#### Abstract

In this paper, we study a class of naturally ordered abundant semigroups with an adequate monoid transversal, namely, naturally ordered concordant semigroups with an adequate monoid transversal. After giving some properties of such semigroups, we obtain a structure theorem for naturally ordered concordant semigroups with an adequate monoid transversal.

#### 1. Introduction

Suppose that is a regular semigroup and is an inverse subsemigroup of . Then is called an inverse transversal of , if contains a unique inverse of each element of . The structure theorems for regular semigroups with an inverse transversal have been given by many authors (see [1โ4]). An analogue of an inverse transversal which is termed an adequate transversal was introduced for abundant semigroups by El-Qallali (see [5]). An ordered semigroup is* naturally ordered* if, for all , implies , where is the natural partial order on . Recently, Blyth and Almeida Santos have investigated naturally ordered regular semigroups with an inverse monoid transversal and determined the structure of such regular semigroups. Since abundant semigroups generalize regular semigroups, it is natural that, in the first instance, we should approach their structure theory by looking for generalization of results from the theory of regular semigroups. The following papers contain some work along these lines: [5โ20]. As we shall discuss below, this paper is one of a sequence in which we concentrate on the structure of a class of naturally ordered abundant semigroups with an adequate monoid transversal.

We proceed as follows: Section 2 presents some necessary notation and known results. In Section 3, we obtain some characterizations for naturally ordered concordant semigroups with an adequate transversal. In Section 4, we give several order properties of naturally ordered concordant semigroups with an adequate transversal. In Section 5, we give some characterizations of the regularity of Green relations. In Section 6, we establish a structure theorem for naturally ordered concordant semigroups with an adequate monoid transversal, which generalizes the main result of [21].

#### 2. Preliminaries

In what follows, we shall use the notion and notation ofโโ [6, 21]. Other undefined terms can be found in [11, 22, 23]. Here we provide some known results repeatedly used without mention in the following. First we recall some of the basic facts about the relations and .

Lemma 1 (see [11]). *Let be elements of a semigroup . Then the following statements are equivalent:**(1) ;**(2) for all , *

As an easy but useful consequence, we have the following.

Corollary 2 (see [11]). *Let . Then the following statements are equivalent:**(1) ;**(2) and for all , *

It is well known that is a right congruence while is a left congruence. In general, and . But when are regular elements, [resp., ] if and only if [resp., ]. For convenience, [resp. ] is denoted the typical idempotent related by the relation [resp. ]. A semigroup is called* abundant* if each -class and each -class contains at least an idempotent. An abundant semigroup is called* quasi-adequate* if the set of idempotents forms a subsemigroup. Moreover, a quasi-adequate semigroup is called* adequate* if its band of idempotents is a semilattice. In this case, each -class and each -class contains exactly one idempotent.

In this paper, denotes the natural partial order on an abundant semigroup S, defined by the rule that if and only if for some idempotents and , . And stands for the set . Following [9], is* idempotent-connected *(for short, IC) provided, for every element of , and for some [for all] , there exists a bijection such that for all of , where () is the subsemigroup of generated by the set (resp., .

Lemma 3 (see [18]). *Let be elements of a semigroup . Then the following statements are equivalent:**(1) is IC;**(2) for each element a of two conditions hold:**(i) for some[for all] and for all there exists an idempotent such that .**(ii) for some[for all] and for all there exists an idempotent such that .*

An abundant semigroup is called* concordant* if it is IC and satisfies the regularity condition.

If is an abundant semigroup and is an abundant subsemigroup of ; then we say that is a -*subsemigroup* of if and .

Lemma 4 (see [19]). *Let be an abundant semigroup. If is an idempotent of , then is a -subsemigroup of .*

Let be an adequate -subsemigroup of and be the idempotent semilattice of . As in [5], is called an* adequate transversal* for if, for each element in , there exists a unique element and idempotents in such that , where , for in . It is straightforward to show that such and are uniquely determined by (see [5]). Hence we normally denote by , by . We define The adequate transversal is said to be a* quasi-ideal* if or, equivalently, and* left simplistic* if or equivalently, . Let be a -adequate subsemigroup of and . Throughout this paper, we denote by [resp., ] the unique idempotent of [resp. ] in . For our purpose, we list the following results.

Lemma 5 (see [6]). *Let be an adequate transversal of an abundant semigroup and . Then**(1) and ;**(2) and ;**(3) , ;**(4) if then , , ;**(5) if then ;**(6) if , then and .*

Lemma 6 (see [7]). *Let be an adequate transversal of an abundant semigroup and . Then**(1) ;**(2) *

Lemma 7 (see [20]). *Let be an abundant semigroup with a quasi-ideal adequate transversal . For any , then**(1) ;**(2) ;**(3) *

Let be an abundant semigroup with an adequate transversal . If , then, by Lemma 5(1) and [23, Theorem 2.3.4], there exists a unique with and .

Theorem 8 (see [6]). *Let be an abundant semigroup with an adequate transversal . If then , and .*

Lemma 9 (see [6]). *Let be an abundant semigroup with a quasi-ideal adequate transversal . If is a subsemigroup of then is a left regular subband of and is a right regular subband of .*

Lemma 10 (see [6]). *Let be an abundant semigroup with an adequate transversal . Then *

Theorem 11 (see [6]). *Let be an abundant semigroup with an adequate transversal . If is a band, then the following statements are equivalent:**(1) is left simplistic;**(2) is a right ideal of ;**(3) is a normal band.*

Theorem 12 (see [6]). *Let be an abundant semigroup with an adequate transversal . Then is both left and right simplistic if and only if is a quasi-ideal of .*

#### 3. Characterization

The object of this section is to give characterizations of naturally ordered concordant semigroups with an adequate transversal.

Lemma 13 (see [17]). *Let be an ordered abundant semigroup. Then is naturally ordered if and only if, for all , is -unipotent, -unipotent, and naturally ordered.*

As a consequence of Lemma 13, we have the following.

Corollary 14. *Let be an ordered abundant semigroup which satisfies the regularity condition. Then is naturally ordered if and only if is locally adequate and, for all , is naturally ordered.*

Lemma 15 (see [16]). *Let be an IC abundant semigroup. Then the following statements are equivalent:**(1) is locally adequate;**(2) is compatible with multiplication.*

Combining with Corollary 14 and Lemma 15, we have the following.

Theorem 16. *A concordant semigroup can be naturally ordered if and only if it is locally adequate.*

Theorem 17. *Let be a concordant semigroup with a quasi-ideal adequate transversal . Then is locally adequate.*

*Proof. *Suppose that is a quasi-ideal adequate transversal of a concordant semigroup . Let and let . It is clear that is a regular subsemigroup of . It is easy to see that . Since is adequate, then we can easily deduce that is inverse and that is an inverse transversal of the regular semigroup . It is reasonably clear that , . Since is a concordant semigroup, . Because is a quasi-ideal, . Thus . This implies that is a quasi-ideal inverse transversal of . By [21, Theorem 3], is locally inverse. Let . Then is an inverse subsemigroup of and, by [18, Lemma 1.5], is concordant. Assume that . Then . Thus . It follows that . Since is concordant, and so . Therefore, is adequate. This shows that is locally adequate.

Furthermore, we have the following result, which generalizes [21, Theorem 3].

Theorem 18. *Let be a concordant semigroup with an adequate transversal . Then the following statements are equivalent:**(1) can be naturally ordered;**(2) is locally adequate;**(3) is a quasi-ideal of .*

*Proof. *We need to only prove that (2) implies (3). Suppose that is locally adequate. Then, by Lemma 15, is compatible with multiplication. Hence, by Lemma 9 and [23, Exercise 4.7.18], is a normal band whence by Theorem 11, is left simplistic. Similarly, is right simplistic. Hence, by Theorem 12, is a quasi-ideal.

#### 4. Order Properties

Let be a naturally ordered concordant semigroup with an adequate transversal . By Theorem 18 above the adequate transversal is a quasi-ideal. We now consider relationships between the imposed order and the natural order on . The proof of the following lemma is essentially the same as that of [21, note].

Lemma 19. *Let be a naturally ordered concordant semigroup with an adequate transversal . Then the orders and coincide on .*

In the following results, we shall see that the order also extends the natural order on the adequate transversal ; and, as a consequence, the assignment is always isotone.

Lemma 20. *Let be a naturally ordered concordant semigroup with an adequate transversal . If are such that then .*

*Proof. *Suppose that are such that ; then there exist such that . Observe that gives by Lemma 7(2), whence we have ; and, by Lemma 9, . Hence we have and so Similarly .

The next results generalize [21, Theorem 5 and its Corollary].

Theorem 21. *Let be a naturally ordered concordant semigroup with an adequate transversal . If are such that , then .*

*Proof. *Suppose that are such that . Then, by Lemma 20, in we have . Consequently, and . Since , there exists such that . Let . Since , and so . Because , . Since is a semilattice, , which implies that . Hence, by Lemma 19. It follows that

Corollary 22. *Let be a naturally ordered concordant semigroup with an adequate transversal . If are such that , then .*

*Proof. *If are such that , then, using the fact that is a quasi-ideal, we have By Theorems 11 and 12, . Writing and , we have . Since is IC, there exists such that . Hence . Similarly . Since by Lemma 5(3) and is a quasi-ideal, by Lemma 5(4) and Theorem 21, .

By the proof of [21, Theorem 8], we have the following result:

Lemma 23. *Let be a naturally ordered concordant semigroup with an adequate transversal . If are such that then .*

Theorem 24. *Let be a naturally ordered concordant semigroup with an adequate transversal . The following conditions are equivalent:**(1) If are such that , then ;**(2) If are such that , then ;**(3) is completely -simple;**(4) The adequate transversal is a cancellative monoid.*

*Proof. *(1)(2) It is trivial.

(2)(3) Let be such that . Then . Hence by the hypothesis and Lemma 23, . Thus . Therefore, is equality on and by [11, Corollary 5.2] is completely -simple.

(3)(4) Suppose that is completely -simple. Then by [11, Corollary 5.2], we may assume is a Rees matrix semigroup without zero over a cancellative monoid where each entry in P is a unit of . Since is an adequate semigroup, contains an idempotent. Let and be idempotents of . Then they are inverses of . Since is an adequate transversal of , by [7, Corollary 2.4], , so that has only one idempotent. Thus is a cancellative monoid.

(4)(1) If are such that then, by the Corollary 22, we have . Because is a cancellative monoid, is a group. Thus .

Theorem 25. *Let be a naturally ordered concordant semigroup with an adequate transversal . The following conditions are equivalent:**(1) The adequate transversal is a monoid;**(2) has a biggest idempotent;**(3) ;**(4) ;**(5) ;**(6) [resp. ] is an idempotent-generated principal left [resp. right] ideal.*

*Proof. *(1)(2) If (1) holds then for every we have . Thus .

(2)(1) Suppose that (2) holds and let . If , then by Lemma 5 (3) . Since coincides with on , . Thus . Similarly, . Consequently, .

(1)(3) If (1) holds then we have whence , where .

(3)(4) Let ; let . It is clear that is a regular subsemigroup of . It is easy to see that . Since is adequate then we can easily deduce that is inverse and that is an inverse transversal of the regular semigroup . It is reasonably clear that , . Since , . By [6, Theorem 18(4)], for all . Hence we have, for all , (4)(1) It is trivial.

(4)(5) If (4) holds, then for every we have . Hence . Similarly . By Lemma 7, we have Thus . Similarly and we have (5).

(5)(3) If (5) holds, then, by Lemma 10,

(5)(6) This is clear.

(6)(5) The proof is essentially the same as that of [21, Theorem 10].

#### 5. The Regularity of and

Lemma 20 and [21, Definition 3.2] motivate the next definition.

*Definition 26. *Let be a naturally ordered concordant semigroup with an adequate transversal . [resp. ] is said to be regular if

*Definition 27 (see [21]). *An equivalence relation on an ordered set is said to satisfy the link property if and the dual link property if

Theorem 28. *Let be a naturally ordered concordant semigroup with an adequate transversal . If [resp. ] is regular then [resp. ] satisfies the dual link property.*

*Proof. *If , then since is regular, we have by Lemma 6 and so . Since , whence . Since also , it follows that and so . If such that , then we have . Hence and so . Thus ay=az. If such that , then we have and so . It follows that . Similarly, when regular, satisfies the dual link property.

*Definition 29. *Let be a naturally ordered concordant semigroup with an adequate transversal . is said to be lower -stable ifand upper -stable if Dually, we define to be lower and upper -stable.

Theorem 30. *Let be a naturally ordered concordant semigroup with an adequate transversal . The following conditions are equivalent:**(1) is regular;**(2) satisfies the dual link property and is lower -stable.*

*Proof. *(1)(2) If is regular then, by Theorem 28, it satisfies the dual link property. Moreover, if , then by Lemma 5(6) and so is lower -stable.

(2)(1) Let be such that Because and satisfies the dual link property, there exists such that . Then . Since is lower -stable, . Thus is regular.

Theorem 31. *Let be a naturally ordered concordant semigroup with an adequate transversal . If is upper directed then the following conditions are equivalent:**(1) is regular;**(2) satisfies the link property and is upper -stable.*

*Proof. *(1)(2) Suppose that and let be such that . Then whence . Also, which, by [21, Theorem 1] and Lemma 9, gives whence, by Lemmas 5 and 7, . Hence It is easy to see that is upper -stable.

(2)(1) Let be such that Since and satisfies the link property, there exists such that Then . Because is upper -stable, Thus is regular.

#### 6. Structure Theorem

The main objective in this section is to give a structure theorem for a naturally ordered concordant semigroup with an adequate monoid transversal.

Let be an abundant semigroup with a set of idempotents . An idempotent of is called* medial* if, for all

We have the following result, which generalizes [21, Theorem 17].

Theorem 32. *Let and be concordant semigroups with a common medial idempotent and a common adequate submonoid . Define a mapping by with the following properties:**(1) ;**(2) ;**(3) ;**(4) ;**(5) ** On the set define a multiplication by Then is a locally adequate concordant semigroup that has an adequate monoid transversal.**Moreover, every such semigroup is obtained in this way. More precisely, if is a locally adequate concordant semigroup with an adequate monoid transversal , that is a monoid with identity, then , is a medial idempotent of both and , the mapping given by satisfies properties (1) to (5) above, and there is a semigroup isomorphism *

*Proof. *It is easy to see that the multiplication on is well-defined.

We proceed in the following stages.

(i) is a semigroup.

For associativity of the multiplication, let . It is easy to see that the first component of is which by (1) is Using the fact that and is an identity of , we see that this reduces to On the other hand, the first component of is which by (2) is . It also reduces to Thus the first components of the products are the same and, similarly, so are the second components. Thus, with the above multiplication, is a semigroup.

(ii) is an abundant semigroup.

Suppose that . Let be such that . It is easy to see and . We have for some , . Similarly, . Since and , it follows that . If are such that then we have for some , . Hence and so . Thus and so whence . We have whence . We also have If are such that , then and . Hence by (5) and , and . Thus we have whence .

Dually, and , where . This shows that is abundant.

(iii) is IC.

Let and where . Then and . It follows that , , and , whence , , and . Hence, we have, for some , and so . Suppose that , it is easy to see that . Thus Dually, let where . Then we have Let We have Since is IC and , by Lemma 3, there exists such that . Hence we have Thus . Similarly, for every , there exists such that . This shows that is IC.

(iv) satisfies the regularity condition.

If , then exists such that . Hence we have whence and . Since is an adequate semigroup, Reg(T) is an inverse semigroup. If , , and , then we can define . Then the first component of the product is as in (i), where . In a similar way we can see that the second component of the product is . Thus we see that Therefore, If , then and . Since and are both concordant, and . Hence . This proves that satisfies the regularity condition. Thus is a concordant semigroup.

(v) is a -adequate submonoid of .

It is easy to see that is a subsemigroup of . Since , by Lemma 4 and is a -adequate submonoid of both and . If , then, by (ii), . Hence is abundant. The proof of the following results is essentially the same as that of [21, Theorem 17]: E() is a semilattice and Therefore, is an adequate semigroup. Let and be such that . Then . Hence we have Similarly, we have . Hence Similarly, . Thus and so . Therefore, by [9, Lemma 1.6], is a -adequate subsemigroup of . It is easy to see that is an identity element for .

(vi) is a quasi-ideal adequate transversal of .

Suppose that and where