On Epiorthodox Semigroups

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 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, 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 ∀𝑒,𝑓∈𝐸𝑒⪯𝑓⟹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 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 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 (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

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 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 semigroup 𝑆 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 𝜃𝜉²≤𝜃(𝜉)⟹𝜃(𝜉)=𝜃𝜉²𝜉^′≤𝜃𝜉𝜉^′=1. (2.1) 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. Let 𝑆 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 set 𝐸 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 𝑒𝛼𝑒=𝑒 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 Blyth [20]. Because 𝜉 is a middle unit, for any 𝑒,𝑓,𝑔,andℎ in 𝐸, we have that 𝑒𝑓𝑔ℎ=𝑒⋅𝜉𝑓𝜉⋅𝜉⋅ℎ=𝑒⋅𝜉𝑔𝜉⋅𝜉𝑓𝜉⋅ℎ=𝑒𝑔𝑓ℎ. (2.2) This shows that 𝐸 is a normal band.

Lemma 2.2. Let 𝑆 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 Lemma 2.1, 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 that 𝑌 is a semilattice and (𝑆_𝛼)_𝛼∈𝑌 is a family of pairwise disjoint semigroups. For 𝛼,𝛽∈𝑌 with 𝛼≥𝛽, let 𝜑_𝛽,𝛼∶𝑆_𝛼→𝑆_𝛽 be a morphism satisfying the following conditions:(a)(∀𝛼∈𝑌)𝜑_𝛼,𝛼=id_𝑆𝛼;(b)if 𝛼≥𝛽≥𝛾, then 𝜑_𝛾,𝛽𝜑_𝛽,𝛼=𝜑_𝛾,𝛼.

Then, it is known that the set ⋃_𝛼∈𝑌𝑆_𝛼 under the following multiplication:∀𝑥∈𝑆_𝛼∀𝑦∈𝑆_𝛽𝑥𝑦=𝜑_{𝛼𝛽,𝛼}(𝑥)𝜑_{𝛼𝛽,𝛽}𝑦 (2.3) 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.

Let 𝑆=⋃_𝛼∈𝑌𝑆_𝛼 be a strong semilattice of semigroups. Suppose that each 𝑆_𝛼 is an ordered semigroup and that each of the structure maps 𝜑_𝛽,𝛼 is isotone. Then the relation “⊑” defined on 𝑆 by ∀𝑥∈𝑆_𝛼∀𝑦∈𝑆_𝛽𝑥⊑𝑦⟺𝛼≤𝛽,𝑥≤𝜑_𝛼,𝛽𝑦 (2.4) is a partial order on 𝑆, and so 𝑆=⋃_𝛼∈𝑌𝑆_𝛼 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 Santos [8].

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∈𝑌𝑆_𝛼 is a strong semilattice of ordered semigroups;(2) the semilattice 𝑌 has the greatest element;(3) every ordered semigroup 𝑆_𝛼 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 𝑆 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 Lemma 2.1, 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 𝑒,𝑓∈𝒴⟺𝑉(𝑒)=𝑉𝑓. (2.5) 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 𝑒=𝑒³=𝑒𝜉𝑒𝜉𝑒=𝑒𝜉𝑓𝜉𝑒=𝑒𝑓𝑒. (2.6) 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 𝑒,𝑓∈𝒟⟺𝑒^∘=𝑓^∘. (2.7) Hence, each 𝒟-class has the greatest element and so 𝑒^∘ is the greatest element of 𝐷_𝑒. 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 𝜑_𝑓^∘_,𝑒∘∶𝐷_𝑒∘→𝐷_𝑓^∘ is given by ∀𝑥∈𝐷_𝑒∘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∈𝐷_𝑒∘ and 𝑦∈𝐷_𝑓^∘ satisfy the relation 𝑥⊑𝑦, then 𝑒^∘≤𝑓^∘ and 𝑥≤𝜑_𝑒∘,𝑓^∘(𝑦)=𝑦𝑒^∘𝑦≤𝑦𝑓^∘𝑦=𝑦 because 𝐷_𝑓^∘ is a rectangular band. Conversely, if 𝑥≤𝑦 then 𝑥=𝑥³=𝑥⋅𝜉𝑥𝜉⋅𝑥≤𝑦⋅𝜉𝑥𝜉⋅𝑦=𝜑_{𝜉𝑥𝜉,𝜉𝑦𝜉}𝑦. (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 map 𝜑_𝑓^∘_,𝑒∘∶𝐷_𝑒∘→𝐷_𝑓^∘ preserves the greatest element in order. In fact, we have that 𝜑_𝑓^∘_,𝑒∘𝑒^∘=𝑒^∘𝑓^∘𝑒^∘≥𝑓^∘𝑓^∘𝑓^∘=𝑓^∘, (2.10) and whence, 𝜑_𝑓^∘_,𝑒∘(𝑒^∘)=𝑓^∘ since 𝑓^∘=𝑓^∘∘=max𝐷_𝑓^∘.

We have already proved in Lemma 2.1 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]). Let 𝑆 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∈𝐸𝐿_𝛼 and 𝑅=⋃_𝛽∈𝐸𝑅_𝛽.
Then on the set 𝐿⊗𝑆⊗𝑅=𝑒,𝑥,𝑓∶𝑥∈𝑆,𝑒∈𝐿_𝑥𝑥−1,𝑓∈𝑅_𝑥−1𝑥 (2.11) the multiplication given by 𝑒,𝑥,𝑓𝑔,𝑦,ℎ=𝑒𝑢,𝑥𝑦,𝑣ℎ, (2.12) where 𝑢∈𝐿_{𝑥𝑦(𝑥𝑦)}⁻¹ and 𝑣∈𝑅_(𝑥𝑦)⁻¹_𝑥𝑦, 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. Let 𝑆 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 1_𝐿 which is a right identity, and let 𝑅 be an ordered right normal band with the greatest element 1_𝑅 which is a left identity. Then the following statements hold.(i)𝐿=⋃_𝛼∈𝐸𝐿_𝛼 is a pointed semilattice of pointed left zero semigroups, and 𝑅=⋃_𝛽∈𝐸𝑅_𝛽 is also a pointed semilattice of pointed right zero semigroups.(ii)Let (𝐿⊗𝑆⊗𝑅)_𝑐 denote the set 𝐿⊗𝑆⊗𝑅=𝑒,𝑥,𝑓∶𝑥∈𝑆,𝑒∈𝐿_𝑥𝑥−1,𝑓∈𝑅_𝑥−1𝑥 (2.13) equipped with the Cartesian order and the multiplication 𝑒,𝑥,𝑓𝑔,𝑦,ℎ=𝑒𝐿^∗_{𝑥𝑦(𝑥𝑦)−1},𝑥𝑦,𝑅^∗_{(𝑥𝑦)−1𝑥𝑦}ℎ, (2.14) where 𝐿^∗_𝛼 is the greatest element of 𝐿_𝛼 and 𝑅^∗_𝛼 is the greatest element of 𝑅_𝛼. 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) Since 1_𝐿 is a right identity for 𝐿, for 𝑒,𝑓∈𝐿, 𝑒⪯𝑓⟹𝑒=𝑓𝑒≤𝑓1_𝐿=𝑓, (2.15) and so 𝐿 is naturally ordered. Since 1_𝐿 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 1_𝐿𝑒1_𝐿=𝑒^∘=𝑓^∘=1_𝐿𝑓1_𝐿, and so 1_𝐿𝑒=1_𝐿𝑓. Consequently, we can deduce that 𝑒=𝑒1_𝐿𝑒=𝑒1_𝐿𝑓=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 that 𝐿 and 𝑅 admit the structure decompositions 𝐿=⋃_𝛼∈𝐸𝐿_𝛼 and 𝑅=⋃_𝛽∈𝐸𝑅_𝛽, 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, 𝜙_𝛽,𝛼(𝐿^∗_𝛼)=𝐿^∗_𝛽. Consequently, 𝐿^∗_𝛽⊑𝐿^∗_𝛼 and by Lemma 2.4, 𝐿^∗_𝛽≤𝐿^∗_𝛼. Similarly, we have that 𝑅^∗_𝛽≤𝑅^∗_𝛼. 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, 𝐿^∗_{𝑥𝑦(𝑥𝑦)−1}≤𝐿^∗_{𝑥1𝑦(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: 𝑒,𝑥,𝑓𝑔,𝑦,ℎ=𝑒𝐿^∗_{𝑥𝑦(𝑥𝑦)−1},𝑥𝑦,𝑅^∗_{(𝑥𝑦)−1𝑥𝑦}ℎ≤𝑒₁𝐿^∗_{𝑥1𝑦(𝑥1𝑦)−1},𝑥₁𝑦,𝑅^∗_{(𝑥1𝑦)−1𝑥1𝑦}ℎ=𝑒₁,𝑥₁,𝑓₁𝑔,𝑦,ℎ. (2.17) By using similar arguments, we can show that 𝑔,𝑦,ℎ𝑒,𝑥,𝑓≤𝑔,𝑦,ℎ𝑒₁,𝑥₁,𝑓₁, (2.18) and so (𝐿⊗𝑆⊗𝑅)_𝑐 forms an ordered semigroup.
Since each 𝐿_𝛼 is a left zero semigroup and each 𝑅_𝛼 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=𝑔𝐿^∗_{𝑦𝑥(𝑦𝑥)−1}≤𝑔1_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∶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 max𝑒,𝑥,𝑓∈(𝐿⊗𝑆⊗𝑅)_𝑐∶𝜓𝑒,𝑥,𝑓²≤𝜓𝑒,𝑥,𝑓=1_𝐿,𝜉,1_𝑅. (2.19) Since (1_𝐿,𝜉,1_𝑅) is an idempotent, 1_𝐿,𝜉,1_𝑅∈𝑒,𝑥,𝑓∈(𝐿⊗𝑆⊗𝑅)_𝑐∶𝜓𝑒,𝑥,𝑓²≤𝜓𝑒,𝑥,𝑓. (2.20) On the other hand, we have that 𝜃𝑥²=𝜓𝑒,𝑥,𝑓²≤𝜓𝑒,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 max𝑒,𝑥,𝑓∈(𝐿⊗𝑆⊗𝑅)_𝑐∶𝜓𝑒,𝑥,𝑓²≤𝜓𝑒,𝑥,𝑓=1_𝐿,𝜉,1_𝑅. (2.22) Consequently, we can see that (1_𝐿,𝜉,1_𝑅) 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)^∘=(1_𝐿,𝜉,1_𝑅)(𝑒,𝑥,𝑓)^′(1_𝐿,𝜉,1_𝑅), where (𝑒,𝑥,𝑓)^′∈𝑉(𝑒,𝑥,𝑓). We now show that (𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔)∈𝑉(𝑒,𝑥,𝑓). In fact, we can deduce the following equalities:𝑒,𝑥,𝑓𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔𝑒,𝑥,𝑓=𝑒𝐿^∗_𝑥𝑥−1,𝑥𝑥⁻¹,𝑅^∗_𝑥𝑥−1𝑔=𝑒,𝑥,𝑓𝑒,𝑥𝑥⁻¹,𝑅^∗_𝑥𝑥−1𝑔=𝑒,𝑥,𝑓𝑒𝐿^∗_𝑥𝑥−1,𝑥,𝑅^∗_𝑥−1𝑥𝑓=𝑒,𝑥,𝑓,𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔𝑒,𝑥,𝑓𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔=𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹𝑥,𝑅^∗_𝑥−1𝑥𝑓𝑒,𝑥⁻¹=,𝑔𝑒𝐿^∗_𝑥−1𝑥𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑅^∗_𝑥𝑥−1𝑔=𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔. (2.23)

So (𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔)∈𝑉(𝑒,𝑥,𝑓). Now, we have that 𝑒,𝑥,𝑓^∘=1_𝐿,𝜉,1_𝑅𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔1_𝐿,𝜉,1_𝑅. (2.24) Hence𝑒,𝑥,𝑓𝑒,𝑥,𝑓^∘=𝑒,𝑥,𝑓1_𝐿,𝜉,1_𝑅𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑔1_𝐿,𝜉,1_𝑅=𝑒𝐿^∗_𝑥𝑥−1,𝑥,𝑅^∗_𝑥−1𝑥1_𝑅𝑒𝐿^∗_𝑥−1𝑥𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑅^∗_𝑥𝑥−11_𝑅=𝑒,𝑥,𝑅^∗_𝑥−1𝑥1_𝑅𝑒𝐿^∗_𝑥−1𝑥,𝑥⁻¹,𝑅^∗_𝑥𝑥−11_𝑅=𝑒𝐿^∗_𝑥𝑥−1,𝑥𝑥⁻¹,𝑅^∗_𝑥𝑥−1𝑅^∗_𝑥𝑥−11_𝑅=𝑒,𝑥𝑥⁻¹,𝑅^∗_𝑥𝑥−11_𝑅. (2.25) 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. If 𝑌=(𝐿⊗𝑆⊗𝑅)_𝑐 has a band of idempotents 𝐸(𝑌) and containing the greatest idempotent 𝜉, then there are ordered semigroup isomorphisms 𝜉𝑌𝜉≃𝑆,𝐸(𝑌)𝜉≃𝐿,𝜉𝐸(𝑌)≃𝑅. (2.26)

Proof. Since 𝜉=(1_𝐿,1,1_𝑅)=(𝐿^∗₁,1,𝑅^∗₁), it can be readily seen that 𝜉𝑒,𝑥,𝑓𝜉=𝐿^∗_𝑥𝑥−1,𝑥,𝑅^∗_𝑥−1𝑥. (2.27) The mapping 𝜃∶𝜉𝑌𝜉⟶𝑆,𝜃𝐿^∗_𝑥𝑥−1,𝑥,𝑅^∗_𝑥−1𝑥=𝑥 (2.28) 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.
For 𝛼,𝛽∈𝐸, since the structure maps preserve the greatest elements, we have that 𝐿^∗_𝛼𝐿^∗_𝛽=𝜑_{𝛼𝛽,𝛼}𝐿^∗_𝛼𝜑_{𝛼𝛽,𝛽}𝐿^∗_𝛽=𝐿^∗_𝛼𝛽𝐿^∗_𝛼𝛽=𝐿^∗_𝛼𝛽. (2.29) Now 𝐸(𝑌)={(𝑒,𝛼,𝑓);𝛼∈𝐸}, and 𝑒,𝑥,𝑓𝜉=𝑒,𝑥,𝑓𝐿^∗₁,1,𝑅^∗₁=𝑒𝐿^∗_𝛼,𝛼,𝑅^∗_𝛼𝑅^∗₁=𝑒,𝛼,𝑅^∗_𝛼. (2.30) Consider the mapping 𝜓∶𝐸(𝑌)𝜉→𝐿,𝜓(𝑒,𝛼,𝑅^∗_𝛼)=𝑒. Because 𝑒,𝛼,𝑅^∗_𝛼𝑔,𝛽,𝑅^∗_𝛽=𝑒𝐿^∗_𝛼𝛽,𝛼𝛽,𝑅^∗_𝛼𝛽𝑅^∗_𝛽=𝑒𝐿^∗_𝛼𝛽,𝛼𝛽,𝑅^∗_𝛼𝛽, (2.31) and since 𝐿_𝛼𝛽 is a left zero semigroup, we can deduce that 𝑒𝐿^∗_𝛼𝛽=𝜑_{𝛼𝛽,𝛼}(𝑒)𝜑_{𝛼𝛽,𝛼𝛽}𝐿^∗_𝛼𝛽=𝜑_{𝛼𝛽,𝛼}(𝑒)𝐿^∗_𝛼𝛽=𝜑_{𝛼𝛽,𝛼}(𝑒),𝑒𝑔=𝜑_{𝛼𝛽,𝛼}(𝑒)𝜑_{𝛼𝛽,𝛽}𝑔=𝜑_{𝛼𝛽,𝛼}(𝑒). (2.32) Thus, 𝜓 is a semigroup homomorphism. Obviously, 𝜓 is surjective. Since for arbitrary 𝑒∈𝐿_𝛼and𝑔∈𝐿_𝛽 the equality 𝑒=𝑔 implies that 𝛼=𝛽,𝜓 is injective. 𝜓 is clearly isotone. Finally, if 𝑒≤𝑔 with 𝑒∈𝐿_𝛼,𝑔∈𝐿_𝛽, then 𝑒⊑𝑔 and therefore 𝛼≤𝛽 by Lemma 2.4. This shows that 𝜓 is an order isomorphism.
Similarly, we can prove that 𝜉𝐸(𝑌)≃𝑅.

3. Main Theorem

In this section, we will give a structure theorem for the epiorthodox semigroups. We establish the converse of Lemma 2.7 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). Let 𝜉 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, 𝑇≃(𝐸𝜉⊗𝜉𝑇𝜉⊗𝜉𝐸)_𝑐. (3.1)

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, 𝜉𝑒𝜉⋅𝜉𝑓𝜉=𝜉⋅𝜉𝑒𝜉⋅𝜉𝑓𝜉⋅𝜉=𝜉⋅𝜉𝑓𝜉⋅𝜉𝑒𝜉⋅𝜉=𝜉𝑓𝜉⋅𝜉𝑒𝜉. (3.2) Hence, we have proved that 𝜉𝑇𝜉 is an inverse subsemigroup of 𝑇.
To show that the Dubreil-Jacotin condition is satisfied by 𝜉𝑇𝜉, 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)=1_𝐺 and so 𝜑|_𝜉𝑇𝜉∶𝜉𝑇𝜉→𝐺 is also an isotone epimorphism. It can be easily verified that max𝑥∈𝜉𝑇𝜉∶𝜑_𝜉𝑇𝜉𝑥²≤𝜑_𝜉𝑇𝜉(𝑥)=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 Lemma 2.4. Similarly, the structure semilattice of 𝜉𝐸 is 𝜉𝐸𝜉.
By Lemma 2.7, 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: 𝜒∶𝑇⟶(𝐸𝜉⊗𝜉𝑇𝜉⊗𝜉𝐸)_𝑐 (3.4) defined by 𝜒(𝑥)=𝑥𝑥^∘,𝜉𝑥𝜉,𝑥^∘𝑥. (3.5) 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 𝑥𝑥^∘∈𝐿_{𝜉𝑥𝑥∘𝜉}=𝐿_{𝜉𝑥𝜉(𝜉𝑥𝜉)}^∘=𝐿_{𝜉𝑥𝜉(𝜉𝑥𝜉)}⁻¹. (3.6)
Because 𝜉 is a middle unit, 𝑥𝑥^∘⋅𝜉𝑥𝜉⋅𝑥^∘𝑥=𝑥 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_{𝜉𝑥𝜉(𝜉𝑥𝜉)}⁻¹=𝐿_{𝜉𝑥𝜉(𝜉𝑥𝜉)}^∘=𝐿_{𝜉𝑥𝑥∘}, (3.7) and hence (𝑒𝜉,𝜉𝑥𝑥^∘)∈𝒟. Therefore, by Lemma 2.2, we have that 𝜉𝑒𝜉=(𝑒𝜉)^∘=𝜉𝑥𝑥^∘∘=𝜉𝑥𝑥^∘. (3.8) Similarly, 𝜉𝑓∈𝑅_{𝑥∘𝑥𝜉} implies that 𝜉𝑓𝜉=𝑥^∘𝑥𝜉. Now, consider 𝜒𝑒𝑥𝑓=𝑒𝑥𝑓𝑒𝑥𝑓^∘,𝜉𝑒𝑥𝑓𝜉,𝑒𝑥𝑓^∘𝑒𝑥𝑓. (3.9) 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)^∘𝑒𝑥𝑓=𝜉𝑓 and 𝜉𝑒𝑥𝑓𝜉=𝜉𝑒𝜉𝑥𝜉𝑓𝜉=𝜉𝑥𝑥^∘𝑥𝑥^∘𝑥𝜉=𝜉𝑥𝜉. (3.11) Thus 𝜒𝑒𝑥𝑓=𝑒𝜉,𝜉𝑥𝜉,𝜉𝑓, (3.12) and so we have proved that 𝜒 is a surjective mapping.
We now deduce that 𝜒(𝑥)𝜒𝑦=𝑥𝑥^∘,𝜉𝑥𝜉,𝑥^∘𝑥𝑦𝑦^∘,𝜉𝑦𝜉,𝑦^∘𝑦=𝑥𝑥^∘𝐿^∗_{𝜉𝑥𝑦𝜉𝜉𝑥𝑦𝜉−1},𝜉𝑥𝑦𝜉,𝑅^∗_{𝜉𝑥𝑦𝜉−1𝜉𝑥𝑦𝜉}𝑦^∘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^∘=𝑥𝑦𝑥𝑦𝑥𝑦^∘,𝜉𝑥𝑦𝜉,𝑥𝑦^∘.𝑥𝑦=𝜒𝑥𝑦 (3.13)
Thus, 𝜒 is a semigroup isomorphism.
Since Green's relations ℛ,ℒ are, respectively, the left and the right regular relations on 𝑇, 𝜒 is isotone; and since 𝜒(𝑥)≤𝜒𝑦⟹𝑥𝑥^∘≤𝑦𝑦^∘,𝜉𝑥𝜉≤𝜉𝑦𝜉,𝑥^∘𝑥≤𝑦^∘𝑦⟹𝑥=𝑥𝑥^∘𝜉𝑥𝜉𝑥^∘𝑥≤𝑦𝑦^∘𝜉𝑦𝜉𝑦^∘𝑦=𝑦, (3.14) it follows that 𝜒 is an order isomorphism between the ordered semigroups 𝑇 and (𝐸𝜉⊗𝜉𝑇𝜉⊗𝜉𝐸)_𝑐, that is, 𝑇≃(𝐸𝜉⊗𝜉𝑇𝜉⊗𝜉𝐸)_𝑐. (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 semigroup 𝑇=(𝐸𝜉⊗𝜉𝑇𝜉⊗𝜉𝐸)_𝑐:(1)(𝑒,𝑥,𝑓)ℛ^𝑇(𝑔,𝑦,ℎ)⇔𝑥ℛ^𝜉𝑇𝜉𝑦,𝑒=𝑔;(2)(𝑒,𝑥,𝑓)ℒ^𝑇(𝑔,𝑦,ℎ)⇔𝑥ℒ^𝜉𝑇𝜉𝑦,𝑓=ℎ;(3)(𝑒,𝑥,𝑓)𝒟^𝑇(𝑔,𝑦,ℎ)⇔𝑥𝒟^𝜉𝑇𝜉𝑦.

Proof. We first deduce the following implications: 𝑒,𝑥,𝑓ℛ^𝑇𝑔,𝑦,ℎ⟺𝑒,𝑥,𝑓𝑒,𝑥,𝑓^∘=𝑔,𝑦,ℎ𝑔,𝑦,ℎ^∘⟺𝑒,𝑥𝑥⁻¹,𝑅^∗_𝑥𝑥−11_𝑅=𝑔,𝑦𝑦⁻¹,𝑅^∗_𝑦𝑦−11_𝑅⟺𝑒=𝑔,𝑥𝑥⁻¹=𝑦𝑦⁻¹⟺𝑒=𝑔,𝑥ℛ^𝜉𝑇𝜉𝑦. (3.16)
Hence (1) holds. As (2) is the dual of (1), (2) holds. It follows from (1) and (2) that (3) 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 Province (2008BS01016), China.