In any U-abundant semigroup with an Ehresmann transversal, two significant components R and L are introduced in this paper and described by Green’s -relations. Some interesting properties associated with R and L are explored and some equivalent conditions for the Ehresmann transversal to be a quasi-ideal are acquired. Finally, a spined product structure theorem is established for a U-abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L.
Suppose that S is a regular semigroup and a subsemigroup of S. We denote the intersection of and by and that and . An inverse transversal of the semigroup S is a subsemigroup that contains exactly one inverse of every element of S, that is, S is regular and inverse with . This important concept was introduced by Blyth and McFadden . Thereafter, this class of regular semigroups excited many semigroup researchers’ attention and a good deal of important results were obtained (see [1,2,3,4] and references therein). Tang  has shown that for S a regular semigroup with an inverse transversal , I and are both bands with I left regular and right regular. These two bands play an important role in the study of regular semigroups with inverse transversals. Other important subsets of S are and . They are subsemigroups with R and L left and right inverse, respectively.
The concept of an adequate transversal was introduced in the class of abundant semigroups by El-Qallali  as the generalisation of the concept of an inverse transversal. Chen, Guo and Shum [6,7] obtained some important results about a quasi-ideal adequate transversal. Kong  explored some properties about adequate transversals. Kong and Wang  considered the product of quasi-ideal adequate transversals. The concept of adequate transversals was generalised, and the product of quasi-ideal adequate transversals was generalised to the refined quasi-adequate transversals and quasi-Ehresmann transversals [10,11,12,13,14]. In 2008, Kong  introduced two important subsets R and L in the adequate transversal case and described them by Green’s ∗-relations. Furthermore, Kong established a spined product structure theorem for an abundant semigroup with a quasi-ideal adequate transversal by means of R and L. It was interesting that this spined product structure theorem was independently reobtained by Al-Bar and Renshaw  and later, they  noticed the fact that this structure theorem had been already obtained by Kong.
To extend the class of regular semigroups and the class of abundant semigroups, a new relation on a semigroup S is introduced in the following way. Let be the set of all idempotents of a semigroup S and suppose that U is a non-empty subset of . Then the relation on S is defined as follows:
It is easy to see that . In particular, if S is an abundant semigroup with , then . Furthermore, if S is a regular semigroup with , then . Dually, a relation is defined, and for any results concerning , we also have the dual results for . Recall that a semigroup S is a weakly U-abundant semigroup if each -class and each -class of S contain an idempotent from U, and we denote such a weakly U-abundant semigroup by . In this case, we call the set U a set of distinguished idempotents of , and an element u of U is called a distinguished idempotent of . It is well known that the relations and are always right congruences on a semigroup S, but the same need not be true for . A weakly U-abundant semigroup is said to satisfy the congruence condition (C) if and are a right congruence and a left congruence on , respectively. We call a weakly U-abundant semigroup having the congruence condition (C) a U-abundant semigroup. Clearly, regular semigroups and abundant semigroups are all U-abundant semigroups with . A U-abundant semigroup is called an Ehresmann semigroup (quasi-Ehresmann semigroup) if U is a semilattice (band). The class of Ehresmann semigroups, named by Lawson , is an important subclass of the class of U-abundant semigroups, which are generalizations of inverse semigroups in the class of regular semigroups and adequate semigroups in the class of abundant semigroups.
The concept of an Ehresmann transversal was introduced almost simultaneously in three papers by Ma, Ren, Gong , Wang  and Yang  in the class of U-abundant semigroups as the generalisation of the concept of an inverse transversal and an adequate transversal. The aforementioned three kinds of Ehresmann transversals were basically the same, with only slight difference. To achieve interesting properties of Ehresmann transversals which parallel to those of adequate transversals, we give the following definition of Ehresmann transversals combining the aforementioned three kinds of Ehresmann transversals. In [19,20,21], some basic properties about Ehresmann transversals are given.
In the present paper, we continue along the lines of [3,15,16] by exploring the properties associated with R and L of Ehresmann transversals of U-abundant semigroups. The main result of this paper is to establish a spined product structure theorem for a U-abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L. The related results concerning adequate transversals are generalised and enriched.
Throughout this paper, a U-abundant semigroup is always denoted as and the -class and the -class of containing the element a as and , respectively. Moreover, for any the distinguished idempotents in and are denoted by and , respectively. In particular, when is an Ehresmann semigroup, then there exists a unique distinguished idempotent in and a unique distinguished idempotent in . Therefore, as for Ehresmann semigroups, for all , if and only if ; if and only if , and consequently and . We remark here that an excellent survey of investigations of restriction semigroups and Ehresmann semigroups was given by Gould  and the class of Ehresmann monoids were deeply investigated by Branco, Gomes and Gould [23,24].
Suppose that S is a semigroup and U a non-empty subset of . Let . Then
If , then if and only if . The following result is immediate from the definition of , and there is a dual result for .
Let a be an element of a weakly U-abundant semigroup and e a distinguished idempotent of . Then the following statements are equivalent:
, and for all implies .
Let be a U-abundant semigroup and a U-abundant subsemigroup of , that is, and . Then the U-abundant subsemigroup is called a -subsemigroup of if for any , there exist and , or equivalently, if and .
An Ehresmann -subsemigroup of a U-abundant semigroup is called an Ehresmann transversal of if U forms an order ideal of , and for any , there exist and a unique such that , where and . It can easily be shown that e and f are uniquely determined by x and T (see ). Hence, we denote e by and f by and we have , .
A subsemigroup of a semigroup is called a quasi-ideal if , and the corresponding Ehresmann transversal is called a quasi-ideal Ehresmann transversal. Let be an Ehresmann transversal of the U-abundant semigroup . Let
Let be a U-abundant semigroup with an Ehresmann transversal . Then
if and only if ; if and only if ;
If and , then ; if and , then ;
If T is a quasi-ideal of S, then .
Similarly to the adequate transversal case, one can easily show the following decomposition formula. It is worth remarking that if the condition “quasi-ideal” is strengthened as “multiplicative,” the proof was obtained in .
Let be a U-abundant semigroup with a quasi-ideal Ehresmann transversal . For any , then ; ; .
The so-called Miller-Clifford theorem is crucial and frequently used in this paper.
Let e and f be -equivalent idempotents of a semigroup S. Then each element a of has a unique inverse in , such that and .
Let . Then if and only if contains an idempotent.
3 Some properties of R and L
The objective in this section is to give some properties about R and L. It is known that R and L play an important role in constructing regular semigroups with quasi-ideal inverse transversals and abundant semigroups with quasi-ideal adequate transversals. For any result concerning R, there is a dual result for L which we list but omit its proof.
Let be an Ehresmann transversal of a U-abundant semigroup . If and are such that , then .
Since and is an Ehresmann semigroup, there exist , such that
From the definition of an Ehresmann transversal, we have , where and . Moreover, . It follows that , consequently . Similarly, and . Thus, .□
Let be an Ehresmann transversal of a U-abundant semigroup . Then for any , and we call the unique U-inverse in of x.
For every , since and are all U-regular, from we deduce that , so by Lemma 2.6 x has an inverse and . Thus, , so by Proposition 3.1 . Now let denote the unique U-inverse of in the Ehresmann semigroup . Then from and Definition 2.3, the uniqueness of is obvious. That is .□
Let be a U-abundant semigroup with an Ehresmann transversal . If , then S is regular if and only if is an inverse semigroup; in this case, is an inverse transversal of S.
Suppose that S is a regular semigroup. Then by and , every regular element has a unique inverse in and, in particular, every element in has a unique U-inverse in itself. Hence, is an inverse semigroup and is an inverse transversal of S.
Conversely, assume that is an inverse semigroup. Then and for every x in S, there exists such that . Consequently,
and so x is regular in S. Thus, S is a regular semigroup.□
Let be a U-abundant semigroup with an Ehresmann transversal . If is a right ideal of S, then for every and . Consequently, and thus .
Dually, if is a left ideal of S, then for every and . Consequently, and thus .
By Definition 2.3, for every , , where and . Since is a right ideal of S, and thus . Let and denote the unique U-inverse in of e. Since , we have . Thus, for every . Consequently, .□
Suppose that is a U-abundant semigroup with an Ehresmann transversal . Let
Consequently, , and .
It is clear that if , there exists such that .
Conversely, for , if there exists such that , then . Hence, . Therefore, .□
Suppose that is a U-abundant semigroup with an Ehresmann transversal and is a quasi-ideal of S. Let and . Then and are both U-abundant semigroups with an Ehresmann transversal , where is a right ideal of R and a left ideal of L.
Moreover, for and , if and , then and .
Obviously, and . For any , by Lemma 2.5, we have and so R is a subsemigroup of .
For any and , we have and consequently since is a quasi-ideal of S. Thus, is a right ideal of R and by Proposition 3.4, . Thus, is an Ehresmann transversal of .
If and for and , then we have
Similarly, we have . Thus, , and consequently . Therefore,
Similarly, we have .□
Let be an Ehresmann transversal of a U-abundant semigroup . Then the following statements are equivalent:
is a quasi-ideal of S;
is a left ideal of L and a right ideal of R;
R is a left ideal and L is a right ideal of S.
(1) (2). For any and , there exist such that and . So we have
(2) (3). This is trivial.
(3) (4). For any and , there exist such that and . Thus,
(4) (5). This is clear since
(5) (6). This is clear since and .
(6) (7). This is obvious.
(7) (8). If (7) holds, then for any , we have
since . Hence, by Proposition 3.5 and . Dually, .
(8) (9). For any , with , we have and R is a left ideal of S. Dually, implies that L is a right ideal of S.
(9) (1). For any and , we have
Consequently, and is a quasi-ideal of S.□
4 The main theorem
The main objective of this paper is to establish a structure theorem for U-abundant semigroups with quasi-ideal Ehresmann transversals. In what follows, denotes a U-abundant semigroup with a right ideal Ehresmann transversal . Then, by Proposition 3.4, for every and . And denotes a U-abundant semigroup with a left ideal Ehresmann transversal , which has a dual result of , especially, for every . Then R and L will be the components in such construction, which satisfy some compatible conditions.
Let and be a pair of U-abundant semigroups sharing a common Ehresmann transversal . Suppose that is a right ideal of R and a left ideal of L. Let described by be a mapping such that for any and for any :
If , then ;
For any , , if , then ; if , then .
Define a multiplication on the set
Then is a U-abundant semigroup with a quasi-ideal Ehresmann transversal which is isomorphic to .
Conversely, every U-abundant semigroup with a quasi-ideal Ehresmann transversal can be constructed in this manner.
To prove this theorem, in the following we will give a sequence of lemmas.
The multiplication on is well-defined.
We only need to prove . It follows from that and . It is obvious that both a right ideal and a left ideal are quasi-ideals, and so we have
On the other hand, we have
Therefore, and .□
is a semigroup.
Let . Then we have
On the other hand, we have
since Thus, and so is a semigroup.□
Let . Then if and only if .
Since , and note that , and , it is clear that if , then
Thus, . Conversely, if , then . It follows from that .□
Let be set of distinguished idempotents of . Then is a weakly abundant semigroup.
Let . Then and and so . Thus,
Suppose that such that . Then, from , we have and . From , we deduce that . Thus, and so . Therefore, and .
For , we have with and so . Thus,
Suppose that such that . Then and . That implies that . Thus, and so . Therefore, and .□
Let . Then
if and only if if and only if .
if and only if if and only if .
To prove the first part of this Lemma, by Lemma 4.4, it then suffices to show that if and only if . If , then and Conversely, if , then and , and this implies
Thus, , and from Lemma 2.4 we deduce that . That if and only if is obvious.
is dual to (1).□
is an -abundant semigroup.
We need only to prove that satisfies the congruence conditions. Suppose that with . It follows from Lemma 4.5 that and . For any , we have
Since and so by (3), thus . Consequently, by Lemma 4.5 again . Therefore, is a right congruence on , and dually, is a left congruence. That is, is an -abundant semigroup.□
Let with the distinguished idempotent set . Then is an Ehresmann -subsemigroup of , and is isomorphic to .
Clearly, . For any , we have and is a subsemigroup. For any , we naturally define a mapping by . It is easy to see that is bijective. For any , . Thus, is a homomorphism and . By Lemmas 4.4 and 4.5, with . Therefore, is a -subsemigroup of .□
is an Ehresmann transversal of .
For any , let . It is obvious that with , .
If can be written in another form , where and . By Lemma 4.7, we have and . From , we deduce that . Similarly, we have . Therefore,
and so . It follows from and Lemma 4.3 that . From , we deduce that since . Thus, since . Note that and , we have and similarly . Consequently, and similarly . Hence,
This implies , and combining with implies that . Premultiplying by , we obtain . Note that . Therefore, and by Definition 2.3, is an Ehresmann Transversal of .□
W is a quasi-ideal of .
For any , we have
since both a right ideal and a left ideal are quasi-ideals.□
Now we shall prove the converse half of Theorem 4.1. Assume that is a U-abundant semigroup with a quasi-ideal Ehresmann transversal . Let
Then by Proposition 3.6, R and L are both U-abundant semigroups with an Ehresmann transversal , where is a right ideal of R and a left ideal of L.
For every , let . Then and so since is a quasi-ideal of S. Obviously, the map satisfies (1), (2) and (3). Therefore, we may acquire a U-abundant semigroup in the way of the direct part of Theorem 4.1. In the following, we prove that is isomorphic to .
For every , we define by . It is clear that is well-defined and by Proposition 3.6, is injective.
For any , we have
and so is a homomorphism.
For any , it is obvious that and with and . Consequently, and , that is, is surjective. Therefore, is an isomorphism.
Funding information: This research was supported by the Project Higher Educational Science and Technology Program of Shandong Province (J18KA248), Shandong Province NSF (ZR2020MA002, ZR2016AM02) and the NSFC (11801081, 11871301).
Author contributions: Xiangjun Kong contributed in an overall design, put forward the main results and gave most of the proof; Pei Wang gave the idea of writing and specific writing of this paper, using latex to compile this paper and give some proof; Jian Tang optimized the structure of this paper, polished the language, and gave the valuable suggestion for Theorem 4.1 which greatly improved the readability of the paper.
Conflict of interest: Authors state no conflict of interest.
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