Abstract

In this paper, the translational hull of a left restriction semigroup is considered. We prove that the translational hull of a left restriction semigroup is still of the same type. This result extends the result of Guo and Shum on translational hulls of type A semigroups given in 2003.

1. Introduction

Let S be a semigroup. A mapping from to itself is called a left (right) translation of S if we have for all A left translation and a right translation are called linked if for all in which case the pair is called a bitranslation of . Denote by the set of left (right) translations of . It is easy to see that and are both semigroups under the composition of mappings. And it is also easy to check that , the set of bitranslations of , constitutes a subsemigroup of . We call the semigroup the translational hull of S. The concept of translational hull of semigroups and rings was first introduced by Petrich in 1970 (see ¹). The translational hull of an inverse semigroup was first studied by Ault ² in 1973.

Later on, Fountain and Lawson ³ further studied the translational hulls of adequate semigroups. Guo and Shum ⁴ investigated the translational hull of type A semigroup, in particular, the result obtained by Ault ² was substantially generalized and extended. Thus, the translational hull of a semigroup plays an important role in the theory of semigroups.

On the other hand, left restriction semigroups are class of semigroups which generalize inverse semigroups and which emerge very naturally from the study of partial transformation of a set. A more detailed description of left restriction semigroups can be found in ⁵ and ⁶.

Following Fountain ⁷, a semigroup S is said to be left abundant if each - class of S contains at least one idempotent. Dually, right abundant semigroup can be defined. The semigroup S is called abundant if S is both left abundant and right abundant. As in ⁸, a left (right) abundant semigroup is called a left (right) adequate semigroup if the set of idempotents of S (i.e. ) forms a semilattice. Regular semigroups are abundant semigroups and inverse semigroups are adequate semigroups.

In this paper, we shall show that the translational hull of a left restriction semigroup is still the same type. Thus, the result obtained by Guo and Shum in ⁴ for the translational hull of type A semigroup will be amplified.

2. Preliminaries

In this section we recall some definitions as well as some known results which will be useful in the sequel. We will use the notions and terminologies in ³, ⁴, ⁸, and ⁹.

Definition 2.1 ⁸. Let be a semigroup. Then is said to be left (right) ample if

i) is a semilattice.

ii) every element is related to an idempotent, denoted by .

iii) for all and all

Definition 2.2 ³. Let be a semigroup and let (E is the distinguished semilattice of idempotents).

Let we have following relations on

Definition 2.3 ⁶. Let be a semigroup and let Then is said to be left (right) restriction semigroup if

i) is a semilattice.

ii) every element is related to an idempotent of denoted by

iii) the relation is a left (right) congruence

iv) the left (right) ample condition holds:

The following Lemmas are due to Fountain ⁸ and Gould ⁶.

Lemma 2.4 ⁷. Let be a semigroup and be an idempotent in Then the following are equivalent for .

i)

ii) and for all , .

Lemma 2.5 ⁶. Let S be a semigroup and let Then the following conditions are equivalent:

i)

ii) and for all .

In a similar way to the *-relations, the ~ -relations are also related to the Green’s relations as follows:

Lemma 2.6. In any semigroup S we have . If S is regular, and then and so

Dually we have , and if is regular, and then and so

We note the following useful Lemma, the proof for which in ⁸ for left adequate semigroups can be easily adapted for left restriction semigroups.

Lemma 2.7. Let S be a left restriction semigroup and let . Then

i) if and only if

ii) for all

iii) and .

iv)

v)

vi)

vii)

viii)

Proof. Clearly, i) holds by definition. For ii), since is a left congruence on S, we have Now, by Lemma 2.5, we have Part iii) follows immediately from ii). iv) – viii) can be easily checked.

Lemma 2.8. Let S be a left restriction semigroup. Suppose that are left (right) translations of S whose restriction to E are equal. Then

Proof. Let and such that . It is known from Lemma 2.5 that and so

Consequently, . Similarly, it follows that .

Lemma 2.9. Let S be a left restriction semigroup. If for then the following statements are equivalent:

i)

ii)

iii) .

Proof. Note that is the dual of and that is trivial. We need to verify .

Now suppose we let . To show , it suffices to verify that . To see this, let then and we have that

Now since is a left congruence and by Lemma 2.5 (i), we have that

thereby, since each –class of a left restriction semigroup contains exactly one idempotent. Similarly, . Hence

Consequently,

and hence , as required.

3. The Translational Hull

Throughout this section, S will denote a left restriction semigroup with distinguished semilattice of idempotents E.

Now let S be a left restriction semigroup with distinguished semilattice E of idempotents. Let and define the mappings and of S to itself as follows;

for all

For the mappings and , we have the following Lemma.

Lemma 3.1. Let Then for all

i) and

ii)

Proof. For all and by the definition of the mappings above we have that

Also, the element is clearly an idempotent.

ii) Since is a left congruence on S, and using Lemma 2.5, we have , as required.

Lemma 3.2. The pair is a member of the translational hull of S.

Proof. Suppose using Lemma 2.7, we have

We now prove that is a right translation of S. For all we first observe that , by Lemma 2.7 (v), we have that . So we have that

So is a right translation of S, as required.

To complete the proof, we proceed to show that the pair are linked. We have that

Consequently, .

Lemma 3.3. Suppose Then is the distinguished semilattice of idempotents of

Proof. Suppose and . Then, and . Thus, we have

and by Lemma 2.8, Similarly, . By Lemma 2.9, it follows that

Conversely, suppose that and then for all

Similarly, it follows that Consequently, so that

An immediate consequence of Lemma 3.1 – 3.3 is the following

Corollary 3.4. Let S be a left restriction semigroup and Then .

Lemma 3.5. The elements and of commute with each other.

Proof. For and we have that

It follows from Lemma 2.8 that . Similarly, .

Consequently, It follows from Lemma 2.9 that , that is we have that , as required.

Lemma 3.6. Let Then

Proof. For all , since

(since

we have that and by Lemma 2.9, . This shows that

Similarly, it follows that .

Lemma 3.7. Let S be a left restriction semigroup and Then

Proof. Let be an idempotent of . That entails showing that

That is .

By Lemma 2.9, it entails showing that

Now suppose that Then employ Lemma 2.7 to obtain the following

(by Lemma 2.7 (viii))

(by Lemma 2.7 (vi))

It follows similarly for

Conversely, let Multiplying both sides by , we immediately have

(by Lemma 2.7 (iv)).

It follows similarly for

Consequently, it can be easily seen that .

Lemma 3.8. is a left congruence on for a left restriction semigroup

Proof. To show that is a left congruence, let . Then

So we have that

for any Thus is a left congruence.

Lemma 3.9. Let Then

Proof. From Lemma 3.6, we know that

Now,

Consequently,

Thus, is a left type A (since the left ample condition holds).

By using the above Lemmas 3.2 – 3.3, Corollary 3.4, Lemmas 3.5 – 3.9, we can easily verify that for any there exists a unique idempotent such that and Hence, is indeed a left restriction semigroup.

So far we have proved the following theorem:

Theorem 3.10. The translational hull of a left restriction semigroup is still a left restriction semigroup.

References