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.
2. Properties of r-Ample Semigroups
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.
3. G-Strong Semilattice Structure of r-Ample Semigroups
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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NOTES
*The research is supported by the national natural science foundation of China (11371174, 11301227) and natural science foundation of Jiansu (BK20130119).