Special subgroups of regular semigroups

Communications in Algebra

ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20

Special Subgroups of Regular Semigroups

T. S. Blyth & M. H. Almeida Santos

To cite this article: T. S. Blyth & M. H. Almeida Santos (2016): Special Subgroups of Regular Semigroups, Communications in Algebra, DOI: 10.1080/00927872.2016.1262385

To link to this article: http://dx.doi.org/10.1080/00927872.2016.1262385

Accepted author version posted online: 01 Dec 2016.

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Special subgroups of regular semigroups

T. S. Blyth1and M. H. Almeida Santos2

1_{Mathematical Institute, University of St Andrews, Scotland}

2_{Centro de Matemática e Aplicações (CMA), Departamento de Matemática, FCT, Universidade}

Nova de Lisboa, Portugal

Abstract

Extending the notions of inverse transversal and associate subgroup, we consider a regular

semigroupSwith the property that there exists a subsemigroupT which contains, for eachx∈ S,

a unique ysuch that both xyand yxare idempotent. Such a subsemigroup is necessarily a group

which we call aspecial subgroup. Here we investigate regular semigroups with this property. In

particular, we determine when the subset of perfect elements is a subsemigroup and describe its

structure in naturally arising situations.

KEYWORDS: Quasi-ideal; regular semigroup; special subgroup

2010 Mathematics Subject Classification: 20M20; 20M17

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1. SPECIAL SUBGROUPS

IfSis a regular semigroup then for eachx∈Swe denote the set of inverses ofxbyV(x), and the set

of associates ofxbyA(x)= {y∈ S|xyx=x}. Well known is the concept of aninverse transversal

ofS, namely an inverse subsemigroupS◦ = {x◦ | x ∈ S}with the property that|S◦∩V(x)| = 1

for everyx ∈ S (1). Corresponding to this is the notion of anassociate subgroup ofS, namely a

subgroupS⋆ = {x⋆ | x ∈ S}with the property that|S⋆ ∩A(x)| = 1 for everyx ∈ S (2). Here we

shall be concerned with the subset

I(x)= {y∈ S|xy,yx∈E(S)}

whereE(S)denotes the set of idempotents ofS. Clearly,V(x)⊆ A(x)⊆ I(x). Moreover, equality

holds throughout if and only ifSis completely simple (5)(6).

Our purpose here is to investigate the more general notion of a subsemigroup T of S with the

property that

(∀x∈ S) |T ∩I(x)| =1.

Given such a subsemigroupT, we definex∼ for everyx∈Sby

T∩I(x)= {x∼}.

Thusxx∼, x∼x ∈ E(S)for every x ∈ S, and we observe first that everyx ∈ T is such that x ∈

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every x ∈ S. Since T is a subsemigroup, for every x ∈ T we have x∼xx∼ ∈ T ∩I(x) whence

x∼_xx∼ _{= x}∼ _{and so xx}∼_{x ∈ V(x}∼_{) ⊆ I(x}∼_{). Since alsoxx}∼_{x ∈ T it follows that everyx ∈ T is}

such that xx∼x = x∼∼ = x and so S∼ = T is regular. Now ify is any inverse ofx∼ in S∼ then

y∈ S∼∩I(x∼)= {x∼∼}whencey=x∼∼and consequentlyS∼ is an inverse semigroup in which

(x∼₎−1 _=x∼∼_{. But ife,f ∈E(S}∼_{)thenef =fe∈E(S}∼_{)givese,f ∈S}∼_{∩I(ef)= {(ef)}∼_}whence

e=(ef)∼ =f. Thus we conclude thatS∼ is a subgroup ofS.

In what follows we shall call such a subgroupS∼ aspecial subgroupofS.

Theorem 1. Let S∼ be a special subgroup of the regular semigroup S. If ξ is the identity element

of S∼_{then S}∼ _=H ξ.

Proof. Since the maximal subgroups ofSare precisely theH_{-classes which contain idempotents,}

it follows thatS∼ ⊆Hξ. To obtain the reverse inclusion, letx ∈ Hξ. Thenxx∼, x∼x∈ E(Hξ)and

soxx∼ =ξ =x∼x, whencex−1 =x∼andx=(x∼)−1=x∼∼ ∈S∼. ThusS∼ =Hξ.

Example 1. Let S be an orthodox completely simple semigroup. As is well-known, we can

represent S as the cartesian product semigroup S = G × B of a group G and a rectangular

band B. Choose and fix an element α ∈ B. The H_{-class of the idempotent} e _{= (1G,α), namely}

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t∼

=

t◦ _{if t∈S;}

e ift=1.

Then(S1)∼=He. For each(g,x)∈Swe have that

(h,y)∈ He∩I(g,x) ⇐⇒ h=g−1, y=α

whenceH_e∩I(g,x)= {(g−1,α)} = {(g,x)◦} = {(g,x)∼}. Since alsoH_e∩I(1) =H_e∩E(S1) =

{e} = {1∼}, it follows thatHeis a special subgroup ofS1. Note that sinceA(1)= {1}the semigroup

S1 _{has no associate subgroups. Indeed, any such subgroup would have to contain 1 and the only}

subgroup ofS1which does so is{1}.

Example 2. Let k > 1 be a fixed integer and for eachn ∈ Z _letn_{k be the biggest multiple of} k

that is less than or equal to n. Consider the cartesian product setS = Z_×Z _{equipped with the}

multiplication

(m,n)(p,q)=(max{m,p},n+qk).

Sincemk +nk = (m+nk)k = (mk +n)k it follows thatS is a semigroup which is regular; for

example,(m,n)(m,−nk)(m,n)=(m,n). HereE(S)= {(m,n)| nk=0}and

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subgroup. Here also

V(m,n)= {(p,q)|p=m, qk+nk =0}; A(m,n)= {(p,q)|p6m, q_k+nk =0}.

The following result, which involves formulae similar to those for inverse transversals, will be used

throughout what follows.

Theorem 2. Let S be a regular semigroup. If S∼ is a special subgroup of S then

(1) (∀x,y∈S) (xy∼)∼ =y∼∼x∼ and (x∼y)∼ =y∼x∼∼;

(2) (∀x,y∈S) (xy)∼ =(x∼xy)∼x∼ =y∼(xyy∼)∼.

Proof.

(1) Letξ be the identity element ofS∼. Then on the one handxy∼·y∼∼x∼ =xξx∼ =xx∼ ∈E(S),

and on the other,

y∼∼_x∼_xy∼_·y∼∼_x∼_xy∼ _=y∼∼_x∼_xξx∼_xy∼ _=y∼∼_x∼_xy∼

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y∼_y∼∼_(xy)∼ _=ξ(xy)∼ _=(xy)∼_.

2. PARTICULAR SUBSETS

In the presence of an inverse transversalS◦, or of an associate subgroupS⋆, Green’s relations are

nicely describable. Indeed, in those situations they are as follows:

(x,y)∈R _⇐⇒ xx◦₌yy◦ _[resp.xx⋆ ₌yy⋆_];

(x,y)∈L ⇐⇒ x◦x=y◦y [resp.x⋆x=y⋆y].

This is not so in general for regular semigroups with special subgroups. For instance, in Example

1 we haveee∼ = e = 11∼ but(e, 1) /∈ Rsince 1 ∈/ eS1. So a separate investigation of the sets

J = {xx∼ | x ∈ S}andK = {x∼x | x ∈ S}is warranted. For this, we note thatJ andK have the

equivalent descriptions

J= {x∈S|x=xx∼}, K = {x∈S| x=x∼x}.

For example, if x ∈ J then there exists y ∈ S such that x = yy∼ whence, by Theorem 2(1),

xx∼ _=yy∼_(yy∼₎∼ _=yy∼_y∼∼_y∼ _=yy∼ _=x.

Likewise, we shall consider the subsemigroups

Sξ = {xξ |x∈S}, ξS= {ξx|x∈S}.

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Theorem 3. E(Sξ )=J and E(ξS)=K.

Proof. If j ∈ J then j = jj∼ whencej = jξ and therefore J ⊆ E(Sξ ). Conversely, ife ∈ E(Sξ )

thene=eξ givesξeξe=ξe, so thateξ,ξe∈E(S)and consequentlyξ ∈ S∼∩I(e)= {e∼}. Then

e∼ _{=ξ ande=eξ =ee}∼ _{∈ J. ThusE(Sξ )=Jand duallyE(ξS)=K.}

A further subset that is of structural importance is

P= {x∈S|x=xx∼x}.

Concordant with the terminology of (5), this may be called the set ofperfect elementsofS.

By the formulae in Theorem 2, it is readily seen that, equivalently,

P= {xx∼x|x∈S},

with, moreover,

(∀x∈S) (xx∼x)∼ =x∼.

Since for every j ∈ J we have j = jj∼jwe see thatJ ⊆ P, and similarlyK ⊆ P. Also, for every

x ∈ S, we have x∼x∼∼x∼ = ξx∼ = x∼ and therefore also S∼ ⊆ P. Moreover, by the above,

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Precisely when they coincide is determined as follows.

Theorem 4. A special subgroup S∼of S is an associate subgroup of S if and only if P =S.

Proof. If P = S then every x ∈ S is such that x = xx∼x whence S∼∩A(x) 6= ∅. But if y ∈

S∼_{∩A(x)theny∈S}∼_{∩I(x)= {x}∼_{}. HenceS}∼_{∩A(x)= {x}∼_{}and soS}∼_{is an associate subgroup.}

Conversely, if the special subgroup S∼ is an associate subgroup then for every x ∈ S we have

x=xx⋆x=xx∼x∈PwhenceP=S.

Precisely whenPis a subsemigroup ofSis the substance of the following result.

Theorem 5. The following statements are equivalent:

(1) P is a(regular)subsemigroup of S;

(2) KJ ⊆P.

Proof.

(1)⇒(2): IfPis a subsemigroup ofSthen, since bothJ andK are contained inP, it follows that

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x∼_xyy∼

=x∼xyy∼(x∼xyy∼)∼x∼xyy∼ =x∼xy(xy)∼xyy∼.

Pre-multiplying by x and post-multiplying by y, we obtain xy = xy(xy)∼xy so thatxy ∈ P and

thereforePis a subsemigroup which is regular sinceS∼ ⊆P.

In the case of an inverse transversalS◦the sets which correspond toJandKare denoted by I and3.

It is natural therefore to consider properties which are analogous to the principal properties listed

in (4).

Recalling thatE(S∼)= {ξ}, we shall say that the special subgroupS∼ is

primeifKJ = {ξ} [cf.S◦multiplicativeif3I⊆E(S◦)];

weakly primeif(KJ)∼ = {ξ} [cf.S◦weakly multiplicativeif (3I)◦ ⊆E(S◦)]; a quasi−idealifS∼SS∼ ⊆S∼ [cf.S◦SS◦⊆S◦or, equivalently,3I⊆S◦].

In what follows we shall consider the characteristic properties of each of these types and how they

are related. For this purpose, throughout what follows, S will denote a regular semigroup with a

special subgroup S∼_{whose identity element isξ, and the subsets J, K, P are as defined above.}

3. S∼ _{WEAKLY PRIME}

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Theorem 6. The following statements are equivalent:

(1) S∼ is weakly prime;

(2) (∀x,y∈ S) (xy)∼ =y∼x∼;

(3) the mappingζ :S→S described byζ :x7→x∼∼is a morphism;

(4) ∀x∈ hE(S)i _x∼

=ξ;

(5) KJ ⊆E(P).

Proof.

(1) ⇒ (2): If S∼ is weakly prime then, for all x,y ∈ S, we have(x∼xyy∼)∼ ∈ (KJ)∼ = {ξ}. It

follows by Theorem 2(2) that(xy)∼ =y∼(x∼xyy∼)∼x∼ =y∼ξx∼ =y∼x∼.

(2)⇒(3): This is clear.

(3)⇒(2): By(3)and Theorem 2(1),(xy)∼ =(xy)∼∼∼ =(x∼∼y∼∼)∼ =y∼x∼.

(2) ⇒ (4): If(2)holds ande ∈ E(S)thene∼ = (ee)∼ = e∼e∼ givese∼ ∈ E(S∼)and therefore

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Theorem3,

kj=kjξ =kj(kj)∼ ∈J ⊆E(P)

and(5)follows.

(5) ⇒ (1): If(5)holds then kjξ = kj ∈ E(P)and likewiseξkj ∈ E(P) whence it follows that

ξ ∈S∼∩I(kj)= {(kj)∼}. Thus(KJ)∼ = {ξ}and soS∼ is weakly prime.

Corollary. If S∼is weakly prime then J, K are sub-bands and P is a regular subsemigroup. Also,

S∼ _{is an associate subgroup of P and its identity elementξ is a medial idempotent of P.}

Proof. By Theorem2, for everyx∈Jwe havex∼ =(xx∼)∼ =x∼∼x∼ =ξ and sox=xx∼ =xξ.

Thus, ifx,y ∈ J then, using Theorem6(2), xy = xyξ = xyy∼x∼ = xy(xy)∼ ∈ J whenceJ, and

likewiseK, is a sub-band ofS. ThatPis a subsemigroup follows from Theorem 6(5) and Theorem

5. Now for everyx∈ Swe have thatx∼ =(xx∼x)∼. ThatS∼ =P∼ is then an associate subgroup

ofPfollows from Theorem 4 applied toP. Finally, by Theorem 6(4), for everyx∈ hE(P)iwe have

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a medial idempotent (and therefore that ofPwhenS∼ is weakly prime) was established in (2); see

also (8).

Example 3. In Example 2, P = {(m,n) | m > 0}, J = {(m,n) | m > 0, nk = 0} and

K = {(m, 0)|m>0}. ThenKJ=Kand, sinceK∼ = {(0, 0)}, it follows thatS∼is a weakly prime

special subgroup. HereS∼ is neither prime nor a quasi-ideal.

Example 4. IfLis a semilattice andGis a group, consider the inverse semigroupS=L×G. Here

E(S)=L× {1G}and, for every(x,g)∈ S,

I(x,g)=L× {g−1}.

For every idempotentξ = (e, 1G)of S, the groupHξ = {e} ×Gis a special subgroup of Swith

(x,g)∼ _=(e,g−1_{)for every(x,g)∈S. With respect to this,}

P= {(x,g)|x6e, g∈ G}, J=K = {(x, 1G)|x6e} =E(P).

It follows by Theorem 6 thatHξis a weakly prime special subgroup ofS. It is prime or a quasi-ideal

if and only ifLhas a bottom element 0 ande=0.

Example 5. Consider the semidirect product semigroup Q ×ζ G of a band Q with an identity

elementξ and a group G. With respect to a given morphism ζ : G → AutQ described byζ :

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(x,g)(ξ,g−1)(x,g)= xζ_g(ξ )ζ1G(x), gg−1g

=(xξx, g)=(x,g).

Since(x,g)(x,g)=(xζg(x),g2)it follows thatE(Q×ζ G)= {(x, 1G)| x∈ Q}. Consider now the

subsetH = {ξ} ×G. This is clearly a subgroup ofQ×ζ G. Moreover,

(ξ,h)∈I(x,g) ⇐⇒ ξ ζh(x),hg, xζg(ξ ),gh∈ E(Q×ζ G) ⇐⇒ h=g−1

so thatH∩I(x,g)= {(ξ,g−1)}. Consequently,His a special subgroup ofQ×ζGunder(x,g)∼ =

(ξ,g−1), andHis isomorphic toG. It follows by Theorem 6(2) thatHis weakly prime. It is readily

seen that ifQ6= {ξ}thenH is neither prime nor a quasi-ideal.

The situation described in Example 5 is highlighted in the following result.

Theorem 7. If S∼is weakly prime then the subsemigroup ξPξof P is isomorphic to a semidirect

product of a band with an identity and a group. More precisely, if 1 = {x ∈ P | x∼ = ξ}then

ξ 1ξ is a sub-band of P with identity ξ. For each g ∈ S∼ the mapping ζg : x 7→ gxg∼ is an

automorphism of ξ 1ξ, andζ :g7→ζgis a morphism. Finally,

ξPξ ≃ ξ 1ξ ×ζ S∼

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Now, for everyx ∈ 1, xξ = xx∼ ∈ J whence it follows that ξxξ ∈ J ⊆ E(P). Also, ifx,y ∈ 1

thenxξy∈ 1givesξxξ·ξyξ ∈ ξ 1ξ and thereforeξ 1ξ is a sub-band ofPwith identity element

ξ. Moreover, for everyg∈S∼ and everyx∈ξ 1ξ, we havegxg∼ ∈ξ 1ξ.

For each g ∈ S∼ consider the mapping ζg : ξ 1ξ → ξ 1ξ given by ζg(x) = gxg∼. We have

ζg(ξ )=gξg∼ =gg∼ =ξ. Moreover, ifx,y∈ ξ 1ξ then

ζg(xy)=gxyg∼ =gxξyg∼ =gxg∼gyg∼ =ζg(x)ζg(y)

and soζg∈ Endξ 1ξ. Sinceg∼xg∈ ξ 1ξwithζg(g∼xg)=gg∼xgg∼ =ξxξ =x, it follows thatζg

is surjective. Moreover, ifζg(x)=ζg(y)then, applyingζg−1to this, we obtainx=ξxξ =ξyξ =y.

Consequently, eachζg ∈Autξ 1ξ.

Sinceζg[ζh(x)] = ghxh∼g∼ = ghx(gh)∼ = ζgh(x), the mappingζ : S∼ → Autξ 1ξ given by

ζ (g)=ζgis a morphism.

It follows from the above that we can construct the semidirect productξ 1ξ×ζ S∼.

Sincexx∼ ∈ 1by Theorem 2, it follows thatxx∼ ∈ ξ 1ξ for every x ∈ ξPξ. Consider therefore

the mappingϑ :ξPξ →ξ 1ξ ×ζ S∼ given by

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ϑ (x)ϑ (y) = (xx∼,x∼∼)(yy∼,y∼∼)

= (xx∼ζx∼∼(yy∼), x∼∼y∼∼)

= (xx∼·x∼∼yy∼x∼, x∼∼y∼∼)

= (xξyy∼x∼, (xy)∼∼)

= (xy(xy)∼, (xy)∼∼)

= ϑ (xy).

That ϑ is injective follows from the fact that ifϑ (x) = ϑ (y) thenxx∼ = yy∼ and x∼∼ = y∼∼

whencex = ξxξ = ξxx∼x∼∼ = ξyy∼y∼∼ = ξyξ = y. Finally, given y ∈ ξ 1ξ andg ∈ S∼ we

haveyg∈ξPξ with

ϑ (yg)=(yg(yg)∼,(yg)∼∼)=(ygg∼y∼, y∼∼g∼∼)=(yξ,ξg)=(y,g)

and soϑ is also surjective. Consequently,ξPξ ≃ ξ 1ξ ×ζ S∼.

4. S∼ _{A QUASI-IDEAL}

The quasi-ideal special subgroups have the following characterisations.

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(3) KJ = [E(S)]∼;

(4) ξPξ =S∼;

(5) (∀x∈S) x∼ =x∼xx∼;

(6) (∀x∈S) x∼∼ =ξxξ.

Proof. We establish(1)⇒(2)⇒(3)⇒(1)and(1)⇒(4)⇒(5)⇒(6)⇒(1).

(1) ⇒ (2): IfS∼ is a quasi-ideal then for allx,y ∈ S we have x∼xyy∼ ∈ S∼SS∼ ⊆ S∼ whence

KJ⊆S∼.

(2)⇒(3): If(2)holds then for allj∈ Jandk ∈K we have thatkj=(kj)∼∼ ∈S∼ ⊆P, whence

it follows that(kj)∼ ∈V(kj).

Givenj∈J andk∈ K, recall (7) that thesandwich set S(k,j)is given by

S(k,j)= {g∈ E(S)| g=jg=gk, kgj=kj} = j V(kj)k.

Consider now the elementg = j(kj)∼kwhich, by the above, belongs toS(k,j). Since kjg = kg ∈

E(S)andgkj = gj ∈ E(S)we have that kj ∈ S∼ ∩I(g)whencekj = g∼ and consequentlyKJ ⊆

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and(3)follows.

(3)⇒(1): If(3)holds thenx∼xyy∼ ∈S∼whence, takingx=ξ, we obtainξyy∼ ∈S∼. It follows

thatξyξ =ξyy∼y∼∼ ∈S∼ whencex∼yz∼ =x∼ξyξz∼ ∈ S∼ and consequentlyS∼SS∼ ⊆S∼.

(1) ⇒ (4): If(1)holds thenS∼= ξS∼ξ ⊆ ξSξ ⊆ S∼SS∼ ⊆ S∼. Thus ξSξ = S∼ ⊆ Pwhence

ξPξ =ξSξ and(4)follows..

(4)⇒(5): If(4)holds then for everyp∈ Pwe have, by Theorem 2,ξpξ =(ξpξ )∼∼ =ξp∼∼ξ =

p∼∼_whencep∼_pp∼ _=p∼_p∼∼_p∼ _=p∼_{. Consequently,}

(∀x∈S) (xx∼x)∼xx∼x(xx∼x)∼ =(xx∼x)∼

which reduces tox∼xx∼ =x∼which is(5).

(5)⇒(6): If(5)holds then on pre- and post-multiplying byx∼∼we obtain(6).

(6) ⇒ (1): If (6)holds then for all x,y,z ∈ S, x∼yz∼ = x∼ξyξz∼ = x∼y∼∼z∼ ∈ S∼ whence

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Example 6. Consider the following sets of real 2×2 matrices:

A= _{x x} x x , _x 0 x 0 , _{x x} 0 0 , _x 0 0 0 x6=0

; B=

_x 0 0 x

x6=0

.

LetS=A∪B. ThenSis a regular monoid with

E(S)= 1 2 1 2 1 2 1 2 , 1 1 0 0 , 1 0 1 0 , 1 0 0 0 , 1 0 0 1 .

For eachX ∈SdefineX∼ =

x−1 ₀

0 0

. ThenS∼ =

_x 0 0 0

x6=0

is a special subgroup. Here

K = 1 1 0 0 , 1 0 0 0

; J =

1 0 1 0 , 1 0 0 0 , and consequently KJ= 2 0 0 0 , 1 0 0 0

= [E(S)]∼.

Thus, by Theorem8, the special subgroupS∼ is a quasi-ideal ofS.

Clearly,S∼ is neither prime nor weakly prime.

WhenS∼is a quasi-ideal, we have the following useful identification of the subsemigroupsξSand

Sξ as, respectively, theR-class and theL-class ofξ.

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(2) J =E(Lξ) and is a left zero semigroup;

(3) K =E(Rξ) and is a right zero semigroup;

(4) J∩K = {ξ}.

Proof.

(1) If xL_{ξ then} Sx ₌ Sξ _{whence there exists} y _∈ S _{such that} x ₌ yξ. Then x ₌ xξ _∈ Sξ_.

Conversely, ifx ∈ Sξ thenx = xξ gives, on the one handSx ⊆ Sξ, and on the other by Theorem

8(5), x∼x = x∼xξ = x∼ξxξ = x∼x∼∼ = ξ, whenceSξ ⊆ Sx. The resulting equality now gives

xLξ. ThusSξ =Lξ and similarlyξS=Rξ.

(2) and (3) follow immediately from(1)and Theorem3.

As for(4), it follows from(2)and(3)that

J∩K =E(Lξ)∩E(Rξ)=E(Lξ ∩Rξ)=E(Hξ)= {ξ}.

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Theorem 5 thatPis a regular subsemigroup ofS. Moreover, ifp∈ Pandx∈ Sthen, by Theorem

9(2),

px(px)∼_px=pp∼_px(px)∼_px=pp∼_px=px

whencePS⊆P. Similarly, by Theorem 9(3),SP⊆Pand consequentlyPis an ideal ofS.

Now since Theorem 8(5) holds, ifx∈ Pthenx∼ ∈V(x)soS∼∩V(x)6= ∅. But ify∈ S∼∩V(x)

theny ∈ S∼ ∩I(x) = {x∼}. Hence |S∼ ∩V(x)| = 1 and soS∼ is a group transversal of Pwith

x◦_=x∼_.

Corollary. If S∼ is a quasi-ideal then P is completely simple.

Proof. SinceS∼ is a group transversal ofP, this follows from a basic theorem of Saito (9); see also

(1, Theorem 4.3).

WhenS∼ is a quasi-ideal, the structure ofPcan be described in terms ofSξ = Lξ andξS = Rξ.

For this purpose, consider thespined product set

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(xx∼_ay)∼ _=(x∼_xx∼_ay)∼_x∼ _=(x∼_ay)∼_x∼ _=(ay)∼_x∼∼_x∼ _=(ay)∼_.

Similarly,(ayb∼b)∼ = (ay)∼. We can therefore define a law of composition onSξ|×|ξSby the

prescription

(x,a)(y,b)=(xx∼ay, ayb∼b).

Then

[(x,a)(y,b)](z,c) = (xx∼ay,ayb∼b)(z,c)

= (xx∼ay(xx∼ay)∼ayb∼bz, ayb∼bzc∼c)

= (xx∼ay(x∼xx∼ay)∼x∼ayb∼bz, ayb∼bzc∼c) by Theorem 2

= (xx∼ay·(x∼ay)∼x∼ay·b∼b·z, ayb∼bzc∼c) by Theorem 8(5)

= (xx∼ayb∼bz, ayb∼bzc∼c); by Theorem 9(3)

(x,a)[(y,b)(z,c)] = (x,a)(yy∼bz,bzc∼c)

= (xx∼ayy∼bz, ayy∼bz(bzc∼c)∼bzc∼c)

= (xx∼ayy∼bz, ayy∼bzc∼(bzc∼cc∼)∼bzc∼c) by Theorem 2

= (xx∼ayy∼bz, a·yy∼·bzc∼(bzc∼)∼ ·bzc∼c) by Theorem 8(5)

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show is isomorphic toP.

Theorem 11. If S∼ is a quasi-ideal then P=JS∼K and P≃Sξ|×|ξS=Lξ|×|Rξ.

Proof. For everyx∈Pwe havex =xx∼x= xx∼x∼∼x∼x∈ JS∼K so thatP⊆JS∼K. The reverse

inclusion is immediate from the fact thatJ, S, K ⊆PandPis a subsemigroup.

Consider now the mappingϑ :P→Sξ|×|ξSgiven byϑ (p)=(pξ,ξp).

(a) ϑ is injective.

Ifϑ (p)=ϑ (q)then, by Theorem2(1),

p=pp∼p=pξ(pξ )∼ξp=qξ(qξ )∼ξq=qq∼q=q.

(b) ϑ is surjective. Let(p,q)∈ Sξ|×|ξS. Thenp∼ =q∼ and, by Theorem8(5),

ϑ (pp∼_q)=(pp∼_qξ,ξpp∼_q)=(pp∼_q∼∼_,p∼∼_p∼_{q)=(pξ,ξq)=(p,q).}

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ϑ (p)ϑ (q)=(pξ,ξp)(qξ,ξq) = pξ(pξ )∼ξpqξ, ξpqξ(ξq)∼ξq

= (pp∼pqξ,ξpqq∼q)

= (pqξ,ξpq)

= ϑ (pq).

It follows by(a),(b),(c)thatϑis a semigroup isomorphism.

Example 7. In Example 6 we haveξ =

1 0 0 0 and Sξ = x 0 x 0 , x 0 0 0

|x6=0

, ξS=

x x 0 0 , x 0 0 0

|x6=0

.

In this example, as can readily be verified,Pcoincides with the subsemigroupAand Theorems10

and11apply.

5. S∼ _PRIME

Finally, we consider the special subgroups that are prime. In relation to the previous properties,

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Theorem 12. The following statements are equivalent:

(1) S∼ is prime;

(2) S∼ is weakly prime and a quasi-ideal of S;

(3) S∼ is a quasi-ideal of S andξ is a middle unit of P.

Proof.

(1)⇒ (2): If(1)holds thenKJ = {ξ}and consequently, by Theorem 6(5) and Theorem 8(2),S∼

is both weakly prime and a quasi-ideal.

(2)⇒ (3): IfS∼ is weakly prime and a quasi-ideal then, by Theorem8(5) and Theorem6(4), for

allx,y∈ P,

xy=xξx∼xyy∼ξy=x(x∼xyy∼)∼∼y=xξy

whenceξ is a middle unit ofP.

(3)⇒(1): IfS∼ is a quasi-ideal andξ is a middle unit ofPthen, by Theorem 9(2,3),

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Theorem 13. If S∼ is prime then

(1)JK =E(P)= {x∈ P|x∼ =ξ};

(2)P is orthodox.

Proof.

(1) Ifx=jk∈ JKthen, sinceJ, K ⊆Pand, by Theorem 12 and the Corollary to Theorem 6,Pis

a subsemigroup, we see thatx∈ P. Moreover, by Theorem9,xξ =jkξ = jξ =jandξx=ξjk=

ξk = k. Consequently, sinceξ is a middle unit ofPby Theorem12,x = jk = xξ ξx = x2. Thus

JK ⊆E(P).

Conversely, ife∈E(P)then, by Theorem6(4) and Theorem9,

e=ee∼e=eξe=eξ ξe∈JK

whenceE(P)⊆JKand we have equality.

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ef ∈JKJK =JξK =JK =E(P)

and soPis orthodox.

IfS∼is prime then, by the Corollary to Theorem 10,Pis a completely simple subsemigroup which,

by Theorem 13, is orthodox. In this case,Pcan therefore be expressed as the cartesian product of

a group and a rectangular band. The identification of these is the substance of the following.

Theorem 14. If S∼ is prime then J×K is a rectangular band and P≃J×S∼×K.

Proof. ThatJ×Kis a rectangular band is immediate from Theorem9(2,3).

Consider the mappingϑ :P→J×S∼×K given byϑ (x)=(xx∼, x∼∼, x∼x).

Since everyx∈Pis such thatx=xx∼x=xx∼·x∼∼·x∼x, it is clear thatϑ is injective.

To see that ϑ is also surjective, let (j,g,k) ∈ J ×S∼×K and consider the element jgk which,

by Theorem 11, belongs to P. By Theorem 9(2), jgk(jgk)∼ = jj∼· jgk(jgk)∼ = jj∼ = j, and

likewise, by Theorem 8(3),(jgk)∼jgk = k. Since, by Theorem 5,(jgk)∼∼ = g∼∼ = g it follows

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ϑ (x)ϑ (y) = (xx∼, x∼∼, x∼x)(yy∼, y∼∼, y∼y)

= (xx∼yy∼, x∼∼y∼∼, x∼xy∼y)

= (xx∼, (xy)∼∼, y∼y)

= (xx∼·xy(xy)∼, (xy)∼∼, (xy)∼xy·y∼y)

= (xy(xy)∼, (xy)∼∼, (xy)∼xy)

= ϑ (xy).

Consequently,ϑis a semigroup isomorphism.

Example 8. Consider the subsemigroupT = −N_×Z _{of the semigroup} S _{in Example 2. Here}

T∼ _{= {(0,n}

k)|n∈Z}withJ = {(0,n)|nk =0}andK= {(0, 0)}. ThenKJ = {(0, 0)}and soT∼

is prime. HereP= {(0,n)|n∈Z_{}and, since}K_{is trivial,}P_≃J_×T∼_.

Example 9. Similar to Example 4, let S = L× Gwhere L is a left zero semigroup and G is a

group. Here E(S) = L× {1G}and I(x,g) = L× {g−1}. For every idempotent ξ = (e, 1G) the

groupHξ = {e} ×Gis a special subgroup with(x,g)∼ =(e,g−1). Here(x,g)(x,g)∼ =(x, 1G)and

(x,g)∼_{(x,g)=(e, 1}

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FUNDING

This work was partially supported by the Portuguese Foundation for Science and Technology

through the grant UID/MAT/00297/2013 (CMA).

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