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Volume 2011, Article ID 181707,14pages doi:10.1155/2011/181707

Research Article

On Epiorthodox Semigroups

Shouxu Du,

1

Xinzhai Xu,

2

and K. P. Shum

3

1Department of Basic Science, Qingdao Binhai University, Qingdao, Shandong 266555, China

2College of Mathematics Science, Shandong Normal University, Jinan, Shandong 250014, China

3The Institute of Mathematics, Yunnan University, Kunming 650091, China

Correspondence should be addressed to K. P. Shum,kpshum@ynu.edu.cn

Received 25 December 2010; Accepted 15 April 2011

Academic Editor: Aloys Krieg

Copyrightq2011 Shouxu Du et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It has been well known that the band of idempotents of a naturally ordered orthodox semigroup satisfying the “strong Dubreil-Jacotin condition” forms a normal band. In the literature, the naturally ordered orthodox semigroups satisfying the strong Dubreil-Jacotin condition were first considered by Blyth and Almeida Santos in 1992. Based on the name “epigroup” in the paper of Blyth and Almeida Santos and also the name “epigroups” proposed by Shevrin in 1955; we now call the naturally ordered orthodox semigroups satisfying the Dubreil-Jacotin condition the

epiorthodox semigroups. Because the structure of this kind of orthodox semigroups has not yet been

described, we therefore give a structure theorem for the epi-orthodox semigroups.

1. Introduction

We recall that an ordered semigroupSis an algebraic systemS,·,≤in which the following conditions are satisfied:1 S,·is a semigroup,2 S,≤is a poset, and3abaxbx and xaxb for alla, bS. An ordered semigroupS,·,≤is said to satisfy the Dubreil-Jacotin condition if there exists an isotone epimorphism which is a surjective homomorphism

θfrom the semigroupSonto an ordered groupGsuch that{xS : θx2 θx}has the greatest element. This kind of ordered semigroups was first studied by Dubreil and Jacotin. We notice that Blyth and Giraldes1first investigated the perfect elements of Dubreil-Jacotin regular semigroups in 1992. In this paper, we call the naturally ordered orthodox semigroups satisfying the Dubreil-Jacotin condition the “epiorthodox semigroups.” Recall that a semigroup

Sis called an orthodox semigroup if the set of its idempotentsEforms a subsemigroup of the semigroup S see 2, 3. It is well known in the theory of semigroups that the class

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Shum et al. in 4–6. The so-called super R∗-unipotent semigroups have been particularly studied by Ren et al. in6. In this paper, we call an ordered semigroupS,·,≤a naturally ordered semigroup if

e, fE efef, 1.1

where “” is a natural order on the subsetEofS.

We notice here that the class of naturally ordered regular semigroups with the greatest idempotent was first considered by Blyth and McFadden7and Blyth and Almeida Santos in 1992 8. A well-known generalized class of regular semigroups is the class of rpp

semigroups. For rpp semigroups and their generalizations, the reader is referred to9. It

was observed by McAlister10that each element in a naturally ordered regular semigroup with the greatest idempotent has the greatest inverse.

An ordered semigroupSis said to satisfy the strong Dubreil-Jacotin condition if there exists an epimorphismffromSonto an ordered groupGsuch thatf:SGis residuated in the sense that the preimage under f of every principal order ideal of G is a principal order ideal of S. The class of orthodox semigroups which are naturally ordered satisfying the strong Dubreil-Jacotin condition was first studied by Blyth and Almeida Santos in 1992 see8,11. In their paper8, they first named a naturally ordered semigroup satisfying the

strong Dubreil-Jacotin condition an “epigroup.” However, a semigroup was also called an “epigroup” by Shevrin since 1955see12,13. An epigroup means a semigroup in which

some power of each of its element lies in a subgroup of a given semigroup. Thus, an epigroup can be regarded as a unary semigroup with the unary operation of pseudoinversion see the articles of Shevrin14,15for more information of epigroups. We emphasize here that

the concept of epigroups initiated by Shevrin is quite different from the naturally ordered semigroup satisfying the strong Dubreil-Jacotin condition described by Blyth and Almeida Santos. For the lattice properties of epigroups, the readers are referred to the recent articles of Shevrin and Ovsyannikov in 200816, 17. In this paper, our purpose is to establish a

structure theorem of an epiorthodox semigroup. Concerning the regular semigroups and their generalizations, the reader is referred to9,18. For other notations and terminologies

not mentioned in this paper, the reader is referred to Shum and Guo19and Howie18.

Throughout this paper, following the terminology “epigroups” proposed by Shevrin and Blyth and Almeida Santos, we call an orthodox semigroup which is naturally ordered satisfying the Dubreil-Jacotin condition an “epiorthodox semigroup.”

2. Preliminaries

LetSbe a naturally ordered regular semigroup. We first assume that every elementxShas the greatest inverse inS. Denote this element byx◦. Then, we call Green’s relationRonSthe

left regular relation ifxyxx◦ ≤ yy◦, for allx, yS. Similarly, Green’s relationLon a semigroupSis called the right regular relation onSifxyxxyy, for allx, yS.

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Consider an epiorthodox semigroupSwith max{xS : θx2 θx} ξ. Because

Sis a regular semigroup, ifξ,ξξis an idempotent thenξξξinS. Consequently, because the semigroupSsatisfies the Dubreil-Jacotin condition, we have 1 θξξθξ and

θξ2θξ θξ θξ2ξθξξ1. 2.1

Thus, 1≤ θξ ≤ 1 so thatθξ 1, where 1 is the identity element of the groupG. It hence follows thatξ ∈ 1θ−1 and so the semigroupShas the greatest elementξ. By Lemma 1.7 in 10, McAlister noticed that if an ordered regular semigroup has the greatest element then its

greatest element must be an idempotent. It follows that theξis the greatest idempotent of the epiorthodox semigroupS.

In view of the above results, we have the following lemma.

Lemma 2.1. LetSbe an epiorthodox semigroup in which max{xS:θx2θx}ξ. Then

1ξis the greatest idempotent ofSand is a middle unit;

2the setEof idempotents ofSforms a normal band.

Proof. Part1of the above lemma follows easily from observation. To prove part2of the

lemma, we first recall a result of Blyth and Almeida Santos11 see11, Theorem 2that ifT is an ordered regular semigroup with the greatest idempotentα, thenT is naturally ordered if and only ifαis a normal medial idempotentin the sense thateαeefor alleE, whereE is the subsemigroup generated byE, andαEαis a semilattice. SinceSis orthodox, we have thatE E. Also, since Sis naturally ordered semigroup,ξEξ is a semilattice. The concept of middle unit in an orthodox semigroup was first introduced by Blyth20. Becauseξ is a middle unit, for anye, f, g, andhinE, we have that

efghe·ξfξ·ξ·he·ξgξ·ξfξ·hegfh. 2.2

This shows thatEis a normal band.

Lemma 2.2. LetSbe an epiorthodox semigroup. Suppose that max{xS:θx2≤θx}ξinS.

Then the following properties hold:

1xξxξis the greatest inverse of anyxS, withxVx;

2x◦◦ξxξ, for everyxS;

3ξxxxξ, for everyxS;

4xyyxxyxyxyxy, xyxyxyyxxyyx, for allx, yS;

5xyyxxyxy, yxxy xyxy, for allx, yS;

6 xyyx, for allx, yS.

Proof. By Lemma2.1, it is known thatξis the greatest idempotent ofS. SinceSis a naturally

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ifeRffor the idempotentse, finS, thenefeand so. Similarly,. Thus, from3, we deduce the equalityxyyxxyxy◦. Similarly, we can also proveyxxy xyxy, and hence5holds. From5, we have thatxyxyxyxyyxxyyx

yx. This implies6holds.

In order to establish a structure theorem for an epiorthodox semigroup, we restate here the notion of strong semilattice of ordered semigroups.

Suppose thatYis a semilattice andα∈Y is a family of pairwise disjoint semigroups.

Forα, βYwithαβ, letϕβ,α:be a morphism satisfying the following conditions:

a ∀αYϕα,αid;

bifαβγ, thenϕγ,βϕβ,αϕγ,α.

Then, it is known that the setα∈Yunder the following multiplication:

xy xyϕαβ,αxϕαβ,βy 2.3

forms a semigroup which is called the only strong semilattice of semigroups.

By using the strong semilattices of semigroups, Blyth and Almeida Santos8 estab-lished the following result.

Let S α∈Y be a strong semilattice of semigroups. Suppose that each is an

ordered semigroup and that each of the structure mapsϕβ,αis isotone. Then the relation “”

defined onSby

xy xy⇐⇒αβ, xϕα,βy 2.4

is a partial order onS, and soSα∈Yforms an ordered semigroup.

We now call an ordered semigroup constructed in the above manner a strong semilattice

of ordered semigroups.

The following definition of “pointed semilattice of pointed semigroups” was given by Blyth and Almeida Santos8.

Definition 2.3. An ordered semigroup S is said to be a pointed semilattice of pointed

semigroups if the following conditions are satisfied:

1Sα∈Yis a strong semilattice of ordered semigroups;

2the semilatticeY has the greatest element;

3every ordered semigrouphas the greatest element.

By the above definition and the notion of strong semilattice of ordered semigroups, we have the following lemma.

Lemma 2.4. LetSbe an epiorthodox semigroup. Then the bandEof idempotents ofSis a pointed

semilattice of pointed rectangular bands on which the order “” coincides with the order “≤” onS.

Proof. By Lemma2.1, the set of idempotentsEof the semigroupSis normal. By applying a

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bands which can be regarded as the D-class ofE. Clearly, the D-classes ofE are the same classes as theY-classes, whereYis the finest inverse semigroup congruence given by

e, f∈ Y ⇐⇒Ve Vf. 2.5

By Lemma2.21, we know that,eξeξ. Thus, if e, f ∈ Dthen eξeξ ξfξ f◦. Conversely, ifξeξξfξthen

ee3eξeξeeξfξeefe. 2.6

Similarly, we can prove thatf fef. Consequently, eVf. This leads toVe Vf, whencee, f∈ D. Thus,Dis defined onEby

e, f∈ D ⇐⇒ef. 2.7

Hence, eachD-class has the greatest element and soe◦is the greatest element ofDe. Thus, the

structure semilattice ofEis the setY ξEξ{e◦:eE}which has the greatest element, that is,ξ. Now ife, f◦∈Y withe◦≥f◦, then the structure mapϕf,e◦:De◦ → Df◦is given by

xDeϕf,ex xfx. 2.8

These maps are clearly isotone. Observe that the order “” coincides with the order “≤” on the semigroupS. In fact, ifxDe◦ andyDf◦satisfy the relationxy, thene◦ ≤ f◦and

xϕe,fy yeyyfyybecauseDf◦is a rectangular band. Conversely, ifxythen

xx3 x·ξxξ·xy·ξxξ·yϕ

ξxξ,ξyξy. 2.9

This shows thatEis a pointed semilattice of pointed rectangular bands.

Remark 2.5. It can be easily seen that the structure mapϕf,e◦ : De◦ → Df◦ preserves the

greatest element in order. In fact, we have that

ϕf,eeefe◦ ≥ffff, 2.10

and whence,ϕf,eef◦sinceff◦◦ maxDf◦.

We have already proved in Lemma2.1that ifSis an epiorthodox semigroup then the bandEofSis normal. Now, if we simply ignore the order onS, thenSis isomorphic to the quasidirect product of a left normal band, an inverse semigroup, and a right normal band. In studying the regular semigroups whose idempotents satisfy some permutation identities, Yamada established an important result in22. To be more precise, we state the following

lemma.

Lemma 2.6see22. LetSbe an inverse semigroup with a semilatticeEof idempotents ofS. Let

LandRbe, respectively, a left normal band and a right normal band with a structural decomposition

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Then on the set

LSRe, x, f:xS, eLxx−1, fRx−1x 2.11

the multiplication given by

e, x, fg, y, heu, xy, vh, 2.12

whereuLxyxy−1andvRxy−1xy, is well defined andLSRforms an orthodox semigroup

with a normal band of idempotents. Conversely, every such semigroup can be constructed in the above manner.

In order to establish a structure theorem for the epiorthodox semigroups, we need to find some suitable conditions satisfying the requirements of the construction method given by Yamada in2so that an epiorthodox semigroup can be so constructed.

We formulate the following Lemma.

Lemma 2.7. LetSbe a naturally ordered inverse semigroup satisfying the Dubreil-Jacotin condition.

Suppose that Green’s relationsR,LonSare, respectively, the left and right regular relations onS. Let

Lbe an ordered left normal band with the greatest element 1Lwhich is a right identity, and letRbe

an ordered right normal band with the greatest element 1Rwhich is a left identity. Then the following

statements hold.

iL α∈ELαis a pointed semilattice of pointed left zero semigroups, andR β∈ERβis

also a pointed semilattice of pointed right zero semigroups.

iiLetLSRcdenote the set

LSRe, x, f:xS, eLxx−1, fRx−1x 2.13

equipped with the Cartesian order and the multiplication

e, x, fg, y, heL

xyxy−1, xy, Rxy−1xyh

, 2.14

whereLαis the greatest element ofLαandRαis the greatest element ofRα. ThenLS

Rcforms an epiorthodox semigroup on which Green’s relationsL,Rare, respectively, the

right and left regular relations.

Proof. iSince 1Lis a right identity forL, fore, fL,

efefef1Lf, 2.15

and soLis naturally ordered. Since 1Lis the greatest element ofL,L satisfies the

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thate e1Le e1Lf ef. Similarly,R is also a pointed semilattice of pointed right zero

semigroups. Recall from Lemma2.4that the order “” coincides with the order “≤” in bothL andR.

iiSuppose that LandR admit the structure decompositionsL α∈Eand R

β∈ERβ, respectively. Observe that, forα, βE, we have that

βαL

βLα, RβRα. 2.16

If βα, then since the structure mapping in L maps the greatest element to the greatest element, φβ,αLα Lβ. Consequently, Lβ Lα and by Lemma2.4,LβLα. Similarly, we

have thatRβRα. By applying the Yamada construction in Lemma2.6, now, we can see that

LSRcis an orthodox semigroup. Thus, under the Cartesian order and the left regularity

ofR and the right regular regularity of Lon S, we can easily see thatLSRc forms an ordered semigroup. At first, we lete, x, fe1, x1, f1. Then,xx1 and soxyx1y for everyyS. Since Ris a left regular relation onS, xyxy−1 ≤ x1yx1y−1 and so by the above observation,L

xyxy−1 ≤ Lx1yx1y−1. By applying the right regularity ofL, we can

similarly show thatR

xy−1xyRx1y−1x1y. Thus, we obtain the following:

e, x, fg, y, heL

xyxy−1, xy, Rxy−1xyh

e1Lx1yx1y−1, x1y, Rx1y−1x1yh

e1, x1, f1

g, y, h.

2.17

By using similar arguments, we can show that

g, y, he, x, fg, y, he1, x1, f1

, 2.18

and soLSRcforms an ordered semigroup.

Since eachis a left zero semigroup and each is a right zero semigroup, we can

easily verify that the idempotents ofLSRcare the elements of the forme, x, f, where

xE. Suppose thate, x, f,g, y, hare idempotents inLSRcwithe, x, fg, y, h. Thene, x, f e, x, fg, y, h g, y, he, x, f. This leads toxxyyxand soxyin

E. SinceSis naturally ordered, we have thatxy. Also, we havee gL

yxyx−1 ≤ g1L g

and similarly,fh. This shows thatLSRcis naturally ordered.

SinceSsatisfies the Dubreil-Jacotin condition, there exists an ordered groupGand an isotone surjective homomorphismθ:SGsuch that{xS:θx2θx}has the greatest elementξ. Define the mappingψ:LSRcGbyψe, x, f θx. Then,ψis an isotone surjective homomorphism.

We now proceed to show that

max e, x, fLSRc:ψe, x, f2

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Since1L, ξ,1Ris an idempotent,

1L, ξ,1Re, x, fLSRc:ψe, x, f2

ψe, x, f. 2.20

On the other hand, we have that

θx2ψe, x, f2ψe, x, fθx xξ, 2.21

ande≤1L, f≤1Rare clear. Hence,e, x, f≤1L, ξ,1R, and so

max e, x, fLSRc:ψe, x, f2≤ψe, x, f 1L, ξ,1R. 2.22

Consequently, we can see that1L, ξ,1Ris the greatest idempotent ofLSRc. This shows

thatLSRcis indeed a semigroup satisfying the Dubreil-Jacotin condition.

Now, we have proved that LSRc is an epiorthodox semigroup. Finally, we consider Green’s relationRon the semigroupLSRc. We need to show that the relation

Ris a left regular relation onLSRc. For this purpose, we need to identifye, x, f◦. By Lemma2.2, we have thate, x, f◦ 1L, ξ,1Re, x, f1L, ξ,1R, wheree, x, fVe, x, f.

We now show thateLx−1x, x−1, gVe, x, f. In fact, we can deduce the following equalities:

e, x, feL

x−1x, x−1, g

e, x, feL

xx−1, xx−1, Rxx−1g

e, x, f

e, xx−1, R

xx−1g

e, x, f

eL

xx−1, x, Rx−1xf

e, x, f,

eL

x−1x, x−1, g

e, x, feL

x−1x, x−1, g

eL

x−1x, x−1x, Rx−1xf

e, x−1, g

eL

x−1xLx−1x, x−1, Rxx−1g

eL

x−1x, x−1, g

.

2.23

SoeLx−1x, x−1, gVe, x, f. Now, we have that

e, x, f◦ 1L, ξ,1R

eL

x−1x, x−1, g

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Hence

e, x, fe, x, fe, x, f1L, ξ,1R

eL

x−1x, x−1, g

1L, ξ,1R

eL

xx−1, x, Rx−1x1R

eL

x−1xLx−1x, x−1, Rxx−11R

e, x, R

x−1x1R

eL

x−1x, x−1, Rxx−11R

eL

xx−1, xx−1, Rxx−1Rxx−11R

e, xx−1, R

xx−11R

.

2.25

Consequently, we can deduce thatRis a left regular relation onLSRc. Similarly,Lis a right regular relation onRSLc.

We formulate the following lemma.

Lemma 2.8. If Y LSRc has a band of idempotents EY and containing the greatest

idempotentξ, then there are ordered semigroup isomorphisms

ξYξS, EYξL, ξEYR. 2.26

Proof. Sinceξ 1L,1,1R L1,1, R1, it can be readily seen that

ξe, x, fξL

xx−1, x, Rx−1x

. 2.27

The mapping

θ:ξYξ−→S, θLxx−1, x, Rx−1x

x 2.28

is a semigroup isomorphism. Since Green’s relationsR,Lare, respectively, the left and right regular relations onS,θis an order isomorphism.

Forα, βE, since the structure maps preserve the greatest elements, we have that

L

αLβϕαβ,αLαϕαβ,β

Lβ

L

αβLαβLαβ. 2.29

NowEY {e, α, f;αE}, and

e, x, fξe, x, fL

1,1, R∗1

eL

α, α, RαR∗1

e, α, R

α. 2.30

Consider the mappingψ :EYξL, ψe, α, Rα e. Because

e, α, R

α

g, β, Rβ

eL

αβ, αβ, RαβRβ

eL

αβ, αβ, Rαβ

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and sinceLαβis a left zero semigroup, we can deduce that

eL

αβϕαβ,αeϕαβ,αβ

Lαβ

ϕαβ,αeLαβϕαβ,αe,

egϕαβ,αeϕαβ,βgϕαβ,αe. 2.32

Thus,ψ is a semigroup homomorphism. Obviously,ψ is surjective. Since for arbitrary e

andgthe equalityegimplies thatαβ, ψis injective.ψis clearly isotone. Finally,

ifegwitheLα, g, thenegand thereforeαβby Lemma2.4. This shows thatψ

is an order isomorphism.

Similarly, we can prove thatξEYR.

3. Main Theorem

In this section, we will give a structure theorem for the epiorthodox semigroups. We establish the converse of Lemma2.7by showing every epiorthodox semigroupS on which Green’s relationsR,Lare, respectively, the left and right regular relations which arise in the way as stated in Lemma2.7. Our Lemma2.8indicates how this goal can be achieved.

Theorem 3.1main theorem. Letξbe the greatest idempotent of an epiorthodox semigroupTand

Ethe band of idempotents ofT. ThenEξ is an ordered left normal band with the greatest element

which is a right identity, andξEis an ordered right normal band with the greatest element which is a left identity. Moreover,ξTξis a naturally ordered inverse semigroup satisfying the Dubreil-Jacotin condition, and its semilattice of idempotentsξEξis the structure semilattice ofEξand ofξE. If Green’s relationsR,Lare, respectively, the left and right regular relations onT, thenT andEξξTξξEc are order isomorphic, that is,

T ξTξξEc. 3.1

Proof. Clearly,ξis the greatest element ofandξis a right identity for. By Lemma2.1,

Eis a normal band and henceefghegfhfor alle, f, g, hE. Takee, f, gand. Thenefgegffor alle, f, gbecauseξis a right identity of. Thus,is a band since

ξis a middle unit and sois a left normal band. Similarly,ξEis a right normal band with the greatest elementξwhich is a left identity. As forξTξ, it is clear that it is a subsemigroup of

Tand is regular becauseT itself is regular andξis a middle unit. IfxE, then it is clear that

ξxξEξTξ. Conversely, ifξxξEξTξ, thenξxξ ξxξ·ξxξξx2ξ. LetxVx. Then, we

have thatxξxξxxξx2ξxand soxxx2x. This leads toxxxxxxx2xxx2and so

xE. Consequently,EξTξ ξEξ. Clearly,EξTξis a semilattice sinceEis a normal band. Thus,

ξeξ·ξfξξ·ξeξ·ξfξ·ξξ·ξfξ·ξeξ·ξξfξ·ξeξ. 3.2

Hence, we have proved thatξTξis an inverse subsemigroup ofT.

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the greatest element. Thenϕξ 1Gand soϕ|ξTξ:ξTξGis also an isotone epimorphism.

It can be easily verified that

max

xξTξ:ϕξTξx2≤ϕ

ξTξx

ξ, 3.3

and hence the Dubreil-Jacotin condition is satisfied onξTξ. Moreover, sinceT is naturally ordered, so is ξTξ. The structure semilattice of is ξEξ. This follows from the proof of Lemma2.4. Similarly, the structure semilattice ofξEisξEξ.

By Lemma2.7, we can construct an epiorthodox semigroupξTξξEc. Now,

suppose that Green’s relationsR,Lare, respectively, the left and right regular relations onT. Then we consider the following mapping:

χ:T −→ξTξξEc 3.4

defined by

χx xx, ξxξ, xx. 3.5

Note thatχis well defined. On the one hand, by Lemma2.2, we can easily deduce thatxx

xxξand, similarly,xxξE; on the other hand, we have that

xxLξxxξL

ξxξξxξLξxξξxξ−1. 3.6

Becauseξis a middle unit,xx◦·ξxξ·xxxand soχis injective.

To show thatχis also surjective, we first leteξ, ξxξ, ξfξTξξEc. Then we have that

Lξxξξxξ−1 LξxξξxξLξxx, 3.7

and henceeξ, ξxx◦∈ D. Therefore, by Lemma2.2, we have that

ξeξ ξxxξxx. 3.8

Similarly,ξfRximplies thatξfξx. Now, consider

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By the above observation, we have that

exfexfexffxe

exξfξxe

exxxξxe

exxe

eξxxe

eξeξξeξ

eξ.

3.10

Similarly,exfexfξfand

ξexfξξeξxξfξξxxxxξxξ. 3.11

Thus

χexfeξ, ξxξ, ξf, 3.12

and so we have proved thatχis a surjective mapping. We now deduce that

χxχy xx, ξxξ, xxyy, ξyξ, yy

xxL

ξxyξξxyξ−1, ξxyξ, R

ξxyξ−1ξxyξy

y

xxL

ξxyyx, ξxyξ, Ryxxyξyy

xxξxyyxξ, ξxyξ, ξyxxyξyy

xyyx, ξxyξ, yxxy

xyxy, ξxyξ,xyxy

χxy.

3.13

Thus,χis a semigroup isomorphism.

Since Green’s relationsR,Lare, respectively, the left and the right regular relations on

T,χis isotone; and since

χxχyxxyy, ξxξξyξ, xxyy

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it follows thatχis an order isomorphism between the ordered semigroupsTandξTξ

ξEc, that is,

T ξTξξEc. 3.15

The proof is completed.

We conclude the above results by the following remark.

Remark 3.2. It is noted that the following statements hold on the semigroupT ξTξ

ξEc:

1 e, x, fRTg, y, hxRξTξy, eg;

2 e, x, fLTg, y, hxLξTξy, f h;

3 e, x, fDTg, y, hxDξTξy.

Proof. We first deduce the following implications:

e, x, fRTg, y, h⇐⇒e, x, fe, x, fg, y, hg, y, h

⇐⇒e, xx−1, R

xx−11R

g, yy−1, R

yy−11R

⇐⇒eg, xx−1 yy−1

⇐⇒eg, xRξTξy.

3.16

Hence1holds. As2is the dual of1,2holds. It follows from1and2that3 holds.

Acknowledgments

The authors would like to thank the referees for giving them many valuable opinions and comments to this paper. The research of X. Xu is supported by an award of Young Scientist fund from Shandong Province2008BS01016, China.

References

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