# On Epiorthodox Semigroups

Authors: Shouxu Du; Xinzhai Xu; K. P. Shum
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

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
theepiorthodox 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.


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## Body

## 1. Introduction

We recall that an ordered semigroupS is an algebraic system (S,·,≤) in which the following conditions are satisfied: (1) (S,·) is a semigroup, (2) (S,≤) is a poset, and (3) 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:θ(x2)≤θ(x)} has the greatest element. This kind of ordered semigroups was first studied by Dubreil and Jacotin. We notice that Blyth and Giraldes [1] first 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 of orthodox semigroups played an important role in the class of regular semigroups. The structure of some special orthodox semigroup has been investigated and studied by Ren, Shum  et al. in [4–6]. The so-called super ℛ*-unipotent semigroups have been particularly studied by Ren et al. in [6]. In this paper, we call an ordered semigroup (S,·,≤) a naturally ordered semigroup if (1.1)(∀e,f∈E)e⪯f⟹e≤f,
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 McFadden [7] and 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 to [9]. It was observed by McAlister [10] that each element in a naturally ordered regular semigroup with the greatest idempotent has the greatest inverse.An ordered semigroupS 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 (see [8, 11]). In their paper [8], 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 1955 (see [12, 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   Shevrin [14, 15] for 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 2008 [16, 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 to [9, 18]. For other notations and terminologies not mentioned in this paper, the reader is referred to Shum and Guo [19] and Howie [18].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

LetS 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 relation ℛ on S the left  regular relation if x≤y⇒xx∘≤yy∘, for all x,y∈S. Similarly, Green's relation ℒ on a semigroup S is called the right  regular relation on S if x≤y⇒x∘x≤y∘y, for all x,y∈S.It was shown by McAlister [10] (see [10, Proposition 1.9]) that if S is an ordered regular semigroup with the greatest idempotent u, then S is a naturally ordered orthodox semigroup if and only if u is a middle unit, that is, xy=xuy for all x,y∈S. In addition, it has been stated in [18] that if u is a middle unit then every x∈S has the greatest inverse, say, x∘=ux′u for every inverse element x′ of x.Consider an epiorthodox semigroupS with max{x∈S:θ(x2)≤θ(x)}=ξ. Because S is a regular semigroup, if ξ′∈V(ξ), ξξ′ is an idempotent then ξξ′≤ξ in S. Consequently, because the semigroup S satisfies the Dubreil-Jacotin condition, we have 1=θ(ξξ′)≤θ(ξ) and (2.1)θ(ξ2)≤θ(ξ)⟹θ(ξ)=θ(ξ2ξ′)≤θ(ξξ′)=1.
Thus, 1≤θ(ξ)≤1 so that θ(ξ)=1, where 1 is the identity element of the group G. It hence follows that ξ∈1θ-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.
LetS be an epiorthodox semigroup in which max{x∈S:θ(x2)≤θ(x)}=ξ. Then(1)
ξ is the greatest idempotent of S and is a middle unit;(2)
the setE of idempotents of S forms a normal band.Proof.
Part (1) of the above lemma follows easily from observation. To prove part (2) of the lemma, we first recall a result of Blyth and Almeida Santos [11] (see [11,Theorem 2]) that 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 idempotent (in the sense that e¯αe¯=e¯ for all e¯∈E¯, where E¯ is the subsemigroup generated by E, and αE¯α 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 Blyth [20]. Because ξ is a middle unit, for any e,f,g,andh in E, we have that
(2.2)efgh=e⋅ξfξ⋅ξ⋅h=e⋅ξgξ⋅ξfξ⋅h=egfh.
This shows that E is a normal band.Lemma 2.2.
LetS be an epiorthodox semigroup. Suppose that max{x∈S:θ(x2)≤θ(x)}=ξ in S. 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 Lemma2.1, it is known that ξ is the greatest idempotent of S. Since S is a naturally ordered regular semigroup with the greatest idempotent ξ, by a result of Blyth and McFadden in [7], we know immediately that (1), (2), and (3) hold. Since S is orthodox, y∘x∘∈V(xy) and hence (4) holds. By (4), the idempotents xyy∘x∘ and xy(xy)∘ are clearly ℛ-related. But if eℛf for the idempotents e,f in S, then e=fe≤fξ and so eξ≤fξ. Similarly, fξ≤eξ. Thus, from (3), we deduce the equality xyy∘x∘=xy(xy)∘. Similarly, we can also prove y∘x∘xy=(xy)∘xy, and hence (5) holds. From (5), we have that (xy)∘=(xy)∘xy(xy)∘= 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 thatY 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∈Y)φα,α=idSα;(b)
ifα≥β≥γ, then φγ,βφβ,α=φγ,α.Then, it is known that the set⋃α∈YSα under the following multiplication:(2.3)(∀x∈Sα)(∀y∈Sβ)xy=φαβ,α(x)φαβ,β(y)
forms a semigroup which is called the only strong semilattice of semigroups.By using the strong semilattices of semigroups, Blyth and Almeida Santos [8] established the following result.LetS=⋃α∈YSα 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 (2.4)(∀x∈Sα)(∀y∈Sβ)x⊑y⟺α≤β,x≤φα,β(y)
is a partial order on S, and so S=⋃α∈YSα forms an ordered semigroup.We now call an ordered semigroup constructed in the above manner astrong semilattice of ordered semigroups.The following definition of“pointed semilattice of pointed semigroups” was given by Blyth and Almeida Santos [8].Definition 2.3.
An ordered semigroupS is said to be a pointed semilattice of pointed semigroups if the following conditions are satisfied:(1)
S=⋃α∈YSα is a strong semilattice of ordered semigroups;(2)
the semilatticeY has the greatest element;(3)
every ordered semigroupSα 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.
LetS be an epiorthodox semigroup. Then the band E of idempotents of S is a pointed semilattice of pointed rectangular bands on which the order “⊑”  coincides with the order “≤” on S.Proof.
By Lemma2.1, the set of idempotents E of the semigroup S is normal. By applying a theorem of Yamada and Kimura [21], E is known to be a strong semilattice of rectangular bands which can be regarded as the 𝒟-class of E. Clearly, the 𝒟-classes of E are the same classes as the 𝒴-classes, where 𝒴 is the finest inverse semigroup congruence given by
(2.5)(e,f)∈Y⟺V(e)=V(f).
By Lemma 2.2(1), we know that, e∘=ξeξ. Thus, if (e,f)∈𝒟 then e∘=ξeξ=ξfξ=f∘. Conversely, if ξeξ=ξfξ then
(2.6)e=e3=eξeξe=eξfξe=efe.
Similarly, we can prove that f=fef. Consequently, e∈V(f). This leads to V(e)=V(f), whence (e,f)∈𝒟. Thus, 𝒟 is defined on E by
(2.7)(e,f)∈D⟺e∘=f∘.
Hence, each 𝒟-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∘:De∘→Df∘ is given by
(2.8)(∀x∈De∘)φf∘,e∘(x)=xf∘x.
These maps are clearly isotone. Observe that the order “⊑” coincides with the order “≤” on the semigroup S. In fact, if x∈De∘ and y∈Df∘ satisfy the relation x⊑y, then e∘≤f∘ and x≤φe∘,f∘(y)=ye∘y≤yf∘y=y because Df∘ is a rectangular band. Conversely, if x≤y then
(2.9)x=x3=x⋅ξxξ⋅x≤y⋅ξxξ⋅y=φξxξ,ξyξ(y).
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∘:De∘→Df∘ preserves the greatest element in order. In fact, we have that
(2.10)φf∘,e∘(e∘)=e∘f∘e∘≥f∘f∘f∘=f∘,
and whence, φf∘,e∘(e∘)=f∘ since f∘=f∘∘=maxDf∘.We have already proved in Lemma2.1 that 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 in [22]. To be more precise, we state the following lemma.Lemma 2.6 (see [22]).
LetS be an inverse semigroup with a semilattice E of idempotents of S. Let L and R be, respectively, a left normal band and a right normal band with a structural decomposition L=⋃α∈ELα and R=⋃β∈ERβ.
Then on the set(2.11)L⊗S⊗R={(e,x,f):x∈S,e∈Lxx-1,f∈Rx-1x}
the multiplication given by
(2.12)(e,x,f)(g,y,h)=(eu,xy,vh),
where u∈Lxy(xy)-1 and v∈R(xy)-1xy, is well defined and L⊗S⊗R forms 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 in [2] so that an epiorthodox semigroup can be so constructed.We formulate the following Lemma.Lemma 2.7.
LetS be a naturally ordered inverse semigroup satisfying the Dubreil-Jacotin condition. Suppose that Green's relations ℛ,ℒ on S are, respectively, the left and right regular relations on S. Let L be an ordered left normal band with the greatest element 1L which is a right identity, and let R be an ordered right normal band with the greatest element 1R which is a left identity. Then the following statements hold.(i)
L=⋃α∈ELα is a pointed semilattice of pointed left zero semigroups, and R=⋃β∈ERβ is also a pointed semilattice of pointed right zero semigroups.(ii)
Let(L⊗S⊗R)c denote the set
(2.13)L⊗S⊗R={(e,x,f):x∈S,e∈Lxx-1,f∈Rx-1x}
equipped with the Cartesian order and the multiplication
(2.14)(e,x,f)(g,y,h)=(eLxy(xy)-1*,xy,R(xy)-1xy*h),
where Lα* is the greatest element of Lα and Rα* is the greatest element of Rα. Then (L⊗S⊗R)c forms an epiorthodox semigroup on which  Green's relations ℒ,ℛ are,  respectively, the right and left regular relations.Proof.
(i) Since1L is a right identity for L, for e,f∈L,
(2.15)e⪯f⟹e=fe≤f1L=f,
and so L is naturally ordered. Since 1L is the greatest element of L, L satisfies the Dubreil-Jacotin condition. Hence, by applying Lemma 2.4 with S=E=L, L is a pointed semilattice of pointed rectangular bands. These 𝒟-class rectangular bands are left zero semigroups, for if (e,f)∈𝒟, then 1Le1L=e∘=f∘=1Lf1L, and so 1Le=1Lf. Consequently, we can deduce that e=e1Le=e1Lf=ef. Similarly, R is also a pointed semilattice of pointed right zero semigroups. Recall from Lemma 2.4 that the order “⊑” coincides with the order “≤” in both L and R.
(ii) Suppose thatL and R admit the structure decompositions L=⋃α∈ELα and R=⋃β∈ERβ, respectively. Observe that, for α,β∈E, we have that
(2.16)β≤α⟹Lβ*≤Lα*,Rβ*≤Rα*.
If β≤α, then since the structure mapping in L maps the greatest element to the greatest element, ϕβ,α(Lα*)=Lβ*. Consequently, Lβ*⊑Lα* and by Lemma 2.4, Lβ*≤Lα*. Similarly, we have that Rβ*≤Rα*. By applying the Yamada construction in Lemma 2.6, now, we can see that (L⊗S⊗R)c is an orthodox semigroup. Thus, under the Cartesian order and the left regularity of ℛ and the right regular regularity of ℒ on S, we can easily see that (L⊗S⊗R)c forms an ordered semigroup. At first, we let (e,x,f)≤(e1,x1,f1). Then, x≤x1 and so xy≤x1y for every y∈S. Since ℛ is a left regular relation on S,xy(xy)-1≤x1y(x1y)-1 and so by the above observation, Lxy(xy)-1*≤Lx1y(x1y)-1*. By applying the right regularity of ℒ, we can similarly show that R(xy)-1xy*≤R(x1y)-1x1y*. Thus, we obtain the following:
(2.17)(e,x,f)(g,y,h)=(eLxy(xy)-1*,xy,R(xy)-1xy*h)≤(e1Lx1y(x1y)-1*,x1y,R(x1y)-1x1y*h)=(e1,x1,f1)(g,y,h).
By using similar arguments, we can show that
(2.18)(g,y,h)(e,x,f)≤(g,y,h)(e1,x1,f1),
and so (L⊗S⊗R)c forms an ordered semigroup.
Since eachLα 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,f)⪯(g,y,h). Then (e,x,f)=(e,x,f)(g,y,h)=(g,y,h)(e,x,f). This leads to x=xy=yx and so x⪯y in E. Since S is naturally ordered, we have that x≤y. Also, we have e=gLyx(yx)-1*≤g1L=g and similarly, f≤h. This shows that (L⊗S⊗R)c is naturally ordered.
SinceS satisfies the Dubreil-Jacotin condition, there exists an ordered group G and an isotone surjective homomorphism θ:S→G such that {x∈S:θ(x2)≤θ(x)} has the greatest element ξ. Define the mapping ψ:(L⊗S⊗R)c→G by ψ(e,x,f)=θ(x). Then, ψ is an isotone surjective homomorphism.
We now proceed to show that(2.19)max⁡{(e,x,f)∈(L⊗S⊗R)c:ψ[(e,x,f)2]≤ψ(e,x,f)}=(1L,ξ,1R).
Since (1L,ξ,1R) is an idempotent,
(2.20)(1L,ξ,1R)∈{(e,x,f)∈(L⊗S⊗R)c:ψ[(e,x,f)2]≤ψ(e,x,f)}.
On the other hand, we have that
(2.21)θ(x2)=ψ[(e,x,f)2]≤ψ(e,x,f)=θ(x)⟹x≤ξ,
and e≤1L,f≤1R are clear. Hence, (e,x,f)≤(1L,ξ,1R), and so
(2.22)max⁡{(e,x,f)∈(L⊗S⊗R)c:ψ[(e,x,f)2]≤ψ(e,x,f)}=(1L,ξ,1R).
Consequently, we can see that (1L,ξ,1R) is the greatest idempotent of (L⊗S⊗R)c. 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 relation ℛ on the semigroup (L⊗S⊗R)c. We need to show that the relation ℛ is a left regular relation on (L⊗S⊗R)c. For this purpose, we need to identify (e,x,f)∘. By Lemma 2.2, we have that  (e,x,f)∘=(1L,ξ,1R)(e,x,f)′(1L,ξ,1R), where (e,x,f)′∈V(e,x,f). We now show that (eLx-1x*,x-1,g)∈V(e,x,f). In fact, we can deduce the following equalities:(2.23)(e,x,f)(eLx-1x*,x-1,g)(e,x,f)=(eLxx-1*,xx-1,Rxx-1*g)(e,x,f)=(e,xx-1,Rxx-1*g)(e,x,f)=(eLxx-1*,x,Rx-1x*f)=(e,x,f),(eLx-1x*,x-1,g)(e,x,f)(eLx-1x*,x-1,g)=(eLx-1x*,x-1x,Rx-1x*f)(e,x-1,g)=(eLx-1x*Lx-1x*,x-1,Rxx-1*g)=(eLx-1x*,x-1,g).So(eLx-1x*,x-1,g)∈V(e,x,f). Now, we have that (2.24)(e,x,f)∘=(1L,ξ,1R)(eLx-1x*,x-1,g)(1L,ξ,1R).
Hence(2.25)(e,x,f)(e,x,f)∘=(e,x,f)(1L,ξ,1R)(eLx-1x*,x-1,g)(1L,ξ,1R)=(eLxx-1*,x,Rx-1x*1R)(eLx-1x*Lx-1x*,x-1,Rxx-1*1R)=(e,x,Rx-1x*1R)(eLx-1x*,x-1,Rxx-1*1R)=(eLxx-1*,xx-1,Rxx-1*Rxx-1*1R)=(e,xx-1,Rxx-1*1R).
Consequently, we can deduce that ℛ is a left regular relation on (L⊗S⊗R)c. Similarly, ℒ is a right regular relation on (R⊗S⊗L)c.We formulate the following lemma.Lemma 2.8.
IfY=(L⊗S⊗R)c has a band of idempotents E(Y) and containing the greatest idempotent ξ, then there are ordered semigroup isomorphisms
(2.26)ξYξ≃S,E(Y)ξ≃L,ξE(Y)≃R.Proof.
Sinceξ=(1L,1,1R)=(L1*,1,R1*), it can be readily seen that
(2.27)ξ(e,x,f)ξ=(Lxx-1*,x,Rx-1x*).
The mapping
(2.28)θ:ξYξ⟶S,θ(Lxx-1*,x,Rx-1x*)=x
is a semigroup isomorphism. Since Green's relations ℛ,ℒ are, respectively, the left and right regular relations on S, θ is an order isomorphism.
Forα,β∈E, since the structure maps preserve the greatest elements, we have that
(2.29)Lα*Lβ*=φαβ,α(Lα*)φαβ,β(Lβ*)=Lαβ*Lαβ*=Lαβ*.
Now E(Y)={(e,α,f);α∈E}, and
(2.30)(e,x,f)ξ=(e,x,f)(L1*,1,R1*)=(eLα*,α,Rα*R1*)=(e,α,Rα*).
Consider the mapping ψ:E(Y)ξ→L,ψ(e,α,Rα*)=e. Because
(2.31)(e,α,Rα*)(g,β,Rβ*)=(eLαβ*,αβ,Rαβ*Rβ*)=(eLαβ*,αβ,Rαβ*),
and since Lαβ is a left zero semigroup, we can deduce that
(2.32)eLαβ*=φαβ,α(e)φαβ,αβ(Lαβ*)=φαβ,α(e)Lαβ*=φαβ,α(e),eg=φαβ,α(e)φαβ,β(g)=φαβ,α(e).
Thus, ψ is a semigroup homomorphism. Obviously, ψ is surjective. Since for arbitrary e∈Lαandg∈Lβ the equality e=g implies that α=β,ψ is injective. ψ is clearly isotone. Finally, if e≤g with e∈Lα,g∈Lβ, then e⊑g and therefore α≤β by Lemma 2.4. This shows that ψ is an order isomorphism.
Similarly, we can prove thatξE(Y)≃R.


## 3. Main Theorem

In this section, we will give a structure theorem for the epiorthodox semigroups. We establish the converse of Lemma2.7 by showing every epiorthodox semigroup S on which Green's relations ℛ,ℒ are, respectively, the left and right regular relations which arise in the way as stated in Lemma 2.7. Our Lemma 2.8 indicates how this goal can be achieved.Theorem 3.1 (main theorem).
Letξ be the greatest idempotent of an epiorthodox semigroup T and E the band of idempotents of T. Then Eξ is an ordered left normal band with  the greatest element which is a right identity, and ξE is an ordered right normal band with the greatest element which is a left identity. Moreover, ξTξ is a naturally ordered inverse semigroup satisfying the Dubreil-Jacotin condition, and its semilattice of idempotents ξEξ is the structure semilattice of Eξ and of ξE. If Green's relations ℛ,ℒ are, respectively, the left and right regular relations on T, then T and (Eξ⊗ξTξ⊗ξE)c are order isomorphic, that is,
(3.1)T≃(Eξ⊗ξTξ⊗ξE)c.Proof.
Clearly,ξ is the greatest element of Eξ and ξ is a right identity for Eξ. By Lemma 2.1, E is a normal band and hence efgh=egfh for all e,f,g,h∈E. Take e,f,g∈Eξ and h=ξ. Then efg=egf 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∈E(ξTξ). Conversely, if ξxξ∈E(ξTξ), then ξxξ=ξxξ·ξxξ=ξx2ξ. Let x′∈V(x). Then, we have that x′ξxξx′=x′ξx2ξx′ and so x′=x′x2x′. This leads to x=xx′x=xx′x2x′x=x2 and so x∈E. Consequently, E(ξTξ)=ξEξ. Clearly, E(ξTξ) is a semilattice since E is a normal band. Thus,
(3.2)ξeξ⋅ξfξ=ξ⋅ξeξ⋅ξfξ⋅ξ=ξ⋅ξfξ⋅ξeξ⋅ξ=ξfξ⋅ξeξ.
Hence, we have proved that ξTξ is an inverse subsemigroup of T.
To show that the Dubreil-Jacotin condition is satisfied byξTξ, we let φ:T→G be an isotone epimorphism from T onto an ordered group G such that {x∈T:φ(x2)≤φ(x)} has the greatest element. Then φ(ξ)=1G and so φ|ξTξ:ξTξ→G is also an isotone epimorphism. It can be easily verified that
(3.3)max⁡{x∈ξTξ:φ|ξTξ(x2)≤φ|ξTξ(x)}=ξ,
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 Lemma 2.4. Similarly, the structure semilattice of ξE is ξEξ.
By Lemma2.7, we can construct an epiorthodox semigroup (Eξ⊗ξTξ⊗ξE)c. Now, suppose that   Green's relations ℛ,ℒ are, respectively, the left and right regular relations on T. Then we consider the following mapping:
(3.4)χ:T⟶(Eξ⊗ξTξ⊗ξE)c
defined by
(3.5)χ(x)=(xx∘,ξxξ,x∘x).
Note that χ is well defined. On the one hand, by Lemma 2.2, we can easily deduce that xx∘=xx∘ξ∈Eξ and, similarly, x∘x∈ξE; on the other hand, we have that
(3.6)xx∘∈Lξxx∘ξ=Lξxξ(ξxξ)∘=Lξxξ(ξxξ)-1.
Becauseξ is a middle unit, xx∘·ξxξ·x∘x=x and so χ is injective.
To show thatχ is also surjective, we first let (eξ,ξxξ,ξf)∈(Eξ⊗ξTξ⊗ξE)c. Then we have that
(3.7)eξ∈Lξxξ(ξxξ)-1=Lξxξ(ξxξ)∘=Lξxx∘,
and hence (eξ,ξxx∘)∈𝒟. Therefore, by Lemma 2.2, we have that
(3.8)ξeξ=(eξ)∘=(ξxx∘)∘=ξxx∘.
Similarly, ξf∈Rx∘xξ implies that ξfξ=x∘xξ. Now, consider
(3.9)χ(exf)=(exf(exf)∘,ξexfξ,(exf)∘exf).
By the above observation, we have that
(3.10)exf(exf)∘=exff∘x∘e∘=exξfξx∘e∘=exx∘xξx∘e∘=exx∘e∘=eξxx∘e∘=eξeξξeξ=eξ.
Similarly, (exf)∘exf=ξf and
(3.11)ξexfξ=ξeξxξfξ=ξxx∘xx∘xξ=ξxξ.
Thus
(3.12)χ(exf)=(eξ,ξxξ,ξf),
and so we have proved that χ is a surjective mapping.
We now deduce that(3.13)χ(x)χ(y)=(xx∘,ξxξ,x∘x)(yy∘,ξyξ,y∘y)=(xx∘Lξxyξ(ξxyξ)-1*,ξxyξ,R(ξxyξ)-1ξxyξ*y∘y)=(xx∘Lξxyy∘x∘*,ξxyξ,Ry∘x∘xyξ*y∘y)=(xx∘ξxyy∘x∘ξ,ξxyξ,ξy∘x∘xyξy∘y)=(xyy∘x∘,ξxyξ,y∘x∘xy)=(xy(xy)∘,ξxyξ,(xy)∘xy)=χ(xy).
Thus,χ is a semigroup isomorphism.
Since Green's relationsℛ,ℒ are, respectively, the left and the right regular relations on T, χ is isotone; and since
(3.14)χ(x)≤χ(y)⟹xx∘≤yy∘,ξxξ≤ξyξ,x∘x≤y∘y⟹x=xx∘ξxξx∘x≤yy∘ξyξy∘y=y,
it follows that χ is an order isomorphism between the ordered semigroups T and (Eξ⊗ξTξ⊗ξE)c, that is,
(3.15)T≃(Eξ⊗ξTξ⊗ξE)c.
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⊗ξE)c:(1)
(e,x,f)ℛT(g,y,h)⇔xℛξTξy,e=g;(2)
(e,x,f)ℒT(g,y,h)⇔xℒξTξy,f=h;(3)
(e,x,f)𝒟T(g,y,h)⇔x𝒟ξTξy.Proof.
We first deduce the following implications:(3.16)(e,x,f)RT(g,y,h)⟺(e,x,f)(e,x,f)∘=(g,y,h)(g,y,h)∘⟺(e,xx-1,Rxx-1*1R)=(g,yy-1,Ryy-1*1R)⟺e=g,xx-1=yy-1⟺e=g,xRξTξy.
Hence (1) holds. As (2) is the dual of (1), (2) holds. It follows from (1) and (2) that (3) holds.



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*Source: 181707-2011-06-08.xml*