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

*Research Article*

**On Epiorthodox Semigroups**

**Shouxu Du,**

**1**

_{Xinzhai Xu,}

_{Xinzhai Xu,}

**2**

_{and K. P. Shum}

_{and K. P. Shum}

**3**

*1 _{Department of Basic Science, Qingdao Binhai University, Qingdao, Shandong 266555, China}*

*2 _{College of Mathematics Science, Shandong Normal University, Jinan, Shandong 250014, China}*

*3 _{The 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 semigroup*S*is an algebraic system*S,*·*,*≤in which the following
conditions are satisfied:1 *S,*·is a semigroup,2 *S,*≤is a poset, and3*a*≤*b*⇒*ax*≤*bx*
and *xa* ≤ *xb* for all*a, b* ∈ *S*. An ordered semigroup*S,*·*,*≤is said to satisfy the
Dubreil-Jacotin condition if there exists an isotone epimorphism which is a surjective homomorphism

*θ*from the semigroup*S*onto an ordered group*G*such that{*x*∈ *S* : *θx*2_{} _{≤} _{θ}_{}_{x}_{}_{}}_{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*

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

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 semigroup*S,*·*,*≤a naturally
ordered semigroup if

∀*e, f*∈*E* *ef*⇒*e*≤*f,* 1.1

where “” is a natural order on the subset*E*of*S*.

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 semigroup*S*is said to satisfy the strong Dubreil-Jacotin condition if there
exists an epimorphism*f*from*S*onto an ordered group*G*such that*f*:*S* → *G*is 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**

Let*S*be a naturally ordered regular semigroup. We first assume that every element*x*∈*S*has
the greatest inverse in*S*. Denote this element by*x*◦. Then, we call Green’s relationRon*S*the

*left regular relation ifx*≤ *y* ⇒ *xx*◦ ≤ *yy*◦, for all*x, y* ∈*S*. Similarly, Green’s relationLon a
semigroup*Sis called the right regular relation onS*if*x*≤*y*⇒*x*◦*x*≤*y*◦*y*, for all*x, y*∈*S*.

Consider an epiorthodox semigroup*S*with max{*x* ∈*S* : *θx*2_{}_{≤} _{θ}_{}_{x}_{}_{}} _{} _{ξ}_{. Because}

*S*is a regular semigroup, if*ξ* ∈ *Vξ*,*ξξ*is an idempotent then*ξξ* ≤ *ξ*in*S*. Consequently,
because the semigroup*S*satisfies the Dubreil-Jacotin condition, we have 1 *θξξ* ≤ *θξ*
and

*θξ*2_{≤}_{θ}_{}_{ξ}_{⇒}_{θ}_{}_{ξ}* _{θ}_{ξ}*2

_{ξ}_{≤}

_{θ}_{ξξ}_{}

_{1}

_{.}_{2.1}

Thus, 1≤ *θξ* ≤ 1 so that*θξ* 1, where 1 is the identity element of the group*G*. It hence
follows that*ξ* ∈ 1*θ*−1 _{and so the semigroup}_{S}_{has the greatest element}_{ξ}_{. 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 the*ξ*is the greatest idempotent of the
epiorthodox semigroup*S*.

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

* Lemma 2.1. LetSbe an epiorthodox semigroup in which max{x*∈

*S*:

*θx*2

_{}

_{≤}

_{θ}_{}

_{x}_{}

_{}}

_{}

_{ξ}_{. Then}1*ξis the greatest idempotent ofSand is a middle unit;*

2*the setEof idempotents ofSforms a normal band.*

*Proof. Part*1of 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 if*T*
is an ordered regular semigroup with the greatest idempotent*α*, then*T* is naturally ordered
if and only if*α*is a normal medial idempotentin the sense that*eαee*for all*e*∈*E*, where*E*
is the subsemigroup generated by*E*, and*αEα*is a semilattice. Since*S*is orthodox, we have
that*E* *E*. Also, since *S*is naturally ordered semigroup,*ξEξ* is a semilattice. The concept
of middle unit in an orthodox semigroup was first introduced by Blyth20. Because*ξ* is a
middle unit, for any*e, f, g,* and*h*in*E*, we have that

*efghe*·*ξfξ*·*ξ*·*he*·*ξgξ*·*ξfξ*·*hegfh.* 2.2

This shows that*E*is a normal band.

* Lemma 2.2. LetSbe an epiorthodox semigroup. Suppose that max{x*∈

*S*:

*θx*2≤

*θx*}

*ξinS.*

*Then the following properties hold:*

1*x*◦_{}_{ξx}_{ξ}_{is the greatest inverse of any}_{x}_{∈}_{S}_{, with}_{x}_{∈}_{V}_{}_{x}_{}_{;}

2*x*◦◦_{}_{ξxξ}_{, for every}_{x}_{∈}_{S}_{;}

3*ξx*◦ _{}* _{x}*◦

_{}

*◦*

_{x}

_{ξ}_{, for every}_{x}_{∈}

_{S}_{;}4*xyy*◦* _{x}*◦

_{xy}_{}

_{xy}_{}◦

_{}

_{xy}_{}

_{xy}_{}◦

_{, xy}_{}

_{xy}_{}◦

*◦*

_{xyy}*◦*

_{x}_{}

*◦*

_{xyy}*◦*

_{x}

_{, for all}_{x, y}_{∈}

_{S}_{;}5*xyy*◦* _{x}*◦

_{}

_{xy}_{}

_{xy}_{}◦

*◦*

_{, y}*◦*

_{x}

_{xy}

_{xy}_{}◦

_{xy}_{, for all}_{x, y}_{∈}

_{S}_{;}6 *xy*◦*y*◦* _{x}*◦

_{, for all}_{x, y}_{∈}

_{S}_{.}*Proof. By Lemma*2.1, it is known that*ξ*is the greatest idempotent of*S*. Since*S*is a naturally

if*e*R*f*for the idempotents*e, f*in*S*, then*efe*≤*fξ*and so*eξ*≤*fξ*. Similarly,*fξ*≤*eξ*. Thus,
from3, we deduce the equality*xyy*◦*x*◦ *xyxy*◦. Similarly, we can also prove*y*◦*x*◦*xy*
*xy*◦*xy*, and hence5holds. From5, we have that*xy*◦ *xy*◦*xyxy*◦*y*◦*x*◦*xyy*◦*x*◦

*y*◦* _{x}*◦

_{. This implies}

_{6}

_{holds.}

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

Suppose that*Y*is a semilattice and*Sα _{α∈Y}* is a family of pairwise disjoint semigroups.

For*α, β*∈*Y*with*α*≥*β*, let*ϕβ,α*:*Sα* → *Sβ*be a morphism satisfying the following conditions:

a ∀*α*∈*Yϕα,α*id*Sα*;

bif*α*≥*β*≥*γ*, then*ϕγ,βϕβ,αϕγ,α*.

Then, it is known that the set* _{α∈Y}Sα*under the following multiplication:

∀*x*∈*Sα*∀*y*∈*Sβ* *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}Sα* be a strong semilattice of semigroups. Suppose that each

*Sα*is an

ordered semigroup and that each of the structure maps*ϕβ,α*is isotone. Then the relation “”

defined on*S*by

∀*x*∈*Sα*∀*y*∈*Sβ* *xy*⇐⇒*α*≤*β, x*≤*ϕα,βy* 2.4

is a partial order on*S*, and so*S _{α∈Y}Sα*forms 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:

1*S _{α∈Y}Sα*is a strong semilattice of ordered semigroups;

2the semilattice*Y* has the greatest element;

3every ordered semigroup*Sα*has the greatest element.

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

**Lemma 2.4. Let**Sbe 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 Lemma*2.1, the set of idempotents*E*of the semigroup*S*is normal. By applying a

bands which can be regarded as the D-class of*E*. Clearly, the D-classes of*E* 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,*e*◦ *ξeξ*. Thus, if *e, f* ∈ Dthen *e*◦ *ξeξ* *ξfξ* *f*◦.
Conversely, if*ξeξξfξ*then

*ee*3_{}_{eξeξe}_{}_{eξfξe}_{}_{efe.}_{2.6}

Similarly, we can prove that*f* *fef*. Consequently, *e* ∈ *Vf*. This leads to*Ve* *Vf*,
whence*e, f*∈ D. Thus,Dis defined on*E*by

*e, f*∈ D ⇐⇒*e*◦_{}* _{f}*◦

_{.}_{2.7}

Hence, eachD-class has the greatest element and so*e*◦is the greatest element of*De*. Thus, the

structure semilattice of*E*is the set*Y* *ξEξ*{*e*◦:*e*∈*E*}which has the greatest element, that
is,*ξ*. Now if*e*◦*, f*◦∈*Y* with*e*◦≥*f*◦, then the structure map*ϕf*◦* _{,e}*◦:

*D*◦ →

_{e}*D*◦is given by

_{f}∀*x*∈*De*◦ *ϕ _{f}*◦

*◦*

_{,e}*x*

*xf*◦

*x.*2.8

These maps are clearly isotone. Observe that the order “” coincides with the order “≤” on
the semigroup*S*. In fact, if*x*∈*De*◦ and*y* ∈*D _{f}*◦satisfy the relation

*xy*, then

*e*◦ ≤

*f*◦and

*x*≤*ϕe*◦* _{,f}*◦

*y*

*ye*◦

*y*≤

*yf*◦

*yy*because

*D*◦is a rectangular band. Conversely, if

_{f}*x*≤

*y*then

*xx*3 _{}_{x}_{·}_{ξxξ}_{·}_{x}_{≤}_{y}_{·}_{ξxξ}_{·}_{y}_{}_{ϕ}

*ξxξ,ξyξy.* 2.9

This shows that*E*is a pointed semilattice of pointed rectangular bands.

*Remark 2.5. It can be easily seen that the structure mapϕf*◦* _{,e}*◦ :

*D*◦ →

_{e}*D*◦ preserves the

_{f}greatest element in order. In fact, we have that

*ϕf*◦* _{,e}*◦

*e*◦

*e*◦

*f*◦

*e*◦ ≥

*f*◦

*f*◦

*f*◦

*f*◦

*,*2.10

and whence,*ϕf*◦* _{,e}*◦

*e*◦

*f*◦since

*f*◦

*f*◦◦ max

*D*◦.

_{f}We have already proved in Lemma2.1that if*S*is an epiorthodox semigroup then the
band*E*of*S*is normal. Now, if we simply ignore the order on*S*, then*S*is 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 2.6**see22. Let*Sbe an inverse semigroup with a semilatticeEof idempotents ofS. Let*

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

*Then on the set*

*L*⊗*S*⊗*Re, x, f*:*x*∈*S, e*∈*Lxx*−1*, f* ∈*R _{x}*−1

*2.11*

_{x}*the multiplication given by*

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

*whereu*∈ *L _{xyxy}*−1

*andv*∈

*R*

_{xy}−1

*⊗*

_{xy}, is well defined andL*S*⊗

*Rforms 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. Let**Sbe a naturally ordered inverse semigroup satisfying the Dubreil-Jacotin condition.

*Suppose that Green’s relations*R*,*L*onSare, 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.*

i*L* _{α∈E}Lαis a pointed semilattice of pointed left zero semigroups, andR_{β∈E}Rβis

*also a pointed semilattice of pointed right zero semigroups.*

ii*LetL*⊗*S*⊗*R _{c}denote the set*

*L*⊗*S*⊗*Re, x, f*:*x*∈*S, e*∈*Lxx*−1*, f* ∈*R _{x}*−1

*2.13*

_{x}*equipped with the Cartesian order and the multiplication*

*e, x, fg, y, heL*∗

*xyxy*−1*, xy, R*∗_{xy}−1_{xy}h

*,* 2.14

*whereL*∗* _{α}is the greatest element ofLαandRα*∗

*is the greatest element ofRα. ThenL*⊗

*S*⊗

*Rcforms an epiorthodox semigroup on which Green’s relations*L*,*R*are, respectively, the*

*right and left regular relations.*

*Proof.* iSince 1*L*is a right identity for*L*, for*e, f*∈*L*,

*ef*⇒*efe*≤*f*1*Lf,* 2.15

and so*L*is naturally ordered. Since 1*L*is the greatest element of*L*,*L* satisfies the

that*e* *e*1*Le* *e*1*Lf* *ef*. Similarly,*R* is also a pointed semilattice of pointed right zero

semigroups. Recall from Lemma2.4that the order “” coincides with the order “≤” in both*L*
and*R*.

iiSuppose that *L*and*R* admit the structure decompositions*L* * _{α∈E}Lα*and

*R*

*β∈ERβ*, respectively. Observe that, for*α, β*∈*E*, we have that

*β*≤*α*⇒*L*∗

*β*≤*L*∗*α, R*∗*β*≤*R*∗*α.* 2.16

If *β* ≤ *α*, then since the structure mapping in *L* maps the greatest element to the greatest
element, *φβ,αL*∗*α* *L*∗*β*. Consequently, *L*∗*β* *L*∗*α* and by Lemma2.4,*L*∗*β* ≤ *L*∗*α*. Similarly, we

have that*R*∗* _{β}*≤

*R*∗

*α*. By applying the Yamada construction in Lemma2.6, now, we can see that

*L*⊗*S*⊗*Rc*is 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 that*L*⊗*S*⊗*R _{c}* forms
an ordered semigroup. At first, we let

*e, x, f*≤

*e*1

*, x*1

*, f*1. Then,

*x*≤

*x*1 and so

*xy*≤

*x*1

*y*for every

*y*∈

*S*. Since Ris a left regular relation on

*S, xyxy*−1 ≤

*x*1

*yx*1

*y*−1 and so by the above observation,

*L*∗

*xyxy*−1 ≤ *L*∗* _{x1yx1y}*−1. By applying the right regularity ofL, we can

similarly show that*R*∗

xy−1* _{xy}* ≤

*R*∗

_{x1y}−1

*. Thus, we obtain the following:*

_{x1y}

*e, x, fg, y, heL*∗

*xyxy*−1*, xy, R*∗_{xy}−1_{xy}h

≤*e*1*L*∗* _{x1yx1y}*−1

*, x*1

*y, R*∗

_{x1y}−1

_{x1y}h

*e*1*, x*1*, f*1

*g, y, h.*

2.17

By using similar arguments, we can show that

*g, y, he, x, f*≤*g, y, he*1*, x*1*, f*1

*,* 2.18

and so*L*⊗*S*⊗*R _{c}*forms an ordered semigroup.

Since each*Lα*is a left zero semigroup and each*Rα* is a right zero semigroup, we can

easily verify that the idempotents of*L*⊗*S*⊗*R _{c}*are the elements of the form

*e, x, f*, where

*x*∈*E*. Suppose that*e, x, f,g, y, h*are idempotents in*L*⊗*S*⊗*R _{c}*with

*e, x, fg, y, h*. Then

*e, x, f*

*e, x, fg, y, h*

*g, y, he, x, f*. This leads to

*xxyyx*and so

*xy*in

*E*. Since*S*is naturally ordered, we have that*x*≤ *y*. Also, we have*e* *gL*∗

*yxyx*−1 ≤ *g*1*L* *g*

and similarly,*f*≤*h*. This shows that*L*⊗*S*⊗*R _{c}*is naturally ordered.

Since*S*satisfies the Dubreil-Jacotin condition, there exists an ordered group*G*and an
isotone surjective homomorphism*θ*:*S* → *G*such that{*x*∈*S*:*θx*2_{}_{≤}_{θ}_{}_{x}_{}_{}}_{has the greatest}
element*ξ*. Define the mapping*ψ*:*L*⊗*S*⊗*R _{c}* →

*G*by

*ψe, x, f*

*θx*. Then,

*ψ*is an isotone surjective homomorphism.

We now proceed to show that

max *e, x, f*∈*L*⊗*S*⊗*Rc*:*ψe, x, f*2

Since1*L, ξ,*1*R*is an idempotent,

1*L, ξ,*1*R*∈ *e, x, f*∈*L*⊗*S*⊗*Rc*:*ψe, x, f*2

≤*ψe, x, f.* 2.20

On the other hand, we have that

*θx*2_{}* _{ψ}_{e, x, f}*2

_{≤}

_{ψ}_{e, x, f}_{}

_{θ}_{}

_{x}_{⇒}

_{x}_{≤}

_{ξ,}_{2.21}

and*e*≤1*L, f*≤1*R*are clear. Hence,*e, x, f*≤1*L, ξ,*1*R*, and so

max *e, x, f*∈*L*⊗*S*⊗*R _{c}*:

*ψe, x, f*2≤

*ψe, x, f*1

*L, ξ,*1

*R.*2.22

Consequently, we can see that1*L, ξ,*1*R*is the greatest idempotent of*L*⊗*S*⊗*Rc*. This shows

that*L*⊗*S*⊗*R _{c}*is indeed a semigroup satisfying the Dubreil-Jacotin condition.

Now, we have proved that *L* ⊗*S*⊗ *R _{c}* is an epiorthodox semigroup. Finally, we
consider Green’s relationRon the semigroup

*L*⊗

*S*⊗

*R*. We need to show that the relation

_{c}Ris a left regular relation on*L*⊗*S*⊗*R _{c}*. For this purpose, we need to identify

*e, x, f*◦. By Lemma2.2, we have that

*e, x, f*◦ 1

*L, ξ,*1

*Re, x, f*1

*L, ξ,*1

*R*, where

*e, x, f*∈

*Ve, x, f*.

We now show that*eL*∗* _{x}*−1

*−1*

_{x}, x*, g*∈

*Ve, x, f*. In fact, we can deduce the following equalities:

*e, x, feL*∗

*x*−1* _{x}, x*−1

*, g*

*e, x, feL*∗

*xx*−1*, xx*−1*, R*∗* _{xx}*−1

*g*

*e, x, f*

*e, xx*−1* _{, R}*∗

*xx*−1*g*

*e, x, f*

*eL*∗

*xx*−1*, x, R*∗* _{x}*−1

_{x}f*e, x, f,*

*eL*∗

*x*−1* _{x}, x*−1

*, g*

*e, x, feL*∗

*x*−1* _{x}, x*−1

*, g*

*eL*∗

*x*−1* _{x}, x*−1

*x, R*∗

*−1*

_{x}

_{x}f

*e, x*−1_{, g}

*eL*∗

*x*−1* _{x}L*∗

*−1*

_{x}*−1*

_{x}, x*, R*∗

*−1*

_{xx}*g*

*eL*∗

*x*−1* _{x}, x*−1

*, g*

*.*

2.23

So*eL*∗* _{x}*−1

*−1*

_{x}, x*, g*∈

*Ve, x, f*. Now, we have that

*e, x, f*◦ 1*L, ξ,*1*R*

*eL*∗

*x*−1* _{x}, x*−1

*, g*

Hence

*e, x, fe, x, f*◦*e, x, f*1*L, ξ,*1*R*

*eL*∗

*x*−1* _{x}, x*−1

*, g*

1*L, ξ,*1*R*

*eL*∗

*xx*−1*, x, R*∗* _{x}*−1

*1*

_{x}*R*

*eL*∗

*x*−1* _{x}L*∗

*−1*

_{x}*−1*

_{x}, x*, R*∗

*−11*

_{xx}*R*

*e, x, R*∗

*x*−1* _{x}*1

*R*

*eL*∗

*x*−1* _{x}, x*−1

*, R*∗

*−11*

_{xx}*R*

*eL*∗

*xx*−1*, xx*−1*, R*∗* _{xx}*−1

*R*∗

*−11*

_{xx}*R*

*e, xx*−1* _{, R}*∗

*xx*−11*R*

*.*

2.25

Consequently, we can deduce thatRis a left regular relation on*L*⊗*S*⊗*R _{c}*. Similarly,Lis a
right regular relation on

*R*⊗

*S*⊗

*Lc*.

We formulate the following lemma.

**Lemma 2.8. If***Y* *L*⊗ *S*⊗*Rc* *has a band of idempotents* *EY* *and containing the greatest*

*idempotentξ, then there are ordered semigroup isomorphisms*

*ξYξS,* *EYξL,* *ξEYR.* 2.26

*Proof. Sinceξ* 1*L,*1*,*1*R* *L*∗_{1}*,*1*, R*∗_{1}, it can be readily seen that

*ξe, x, fξL*∗

*xx*−1*, x, R*∗* _{x}*−1

_{x}*.* 2.27

The mapping

*θ*:*ξYξ*−→*S, θL*∗* _{xx}*−1

*, x, R*∗

*−1*

_{x}

_{x}*x* 2.28

is a semigroup isomorphism. Since Green’s relationsR*,*Lare, respectively, the left and right
regular relations on*S*,*θ*is an order isomorphism.

For*α, β*∈*E*, since the structure maps preserve the greatest elements, we have that

*L*∗

*αL*∗*βϕαβ,αL*∗*αϕαβ,β*

*L*∗
*β*

*L*∗

*αβL*∗*αβL*∗*αβ.* 2.29

Now*EY* {*e, α, f*;*α*∈*E*}, and

*e, x, fξe, x, fL*∗

1*,*1*, R*∗1

*eL*∗

*α, α, R*∗*αR*∗1

*e, α, R*∗

*α.* 2.30

Consider the mapping*ψ* :*EYξ* → *L, ψe, α, R*∗*α* *e*. Because

*e, α, R*∗

*α*

*g, β, R*∗
*β*

*eL*∗

*αβ, αβ, R*∗*αβR*∗*β*

*eL*∗

*αβ, αβ, R*∗*αβ*

and since*Lαβ*is a left zero semigroup, we can deduce that

*eL*∗

*αβϕαβ,αeϕαβ,αβ*

*L*∗
*αβ*

*ϕαβ,αeL*∗_{αβ}ϕαβ,αe,

*egϕαβ,αeϕαβ,βgϕαβ,αe.* 2.32

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

*Lα* and*g* ∈*Lβ*the equality*eg*implies that*αβ, ψ*is injective.*ψ*is clearly isotone. Finally,

if*e*≤*g*with*e*∈*Lα, g*∈*Lβ*, then*eg*and therefore*α*≤*β*by Lemma2.4. This shows that*ψ*

is an order isomorphism.

Similarly, we can prove that*ξEYR*.

**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 semigroup*S* 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 3.1**main theorem. Let*ξbe 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*
*relations*R*,*L*are, respectively, the left and right regular relations onT, thenT* *andEξ*⊗*ξTξ*⊗*ξE _{c}*

*are order isomorphic, that is,*

*T* *Eξ*⊗*ξTξ*⊗*ξEc.* 3.1

*Proof. Clearly,ξ*is the greatest element of*Eξ*and*ξ*is a right identity for*Eξ*. By Lemma2.1,

*E*is a normal band and hence*efghegfh*for all*e, f, g, h*∈*E*. Take*e, f, g* ∈*Eξ*and*hξ*.
Then*efgegf*for all*e, f, g* ∈*Eξ*because*ξ*is a right identity of*Eξ*. Thus,*Eξ*is a band since

*ξ*is a middle unit and so*Eξ*is a left normal band. Similarly,*ξE*is a right normal band with
the greatest element*ξ*which is a left identity. As for*ξTξ*, it is clear that it is a subsemigroup of

*T*and is regular because*T* itself is regular and*ξ*is a middle unit. If*x*∈*E*, then it is clear that

*ξxξ*∈*EξTξ*. Conversely, if*ξxξ*∈*EξTξ*, then*ξxξ* *ξxξ*·*ξxξξx*2_{ξ}_{. Let}_{x}_{∈}_{V}_{}_{x}_{. Then, we}

have that*xξxξxxξx*2_{ξx}_{and so}_{x}_{}* _{x}_{x}*2

_{x}_{. This leads to}

_{x}_{}

_{xx}_{x}_{}

*2*

_{xx}_{x}

_{x}_{x}_{}

*2*

_{x}_{and so}

*x*∈*E*. Consequently,*EξTξ* *ξEξ*. Clearly,*EξTξ*is a semilattice since*E*is a normal band.
Thus,

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

Hence, we have proved that*ξTξ*is an inverse subsemigroup of*T*.

the greatest element. Then*ϕξ* 1*G*and so*ϕ*|*ξTξ*:*ξTξ* → *G*is also an isotone epimorphism.

It can be easily verified that

max

*x*∈*ξTξ*:*ϕ _{ξTξ}x*2≤

*ϕ*

*ξTξx*

*ξ,* 3.3

and hence the Dubreil-Jacotin condition is satisfied on*ξTξ*. Moreover, since*T* is naturally
ordered, so is *ξTξ*. The structure semilattice of *Eξ* is *ξEξ*. This follows from the proof of
Lemma2.4. Similarly, the structure semilattice of*ξE*is*ξEξ*.

By Lemma2.7, we can construct an epiorthodox semigroup*Eξ*⊗*ξTξ*⊗*ξEc*. Now,

suppose that Green’s relationsR*,*Lare, respectively, the left and right regular relations on*T*.
Then we consider the following mapping:

*χ*:*T* −→*Eξ*⊗*ξTξ*⊗*ξE _{c}* 3.4

defined by

*χx* *xx*◦* _{, ξxξ, x}*◦

_{x}_{}

_{.}_{3.5}

Note that*χ*is well defined. On the one hand, by Lemma2.2, we can easily deduce that*xx*◦

*xx*◦_{ξ}_{∈}_{Eξ}_{and, similarly,}* _{x}*◦

_{x}_{∈}

_{ξE}_{; on the other hand, we have that}

*xx*◦ _{∈}_{L}_{ξxx}_{◦}_{ξ}_{}_{L}

*ξxξξxξ*◦ *L _{ξxξξxξ}*−1

*.*3.6

Because*ξ*is a middle unit,*xx*◦·*ξxξ*·*x*◦*xx*and so*χ*is injective.

To show that*χ*is also surjective, we first let*eξ, ξxξ, ξf* ∈*Eξ*⊗*ξTξ*⊗*ξE _{c}*. Then we
have that

*eξ*∈*L _{ξxξξxξ}*−1

*L*◦

_{ξxξξxξ}*L*◦

_{ξxx}*,*3.7

and hence*eξ, ξxx*◦∈ D. Therefore, by Lemma2.2, we have that

*ξeξ* *eξ*◦ *ξxx*◦_{}◦_{}* _{ξxx}*◦

_{.}_{3.8}

Similarly,*ξf* ∈*Rx*◦* _{xξ}*implies that

*ξfξx*◦

*xξ*. Now, consider

By the above observation, we have that

*exfexf*◦ *exff*◦* _{x}*◦

*◦*

_{e}*exξfξx*◦* _{e}*◦

*exx*◦* _{xξx}*◦

*◦*

_{e}*exx*◦* _{e}*◦

*eξxx*◦* _{e}*◦

*eξeξξeξ*

*eξ.*

3.10

Similarly,*exf*◦*exfξf*and

*ξexfξξeξxξfξξxx*◦* _{xx}*◦

_{xξ}_{}

_{ξxξ.}_{3.11}

Thus

*χexfeξ, ξxξ, ξf,* 3.12

and so we have proved that*χ*is a surjective mapping.
We now deduce that

*χxχy* *xx*◦* _{, ξxξ, x}*◦

_{x}_{}

*◦*

_{yy}*◦*

_{, ξyξ, y}

_{y}

*xx*◦* _{L}*∗

*ξxyξξxyξ*−1*, ξxyξ, R*
∗

*ξxyξ*−1*ξxyξy*

◦_{y}

*xx*◦* _{L}*∗

*ξxyy*◦* _{x}*◦

*, ξxyξ, R*∗

*◦*

_{y}*◦*

_{x}*◦*

_{xyξ}y*y*

*xx*◦* _{ξxyy}*◦

*◦*

_{x}*◦*

_{ξ, ξxyξ, ξy}*◦*

_{x}*◦*

_{xyξy}

_{y}*xyy*◦* _{x}*◦

*◦*

_{, ξxyξ, y}*◦*

_{x}

_{xy}*xyxy*◦*, ξxyξ,xy*◦*xy*

*χxy.*

3.13

Thus,*χ*is a semigroup isomorphism.

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

*T*,*χ*is isotone; and since

*χx*≤*χy*⇒*xx*◦ _{≤}* _{yy}*◦

_{, ξxξ}_{≤}

*◦*

_{ξyξ, x}

_{x}_{≤}

*◦*

_{y}

_{y}it follows that*χ*is an order isomorphism between the ordered semigroups*T*and*Eξ*⊗*ξTξ*⊗

*ξEc*, that is,

*T* *Eξ*⊗*ξTξ*⊗*ξEc.* 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* *Eξ*⊗*ξTξ*⊗

*ξEc*:

1 *e, x, f*R*T*_{}_{g, y, h}_{}_{⇔}_{x}_{R}*ξTξ _{y, e}*

_{}

_{g}_{;}

2 *e, x, f*L*T*_{}_{g, y, h}_{}_{⇔}_{x}_{L}*ξTξ _{y, f}*

_{}

_{h}_{;}

3 *e, x, f*D*T*_{}_{g, y, h}_{}_{⇔}_{x}_{D}*ξTξ _{y}*

_{.}

*Proof. We first deduce the following implications:*

*e, x, f*R*T _{g, y, h}*

_{⇐⇒}

*◦*

_{e, x, f}_{e, x, f}_{}

*◦*

_{g, y, h}_{g, y, h}⇐⇒*e, xx*−1* _{, R}*∗

*xx*−11*R*

*g, yy*−1* _{, R}*∗

*yy*−11*R*

⇐⇒*eg, xx*−1 _{}* _{yy}*−1

⇐⇒*eg, x*R*ξTξ _{y.}*

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.

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