi∈S1for alli=1, 2,...,n. Now,Nj+1being a normal subgroup ofS1implies that

y−1

i xiyi∈Nj+1, for alli=1, 2,...,n. Therefore,s∈Nj+1, a contradiction.

(ii) Suppose thatS1 satisfies min−nbutSdoes not satisfy min−qn. There exists an

infinite properly descending chain

K1⊃K2⊃ ··· in QN(S). (3.5)

For eachi=1, 2,..., (Ki)1 is a normal subgroup ofS1, where (Ki)1 is the maximal

sub-group ofKiwith identity 1. Thus, the above chain induces a descending chain

K1 1

K2 1⊃ ··· in N

S1 . (3.6)

SinceS1satisfies min−n, we have for somej

Kj 1=

Kj+1 1. (3.7)

By assumption,Kj⊃Kj+1andKj=Kj+1. Thus we must have, for somee∈E(S),

Kj e⊃Kj+1 e, Kj e=Kj+1 e. (3.8)

There existsx∈(Kj)esuch thatx /∈(Kj+1)e. SinceKjis aqpartial group, we havex=se

for somes∈(Kj)1=(Kj+1)1. Now,s∈(Kj+1)1, which implies that se∈(Kj+1)e and so

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Using Lemmas3.1(ii),2.2, and proceeding as in the proof ofLemma 3.1(i), we obtain the following lemma.

Lemma 3.2. Aqpartial groupSsatisfies min−qnif and only ifS1satisfies min−n.

LetSbe an arbitrary partial group. A wide subpartial groupHofSis said to have a finite index in S ifS=HT, for some finite subsetTofS.

Lemma 3.3. LetSbe a partial group and letH be a wide subpartial group ofS. IfH has a finite index inS, thenHehas a finite index inSefor everye∈E(S).

Proof. By assumption,S=HT, for some finite subsetT ofS. Lete∈E(S) be fixed but arbitrary. For eachx∈Se, we havex∈Se⊂S=HT. Thusx=htx, for someh∈H and tx∈T. We have,e=ex=ehetx, and soe≤ehande≤etx, that is,eeh=eandeetx=e. Now,

x=htx=(he)tx∈Hetx=He(etx). SettingxT= {etx:x∈Se}, then clearlyxT is a finite subset ofSeandSe ⊂HexT. Also,HexT⊂SeSe=Se. Therefore,Se=HexT, which proves

thatHehas a finite index inSe.

LetSbe a partial group with identity 1 and letH be a wide subpartial group ofS. We say thatHhas a local finite index in SifH1has a finite index inS1. Clearly, byLemma 3.3,

for a wide subpartial groupHof a partial groupSwith identity, we have a finite index in Swhich implies a local finite index inS.

Forqpartial groups, the implication ofLemma 3.3may be refined as follows.

Lemma 3.4. LetSbe aqpartial group and letHbe a wide subpartial group ofS. IfHhas a local finite index inS, thenHehas a finite index inSefor everye∈E(S).

Proof. We haveS1=H1T, for some finite subsetT⊂S1. Lete∈E(S), sinceSis aqpartial

group, thenSe=S1e. Thus

Se=H1Te=

H1e (Te)=

H1e eT, (3.9)

whereeT is the finite setTe⊂Se. Since H is wide,H1e⊂H, in particular, H1e⊂He.

Therefore,

Se=H1e eT⊂He eT⊂Se. (3.10)

Thus,Se=He eTand the result follows.

Now we give our main result which is an extension of [6, Theorem 3.1.8].

Theorem 3.5. If aqpartial groupSsatisfies min−qnandH is aqsubpartial group ofS with local finite index, thenHsatisfies min−qn.

Proof. By definition (of local finite index),H1has a finite index inS1, and byLemma 3.2,

S1satisfies min−n. SinceH1is a subgroup ofS1,Wilson theorem [6, Theorem 3.1.8]

im-plies thatH1satisfies min−n. Again byLemma 3.2, we haveH satisfying min−qn.

For partial groups with identities, we have the following version ofTheorem 3.5.

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The proof follows similarly by applying Lemmas3.1(i),3.3, [6, Theorem 3.1.8], and

Lemma 3.1(ii).

In view of Theorems1.7and1.1, the above two Theorems (3.5and3.6) can be for-mulated in terms ofqcongruences and idempotent-separating congruences, respectively. For instance, by Theorems1.7and3.5, we have the following corollary.

Corollary 3.7. If aqpartial groupSsatisfies the minimal condition onqcongruences and His aqsubpartial group ofSwith local finite index, thenHsatisfies the minimal condition onqcongruences.

References

[1] M. E.-G. M. Abdallah, L. N. Gab-Alla, and S. K. M. Elagan, On semilattices of groups whose arrows are epimorphisms, to appear in International Journal of Mathematics and Mathematical Sciences.

[2] A. M. Abd-Allah and M. E.-G. M. Abdallah, Congruences on Cliﬀord semigroups, Pure Mathe-matics Manuscript 7 (1988), 19–35.

[3] , On Cliﬀord semigroups, Pure Mathematics Manuscript 7 (1988), 1–17. [4] J. M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976.

[5] M. Petrich, Inverse Semigroups, Pure and Applied Mathematics (New York), John Wiley & Sons, New York, 1984.

[6] D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80, Springer, New York, 1982.

M. El-Ghali M. Abdallah: Department of Mathematics, Faculty of Science, Menoufiya University, Shebin El-kom 32511, Egypt