It is well known that the usual Green’s relations on a semigroup play an important role in the study of the structure of regular semigroups [1-7]. Especially, the well-known theorem of A. H. Clifford states that a semigroup is a completely regular semigroup if and only if it can be expressed as a semilattice of completely simple semigroups (see [1]), where a completely regular semigroup is a semigroup whose -class contains an idempotent. By using this result, A. H. Clifford, M. Petrich both showed that a completely regular semigroup with its Green’s relation a normal band congruence if and only if is a strong semilattice of complete simple semigroups [3]. On the other hand, J. B. Fountain generalized the Clifford theorem by showing that an abundant semigroup is a superabundant semigroup, that is, an abundant semigroup with every -class of contains an idempotent of if and only if is a semilattice of completely -simple semigroups.

We first recall some of the generalized Green’s relations which are frequently used to study the structure of abundant semigroups. The following Green -relations on a semigroup were originally due to F. Pastijn [8] and were extensively used by J. B. Fountain to study the so called abundant semigroups in [9]. Let be an arbitrary semigroup. Then, we define. Dually, we define and define, while

where is the smallest ideal containing element saturated by

and, that is, is a union of some -classes and also a union of some -classes of.

It was given by M. V. Lawson in [10] the definition of on a semigroup as

where is the idempotents set of. It can be easily seen that and for any regular elements of a semigroup, if and only if.

In order to further investigate the structure of non-regular semigroups, we have to generalize the usual Green’s relations. For this purpose, J. B. Fountain and F. Pastijn both generalized the Green’s relations to the so called Green -relations in [8] and [9], respectively and by using these Green -relations, many new results of -semigroups and abundant semigroups have been obtained by many authors in [10-18]. For the results of all other generalized Green’s relations and their mutual relationships, the reader is referred to a recent paper of Shum, Du and Guo [17].

In this paper, we introduce the concept of the Green -relations which is a common generalization of the Green -relations and the relation. We also introduce the concept of the -strong semilattice of semigroups and give the semilattice decomposition of a -ample semigroup whose is a congruence. By using this decomposition, we will show that a semigroup is a -ample semigroup whose is a regular band congruence if and only if is a -strong semilattice of completely -simple semigroups. Our result extends and enriches the results of A. H. Clifford, M. Petrich and J. B. Fountain in the literature.

we first generalize the usual Green’s relations and the Green -relations to the Green -relations on a semigroup.

where is the smallest ideal containing saturated by and. We can easily see that is a right congruence on while is only an equivalence relation on. One can immediately see that there is at most one idempotent contained in each -class. If, for some, then we write as, for any. Clearly, for any with, we have.

If a semigroup is a regular semigroup, then every -class of contains at least one idempotent, and so does every -class of. If is a completely regular semigroup, then every -class of contains an idempotent, in such a case, every -class is a group. A semigroup is called an semigroup by J. B. Fountain in [9] if every - and -class of contains an idempotent. One can easily see that on the regular elements of a semigroup. Therefore, all regular semigroups are obviously abundant semigroups. As an analogy of the orthodox subsemigroup in a regular semigroup, a subsemigroup in an abundant semigroup is called a semigroup [9] if each of the -classes of the abundant semigroup contains an idempotent, in such a case, every -class of is a cancellative monoid, which is the generalization of completely regular semigroups within the classes of abundant semigroups. The concept of ample semigroups was first mentioned in the paper of G. Gomes and V. Gould [19]. We now call a semigroup -ample if each -class and each -class contain an idempotent, the concept was first mentioned by Y. Q. Guo, K. P. Shum and C. M. Gong [3]. Certainly, an abundant semigroup is a -ample semigroup, but the converse is not true, and an example can be found in [3]. We now call a semigroup a super -ample semigroup if each -class of contains an idempotent and is a congruence. It is clear that every -class of such super -ample semigroup forms a left cancellative monoid which is a generalization of the completely regular semigroups and the superabundant semigroups in the classes of -ample semigroups.

It is recalled that a regular band is a band that satisfies the identity. For further notations and terminology, such as strong semilattice decomposition of semigroups, the readers refer to [2,3]. For some other concepts that have already appeared in the literature, we occasionally use its alternatives, though equivalent definitions.

A completely simple semigroup is a -simple completely regular semigroup whose Green’s relation is a congruence on, as a natural generalization of this concept, we call a -ample semigroup a completely -simple semigroup if it is a -simple semigroup and the Green -relation is a congruence on.

We first state the following crucial lemma.

Lemma 1 Let be a -ample semigroup with each -class contains an idempotent. Then the Green

relation on is a congruence on if and only if for any,.

Proof. Necessity. Let. Then, and. Since is a congruence on,

. But and so, since every -class contains a unique idempotent.

Sufficiency. Since is an equivalence on, we only need to show that is compatible with the multiplication of. Let and. Then and so that

is left compatible with the multiplication on. Similarly, is right compatible wit the multipication on and thus is a congruence on.

Lemma 2 If are -related idempotents of a -ample semigroup with each -class contains an idempotent, then.

Proof. Since, there are elements of such that

Since is -ample,. Thus, since for regular elements and.

Corollary 3 If is a -ample semigroup with each -class contains an idempotent, then

Proof. Let and. Then, by Lemma 2,. Thus, there exist elements in with and. Then and and the result follows.

Lemma 4 Let be idempotents in a -ample semigroup with each -class contains an idempotent. if, then.

Proof. Since, there are elements in such that. Let and

. Then so that, and so that. It follows that are idempotents with and. Hence and. Now and so that, that is,.

Proposition 5 If is an element of a -ample semigroup, then.

Proof. Certainly, so that. We now show that the ideal is actually an ideal which is saturated by and, since, the result follows. Let

and. Then so that. Also since

is a congruence on,. Now let. Then so that

. Hence if, then so that is indeed an ideal saturated by and, as required.

Proposition 6 On a completely -simple semigroup,.

Proof. Suppose that with. Then, by Proposition 5,. By Lemma 4,

and so, which implies that and hence. Conversely, let with. Now, by Corollary 3, there exists such that. Thus and so. By Proposition 5, and hence. Now we have.

Proposition 7 A completely -simple is primitive for idempotents.

Proof. Let be idempotents in with. Since is a completely -simple semigroup, it follows from Proposition 5 that. Now by the first part of Exercise 3 of [1, 8.4] there is an idempotent of such that and . Let be such that. Then and since we have

Now we have and and so But so that and all idempotent of are primitive.

Lemma 8 In a completely -simple semigroupm, the regular elements of generate a completely simple subsemigroup.

Proof. Let be regular elements of. Since consists of a single -class(by Proposition 6), it follows from Corollary 3 that there is an element with. Hence, we have. Thus, and since is regular. Now we see that and the regularity of follows from that of. The property of completely simple of the subsemigroup generated by regular elements follows Proposition 6, lemma 2 and Corollary 3 easily.

Theorem 9 Let be a -ample semigroup.Then is a semilattice of completely -simple semigroups such that for and, ,.

Proof. If, then so that by Proposition 5,. Now for ,

, and so

Now, by symmetry, we get. By Proposition 5, , so that if, we have for some. Now

and and by the preceding paragraph.

we have and since, Since, and is a congruence on, and so. Now, , we have and since the opposite inclusion is clear, we conclude that.

Because the set of all ideals forms a semilattice under the usual set intersection and that the map is a homomorphism from onto. The inverse image of is just the -class which is thus a subsemigroup of. Hence is a semilattice of the semigroups.

Now let be elements of -class and suppose that. Certainly

so that we have, that is, and. It follows that

and consequently, since, we have. A similar argument shows that.

From the last paragraph, we have so that is a -ample semigroup.

Furthermore, if, then by Proposition 6, so that, by Corollary 3, there is an element

in. Thus, are -related in so that is a

-simple semigroup.

We need the following crucial lemma.

Lemma 10 Let be a -ample semigroup.

1) Let and. Then, there exists with;

2) Let, and. Then,;

3) Let and be such that. Then,.

Proof. 1) Let. Then, by Lemma 1, and are in the same -class and so. Let. Then and.

2) By the definition of “”, there exist such that,. From and, we have. Similarly,. Thus,. Similarly, and so as required.

3) We have for some whence

Following Proposition 7, we can easily prove the following lemma Lemma 11 Let be a homomorphism from a completely -simple semigroup into another completely -simple semigroup. Then.

If is a homomorphism between two completely -simple semigroups. Then the Green -relations, are preserved, so that is preserved. We call a homomorphism preserving, are good. By Proposition 7 and Lemma 10, we can show that a completely -simple semigroup is primitive.

In this section, we introduce the -strong semilattice of semigroups which is a generalization of the well known strong semilattice of semigroups.

Definition 12 Let be a semilattice decomposition of semigroup into subsemigroups

. Suppose that the following conditions hold in the semigroup.

(C1) for any, there is a band congruence on with congruence classes

, where is the index set and for, is the universal relation

;

(C2) for on and any, there is a homomorphism from into

. Let. Then 1) for, the homomorphism is the identity automorphism of the semigroup.

2) for on, , where is the set

3) for, there exists, for all,

If the semigroup satisfies the above conditions, then we call a -strong semilattice of subsemigroups and write. One can easily see that a -strong semilattice is the ural strong semilattice if and only if all for all on.

Following Theorem 9, we can easily see that a -ample semigroup is a semilattice of completely simple semigroups. In this section, we introduce the band congruence on a regular -ample semigroup and the structure homomorphisms set. Finally, we will show the main result of the paper, that is, a -ample semigroup is a regular -ample semigroup if and only if it is a -strong semilattice of completely -simple semigroups.

Lemma 13 Let be a regular -ample semigroup, that is, is a -ample semigroup with the Green -relation as a regular band congruence on. Then, for any element, we define on as the following:

for some. Then 1) is a band congruence on and for, if and only if for any,

.

2) for on, and is the universal relation on.

3) for on and, ,.

Proof. We only prove 1), 2) and 3) can be proved similarly. Let with, then there exists such that. For any element, we have. Thus

. By the property of regular bands and Lemma 1 and Lemma 8, we easily have

. Now the proof is completed.

We denote the -congruence classes by, following Lemma 13,

is a singleton.

Lemma 14 Let be a regular -ample semigroup.

1) For any on and. Let, there exists a unique element

such that.

2) For any on and, , if for some idempotent, then

, and.

Proof. 1) By Lemma 13 2) and Lemma 10 1), for any, the element

such that. Easily see. If there is another

such that, then there are idempotents such that and so

, thus since, which implies and hence

, that is,. Thus by Lemma 10 (ii), is required.

2) Since

and, we have, that is,

. Also, since and, we have and so that

by (i). Thereby, we have. Similarly, we have

. Since is arbitrarily chosen element in, we can particularly choose. In this way, we obtain that and consequently, by Lemma 1, we have.

Lemma 15 Let be a regular -ample semigroup. For any on and

, define a mapping from into with, where

is defined in Lemma 14. Write. Then 1) is a homomorphism.

2) for, is the identity homomorphism of.

3) for on,.

4) for, there exists, for all,

Proof. 1) Following Lemma 14, is well defined. For and, by Lemma 14 again,

and so

2) It follows easily since is primitive.

3) We only need to show that for any , for some. Let, and, we have and

, and so

which implies for some.

4) For, we need to prove that. In fact, it suffices to show that for any and, we have. For this purpose, we let and. Then, by (i), we have and

. Let, then, because is a completely -simple semigroup, and,

, are elements in. We obtain that and

. By Lemma 1, we conclude that

In other words,. Thus, by the regularity of the band

, we can further simplify the above equality to, that is,

. It hence follows, by the definition of, that is

.

Now let,. Then because is a -equivalence class of. Now, by (i), and for and

. Since we assume that, we have. Similarlywe have. Thus, we have

and also

However, by the definition of the natural partial order “” on semigroup, we have

. On the other hand, because every is a completely -simple semigroup,

is a primitive semigroup. Hence, we obtain that.

Finally, we can easily see that for any and, if and are all subsets of the same -class, then and determine the same mapping and hence for any, we have

Theorem 16 A -ample semigroup is a regular -ample semigroup if and only if it is a -strong semilattice of completely -simple semigroups.

Proof. We have already proved the necessity from Lemma 14 and Lemma 15. We now prove the sufficiency part of the theorem. We first show that the Green’s -relation is a congruence on. In fact, if, then by the definition of -strong semilattice and that each is a completely

simple semigroup, we see that there exist and satisfying the following equalities

since is a band congruence on. Hence, we deduce that

Now by Lemma 1, is a congruence on.

To see that is a regular band, by a result of [18], we only need to show that the Green’s relations and are both congruences on. We only show that is a congruence in as is a congruence in which can be proved in a similar fashion. Since is a -ample semigroup, we can let and, where, with. Thenwe have and. By the definition of -strong semilattice, we can find homomorphisms

and, such that

and

Thereby,. Analogously, we can also prove that. This proves that is left compatible on. Since is always right compatible, we see that is a congruence on, as required. Dually, is also a congruence on. Thus by [18] (see II. 3.6 Proposition), is a regular band and hence is a regular cryptic -ample semigroup. Our proof is completed.

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