Semigroup Forum

, Volume 94, Issue 3, pp 738–776 | Cite as

On \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups and their subclasses

Research Article

Abstract

We study semigroups that behave nicely with respect to a distinguished subset of idempotents E, both in terms of the extended Green’s relations \(\widetilde{\mathcal {K}}_E\) and as unary semigroups. New structure theorems are given, notably in the case of central idempotents. Finally, the decomposition theorems are applied to the study of regular semigroups with particular generalized inverses.

Keywords

Extended Green’s relations Abundant semigroups Restriction semigroups Generalized inverses 

1 Introduction and notation

Classical classes of regular semigroups (regular, completely regular, inverse, Clifford semigroups) were firstly defined by means of generalized inverses. But rapidly, alternative characterizations in terms of the original Green’s relations [21] appeared. Then it was observed that other classes of semigroups, for instance right PP (principal projective) semigroups [12], could also be characterized in terms of relations in the semigroup. But this time, extensions of the classical Green’s relations were needed. And yet other extensions appeared [9, 34, 35].

A fruitful program in semigroup theory has then been the study of non-regular analogs of regular, completely regular, or inverse semigroups in terms of extended Green’s relations (York’s school [9, 10, 12, 13, 14, 15, 16, 17, 32, 35, 50] and more recently Shum and his students [4, 23, 39, 47, 52]).

We pursue this program in the present article, and study analogs to completely regular, completely simple and Clifford semigroups, defined in terms of a distinguished subset of idempotents E (whose elements act as minimal left and right identities). The main tool for their study as plain semigroups (which is the most classical and most developed approach) are the extended Green’s relations \(\widetilde{\mathcal {K}}_E\). The first two cases are not new and have already been the subject of research [39, 47, 50, 52]. But we hopefully present some interesting new properties of these semigroups. The third case is, to a large extent, new. Also, a major part of the article is devoted to another, more algebraic aspect of these semigroups: their structure as unary semigroups. Whereas this approach is largely developed in the literature dealing with extensions of inverse semigroups (the restriction semigroups, or weakly E-ample semigroups, see [26]), such a study is not standard for extensions of completely regular semigroups.

1.1 Classical notions, Green’s relations

We first recall the main notions of semigroup theory that will be used throughout the paper, and shall use [27] as a general reference. In this paper, S denotes a semigroup, M a monoid and E(S) the set of idempotents of S. By \(S^1\) we mean the monoid generated by S (\(M^1=M\)), and by \(S^0\) the semigroup (with or without zero) with an extra 0 added. By E we always mean a distinguished subset of E(S). For A subset of S, \(Z(A)=\{x\in S|\, (\forall a\in A) \; xa=ax \}\) denotes its centralizer (commutant).

Let \(a\in S\). We say that a is regular if there exists \(x\in S\) such that \(axa=a\), in which case x is called an associate, or inner inverse, of a. If moreover x satisfies \(xax=x\) then it is called a reflexive inverse. A particular solution to \(axa=a, xax=x, ax=xa\) is unique if it exists and usually called the group inverse of a, and denoted by \(a^\#\). We denote the set of group invertible elements (also called completely regular elements) by H(E(S)). Finally, in a monoid M, a is unit regular if \(a\in aM^{-1}a\). A semigroup S is regular (completely regular) if all its elements are regular (completely regular). It is inverse if every element admits a unique reflexive inverse.

We will make use of the natural partial order on regular semigroups, defined by \(a\omega b\) if \((\exists e,f\in E(S)), a=eb=bf\). We will note \(a<b\) if \(a\omega b\) and \(a\ne b\). On E(S), \(\omega \) is the intersection of preorders \(\omega _l\) and \(\omega _r\), where \(e\omega _l f\Leftrightarrow ef=e\) and \(e\omega _r f\Leftrightarrow ef=f\).

Of major importance in this article are certain preorders and relations in a semigroup. If \(\le _{\mathcal {K}}\) is a preorder, then \(a\mathcal {K}b\Leftrightarrow \{a\le _{\mathcal {K}}b \text { and } b\le _{\mathcal {K}}a\}\), and \(\mathcal {K}(a)=\{b\in S|\, b\mathcal {K}a\}\) denotes the \(\mathcal {K}\)-class of a (this notation is preferred to the most classical \(\mathcal {K}_a\) to avoid multiple subscripts). For any equivalence relation \(\sigma \) on S, \(A\subseteq S\) is \(\sigma \) -saturated (or saturated by \(\sigma \)) if A is a union of \(\sigma \)-classes, or equivalently if \((a,b)\in \sigma \) and \(a\in \sigma \) implies \(b\in \sigma \). A semigroup S is \(\sigma \) -abundant ( \((E,\sigma )\) -abundant) if every \(\sigma \)-class contains idempotents of S (intersects E). It is \(\sigma \) -simple if it contains a single \(\sigma \)-class.

Green’s preorders and relations [21] have proved fundamental in the early development of semigroup theory, notably in the study of regular semigroups. They are defined upon principal (left, right, two-sided) ideals—or mutual divisibility—as follows. For elements a and b of S
$$\begin{aligned} a \le _{\mathcal {L}} b&\Longleftrightarrow S^1 a\subseteq S^1 b\Longleftrightarrow \left\{ (\exists x\in S^1)\; a = xb\right\} ;\\ a \le _{\mathcal {R}} b&\Longleftrightarrow aS^1\subseteq bS^1\Longleftrightarrow \left\{ (\exists x\in S^1)\; a = bx\right\} ;\\ a \le _{\mathcal {J}} b&\Longleftrightarrow S^1aS^1\subseteq S^1bS^1\Longleftrightarrow \left\{ (\exists x,y\in S^1)\; a = xby\right\} . \end{aligned}$$
The intersection of the preorders (resp. equivalence relations) \(\le _{\mathcal {L}}\) and \(\le _{\mathcal {R}}\) is also a preorder (resp. equivalence relation), denoted by \(\le _{\mathcal {H}}\) (resp. \(\mathcal {H}\)). The relations \(\mathcal {L}\) and \(\mathcal {R}\) commute, so that their join \(\mathcal {D}=\mathcal {L}\vee \mathcal {R}\) is just \(\mathcal {R}\circ \mathcal {L}=\mathcal {L}\circ \mathcal {R}\).
Table 1 summarizes the main classes of regular semigroups and their characterization in terms of generalized inverses and Green’s relations.
Table 1

Classes of regular semigroups

Usual name

Generalized inverses

Green’s relations

Article’s terminology

Regular

\((\forall a \in S, \exists x\in S)\,axa=a\)

\((\forall a \in S, \exists e\in E(S))\,a\mathcal {L}e\) (resp. \(a\mathcal {R}e\))

\((E(S),\mathcal {L})\)-abundant or simply \(\mathcal {L}\)-abundant

Completely regular

\((\forall a \in S, \exists x\in S)\,axa=a, ax=xa\)

\((\forall a \in S, \exists e\in E(S))\, a\mathcal {H}e\)

\((E(S),\mathcal {H})\)-abundant or simply \(\mathcal {H}\)-abundant

Completely simple

 

\((\forall a \in S, \exists e\in E(S))\, a\mathcal {H}e\) and \((\forall a,b \in S)\, a\mathcal {J}b\) (or \(a\mathcal {D}b\))

\((E(S),\mathcal {H})\)-abundant and \(\mathcal {J}\)-simple

Inverse

\((\forall a \in S, \exists ! x\in S)\, axa=a, xax=x\)

\((\forall a \in S, \exists ! e,f\in E(S))\,a\mathcal {L}e\) and \(a\mathcal {R}e\)

 

Clifford

\((\forall a \in S, \exists ! x\in S)\, axa=a, xax=x ax=xa\)

\((\forall a \in S, \exists e\in E(S))\, a\mathcal {H}e\) and \((\forall e,f\in E(S))\, ef=fe\)

\((E(S),\mathcal {H})\)-abundant and E(S) semilattice

We assume the reader familiar with the following facts:
  1. (1)

    Completely regular semigroups are exactly union of groups, and semilattices of completely simple semigroups;

     
  2. (2)

    Completely simple semigroups are completely regular semigroups with primitive idempotents, and by the Rees–Suschkewitsch Theorem they are also matrix semigroups over a group;

     
  3. (3)

    Clifford semigroups are completely regular and inverse semigroups. They are also (strong) semilattices of groups, regular semigroups with central idempotents or regular subdirect products of groups and groups with zero.

     

1.2 Extended Green’s relations, and associated classes of semigroups

We now introduce the following extended Green’s preorders [12, 13]. The associated equivalence relations appeared in link with the homological classification of monoids (left PP monoids [30]), but also via potential properties [38, 48]. They proved useful in the study of non-regular semigroups and led to the definition of new classes of semigroups (adequate semigroups [12], abundant semigroups [13], ample semigroups [32], and their one-sided versions).

For elements a and b of S, the extended Green’s preorders \(\le _{\mathcal {L}^*}\) and \(\le _{\mathcal {R}^*}\) are defined by
$$\begin{aligned} a \le _{\mathcal {L}^*} b&\Longleftrightarrow \left\{ (\forall x,y\in S^1)\; bx=by \Rightarrow ax=ay\right\} ;\\ a \le _{\mathcal {R}^*} b&\Longleftrightarrow \left\{ (\forall x,y\in S^1)\; xb=yb \Rightarrow xa=ya \right\} . \end{aligned}$$
For \(E\subseteq E(S)\), the preorders \(\le _{\widetilde{\mathcal {L}}_E}\) and \(\le _{\widetilde{\mathcal {R}}_E}\) are defined (see for instance [19, 34]) by
$$\begin{aligned} a \le _{\widetilde{\mathcal {L}}_E} b&\Longleftrightarrow \left\{ (\forall e\in E)\; be=b\Rightarrow ae=a\right\} ;\\ a \le _{\widetilde{\mathcal {R}}_E} b&\Longleftrightarrow \left\{ (\forall e\in E)\; eb=b\Rightarrow ea=a\right\} . \end{aligned}$$
(For \(E=E(S)\) we forget the subscript.)

The origin of the relations \({\widetilde{\mathcal {K}}}\) lies in the thesis of El-Qallali [9]. Inspired by this work and the article of de Barros [6], Lawson introduced the relations \({\widetilde{\mathcal {K}}_E}\) in [34] and [35].

Let \(a\in S\) and \(e\in E\). Then \(a\widetilde{\mathcal {L}}_Ee\) if and only if \(ae=a\) (e is a right identity of a) and \(f\in E,\; af=a\) implies \(ef=e\). In other words, \(a\widetilde{\mathcal {L}}_Ee\) if and only if e is minimal, with respect to \(\omega _l\), within the set \(E_r(a)=\{e\in E|\, ae=a\}\) of right identities of a that belong to E.

It is well known that \(a\le _{\mathcal {L}^*} b\) in S if and only if \(a\le _{\mathcal {L}} b\) in an oversemigroup of S [13], that \(\mathcal {L}\subseteq \mathcal {L}^*\subseteq \widetilde{\mathcal {L}}\subseteq \widetilde{\mathcal {L}}_E\) (Lemma 4.1 in [26]) and that \(\mathcal {L}\), \(\mathcal {L}^*\) and \(\widetilde{\mathcal {L}}(=\widetilde{\mathcal {L}}_{E(S)})\) coincide for regular semigroups (Lemma 4.14 in [26]). Moreover, \(\mathcal {L}, \mathcal {L}^*\) are right congruences while \(\mathcal {R}, \mathcal {R}^*\) are left congruences. It was noticed by El-Qallali [9] that \(\widetilde{\mathcal {L}}\) (\(\widetilde{\mathcal {R}}\)) is not a right (left) congruence in general. In particular, \(\widetilde{\mathcal {L}}_E\) (\(\widetilde{\mathcal {R}}_E\)) may not be a right (left) congruence. If this is the case, we will say that S is \(\widetilde{\mathcal {L}}_E-\) (resp. \(\widetilde{\mathcal {R}}_E-\) ) congruent, or, following Fountain et al. [16], that S satisfies (CL) (resp. (CR)). A semigroup which satisfies condition (CL) and (CR) is also said to satisfy the congruence condition (C) [35].

The preorders \(\le _{\mathcal {H}^*}, \le _{\widetilde{\mathcal {H}}}\) and \(\le _{\widetilde{\mathcal {H}}_E}\) (resp. relations \(\mathcal {H}^*, \widetilde{\mathcal {H}}\) and \(\widetilde{\mathcal {H}}_E\)) are defined analogously to \(\le _{\mathcal {H}}\) (resp. \(\mathcal {H}\)), as the meet of the left and right preorders (resp. relations). Also, the relations \(\mathcal {D}^*, \widetilde{\mathcal {D}}\) and \(\widetilde{\mathcal {D}}_E\) are defined analogously as the join of the left and right extended relations, but as these relations do not commute in general, their join may differ from their product.

Following Ren et al. [47] (see also [50]), we finally introduce a relation \(\widetilde{\mathcal {J}}_E\) as follows: let \(\widetilde{J}_E[a]\) be the smallest ideal containing a, saturated by \(\widetilde{\mathcal {L}}_E\) and \(\widetilde{\mathcal {R}}_E\). We pose \(a \widetilde{\mathcal {J}}_E b \Leftrightarrow \widetilde{J}_E[a]=\widetilde{J}_E[b]\). This characterization actually extends the characterization of \(\widetilde{\mathcal {L}}_E\) (resp. \(\widetilde{\mathcal {R}}_E\)) in terms of saturated ideals by Lawson [34]: \(a \widetilde{\mathcal {L}}_E b \Leftrightarrow \widetilde{L}_E[a]=\widetilde{L}_E[b]\), where \(\widetilde{L}_E[a]\) is the smallest left ideal containing a, saturated by \(\widetilde{\mathcal {L}}_E\). For instance, it is proved by Lawson [34] that \((\forall e\in E) \widetilde{L}_E[e]=Se\).

We are now in position to define the classes of semigroups of interest in this study. Let S be a semigroup and \(E\subseteq E(S)\). We will say that:
  • S is \((E,\widetilde{\mathcal {H}}_E)\) -abundant if any element of S is \(\widetilde{\mathcal {H}}_E\)-related to an element of E. These semigroups (with \(E=U\)) were formerly named weakly U-superabundant semigroups;

  • S is completely \((E,\widetilde{\mathcal {H}}_E)\) -abundant if it is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and, in addition, \(\widetilde{\mathcal {L}}_E\) and \(\widetilde{\mathcal {R}}_E\) are right and left congruences. These semigroups (with \(E=U\)) were formerly named U-superabundant semigroups, or weakly U-superabundant semigroups with (C);

  • S is completely E -simple if it is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {D}}_E\)-simple;

  • S is an E -Clifford restriction semigroup if it is \((E,\widetilde{\mathcal {H}}_E)\)-abundant, \(\widetilde{\mathcal {H}}_E\)-congruent and the elements of E are central (\(E\subseteq Z(S)\)).

Table 2 summarizes the main classes of (non-regular) semigroups defined in terms of extended Green’s relations. Our objects of study are the semigroups in the third family, defined in terms of relations \(\widetilde{\mathcal {K}}_E\).
Table 2

Classes of non-regular semigroups

“Usual” name

Extended Green’s relations

Article’s terminology

Left abundant

\((\forall a \in S, \exists e\in E(S))\,a\mathcal {L}^* e\)

\((E(S),\mathcal {L}^*)\)-abundant or simply \(\mathcal {L}^*\)-abundant

Abundant

\((\forall a \in S, \exists e\in E(S))\,a\mathcal {L}^* e\)

\((E(S),\mathcal {L}^*,\mathcal {R}^*)\)-abundant or simply \(\mathcal {L}^*,\mathcal {R}^*\)-abundant

Superabundant

\((\forall a \in S, \exists e\in E(S))\,a\mathcal {H}^* e\)

\((E(S),\mathcal {H}^*)\)-abundant or simply \(\mathcal {H}^*\)-abundant

Left semiabundant

\((\forall a \in S, \exists e\in E(S))\,a\widetilde{\mathcal {L}}e\)

\((E(S),\widetilde{\mathcal {L}})\)-abundant or simply \(\widetilde{\mathcal {L}}\)-abundant

Semisuperabundant

\((\forall a \in S, \exists e\in E(S))\,a\widetilde{\mathcal {H}}e\)

\((E(S),\widetilde{\mathcal {H}})\)-abundant or simply \(\widetilde{\mathcal {H}}\)-abundant

Left E-semiabundant (weakly left E-abundant)

\((\forall a \in S, \exists e\in E)\,a\widetilde{\mathcal {L}}_Ee\)

\((E,\widetilde{\mathcal {L}}_E)\)-abundant

Weakly E-superabundant

\((\forall a \in S, \exists e\in E)\,a\widetilde{\mathcal {H}}_Ee\)

\((E,\widetilde{\mathcal {H}}_E)\)-abundant

Weakly E-superabundant with (C)

\((\forall a \in S, \exists e\in E)\; a\widetilde{\mathcal {H}}_Ee\) \(\widetilde{\mathcal {L}}_E\) right congruence \(\widetilde{\mathcal {R}}_E\) left congruence

Completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant

 

\((\forall a \in S, \exists e\in E)\; a\widetilde{\mathcal {H}}_Ee\,(\forall a,b\in S)\; a\widetilde{\mathcal {D}}_Eb\)

Completely E-simple

 

\((\forall a \in S, \exists e\in E)\; a\widetilde{\mathcal {H}}_Ee\,(\forall a,b,c,d\in S)\; a\widetilde{\mathcal {H}}_Eb, c\widetilde{\mathcal {H}}_Ed \Rightarrow ac\widetilde{\mathcal {H}}_Ebd\,(\forall a\in S, \forall e\in E)\; ae=ea\)

E-Clifford restriction semigroup

The choice of terminology and notations is by no means standard, and probably arguable. However, it must be understood that most of the concepts discussed in this paper appear here and there in the literature in various forms and names, and sometimes regarding a priori very different topics (category theory, homology, function systems...). For instance, the distinguished subset of idempotents is denoted by U in [34, 47], but E in [17, 26]. The extended Green’s relations \(\widetilde{\mathcal {L}}_E\) are therefore sometimes written \(\widetilde{\mathcal {L}}^U\). Also \(\sigma \)-abundant semigroups [8] are sometimes call \(\sigma \)-surjective [4], and the classical abundant semigroups of Fountain [13] refer to \(\mathcal {L}^*,\mathcal {R}^*\)-abundant semigroups. \((E,\widetilde{\mathcal {L}}_E)\)-abundant semigroups were at a time called left E-semiabundant semigroups, but also weakly left E-abundant semigroups, the addition of the prefix“semi-” or the word “weakly” meaning moving from \(\mathcal {K}^*\) to \(\widetilde{\mathcal {K}}_E\) [26]. For more on the terminology, we refer to [26].

The article is organized as follows. In Sect. 2 we focus on the properties of completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and completely E-semigroups, and improve some existing decomposition theorems. Whereas much is known about \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups with the assumption of congruence, less is known otherwise. Section 3 tries to fill the gap. In this section, we also study completely E-semigroups from the perspective of primitive idempotents. In Sect. 4, we describe all the previous classes of semigroups as varieties of unary semigroups. Section 5 recalls the basics of the theory of restriction semigroups (weakly E-ample semigroups), with a special emphasis on central idempotents (Clifford restriction semigroups). We then get an analog of Clifford’s decomposition Theorem, and a “P-theorem” for proper restriction semigroups. Finally, these decompositions are used in Sect. 6 to study regular semigroups with associates (inner inverses) in a distinguished subset T of S.

2 \((E,\widetilde{\mathcal {H}}_E)\)-abundant, completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and completely E-simple semigroups

In this section, we first recall the existing results regarding \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups, completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups and completely E-simple semigroups. Then we improve these results using a simple additional property.

2.1 State of the art

The following results can be found in [34, 39, 47, 50, 52]. They have been rewritten according to the terminology of the article. They mainly concern completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups, that is \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups with \(\widetilde{\mathcal {L}}_E\) a right congruence and \(\widetilde{\mathcal {R}}_E\) a left congruence. To some extent, they show that completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups behave like their classical counterpart.

First, there is a semilattice decomposition.

Lemma 2.1

[47] Let S be completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. Then \(\widetilde{\mathcal {J}}_E=\widetilde{\mathcal {D}}_E\) and it is a semilattice congruence.

Theorem 2.2

[47] A semigroup S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant if and only if it is a semilattice Y of completely \((E_{\alpha },\widetilde{\mathcal {H}}_{E_{\alpha }})\)-abundant, \(\widetilde{\mathcal {J}}_{E_{\alpha }}\)-simple semigroups such that:
  1. (1)

    For each \(\alpha \in Y\) and all \(a\in S_{\alpha }\), \(\widetilde{L}[a](S_{\alpha })= \widetilde{L}[a](S)\) and \(\widetilde{R}[a](S_{\alpha })= \widetilde{R}[a](S)\);

     
  2. (2)

    For all \(a,b\in S_{\alpha }\) and \(x\in S_{\beta }\), \(a\widetilde{\mathcal {L}}_{E_{\alpha }} b\Rightarrow ax\widetilde{\mathcal {L}}_{E_{\alpha }} bx\) and \(a\widetilde{\mathcal {R}}_{E_{\alpha }} b\Rightarrow xa\widetilde{\mathcal {R}}_{E_{\alpha }} xb\).

     

Also, the components of this decomposition are Rees matrix semigroups.

Theorem 2.3

[39] Let M be a monoid, \(I, \Lambda \) be non-empty sets. Let \(P=(p_{\lambda i})\) be a \(\Lambda \times I\) matrix where each entry in P is a unit of M. Suppose that P is normalized at \(1\in I\cap \Lambda \). Then the (normalized) Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) is a completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {J}}_E\)-simple semigroup.

Conversely, every completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {J}}_E\)-simple semigroup is isomorphic to such a Rees matrix semigroup.

And finally, one can construct a completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup from a given semilattice Y and a family \(\{J_{\alpha }|\, \alpha \in Y\}\) of completely \((E_{\alpha },\widetilde{\mathcal {H}}_{E_{\alpha }})\)-abundant, \(\widetilde{\mathcal {J}}_{E_{\alpha }}\)-simple semigroups.

Theorem 2.4

[52] Let Y be a semilattice. For every \(\alpha \in Y\), let \(S_{\alpha }=\mathcal {M}(M_{\alpha },I_{\alpha },\Lambda _{\alpha },P_{\alpha })\) be a Rees matrix semigroup over a monoid \(M_{\alpha }\) such that the entries of \(P_{\alpha }\) are invertible, and \(P_{\alpha }\) is normalized at an element also denoted by \(\alpha \). Suppose also that \(S_{\alpha }\cap S_{\beta }=\emptyset \) if \(\alpha \ne \beta \). Let \(E_{\alpha }=\{(i,p_{\lambda i}^{-1},\lambda ) |\, i\in I_{\alpha }, \lambda \in \Lambda _{\alpha }\}\) and let
$$\begin{aligned} \langle .,.\rangle :&S_{\alpha }\times I_{\beta }\rightarrow I_{\beta }\\ \phi _{\alpha ,\beta }:&S_{\alpha }\rightarrow M_{\beta }\\ a&\mapsto a_{\beta }\\ \left[ .,.\right] :&\Lambda _{\beta }\times S_{\alpha }\rightarrow \lambda _{\beta } \end{aligned}$$
be functions defined whenever \(\alpha \ge \beta \) and satisfying the following conditions.
Let \(a\in S_{\alpha }, b\in S_{\beta }\) and \(c\in S_{\gamma }\):
  1. (I)
    If \(\alpha \beta \gamma =\delta \) then
    $$\begin{aligned} \displaystyle a_{\delta }p_{[\delta ,a]\left\langle b,\left\langle c,\delta \right\rangle \right\rangle } b_{\delta } p_{[\delta ,b]\langle c,\delta \rangle }c_{\delta }=a_{\delta }p_{[\delta ,a]\langle b,\delta \rangle } b_{\delta } p_{\left[ \left[ \delta ,a \right] ,b\right] \langle c,\delta \rangle }c_{\delta } \end{aligned}$$
     
  2. (II)

    If \(i\in I_{\alpha }, \lambda \in \Lambda _{\alpha }\) then \(a=\displaystyle \left( \left\langle a,i\right\rangle , a_{\alpha }, \left[ \lambda ,a\right] \right) \)

    On \(S=\bigcup _{\alpha \in Y} S_{\alpha }\) define a multiplication by
    $$\begin{aligned} a\circ b=\left( \left\langle a, \left\langle b,\alpha \beta \right\rangle \right\rangle , a_{\alpha \beta }p_{[\alpha \beta ,a]\langle b,\alpha \beta \rangle } b_{\alpha \beta }, \left[ \left[ \alpha \beta ,a\right] ,b\right] \right) \end{aligned}$$
     
  3. (III)
    If \(\gamma \le \alpha \beta , i\in I_{\gamma }, \lambda \in \Lambda _{\gamma }\) then
    $$\begin{aligned} \left( \left\langle a, \left\langle b,i\right\rangle \right\rangle , a_{\gamma }p_{[\gamma ,a]\langle b,\gamma \rangle } b_{\gamma }, \left[ \left[ \gamma ,a\right] ,b\right] \right) =\left( \langle a\circ b, i\rangle , (a\circ b)_{\gamma }, [\lambda , a\circ b]\right) \end{aligned}$$
     
  4. (IV)
    (i) If \(a=(i,x,\lambda ), b=(j,y,\lambda )\in S_{\alpha }\) then for all \(e\in E\) and \(d\in S_{\beta }\)
    1. (a)

      \(a\circ e=a\Leftrightarrow b\circ e=b\),

       
    2. (b)

      \((a\circ d)\circ e=a\circ d\Leftrightarrow (b\circ d)\circ e=b\circ d\).

       
    (ii) If \(a=(i,x,\lambda ), b=(i,y,\mu )\in S_{\alpha }\) then for all \(e\in E\) and \(d\in S_{\beta }\)
    1. (a)

      \(e\circ a=a\Leftrightarrow e\circ b=b\),

       
    2. (b)

      \(e\circ (d\circ a)=d\circ a\Leftrightarrow e\circ (d\circ b)=d\circ b\).

       
     
Then \((S,\circ )\) is a completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup with \(E=\bigcup _{\alpha \in Y} E_{\alpha }\) whose multiplication restricted to each \(S_{\alpha }\) coincides with the given multiplication. Conversely, every completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup is isomorphic to one so constructed.

We will also need some technical results (see [50] and references therein), listed as follows:

Lemma 2.5

Let S be \((E,\widetilde{\mathcal {H}}_E)\)-abundant. Then
  1. (1)

    The relations \(\widetilde{\mathcal {L}}_E\) and \(\widetilde{\mathcal {R}}_E\) commute. In particular \(\widetilde{\mathcal {D}}_E=\widetilde{\mathcal {L}}_E\circ \widetilde{\mathcal {R}}_E=\widetilde{\mathcal {R}}_E\circ \widetilde{\mathcal {L}}_E\).

     
  2. (2)

    For any \(e,f\in E\), \(e\widetilde{\mathcal {D}}_Ef\) if and only if exists \(h\in E, e\mathcal {L}h\mathcal {R}f\).

     
If moreover \(\widetilde{\mathcal {L}}_E\) and \(\widetilde{\mathcal {R}}_E\) are right and left congruences then:
  1. (1)

    For all \(e\in E\), \(\widetilde{\mathcal {H}}_E(e)\) is a monoid.

     
  2. (2)

    Green’s lemmas hold. In particular any two \(\widetilde{\mathcal {H}}_E\)-classes in the same \(\widetilde{\mathcal {D}}_E\)-class are isomorphic.

     
  3. (3)

    For all \(e\in E, \widetilde{\mathcal {J}}_E[e]=SeS\).

     

2.2 Improving the previous structure theorems

Regarding the previous results, we see that the hypothesis involved in both the semilattice decomposition theorem and the semilattice composition theorem are rather strong. Notably, the congruence condition has to be checked. We show below that using bisimplicity (\(\widetilde{\mathcal {D}}_E\)-simplicity) instead of simplicity (\(\widetilde{\mathcal {J}}_E\)-simplicity), and a simple property (named \((\Pi )\) and defined afterwards), we can considerably simplify the previous theorems. Type \((\Pi )\) properties will also prove useful in the next sections.

First, we have the following equivalences.

Theorem 2.6

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. Then the following statements are equivalent:
  1. (1)

    The relations \(\widetilde{\mathcal {L}}_E\) and \(\widetilde{\mathcal {R}}_E\) are right and left congruences;

     
  2. (2)

    The relation \(\widetilde{\mathcal {D}}_E\) is a semilattice congruence;

     
  3. (3)

    The relation \(\widetilde{\mathcal {D}}_E\) is a congruence.

     

Proof

That \((1)\Rightarrow (2)\) is Lemma 2.1, and \((2)\Rightarrow (3)\) is straightforward. We prove that \((3)\Rightarrow (1)\) Let \(a,b,c\in S\), where S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {D}}_E\)-congruent. Let e (resp. fg) be the unique idempotent of E in \(\widetilde{\mathcal {H}}_E(a)\) (resp. \(\widetilde{\mathcal {H}}_E(b)\), \(\widetilde{\mathcal {H}}_E(c)\)). Assume \(a\widetilde{\mathcal {R}}_Eb\). As \(\widetilde{\mathcal {D}}_E\) is a congruence, then \(ca\widetilde{\mathcal {D}}_Ece\) and as \(\widetilde{\mathcal {D}}_E=\widetilde{\mathcal {R}}_E\circ \widetilde{\mathcal {L}}_E\) by Lemma 2.5 there exists \(d\in S\) such that \(ca\widetilde{\mathcal {R}}_Ed \widetilde{\mathcal {L}}_Ece\). Let h be the unique idempotent of E in \(\widetilde{\mathcal {H}}_E(d)\) and k be the unique idempotent of E in \(\widetilde{\mathcal {H}}_E(ce)\). Then \(h\widetilde{\mathcal {L}}_Ek\), and as \(h^2=h\) then \(kh=k\). As \(ea=a\), then \(a\le _{\mathcal {R}}e\) and as \(\mathcal {R}\) is a left congruence, then \(ca\le _{\mathcal {R}} ce\). This implies that \(ca\le _{\widetilde{\mathcal {R}}_E} ce\) and \(h\le _{\widetilde{\mathcal {R}}_E} ce\le _{\widetilde{\mathcal {R}}_E} k\). As \(k^2=k\) then \(kh=h\) and finally, \(k=kh=h\). Thus \(ca\widetilde{\mathcal {R}}_Ece\). The same arguments give \(cb\widetilde{\mathcal {R}}_Ecf\). But \(e\widetilde{\mathcal {R}}_Ea\widetilde{\mathcal {R}}_Eb\widetilde{\mathcal {R}}_Ef\) hence \(e\widetilde{\mathcal {R}}_Ef\), and as \(e^2=e\) and \(f^2=f\) then \(fe=e\) and \(ef=f\). It follows that \(e\mathcal {R}f\) and as \(\mathcal {R}\) is a left congruence, \(ce\mathcal {R}cf\) and in particular \(ce\widetilde{\mathcal {R}}_Ecf\). Finally \(ca\widetilde{\mathcal {R}}_Ece\widetilde{\mathcal {R}}_Ecf\widetilde{\mathcal {R}}_Ecb\) and \(\widetilde{\mathcal {R}}_E\) is a left congruence. Dually \(\widetilde{\mathcal {L}}_E\) is a right congruence. \(\square \)

The main consequence of this theorem is that a \((E,\widetilde{\mathcal {H}}_E)\)-abundant, \(\widetilde{\mathcal {D}}_E\)-simple semigroup is automatically completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant. This explain our choice to name them completely E-simple semigroups. Obviously, they are also the completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant, \(\widetilde{\mathcal {J}}_E\)-simple semigroups of Theorems 2.22.3 and 2.4 by Lemma 2.1.

Second, we define the following general property \((\Pi )\). For any \(g\in E\), \(M_g\) denotes a specific submonoid of S with identity g. Property \((\Pi )\) states that these specific submonoids lie entirely within corner semigroups eSf, \(e,f\in E\).
$$\begin{aligned} (\Pi ) \qquad \left( \forall e,f,g\in E\right) M_g\cap eSf\ne \emptyset \Rightarrow M_g \subseteq eSf. \end{aligned}$$
This property admits equivalent characterizations, among them:
$$\begin{aligned} \left( \forall a\in S, \forall e,f,g\in E\right) eaf\in M_g\Rightarrow egf=g \end{aligned}$$
or
$$\begin{aligned} \left( \forall a\in S, \forall e,f\in E\right) ae\in M_f\Rightarrow fe=f \text { and } ea\in M_f\Rightarrow ef=f. \end{aligned}$$
That property \((\Pi )\) is of interest relies on the following interpretation of relation \(\widetilde{\mathcal {H}}_E\):
$$\begin{aligned} a\widetilde{\mathcal {H}}_Eb\Leftrightarrow \left\{ \left( \forall e,f\in E^1\right) eaf=a\Leftrightarrow ebf=b\right\} \end{aligned}$$
Weaker versions may also be useful:Obviously, \((\Pi )\Rightarrow (\Pi ')\Rightarrow (\Pi '')\) (set \(e=g\) or \(f=g\)). Property \((\Pi ')\) is for instance always satisfied when E is a band.

We then improve the previous decomposition and composition theorems.

Theorem 2.7

A semigroup S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant if and only if it is a semilattice Y of \((E_{\alpha },\widetilde{\mathcal {H}}_{E_{\alpha }})\)-abundant, \(\widetilde{\mathcal {D}}_{E_{\alpha }}\)-simple semigroups with
$$\begin{aligned} (\Pi )\qquad \left( \forall a\in S_{\gamma }, e\in E_{\alpha }, f\in E_{\beta }, g\in E_{\alpha \gamma \beta }\right) \; eaf\in \widetilde{\mathcal {H}}_{E_{\alpha \gamma \beta }}(g)\Rightarrow egf=g \end{aligned}$$
or
$$\begin{aligned}&(\Pi '')\;\quad \left( \forall a\in S_{\gamma }, e\in E_{\alpha }, f\in E_{\alpha \gamma }\right) \; ae\in \widetilde{\mathcal {H}}_{E_{\alpha \gamma }}(f)\\&\qquad \qquad \qquad \Rightarrow fe\in E_{\alpha \gamma } \text { and } ea\in \widetilde{\mathcal {H}}_{E_{\alpha \gamma }}(f)\Rightarrow ef\in E_{\alpha \gamma }. \end{aligned}$$

Proof

\(\Rightarrow \)

Assume S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant. Then it is a semilattice of its \(\widetilde{\mathcal {D}}_E\)-classes. We prove that this semilattice satisfies \((\Pi )\). Let \(a\in S_{\gamma }, e\in E_{\alpha } ,f\in E_{\beta }\). Then \(g\in E_{\alpha \gamma \beta }\cap \widetilde{\mathcal {H}}_{E_{\alpha \gamma \beta }}(eaf)\) is actually the only idempotent \(\widetilde{\mathcal {H}}_E\)-related to eaf in S. Indeed, let \(g'\in S_{\delta }\) be this idempotent. As \(g(eaf)=eaf=(eaf)g\) then \(gg'=g'=g'g\), and \(\delta \le \alpha \gamma \beta \). But also \(eafg'=eaf=g'eaf\) and \(\alpha \gamma \beta \le \delta \). Thus \(\delta =\alpha \gamma \delta \), and the equalities \(eafg'=eaf=g'eaf\) in \(S_{\alpha \gamma \beta }\) give \(gg'=g=g'g\). Finally, as \(e(eaf)=eaf\) then \(eg=g\) and as \((eaf)f=eaf\) then \(gf=g\). Hence \(egf=g\) and the semilattice satisfies \((\Pi )\) (it then satisfies a fortiori \((\Pi '')\)).

\(\Leftarrow \)

Conversely, let S be a semilattice Y of \((E_{\alpha },\widetilde{\mathcal {H}}_{E_{\alpha }})\)-abundant, \(\widetilde{\mathcal {D}}_{E_{\alpha }}\)-simple semigroups with \((\Pi '')\). Let \(a\in S_{\alpha }, b\in S_{\beta }\). We first prove the equivalence \(a\widetilde{\mathcal {L}}_Eb\) if and only if \(\alpha =\beta \) and \(a\in \widetilde{\mathcal {L}}_{E_{\alpha }} b\). Assume \(a\widetilde{\mathcal {L}}_Eb\) and let e be the unique idempotent of \(E_{\alpha }\) \(\widetilde{\mathcal {H}}_{E_{\alpha }}\)-related to a in \(S_{\alpha }\). As \(ae=a\) and \(a\widetilde{\mathcal {L}}_Eb\) then \(be=b\) and \(\beta \le \alpha \). Symmetrically \(\alpha \le \beta \). As \(E_{\alpha }\subseteq E\) then \(af=a\Leftrightarrow bf=b\) for all \(f\in E_{\alpha }\) and \(a \widetilde{\mathcal {L}}_{E_{\alpha }} b\). Conversely, assume \(\alpha =\beta \) and \(a\in \widetilde{\mathcal {L}}_{E_{\alpha }} b\). Let as before \(e\in E_{\alpha }\cap \widetilde{\mathcal {H}}_{E_{\alpha }}(a)\). As \(ae=a\) then \(be=b\). Let \(f\in E_{\gamma }\) such that \(af=a\). Then \(\alpha \le \gamma \). By property \((\Pi '')\), as \(af(=a)\in \widetilde{\mathcal {H}}_{E_{\alpha }}(e)\) then \(ef\in E_{\alpha }\). As \(a(ef)=af=a\) with \(ef\in E_{\alpha }\) then \(eef=e\). It follows that \(bf=bef=be=b\) and finally \(a\widetilde{\mathcal {L}}_Eb\). Fromm the equivalence \(\{a\widetilde{\mathcal {L}}_Eb \Leftrightarrow \alpha =\beta \) and \(a\in \widetilde{\mathcal {L}}_{E_{\alpha }} b\}\) and its dual, S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant. By Lemma 2.5, \(\widetilde{\mathcal {D}}_E=\widetilde{\mathcal {R}}_E\circ \widetilde{\mathcal {L}}_E\) and using the previous equivalences and the fact that each \(S_{\alpha }\) is \(\widetilde{D}_{E_{\alpha }}\)-simple, we get that \(a\widetilde{\mathcal {D}}_Eb\) if and only if a and b belong to the same \(S_{\alpha }\). By construction of S, \(\widetilde{\mathcal {D}}_E\) is then a semilattice congruence and by Theorem 2.6, S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant. \(\square \)

Using Theorem 2.3, Petrich’s work on completely regular semigroups [44] and property \((\Pi )\) of Theorem 2.7, we get the following composition theorem, where the last condition is easier to handle than in Theorem 2.4.

Theorem 2.8

Let Y be a semilattice. For every \(\alpha \in Y\), let \(S_{\alpha }=\mathcal {M}(M_{\alpha },I_{\alpha },\Lambda _{\alpha },P_{\alpha })\) be a Rees matrix semigroup over a monoid \(M_{\alpha }\) such that the entries of \(P_{\alpha }\) are invertible, and \(P_{\alpha }\) is normalized at an element also denoted by \(\alpha \). Suppose also that \(S_{\alpha }\cap S_{\beta }=\emptyset \) if \(\alpha \ne \beta \). Let \(E_{\alpha }=\{(i,p_{\lambda i}^{-1},\lambda ) |\, i\in I_{\alpha }, \lambda \in \Lambda _{\alpha }\}\) and let
$$\begin{aligned} \langle .,.\rangle :&S_{\alpha }\times I_{\beta }\rightarrow I_{\beta }\\ \phi _{\alpha ,\beta }:&S_{\alpha }\rightarrow M_{\beta }\\ a&\mapsto a_{\beta }\\ \left[ .,.\right] :&\Lambda _{\beta }\times S_{\alpha }\rightarrow \lambda _{\beta } \end{aligned}$$
be functions defined whenever \(\alpha \ge \beta \) and satisfying the following conditions.
Let \(a\in S_{\alpha }, b\in S_{\beta }\):
  1. (i)

    If \(\alpha \ge \beta , i\in I_{\beta }, \lambda \in \Lambda _{\beta }\) then \(p_{\lambda \langle a,i\rangle } a_{\beta } p_{[\beta ,a]i}=p_{\lambda \langle a,\beta \rangle } a_{\beta } p_{[\lambda ,a]i}\)

     
  2. (ii)

    If \(i\in I_{\alpha }, \lambda \in \Lambda _{\alpha }\) then \(a=\displaystyle \left( \left\langle a,i\right\rangle , a_{\alpha }, \left[ \lambda ,a\right] \right) \)

    On \(S=\bigcup _{\alpha \in Y} S_{\alpha }\) define a multiplication by
    $$\begin{aligned} a\circ b=\left( \left\langle a, \left\langle b,\alpha \beta \right\rangle \right\rangle , a_{\alpha \beta }p_{[\alpha \beta ,a]\langle b,\alpha \beta \rangle } b_{\alpha \beta }, \left[ \left[ \alpha \beta ,a\right] ,b\right] \right) \end{aligned}$$
     
  3. (iii)
    If \(\gamma \le \alpha \beta , i\in I_{\gamma }, \lambda \in \Lambda _{\gamma }\) then
    $$\begin{aligned} \left( \left\langle a, \left\langle b,i\right\rangle \right\rangle , a_{\gamma }p_{[\gamma ,a]\langle b,\gamma \rangle } b_{\gamma }, \left[ \left[ \gamma ,a\right] ,b\right] \right) =\left( \langle a\circ b, i\rangle , (a\circ b)_{\gamma }, [\lambda , a\circ b]\right) \end{aligned}$$
     
  4. (iv)

    If \(\gamma \le \beta , e\in E_{\beta }\) then \(e_{\gamma }=p^{-1}_{[\gamma ,e]\langle e,\gamma \rangle }\)

     
Then \((S,\circ )\) is a completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup with \(E=\bigcup _{\alpha \in Y} E_{\alpha }\) such that \(S/\widetilde{\mathcal {D}}_E\sim Y\) whose multiplication restricted to each \(S_{\alpha }\) coincides with the given multiplication. Conversely, every completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup is isomorphic to one so constructed.

Conditions (i) to (iii) are exactly those of Petrich’s Theorem [44]. In the regular setting, condition (iv) does not appear since it is always satisfied. Following Petrich’s initial proof, we get that \((S,\circ )\) is a semigroup whose multiplication restricted to each \(S_{\alpha }\) coincides with the given multiplication. By construction, it is a semilattice of completely E-simple semigroups. If it satisfies \((\Pi )\), then by Theorem 2.7 it will be completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant. For the converse, all we have to do is to check that the mappings initially defined by Petrich also satisfy (iv). We prove below that in the context, (iv) is equivalent to \((\Pi )\).

Proof

We first prove that the semilattice composition of \((S,\circ )\) satisfies \((\Pi )\). Let \(a\in S_{\alpha }\) and \(e\in E_{\beta }\). By (i), for any \(i\in I_{\alpha \beta }, \lambda \in \Lambda _{\alpha \beta }\) it holds that
$$\begin{aligned} p_{\lambda \langle e,i\rangle }e_{\alpha \beta }p_{[\alpha \beta ,e]i}=p_{\lambda \langle e,\alpha \beta \rangle }e_{\alpha \beta }p_{[\lambda ,e]i}. \end{aligned}$$
Then
$$\begin{aligned} p_{\lambda \langle e,\alpha \beta \rangle }&= p_{\lambda \langle e,\langle e,\alpha \beta \rangle \rangle } \text { by }(ii)\\&= p_{\lambda \langle e,\langle e,\alpha \beta \rangle \rangle } e_{\alpha \beta }p_{[\alpha \beta ,e]\langle e,\alpha \beta \rangle }\text { by }(iv)\\&=p_{\lambda \langle e,\alpha \beta \rangle } e_{\alpha \beta } p_{[\lambda ,e]\langle e,\alpha \beta \rangle } \text { by the previous equation with }i=\langle e,\alpha \beta \rangle . \end{aligned}$$
And as the elements of the sandwich matrix are invertible then
$$\begin{aligned} 1_{\alpha \beta }=e_{\alpha \beta } p_{[\lambda ,e]\langle e,\alpha \beta \rangle }=p_{[\lambda ,e]\langle e,\alpha \beta \rangle }e_{\alpha \beta }. \end{aligned}$$
As also \(a\circ e=\left( \langle a, \langle e,\alpha \beta \rangle \rangle , a_{\alpha \beta } p_{[\alpha \beta ,a]\langle e,\alpha \beta \rangle }e_{\alpha \beta },[[\alpha \beta ,a],e] \right) \), then the only idempotent in \(E_{\alpha \beta }\cap \widetilde{\mathcal {H}}_{E_{\alpha \beta }}(a\circ e)\) is \(f=\left( \langle a, \langle e,\alpha \beta \rangle \rangle , p_{[[\alpha \beta ,a],e]\langle a,\langle e,\alpha \beta \rangle \rangle }^{-1},[[\alpha \beta ,a],e] \right) \) and
$$\begin{aligned} f\circ e&= \left( \langle f, \langle e,\alpha \beta \rangle \rangle , f_{\alpha \beta } p_{[\alpha \beta ,f]\langle e,\alpha \beta \rangle }e_{\alpha \beta },[[\alpha \beta ,f],e] \right) \\&= \left( \langle a, \langle e,\alpha \beta \rangle \rangle , f_{\alpha \beta } p_{[[\alpha \beta ,a],e]\langle e,\alpha \beta \rangle }e_{\alpha \beta },[[\alpha \beta ,a],e] \right) \text { from } (ii) \\&= \left( \langle a, \langle e,\alpha \beta \rangle \rangle , f_{\alpha \beta },[[\alpha \beta ,a],e] \right) =f \text { since } p_{[\alpha \beta ,a],e\langle e,\alpha \beta \rangle }e_{\alpha \beta }=1_{\alpha \beta }. \end{aligned}$$
Property \((\Pi )\) follows by duality.

For the converse, we follow Petrich initial proof and get that the mappings he defined satisfy (i), (ii) and (iii) and that the multiplications . and \(\circ \) coincide. Let \(\gamma \le \beta \) and let \(e\in E_{\beta }\). Pose \(f=(\gamma ,1_{\gamma },\gamma )\) and \(g=(\langle e,\gamma \rangle , p^{-1}_{\gamma \langle e,\gamma \rangle }, \gamma )\). Then \(g\in E_{\gamma }\cap \widetilde{\mathcal {H}}_{E_{\gamma }}(ef)\) and as \(e(ef)=ef\) then \(eg=g\) by \((\Pi )\). But it follows from (ii) and (iii) that \(eg=e\circ g= \left( \langle e,\langle e,\gamma \rangle \rangle ,e_{\gamma }p_{[\gamma ,e]\langle e,\gamma \rangle }p^{-1}_{\gamma \langle e,\gamma \rangle },\gamma \right) \). Thus \(e_{\gamma }p_{[\gamma ,e]\langle e,\gamma \rangle }p^{-1}_{\gamma \langle e,\gamma \rangle }=p^{-1}_{\gamma \langle e,\gamma \rangle }\) and \(e_{\gamma }=p^{-1}_{[\gamma ,e]\langle e,\gamma \rangle }\). \(\square \)

Combining Theorem 2.8 and the classical theorem of Petrich [44] we get:

Corollary 2.9

Let S be a completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. Then \(H(E)=\bigcup _{e\in E} \mathcal {H}(e)\) is a completely regular subsemigroup of S.

The proof goes as follows: first, decompose S as a semilattice Y of Rees matrix semigroups \(S_{\alpha }=\mathcal {M}(M_{\alpha },I_{\alpha },\Lambda _{\alpha },P_{\alpha })\) by Theorem 2.8. Then define \(J_{\alpha }=\mathcal {M}(M^{-1}_{\alpha },I_{\alpha },\Lambda _{\alpha },P_{\alpha })\) and reconstruct a completely regular semigroup by [44]. Finally, check that this semigroup is H(E).

3 \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups: new results

As we have seen, many results regarding completely regular semigroups can be extended to completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups, in particular the semilattice decomposition. However, some others cannot be, and appear to be actually linked with \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups without the congruence property. This is for instance the case for union of monoids or subdirect products of monoids, decompositions that we will study in the first subsection. Primitive idempotents play a special role in these decompositions. In the second subsection, we will thus study \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups with primitive idempotents (within E).

3.1 Unions of monoids, local submonoids, subdirect products

We first consider (disjoint) union of monoids. As in completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups each \(\widetilde{\mathcal {H}}_E\)-class is a monoid, these semigroups are union of monoids, with E the set of identities of the monoid. But contrary to the regular case, where completely regular semigroups are exactly union of groups, the converse is not true for monoids. Adding extra structure on the decomposition (such as demanding a band or semilattice decomposition instead of a mere union) also fails.

Example 3.1

Let S be a semilattice with three elements efg such that \(ef=fe=g\). Then \(\rho =\left\{ (g,g); (g,f);(f,g);(e,e);(f,f)\right\} \) is a semilattice congruence with each \(\rho \)-class (\(\rho (f)=\{f,g\}\) and \(\rho (e))=\{e\}\)) a monoid. Pose \(E=\{e,f\}\) set of identities of these monoids. As \(gf=g=eg\) and \(ef=g\) then g cannot be \(\widetilde{\mathcal {H}}_E\)-related to an idempotent of E, and S is not \((E,\widetilde{\mathcal {H}}_E)\)-abundant.

There is actually a result for union of monoids, but it involves \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups without necessary the congruence property. But a type \((\Pi )\) property is needed.

Theorem 3.2

Let \(S=\dot{\bigcup }_{e\in E} M_e\) be a disjoint union of monoids \(M_e\) with identity e such that
$$\begin{aligned} (\Pi ) \qquad (\forall e,f,g\in E) M_g\cap eSf\ne \emptyset \Rightarrow M_g \subseteq eSf. \end{aligned}$$
Then S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant.

Conversely, any \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup such that each \(\widetilde{\mathcal {H}}_E\)-class is a monoid is a union of monoids with \((\Pi )\).

Proof

Let \(S=\dot{\bigcup }_{e\in E} M_e\) be a disjoint union of monoids with identity e such that
$$\begin{aligned} (\forall e,f,g\in E) M_g\cap eSf\ne \emptyset \Rightarrow M_g \subseteq eSf, \end{aligned}$$
and let \(a\in M_g\). Then \(ag=ga=a\). We now prove that g is minimal with respect to \(\omega _r\) and \(\omega _l\) (among left and right identities of a). Let \(e,f\in E\), \(ea=a=af\). Then \(a\in M_g\cap eSf\) hence \(M_g \subseteq eSf\), and \(g=eg=gf\). It follows that \(a\widetilde{\mathcal {H}}_Eg\) and S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant.

Conversely, let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup such that each \(\widetilde{\mathcal {H}}_E\)-class is a monoid. Then S is the disjoint union of the monoids \(M_e=\widetilde{\mathcal {H}}_E(e),e\in E\). Let \(e,f,g\in E\) such that \(\widetilde{\mathcal {H}}_E(g)\cap eSf\ne \emptyset \). Choose \(a\in \widetilde{\mathcal {H}}_E(g)\cap eSf\). As \(ea=a=af\) then \(eg=g=gf\) and for all \(x\in \widetilde{\mathcal {H}}_E(g)\), \(x=gxg=egxgf\in eSf\). This ends the proof. \(\square \)

The extra condition is always fulfilled in case of union of groups, because \(eaf\in H(g)\) gives \(g=eaf\left[ (eaf)^\#\right] ^2 eaf=egf\). A union of monoids with \((\Pi )\) is not completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant in general.

Example 3.3

Let \(S=\{0,e,1,a,a^2,\ldots \}\) such that \(E=\{0,e,1\}=E(S)\) with \(0<e<1\), and relations \(a1=1a=a\), \(ae=ea=0\). It satisfies the assumptions of Theorem 3.2, but \(a\widetilde{\mathcal {L}}_E1\) whereas \(0=ae\notin \widetilde{\mathcal {L}}_E(1e=e)\). \(\widetilde{\mathcal {L}}_E\) is not a right congruence.

Next lemma gives an insight of the multiplicative structure of \(\widetilde{\mathcal {H}}_E\)-classes of \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups.

Lemma 3.4

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup, and let \(e\in E\). Pose \(I_e=\bigcup _{f\in E, f<e} fSf\). Then
  1. (1)

    If \(a,b\in eSe\) and \(f\in E\cap \widetilde{\mathcal {H}}_E(ab)\) then \(f\omega e\);

     
  2. (2)

    \(I_e\) is an ideal of eSe;

     
  3. (3)

    \(\widetilde{\mathcal {H}}_E(e)=eSe\backslash I_e\);

     
  4. (4)

    \(\widetilde{\mathcal {H}}_E(e)\) is a monoid if and only if the ideal \(I_e\) is prime (in eSe).

     

In any case, we can form the Rees quotient \(\widetilde{\mathcal {H}}_E^0(e)=eSe/I_e\) which is a monoid. We will say that \(e\in E\) is primitive (within E) if e is minimal in E with respect to \(\omega \) (\((\forall f\in E) ef=fe=f\Rightarrow e=f\)). For a primitive idempotent \(e\in E\), \(I_e=\emptyset \) and the Rees quotient is simply the monoid \(\widetilde{\mathcal {H}}_E(e)=eSe\).

Proof

We pose \(I_e=\bigcup _{f\in E, f<e} fSf\).
  1. (1)

    Let \(a,b\in eSe\) and \(f\in E\cap \widetilde{\mathcal {H}}_E(ab)\). As \(ea=a\) then \(eab=ab\) and \(ef=f\). Dually \(fe=f\) and \(f\omega e\).

     
  2. (2)

    Assume \(I_e\) is not empty, and let \(f\in E\). If \(f<e\), then \(fSf=efSfe\subset eSe\), hence \(I_e\subset eSe\). Let \(a\in I_e, b\in eSe\) and \(f\in E\cap \widetilde{\mathcal {H}}_E(ab)\). By the first statement of the lemma, \(f\omega e\). Let \(g\in E\cap \widetilde{\mathcal {H}}_E(a)\). By hypothesis \(ge\ne e\). As \(ga=a\) then \(gab=ab\) hence \(gf=f\), and \(f\ne e\). Finally \(f<e\) and \(ab\in I_e\).

     
  3. (3)

    Let \(a\in \widetilde{\mathcal {H}}_E(e)\). As \(ea=a=ae\) then \(a\in eSe\). Let \(f\in E\), such that \(a\in fSf\). Then \(fa=a=af\) hence \(fe=e=ef\) and \(e\omega f\). Finally, \(a\in eSe\backslash \left( \bigcup _{f\in E, f<e} fSf\right) \).

     
  4. (4)

    The equivalence is a direct consequence of the previous equality \(\widetilde{\mathcal {H}}_E(e)=eSe\backslash I_e\). \(\square \)

     

We consider two special instances of sets E: E is a chain, and \(E\subseteq Z(S)\).

Corollary 3.5

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup, with \(E=\{e_1>e_2>\cdots >e_m\}\) a chain of idempotents. Then S is an ideal series of the local submonoids \((eSe, e\in E)\), which are ideals of S. Conversely, any semigroup which is an ideal series of local submonoids \((eSe, e\in E)\) is \((E,\widetilde{\mathcal {H}}_E)\)-abundant, with E a chain.

Proof

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup, with \(E=\{e_1>e_2>\cdots >e_m\}\) a chain of idempotents. Then
$$\begin{aligned} S=e_1Se_1\supset e_2Se_2\supset \cdots \supset e_mSe_m\supset \emptyset \end{aligned}$$
where the first equality holds by \((E,\widetilde{\mathcal {H}}_E)\)-abundance. By Lemma 3.4 each local submonoid \(e_{i+1}Se_{i+1}\) is an ideal of \(e_{i}Se_{i}\), \(i=1,\ldots , m-1\). It follows that for any \(x\in S, e=e_i\in E\), \(xe_i=xe_1\ldots e_i\in e_iSe_i\) by induction and the ideal property. Finally eSe is an ideal of S.
Conversely, let S be an ideal series of local submonoids \((eSe, e\in E)\),
$$\begin{aligned} S=e_1Se_1\supset e_2Se_2\supset \cdots \supset e_mSe_m\supset \emptyset \end{aligned}$$
with \(E=\{e_1,\ldots ,e_m\}\subseteq E(S)\). Let \(1\le i\le m\). Then \(e_{i+1}=e_ie_{i+1}e_i=e_ie_{i+1}=e_{i+1}e_i\) by the ideal property and E is a chain. Let \(a\in S\). Then either \(a\in e_mSe_m\) or exists \(1\le i<m\), \(a\in e_iSe_i\backslash e_{i+1}Se_{i+1}\). If \(a\in e_mSe_m\) then \(a\widetilde{\mathcal {H}}_Ee_m\) as \(e_m e=ee_m=e_m\) for all \(e\in E\) and \(ae=ea=a\) for all \(e\in E\). If \(a\in e_iSe_i\backslash e_{i+1}Se_{i+1}\) with \(1\le i< m\) we claim that \(a\widetilde{\mathcal {H}}_Ee_i\). Indeed, let \(1\le k\le m\). If \(e_ie_k=e_i\) then \(a_ek=ae_ie_k=ae_i=a\). Conversely, if \(ae_k=a\) then \(a\in e_kSe_k\) by the ideal property, and as \(a\in e_iSe_i\backslash e_{i+1}Se_{i+1}\) then \(k\le i\), whence \(e_ie_k=e_i\). Finally \(a\widetilde{\mathcal {L}}_Ee_i\) and we conclude by duality that S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant. \(\square \)

In this case, idempotents are central. Indeed, let \(a\in e_iSe_i\) and \(e=e_j\in E\). If \(j>i\) then \(ae_j,e_ja\in e_jSe_j\) as \(e_jSe_j\) is an ideal of \(e_iSe_i\), and \(ae_j=e_jae_j=e_ja\). If \(j\le i\), then \(ae_j=ae_ie_j=ae_i=a\) and dually. Finally \(E\subseteq Z(S)\).

Example 3.6

Consider Example 3.3. Then \(E=\{1>e>0\}\) is a chain of central idempotents and \(S=1S1\supset eSe=\{e,0\}\supset 0S0=\{0\}\supset \emptyset \) is an ideal series.

When E is not a chain, but still a set of central idempotents, we obtain a particular subdirect decomposition which is special in being also a partition (\(S=\dot{\bigcup }_{E\in E} \widetilde{\mathcal {H}}_E(e)\)).

Corollary 3.7

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup with \(E\subseteq Z(S)\). Then S is a subdirect product of the factors \(\widetilde{\mathcal {H}}_E(e)=Se\), \(e\in E\) primitive (within E) and \(\widetilde{\mathcal {H}}_E^0(e)=Se/\left( \bigcup _{ f\in E, f<e} Sf\right) \), \(e\in E\) not primitive.

Proof

For all \(e\in E\), we define canonical projections \(\pi _e\) as follows. If e is minimal, then \(\pi _e:S\rightarrow \widetilde{\mathcal {H}}_E(e)=Se\) is defined by \(\pi _e(x)=xe\). If e is not minimal, then \(\pi _e:S\rightarrow \widetilde{\mathcal {H}}_E^0(e)\) is defined by \(\pi _e(x)=xe\) if \(xe\widetilde{\mathcal {H}}_Ee\) and 0 otherwise. By Lemma 3.4, \(\pi _e\) is the composition of the surjective projection \(x\mapsto xe\) from S to Se followed by the quotient map \(Se\rightarrow Se/\left( \bigcup _{ f\in E, f<e} Sf\right) \), and \(\pi _e\) is a surjective homomorphism. We now prove that the projections \((\pi _e, e\in E)\) separate points. Let \(x,y\in S\) such that \(\pi _e(x)=\pi _e(y)\) for all \(e\in E\). By \((E,\widetilde{\mathcal {H}}_E)\)-abundance, there exists \(e,f\in E\), \(x\widetilde{\mathcal {H}}_Ee\) and \(y\widetilde{\mathcal {H}}_Ef\). As \(\pi _e(x)=xe=x\) then \(ye=x\), and as \(fy=y\) then \(fx=x\). But \(x\widetilde{\mathcal {R}}_Ee\) hence \(fe=e\). Dually \(ef=f\). But \(ef=fe\) hence \(e=f\), which in turns implies \(x=\pi _e(x)=\pi _f(y)=y\). \(\square \)

Let S be a commutative semigroup. Then S is called complete if [22]:
  1. (1)

    Every element \(a\in S\) has a power in a subgroup of S;

     
  2. (2)

    For any \(a\in S\), the set of identities of a \(E_a=\{e\in E(S^1)|\, ae=a\}\) admits a minimal element.

     
If S is a monoid, then the second condition is equivalent with \((E(S),\widetilde{\mathcal {H}}_{E(S)})\)-abundance, and Corollary 3.7 gives a subdirect product decomposition. This is actually Theorem 4.3 in [22], as \(\widetilde{\mathcal {H}}_E(e)\) coincide in this case with the partial Ponizovsky factor \(P^*_e\) (Proposition 4.4 in [22]) and \(\widetilde{\mathcal {H}}_E^0(e)\) with the Ponizovsky factor (Rees quotient) \(P_e\).

Once again, the semigroup S may not satisfy the congruence condition (Consider Example 3.3).

3.2 \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups and primitive idempotents

The previous results show that primitive idempotents (within E) play a special role. Indeed, the \(\widetilde{\mathcal {H}}_E\)-class \(\widetilde{\mathcal {H}}_E(e)\) for e primitive is always a monoid, the local submonoid eSe. We thus explore a direction suggested by the classical case of completely simple semigroups, but apparently not studied in the literature on \(\widetilde{\mathcal {J}}_E\)-simple semigroups: \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups with primitive idempotents. By primitive idempotents, we always mean primitive idempotents within E.

Theorem 3.8

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. Then S is \(\widetilde{\mathcal {D}}_E\)-simple if and only if the idempotents of E are primitive.

Proof

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup.
\(\Rightarrow \)

Let \(e,f\in E\) such that \(ef=fe=f\). AS S is \(\widetilde{\mathcal {D}}_E\)-simple, then by Lemma 2.5 there exists an idempotent \(g\in E\) such that \(e\mathcal {R}g\mathcal {L}f\). Then \(eg=g=gf, ge=e\) and \(fg=f\). It follows that \(f=ef=gef=gf=g\), which in turns implies \(e=ge=fe=f\), and e is primitive.

\(\Leftarrow \)

We prove that any two idempotents are \(\widetilde{\mathcal {D}}_E\)-related. Let \(e,f\in E\), and let \(g\in E\cap (efe)\). As \((efe)e=efe=e(efe)\) then \(ge=g=eg\) and as e is primitive, then \(e=g\). Let \(h\in E\cap wHE(ef)\). As \(hef=ef\) then \(hefe=efe\) hence \(he=e\). As \(ee=e\) then \(e(ef)=ef\) hence \(eh=h\). It follows that \(e\mathcal {R}h\widetilde{\mathcal {R}}_Eef\) and \(e\widetilde{\mathcal {R}}_Eef\). Dually, \(e\widetilde{\mathcal {L}}_Efe\). Symmetrically, \(f\widetilde{\mathcal {R}}_Efe\) and \(f\widetilde{\mathcal {L}}_Eef\) and \(e\widetilde{\mathcal {D}}_Ef\). \(\square \)

Together with Theorem 2.6 we get:

Corollary 3.9

Let S be a semigroup. The following statements are equivalent:
  1. (1)

    S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {D}}_E\)-simple (completely E-simple);

     
  2. (2)

    S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {J}}_E\)-simple;

     
  3. (3)

    S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and the idempotents of E are primitive.

     

This allows us to characterize exactly the set E for completely E-simple semigroups.

Corollary 3.10

Let S be a completely E-simple semigroup. Then \(E=\{e\in E(S)|\, (\forall f\in E(S)) e\omega f\Rightarrow e=f\}=Max\), set of maximal idempotents of E.

Proof

Let \(e\in E\), \(f\in E(S)\) such that \(e\omega f\) and let g be the unique idempotent of E \(\widetilde{\mathcal {H}}_E\)-related to f. As \(ef=fe=e\) and \(fg=gf=f\) then \(eg=efg=ef=e\) and \(ge=gfe=fe=e\), and \(e\omega g\). But by Theorem 3.8, elements of E are primitive and \(e=g\). As a consequence, \(e=ef=gf=f\) and e is maximal among idempotents of S. Conversely, let e be a maximal element of E(S), and let f be the unique idempotent of E \(\widetilde{\mathcal {H}}_E\)-related to e. Then \(ef=fe=f\) and \(f\omega e\), and by maximality \(e=f\in E\). \(\square \)

Thus, being completely E-simple is actually an intrinsic property of S. Indeed, the only possible set E is then \(E=Max\).

Finally, we are able to extend a last characterization of completely simple semigroups to completely E-simple ones. It is well known that regular semigroups that are disjoint union of their local submonoids are precisely the completely simple ones. This extends to completely E-simple semigroups. We start with a lemma.

Lemma 3.11

Let S be a completely E-simple semigroup. For any \(e\in E(=Max)\), \(\widetilde{\mathcal {R}}_E(e)=eS, \widetilde{\mathcal {L}}_E(e)=Se\) and \(\widetilde{\mathcal {H}}_E(e)=eSe\). In particular, S is the disjoint union of its local submonoids \(eSe=\widetilde{\mathcal {H}}_E(e), e\in E\).

Proof

Let \(a\widetilde{\mathcal {R}}_Ee\), \(e\in E\). As \(ee=e\) then \(ea=a\) and \(a\in eS\). Conversely, let \(a\in eS\) and let f be the unique idempotent of E \(\widetilde{\mathcal {H}}_E\)-related to a. As \(ea=a\) then \(ef=f\). As S is \(\widetilde{\mathcal {D}}_E\)-simple, then by Lemma 2.5 exists \(g\in E\) such that \(e\mathcal {R}g\mathcal {L}f\), that is \(eg=g=gf\), \(ge=e\) and \(fg=f\). It follows that \(f=ef=gef=gf=g\), and \(e\widetilde{\mathcal {R}}_Ef\widetilde{\mathcal {R}}_Ea\). Dually \(\widetilde{\mathcal {L}}_E(e)=Se\) and therefore \(\widetilde{\mathcal {H}}_E=eS\cap Se=eSe\). Note that this result also follows from Lemma 3.4. \(\square \)

Corollary 3.12

Let S be a semigroup, and E a distinguished set of idempotents such that \(S=\dot{\bigcup }_{e\in E} eSe\). Assume moreover that E is such that
$$\begin{aligned} (\Pi '')\qquad (\forall e,f\in E) fef=fe\Rightarrow fe\in E \text { and } fef=ef\Rightarrow ef\in E. \end{aligned}$$
Then S is completely E-simple.
Conversely, every completely simple semigroup is the disjoint union of its local submonoids \(eSe, e\in E\), and satisfies
$$\begin{aligned} (\Pi )\qquad (\forall e,f\in E) fef=fe\Rightarrow fe=f \text { and } fef=ef\Rightarrow ef=f. \end{aligned}$$

Proof

Let \(e,f\in E\) such that \(ef=fe=e\). Then \(e=e^3\in eSe\) and \(e=fef\in fSf\). It follows that \(e=f\) and idempotents of E are primitive. Let \(a\in fSf\) and \(e\in E\) such that \(ea=a\). Let \(g\in E\) be the unique idempotent of E such that \(ef\in gSg\). As \(gefaefg=efaef=aef=faef\in gSg\cap fSf\), then \(g=f\) and in particular, \(f(ef)=ef\). By \((\Pi '')\), \(ef\in E\). But \(f(ef)=ef=(ef)f\) (\(ef\omega f\)) and as idempotents of E are primitive, \(ef=f\). Finally, \(a\widetilde{\mathcal {R}}_Ef\). Dually \(a\widetilde{\mathcal {L}}_Ef\) and S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant. By Theorem 3.8, S is completely E-simple.

Conversely, let S be completely E-simple. By Lemma 3.11 \(S=\dot{\bigcup }_{e\in E} eSe\). Let \(e,f\in E\) such that \(fef=fe\). Then \(fe\in fSf=\widetilde{\mathcal {H}}_E(f)\) by Lemma 3.11 and as \((fe)e=fe\) then \(fe=f\). The other statement is dual. \(\square \)

Property \((\Pi '')\) is for instance satisfied when \(E=E(S)\), E is a band or \(E=E(J)\) is the set of idempotents of a completely simple subsemigroup J of S. It cannot be removed as shows next example.

Example 3.13

Consider the semigroup \(S=\{0_a,1_a, 0_b,1_b\}\) with multiplication table (Table 3).

Pose \(E=\{1_a,1_b\}\). Then \(S=M_a\dot{\bigcup } M_b\), with \(M_a=1_a S 1_a\) and \(M_b=1_b S1_b\). As \(1_b1_a1_b=0_b=1_a1_b\notin E\) then property \((\Pi '')\) is not satisfied. S is not completely E-simple. Indeed, it is even not \((E,\widetilde{\mathcal {H}}_E)\)-abundant. For instance, \(0_a\) is not \(\widetilde{\mathcal {H}}_E\)-related to an idempotent of E (\(1_b0_a=0_a\) but \(1_b1_a=0_a\ne 1_a\), and \(0_a1_b=0_b\ne 0_a\)).
Table 3

Cayley table for S

 

\(0_a\)

\(1_a\)

\(0_b\)

\(1_b\)

\(0_a\)

\(0_a\)

\(0_a\)

\(0_b\)

\(0_b\)

\(1_a\)

\(0_a\)

\(1_a\)

\(0_b\)

\(0_b\)

\(0_b\)

\(0_a\)

\(0_a\)

\(0_b\)

\(0_b\)

\(1_b\)

\(0_a\)

\(0_a\)

\(0_b\)

\(1_b\)

As another corollary to Lemma 3.11 we also get:

Corollary 3.14

Let S be a completely E-simple semigroup, and let \(a,b\in S\). Then
  1. (1)

    \(ab\in \widetilde{\mathcal {R}}_E(a)\cap \widetilde{\mathcal {L}}_E(b)\);

     
  2. (2)

    \(aba\widetilde{\mathcal {H}}_Ea\);

     
  3. (3)

    \(\widetilde{\mathcal {H}}_E\) is a congruence.

     

Proof

Let \(a,b\in S\), and let \(e\in E\cap \widetilde{\mathcal {H}}_E(a)\) and \(f\in \widetilde{\mathcal {H}}_E(b)\).
  1. (1)

    As \(ea=a\) then \(aS=eaS\subseteq eS=\widetilde{\mathcal {R}}_E(a)\), and dually \(Sb\subseteq \widetilde{\mathcal {L}}_E(b)\).

     
  2. (2)

    \((ab)a\in \widetilde{\mathcal {R}}_E(ab)\cap \widetilde{\mathcal {L}}_E(a)\) and \(ab\in \widetilde{\mathcal {R}}_E(a)\cap \widetilde{\mathcal {L}}_E(b)\) by the previous result, whence \(aba\in \widetilde{\mathcal {R}}_E(a)\cap \widetilde{\mathcal {L}}_E(a)=\widetilde{\mathcal {H}}(a)\).

     
  3. (3)

    We have to prove that \(ab\widetilde{\mathcal {H}}_Eef\). As \(ab=eabf\) then \(ab, ef\in eS\cap Sf=\widetilde{\mathcal {R}}_E(a)\cap \widetilde{\mathcal {L}}_E(b)\) by Lemma 3.11, and ab and ef belong to the same \(\widetilde{\mathcal {H}}_E\)-class. \(\square \)

     

Surprisingly, a result combining Theorem 2.3 and Corollary 3.12 was obtained by Hickey [25] in the regular case, while studying regularity preserving elements of a regular semigroup and without any reference to the extended Green’s relations.

Theorem 3.15

[25, Theorems 5.2 and 5.5] Let S be a regular semigroup and J a completely simple subsemigroup of S. S is the disjoint union of the local submonoids \(eSe, e\in E(J)\) if and only if it is isomorphic to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) where M is a regular monoid and the entries of the matrix P lie in the group of units of M. In this case, \(J\subseteq RP(S)\) and \(E(J)=E(RP(S))\) where RP(S) denotes the (completely simple) subsemigroup of regularity preserving elements of S.

The link with the previous results is as follows. By Corollary 3.12 and remark below, if S is the disjoint union of the local submonoids \(eSe, e\in E=E(J)\) with J completely simple, then S is completely E-simple with \(E=Max\) intrinsic. Also by Theorem 2.3 S is isomorphic to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) with M a monoid and such that the entries of the matrix P lie in the group of units \(G=M^{-1}\) of M. If moreover S is assumed regular, then M has to be regular. Direct calculations show that the set of regularity preserving elements of \(\mathcal {M}(M,I,\Lambda ,P)\) is the completely simple semigroup \(\mathcal {M}(G,I,\Lambda ,P)\), whose set of idempotents is precisely \(E=\{(i,p_{\lambda i}^{-1},\lambda )|\, i\in I, \lambda \in \Lambda \}=Max\). Conversely, if S is isomorphic to such a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\), then it is completely E-simple hence by Corollary 3.12 the disjoint union of the local submonoids \(eSe, e\in E\), and \(E=E(J)\) with \(J\sim \mathcal {M}(G,I,\Lambda ,P)\) completely simple.

We have proved that:

Corollary 3.16

Let S be a semigroup. Then the following statements are equivalent.
  1. (1)

    S is a completely E-simple semigroup;

     
  2. (2)

    S is isomorphic to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) with M a monoid and such that the entries of the matrix P lie in the group of units of M;

     
  3. (3)

    S is the disjoint union of the local submonoids \(eSe, e\in E(J)\) for a completely simple semigroup J.

     
In this case, \(E=Max=E(J)\sim \{(i,p_{\lambda i}^{-1},\lambda )|\, i\in I, \lambda \in \Lambda \}\).

Moreover, S is regular if and only if M is and in this case \([J\subseteq RP(S)=\bigcup _{e\in E} \mathcal {H}(e) \textit{ and } E=Max=E(J)=E(RP(S))]\).

While the extended relations \(\mathcal {K}^*\) and \(\widetilde{\mathcal {K}}\) are only interesting for non-regular semigroups, the extended relations \(\widetilde{\mathcal {K}}_E\) may thus also be valuable tools for the study of regular semigroups. Section 6 will also illustrate this fact.

4 The variety of \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups and its subvarieties

In order to deal with varieties of semigroups, we consider any \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup as a unary semigroup (that is as a particular (2, 1)-algebra), with the additional unary operation that takes any element to the unique idempotent of E in its \(\widetilde{\mathcal {H}}_E\)-class. We denote by \(^+\) this unary operation. We also pose \(S^+=\{x^+|\, x\in S\}\) and \(\sigma ^+=\{(x,y)\in S\times S|\, x^+=y^+\}\).

For any set of identities \(\{1,\ldots , n\}\), \(\mathcal {V}(i_{1},\ldots i_{k})\) denotes the variety of (2, 1)-algebras that satisfies the identities \((i_{1},\ldots i_{k})\).

Lemma 4.1

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. Then the unary operation \(^+\) satisfies:
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} xx^+&=x \end{aligned}$$
(2)
Conversely, any unary semigroup \((S,.,\,^+)\) that satisfies these quasi-identities is a \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant semigroup.

Moreover, in this case \(\widetilde{\mathcal {H}}_{S^+}=\sigma ^+\).

Proof

The implication follows from the definition of the extended Green’s relations. For the converse, we first note that \(S^+\) is a set of idempotents by Eqs. (1) and (R\(\Rightarrow \)) with \(y=x\). Second, Eq. (1) states that \(x\le _{\widetilde{\mathcal {R}}_{S^+}} x^+\) whereas (R\(\Rightarrow \)) states that \(x^+\le _{\widetilde{\mathcal {R}}_{S^+}} x\), and dually. \(\square \)

We can actually replace these quasi-identities by identities suggested by property \((\Pi )\), and consequently prove that \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups form a variety of unary semigroups.

Proposition 4.2

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. Then the unary operation \(^+\) satisfies:
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} xx^+&=x \end{aligned}$$
(2)
$$\begin{aligned} (xy^+)^+y^+&=(xy^+)^+ \end{aligned}$$
(3)
$$\begin{aligned} y^+(y^+x)^+&=(y^+x)^+ \end{aligned}$$
(4)
Conversely, any unary semigroup \((S,.,\,^+)\) that satisfies these identities is a \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant semigroup.

Proof

For the implication, we apply the quasi-identities of Lemma 4.1 to \(y^+(y^+x)=(y^+x)\) and \((xy^+)y^+=(xy^+)\). For the converse, we prove that the new identities imply the quasi-identities of Lemma 4.1. Let \(x,y\in S\) such that \(y^+x=x\). Then \(y^+x^+=y^+(y^+x)^+=(y^+x)^+=x^+\) by Eqs. (4) and (R\(\Rightarrow \)) is satisfied. We conclude by duality. \(\square \)

In the following corollary, we give two other descriptions of \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups. The first one uses the fact that \(S^+\) is a set of idempotents, and the second one that \(^+\) is a projection operator.

Corollary 4.3

A unary semigroup \((S,.,\,^+)\) is \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant if and only if the unary operation \(^+\) satisfies:
$$\begin{aligned} x^+x^+&=x^+ \end{aligned}$$
(5)
$$\begin{aligned} x^+xx^+&=x \end{aligned}$$
(6)
$$\begin{aligned} x^+(x^+zy^+)^+y^+&=(x^+zy^+)^+ \end{aligned}$$
(7)
or
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} xx^+&=x \end{aligned}$$
(2)
$$\begin{aligned} x^{++}&=x^+ \end{aligned}$$
(8)
$$\begin{aligned} x^+(xy)^+y^+&=(xy)^+ \end{aligned}$$
(9)

Proof

Set \(A=\{(1), (2), (3), (4)\}\), \(B=\{(5), (6), (7)\}\) and \(C=\{(1), (2), (8), (9)\}\). We prove that \(A\Rightarrow B\Rightarrow C \Rightarrow A\).
\(A\Rightarrow B\)
From Eqs. (1) and (4) with \(y=x\) we get \(x^+x^+=x^+(x^+x)^+=(x^+x)^+=x^+\). From Eqs. (1) and (2) we deduce (6). Also
$$\begin{aligned} x^+(x^+zy^+)^+y^+=(x^+zy^+)^+y^+=(x^+zy^+)^+ \end{aligned}$$
by Eqs. (4) and (3).
\(B\Rightarrow C\)

Equations (5) and (6) imply \(x^+x=x^+(x^+xx^+)=x^+xx^+=x=(x^+xx^+)x^+=xx^+\). Equation (9) then follows from Eq. (7) with \(z=xy\). Finally from Eq. (7) we deduce \((x^+)^+((x^+)^+zy^+)^+y^+=((x^+)^+zy^+)^+\) and with \(y=x, z=x^+\) we finally get \(x^{++}(x^{++}x^+x^+)^+x^+=(x^{++}x^+x^+)^+\) whence \(x^+=x^{++}\), and \(^+\) is a projection.

\(C\Rightarrow A\)
First, \(x^+=x^+x^{++}=x^+x^+\). Then
$$\begin{aligned} (xy^+)^+y^+=(xy^+)^+(xy^+y^+)^+y^{++}=(xy^+y^+)^+=(xy^+)^+ \end{aligned}$$
by Eq. (9), and dually. \(\square \)

Among these semigroups, \((\widetilde{\mathcal {L}}_E,\widetilde{\mathcal {R}}_E)\)-congruent semigroups need a priori additional quasi-identities.

Lemma 4.4

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. It is \((\widetilde{\mathcal {L}}_E,\widetilde{\mathcal {R}}_E)\)-congruent if and only if the unary operation \(^+\) satisfies the identities (1), (2), (3), (4) and the additional quasi-identities:
$$\begin{aligned} z^+yx=yx&\Rightarrow z^+yx^+=yx^+ \\ xyz^+=xy&\Rightarrow x^+yz^+=x^+y \end{aligned}$$

Proof

Assume S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and let \(x,y\in S\). As \(x\widetilde{\mathcal {R}}_Ex^+\) then \(yx\widetilde{\mathcal {R}}_Eyx^+\) and for all \(z^+\in S^+=E\), \(z^+yx=yx\Rightarrow z^+yx^+=yx^+\). The other quasi-identity is dual.

Conversely, assume that S is \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant and satisfies these quasi-identities. Let \(x,x',y,z\in S\) such that \(x'\widetilde{\mathcal {R}}_{S^+}x\) and \(z^+yx=yx\). Then \(z^+yx^+=yx^+\). As also \(x^+x=x\) by Proposition 4.2 then \(x^+x'=x'\) and \(z^+yx'=z^+yx^+x'=yx^+x'=yx'\). Finally \(yx'\le _{\widetilde{\mathcal {R}}_{S^+}} yx\). By symmetry, \(yx'\widetilde{\mathcal {R}}_{S^+}yx\) and S is \(\widetilde{\mathcal {R}}_{S^+}\)-congruent. The conclusion follows by dual arguments. \(\square \)

Again, we can use identities only.

Proposition 4.5

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. It is \((\widetilde{\mathcal {L}}_E,\widetilde{\mathcal {R}}_E)\)-congruent if and only if the unary operation \(^+\) satisfies the identities (1), (2), (3), (4) and the additional identities:
$$\begin{aligned} (x^+y)(xy)^+&=x^+y \end{aligned}$$
(10)
$$\begin{aligned} (yx)^+(yx^+)&=yx^+ \end{aligned}$$
(11)

Proof

For the implication, we apply the quasi-identities of Lemma 4.4 to \((yx)^+(yx)=(yx)\) and \((xy)(xy)^+=(xy)\). For the converse we prove that these identities imply the quasi-identities of Lemma 4.4. Let \(x,y,z\in S\) such that \(z^+yx=yx\). Then \((yx)^+=(z^+yx)^+=z^+(z^+yx)^+=z^+(yx)^+\) by (4). Thus by Eq. (11) \(z^+yx^+= z^+(yx)^+(yx^+)=(yx)^+(yx^+)=yx^+\). The other statement is dual. \(\square \)

Other identities are possible.

Corollary 4.6

A unary semigroup \((S,.,\,^+)\) is completely \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant if and only if the unary operation \(^+\) satisfies:
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} xx^+&=x \end{aligned}$$
(2)
$$\begin{aligned} x^+(xy)^+y^+&=(xy)^+ \end{aligned}$$
(9)
$$\begin{aligned} (xy)^+&=(x^+y)^+ \end{aligned}$$
(12)
$$\begin{aligned} (yx)^+&=(yx^+)^+ \end{aligned}$$
(13)

Proof

Let \(x,y\in S\). If S is completely \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant, then it satisfies Eqs. (1), (2) and (9) by Corollary 4.3. Also \(y\widetilde{\mathcal {R}}_{S^+}y^+\) implies \(xy\widetilde{\mathcal {R}}_{S^+}xy^+\) hence \((xy)^+=(xy^+)^+\), and dually. Conversely, assume that \((S,.,\,^+)\) satisfies the previous identities. From Eq. (9) with \(y=x^+\) and Eq. (2) we get that \(x^+x^+=x^+\). Thus Eq. (9) gives \(x^+(xy)^+=(xy)^+\) and Eq. (12) gives \(x^+(x^+y)^+=x^+(xy)^+=(xy)^+=(x^+y)^+\), that is Eq. (4). Dually Eq. (3) is satisfied and S is \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant. Finally \((x^+y)(xy)^+=(x^+y)(x^+y)^+ =x^+y\) by Eqs. (12) and (2) and Eq. (10) is satisfied. We conclude by duality and Proposition 4.5. \(\square \)

Also, we have an additional identity for \(\widetilde{\mathcal {H}}_E\)-congruence, whose proof is straightforward.

Lemma 4.7

Let S be a \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroup. It is \(\widetilde{\mathcal {H}}_E\)-congruent if and only if the unary operation \(^+\) satisfies the additional identity:
$$\begin{aligned} (xy)^+=(x^+y^+)^+ \end{aligned}$$
(14)

From Corollary 4.3, Proposition 4.5 and Lemma 4.7 we get:

Corollary 4.8

A \((E,\widetilde{\mathcal {H}}_E)\)-abundant, \(\widetilde{\mathcal {H}}_E\)-congruent semigroup is \((\widetilde{\mathcal {L}}_E,\widetilde{\mathcal {R}}_E)\)-congruent.

Proof

Let \(x,y\in S\). By Proposition 4.5 we have to prove that \((x^+y)(xy)^+=x^+y\) (and dually). As \((xy)^+=(x^+y^+)^+\) then \((x^+y)(xy)^+=(x^+y)(x^+y^+)^+\). Also by Corollary 4.3 \((x^+y)^+=(x^{++}y^+)^+=(x^+y^+)^+\). Finally \((x^+y)(xy)^+=(x^+y)(x^+y)^+=x^+y\). We conclude by duality. \(\square \)

We finally deduce from the previous results a characterization of completely E-simple semigroups.

Proposition 4.9

Let S be a completely E-simple semigroup. Then the unary operation \(^+\) satisfies:
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} xx^+&=x \end{aligned}$$
(2)
$$\begin{aligned} x^+(yx)^+&=x^+ \end{aligned}$$
(15)
$$\begin{aligned} (xy)^+x^+&=x^+ \end{aligned}$$
(16)
$$\begin{aligned} y^+(yx)^+&=(yx)^+ \end{aligned}$$
(17)
$$\begin{aligned} (xy)^+y^+&=(xy)^+ \end{aligned}$$
(18)
Conversely, any unary semigroup \((S,.,\,^+)\) that satisfies these identities is a completely \(S^+\)-simple semigroup.

Proof

Assume that S is completely E-simple. Then it is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and the first two identities follow from Proposition 4.2. Let \(x,y\in S\). By Corollary 3.14 \((yx)^+\widetilde{\mathcal {R}}_Eyx\widetilde{\mathcal {R}}_Ey\widetilde{\mathcal {R}}_Ey^+\). As \(y^+y=y\) then \(y^+(yx)^+=(yx)^+\) and as \((yx)^+(yx)=yx\) then \((yx)^+y^+=y^+\), or interchanging x and y, \((xy)^+x^+=x^+\). The two other equations follow by duality.

For the converse, assume that \((S,.,\,^+)\) is a unary semigroup that satisfies these identities and let \(x\in S\). From Eq. (15) we get \(x^{++}(yx^+)^+=x^{++}\) and with \(y=x\) we get \(x^+=x^{++}x^+=x^{++}(xx^+)^+=x^{++}\) and the operation is a projection. Let \(x,y\in S\). Then \(x^+(xy)^+y^+=(xy)^+\) by Eqs. (17) and (18), and by Corollary 4.3, S is \((S^+, \widetilde{\mathcal {H}}_{S^+})\)-abundant. Let \(x,y\in S\). As \((xy)^+x^+=x^+\) and \(x^+(xy)^+=(xy)^+\) then \(x^+\mathcal {R}(xy)^+\), and dually \(y^+\mathcal {L}(xy)^+\). Finally \(x\widetilde{\mathcal {R}}_Ex^+\widetilde{\mathcal {R}}_E(xy)^+\widetilde{\mathcal {L}}_Ey^+ \widetilde{\mathcal {L}}_Ey\) and S is \(\widetilde{\mathcal {D}}_{S^+}\)-simple. \(\square \)

This set of identities is not minimal. We can for instance replace the identities (17) and (18) by the single identity (9).

We summarize all the previous results in the following theorem.

Theorem 4.10

Let \((S,.,^+)\) be a unary semigroup, and let \(S^+=\{x^+|\, x\in S\}\). We consider the following identities on \((S,.,^+)\).
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} xx^+&=x \end{aligned}$$
(2)
$$\begin{aligned} (xy^+)^+y^+&=(xy^+)^+ \end{aligned}$$
(3)
$$\begin{aligned} y^+(y^+x)^+&=(y^+x)^+ \end{aligned}$$
(4)
$$\begin{aligned} x^+x^+&=x^+ \end{aligned}$$
(5)
$$\begin{aligned} x^+xx^+&=x \end{aligned}$$
(6)
$$\begin{aligned} x^+(x^+zy^+)^+y^+&=(x^+zy^+)^+ \end{aligned}$$
(7)
$$\begin{aligned} x^{++}&=x^+ \end{aligned}$$
(8)
$$\begin{aligned} x^+(xy)^+y^+&=(xy)^+ \end{aligned}$$
(9)
$$\begin{aligned} (x^+y)(xy)^+&=x^+y \end{aligned}$$
(10)
$$\begin{aligned} (yx)^+(yx^+)&=yx^+ \end{aligned}$$
(11)
$$\begin{aligned} (xy)^+&=(x^+y)^+ \end{aligned}$$
(12)
$$\begin{aligned} (yx)^+&=(yx^+)^+ \end{aligned}$$
(13)
$$\begin{aligned} (xy)^+&=(x^+y^+)^+ \end{aligned}$$
(14)
$$\begin{aligned} x^+(yx)^+&=x^+ \end{aligned}$$
(15)
$$\begin{aligned} (xy)^+x^+&=x^+ \end{aligned}$$
(16)
$$\begin{aligned} y^+(yx)^+&=(yx)^+ \end{aligned}$$
(17)
$$\begin{aligned} (xy)^+y^+&=(xy)^+ \end{aligned}$$
(18)
Then
  1. (1)

    \(S^+\mathcal {A}=\mathcal {V}(1,2,3,4)=\mathcal {V}(5,6,7)=\mathcal {V}(1,2,8,9)\) is the variety of unary \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups;

     
  2. (2)

    \(\mathcal {C}S^+\mathcal {A}=\mathcal {V}(1,2,3,4,10,11)=\mathcal {V}(1,2,9,12,13)\) is the subvariety of unary completely \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups;

     
  3. (3)

    \(S^+ \mathcal {C}\mathcal {G}=\mathcal {V}(1,2,3,4,14)\) is the subvariety of unary completely \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant, \(\widetilde{\mathcal {H}}_{S^+}\)-congruent semigroups (\(S^+\)-cryptogroups);

     
  4. (4)

    \(\mathcal {C}S^+\mathcal {S}=\mathcal {V}(1,2,15,16,17,18)=\mathcal {V}(1,2,9,15,16)\) is the subvariety of unary completely \(S^+\)-simple semigroups;

     
Moreover, \(\mathcal {C}S^+\mathcal {S}\subseteq S^+\mathcal {C}\mathcal {G}\subseteq \mathcal {C}S^+\mathcal {A}\subseteq S^+\mathcal {A}\).

This theorem suggests a closer study of the lattice of subvarieties of \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups.

By Birkhoff’s HSP Theorem, homomorphic images, subalgebras and products of unary \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups (resp. unary completely \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups, unary \(S^+\)-cryptogroups, unary completely \(S^+\)-simple semigroups) are unary \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups (resp. unary completely \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroups, unary \(S^+\)-cryptogroups, unary completely \(S^+\)-simple semigroups).

We finally close this section with the definition of a partial order on unary \((E,\widetilde{\mathcal {H}}_{E})\)-abundant semigroups, in the spirit of [7] and the partial order on restriction semigroups (see [26] and next section).

Proposition 4.11

Let S be a unary \((S^+,\widetilde{\mathcal {H}}_{S^+})\)-abundant semigroup. Then \(\sigma _L=\{(a,b)|\, a=a^+b\}\) and \(\sigma _R=\{(a,b)|\, a=ba^+\}\) are reflexive and transitive relations compatible with \(\,^+\), and \(\sigma _{H}=\sigma _L\cap \sigma _R=\{(a,b)|\, a=a^+b=ba^+\}\) is a partial order on S compatible with \(\,^+\).

Proof

Let \(a,b,c\in S\) such that \(a=a^+b\), \(b=b^+c\). As \(a^+a=a\) then \(\sigma _L\) is reflexive. As \(bb^+=b\) then \(ab^+=a\), and as \(a\widetilde{\mathcal {H}}_{S^+}a^+\), then \(a^{++}b^+=a^+b^+=a^+\), and \(\sigma _L\) is compatible with \(\,^+\). It follows that \(a=a^+b^+c=a^+ c\), and \(\sigma _L\) is transitive. Dually, \(\sigma _R\) is reflexive, compatible with \(\,^+\) and transitive.

We finally have to check antisymmetry of \(\sigma _H\). Assume that \(a=a^+b=ba^+\) and \(b=b^+a=ab^+\). Then by the previous arguments, \(a^+=a^+ b^+=b^+\) and finally \(a=b^+b=b\). \(\square \)

The partial order is finer than the natural partial order, as \(S^+\subseteq E(S)\).

5 Clifford restriction semigroups

In the previous sections, we have studied what could be considered as analogs to completely regular and completely simple semigroups, with respect to a distinguished subset E of idempotents. And we have shown that these semigroups may be fruitfully studied as unary semigroups. These two different approaches are already present in the literature in the context of ample semigroups and restriction semigroups. Ample semigroups may be considered as analogs of inverse semigroups with respect to a certain semilattice E of idempotents, and restriction semigroups as unary semigroups of partial functions with an operation \(a\mapsto a^+\) of restriction on the domain. We present rapidly the theory of ample and restriction semigroups in this section, together with the special case of central idempotents. This allows us to make the link with the previous sections. One can find a very nice introduction to the topic of ample and restriction semigroups in [26], and many references therein.

5.1 Restriction and ample semigroups

A unary semigroup \((S,.,\,^+)\) is a left restriction semigroup if its unary operation \(\,^+\) satisfies the identities:
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} x^+y^+&=y^+x^+ \end{aligned}$$
(S)
$$\begin{aligned} (x^+y)^+&=x^+y^+ \end{aligned}$$
(LC)
$$\begin{aligned} xy^+&=(xy)^+x \end{aligned}$$
(LA)
In this case \(E=S^+=\{x^+|\,x\in S\}\) is a semilattice (\((\forall x,y\in S) x^+x^+=x^+\) and \(x^+y^+=y^+x^+\)), and \(\,^+\) is a projection, that is the identity on \(E=S^+\) (\((\forall x\in S) (x^+)^+=x^+\)). These identities first appeared in [28]. A left restriction semigroup carries a partial order compatible with the multiplication (thanks to (LA)), defined by \(a\le b\Leftrightarrow a=a^+b\). This partial order extends the natural partial order on the semilattice E. It is folklore that a unary semigroup is a left restriction semigroup if and only if it embeds (as an algebra of type (2, 1)) in the unary semigroup of partial mappings on a set X, with unary operation \(\alpha ^+=I_{dom \alpha }\). The term restriction semigroup comes indeed from this concrete representation. The set E then corresponds to partial identities. A right restriction semigroup \((S,.,\,^*)\) is defined dually. A restriction semigroup is a bi-unary semigroup \((S,.,\,^+,\,^*)\) such that \((S,.,\,^+)\) is a left restriction semigroup, \((S,.,\,^*)\) a right restriction semigroup and \((\forall x\in S) (x^+)^*=x^+\) and \((x^*)^+=x^*\). In case the semigroup S is a restriction semigroup, then the two partial orders (defined upon each unary operation) coincide since \(a=a^+b\) implies \(a=b(a^+b)^*=ba^*\) and dually.
Let S be a semigroup and \(E\subseteq E(S)\) a semilattice. Then S is weakly left E-ample [17] if:
  1. (1)

    Every \(\widetilde{\mathcal {R}}_E\)-class \(\widetilde{\mathcal {R}}_E(a)\) contains a (necessarily unique) idempotent, that we denote by \(a^+\);

     
  2. (2)

    The relation \(\widetilde{\mathcal {R}}_E\) is a left congruence;

     
  3. (3)

    The left ample condition \((\forall a\in S, \forall e\in E) ae=(ae)^+a \) is satisfied.

     
Note that the idempotent \(a^+\) is then the least right identity of a in E. The left ample condition is equivalent with the following (type A) condition: \((\forall e\in E, \forall a\in S) Se\cap Sa=Sae\).

Weakly right E-ample semigroups are defined dually. Weakly E-ample semigroups are those semigroups both left and right weakly E-ample.

It is well known that left restriction semigroups are precisely weakly left E-ample semigroups, see for instance [26] Theorem 4.13. Precisely, a left restriction semigroup is weakly left \(S^+\)-ample and a weakly E-ample semigroup is a left restriction semigroup when equipped with the additional unary operation \(a\mapsto a^+\).

Other classes of semigroups will be of interest. A semigroup is E -semiadequate if it is \((E,\widetilde{\mathcal {L}}_E)\)-abundant, \((E,\widetilde{\mathcal {R}}_E)\)-abundant and E is a semilattice [35]. If moreover \(\widetilde{\mathcal {L}}_E\) is a right congruence and \(\widetilde{\mathcal {R}}_E\) a left congruence, then S is an Ehrehsmann semigroup (see [35] and the connection with Ehresmann categories). Restriction semigroups are thus Ehresmann semigroups with the left and right ample conditions.

5.2 Clifford restriction semigroups

Proposition 5.1

Let \((S,.,\,^+)\) be a unary semigroup that satisfies the following identities:
$$\begin{aligned} x^+x&=x \end{aligned}$$
(1)
$$\begin{aligned} x^+y&=yx^+ \end{aligned}$$
(19)
$$\begin{aligned} (xy)^{++}&=x^+y^+ \end{aligned}$$
(20)
Then
  1. (1)

    \((S,.,\,^+)\) is a left restriction semigroup and its set of projection \(S^+=\{x^+|\, x\in S\}\) is a semilattice of central idempotents (in S);

     
  2. (2)

    \((S,.,\,^+)\) is a right restriction semigroup and its set of projection \(S^+=\{x^+|\, x\in S\}\) is a semilattice of central idempotents (in S);

     
  3. (3)

    \((S,.,\,^+,\,^+)\) is a restriction semigroup and its set of projection \(S^+=\{x^+|\, x\in S\}\) is a semilattice of central idempotents (in S).

     
Conversely, any left restriction semigroup S with \(S^+=\{x^+|\, x\in S\}\) semilattice of central idempotents (in S) satisfies these axioms.

Proof

We first prove that \((S,., \,^+)\) is a left restriction semigroup. Let \(x,y\in S\). Equations (1) and (19) give \(xx^+=x\). In particular \(x^{+}x^{++}=x^+\). By letting \(y=x^+\) in Eq. (20) we get \(x^{++}=x^+x^{++}\), whence \(x^{++}=x^+x^{++}=x^+\), and \(\,^+\) is a projection. It then follows that\(x^+x^+=x^+\), and \(S^+\subseteq E(S)\). Also Eq. (20) then gives \((xy)^+=x^+y^+\). By letting \(x=x^+\) in this equation we get \((x^+y)^+=x^+y^+\) and the congruence condition (LC) is satisfied. Also \((xy)^+x=x^+y^+x=x^+xy^+=xy^+\) and the ample condition (LA) is satisfied. Finally \((S,., \,^+)\) is a left restriction semigroup. As also \(xx^+=x\) and Eqs. (19) and (20) are self-dual, \((S,., \,^+)\) is a right restriction semigroup by duality. It then follows that \((S,., \,^+,\,^+)\) is a restriction semigroup with \(E=\{x^+, x\in S\}\) semilattice of central idempotents (in S) since \((x^+)^+=x^+\).

Conversely, let S be a left restriction semigroup with \(E=\{x^+|\, x\in S\}\) a semilattice of central idempotents, and let \(x,y\in S\). As \(x^{++}=x^+\) in left restriction semigroups, then \((xy)^+=(x^+xy)^+=x^+(xy)^+=x^+(xyy^+)^+=x^+(y^+xy)^+=x^+y^+(xy)^+\) by Eq. (LC). By Eqs. (LA) and (LC), \((xy^+)^+=((xy)^+x)^+=(xy)^+x^+\). By centrality of \(y^+\) and Eq. (LC) we also have \((xy^+)^+=(y^+x)^+=y^+x^+\) and as \(y^+\) is idempotent, \((xy)^+x^+=y^+x^+ =(xy)^+x^+y^+\). Finally \((xy)^{++}=(xy)^+=x^+y^+(xy)^+=x^+y^+\) and Eq. (20) is satisfied. \(\square \)

Such a semigroup enjoys the following properties.
  1. (1)

    Elements of \(S^+\) are central idempotents;

     
  2. (2)

    Every \(\widetilde{\mathcal {H}}_{S^+}\)-class \(\widetilde{\mathcal {H}}_{S^+}(a)\) contains a (necessarily unique) idempotent \(a^+\);

     
  3. (3)

    The relation \(\widetilde{\mathcal {H}}_{S^+}\) is a congruence.

     
Conversely, we have the following proposition:

Proposition 5.2

Let S be a semigroup, and \(E\subseteq E(S)\) be a set of idempotents such that:
  1. (1)

    Elements of E are central idempotents;

     
  2. (2)

    Every \(\widetilde{\mathcal {H}}_E\)-class \(\widetilde{\mathcal {H}}_E(a)\) contains a (necessarily unique) idempotent, that we denote by \(a^+\);

     
  3. (3)

    The relation \(\widetilde{\mathcal {H}}_E\) is a congruence.

     
Then it is a restriction semigroup when equipped with the additional unary operations \(a\mapsto a^+=a^*\), and its set of projections \(E=\{x^+|\, x\in S\}\) is a semilattice of central idempotents (in S).

Proof

As elements of E are central, then E is a semilattice. Indeed, let \(e,f,g\in E\) such that \(ef=fe\widetilde{\mathcal {H}}_E g\). As \(efe=ef=fef\) then \(ge=fg=g\). But also \(gef=ef\) and finally \(ef=gef=gf=fg=g\in E\), whence E is a subsemigroup. Also \(\widetilde{\mathcal {L}}_E=\widetilde{\mathcal {R}}_E=\widetilde{\mathcal {H}}_E\) are congruences. Let \(a\in S\) and \(e\in E\). As \(a^+\widetilde{\mathcal {H}}_E a\), then \(a^+e\widetilde{\mathcal {H}}_E ae\widetilde{\mathcal {H}}_E (ae)^+\). As \((a^+e)e=a^+e\) then \((ae)^+e=(ae)^+\) and \(ae=(ae)^+(ae)=(ae)^+ea=(ae)^+a\). Finally the (left) ample condition \(ae=(ae)^+a\) is satisfied and \((S,., \,^+)\) is a left restriction semigroup (it is weakly E-ample). Dually it is a right restriction semigroup with the same unary operation, hence a restriction semigroup. \(\square \)

Centrality of the idempotents actually follows from the sole identity \(x^*=x^+\), or from the congruence of \(\widetilde{\mathcal {H}}_E\). The following theorem, which summarizes the previous results, should be compared with [45, Theorem 3.10].

Theorem 5.3

Let \((S,., \,^+)\) be a unary semigroup. Then the following statements are equivalent:
  1. (1)

    S is a left restriction semigroup with \(a\mapsto a^+\) a retraction from S onto \(S^+\);

     
  2. (2)

    S is a left restriction semigroup with \((xy)^+=x^+y^+\);

     
  3. (3)

    S is a left restriction semigroup with \(S^+=\{x^+|\, x\in S\}\) semilattice of central idempotents;

     
  4. (4)

    S satisfies the identities \(x^+x=x\), \(x^+y=yx^+\) and \((xy)^{++}=x^+y^+\);

     
  5. (5)

    \((S,., \,^+, \,^+)\) is a restriction semigroup.

     

Proof

\((1)\Rightarrow (2)\)

This is the homomorphism property of the retraction.

\((2)\Rightarrow (3)\)

As S is a left restriction semigroup, it is left ample and \(xy^+=(xy)^+x\). But \((xy)^+=x^+y^+\) by assumption and since \(S^+\) is a semilattice and \(x^+x=x\), then \(xy^+=x^+y^+x=y^+x^+x=y^+x\) and the elements of \(S^+\) are central idempotents.

\((3)\Rightarrow (4)\)

This follows from Proposition 5.1.

\((4)\Rightarrow (5)\)

This follows from Proposition 5.1.

\((5)\Rightarrow (1)\)
Assume \((S,., \,^+, \,^+)\) is a restriction semigroup. Then
$$\begin{aligned} x^+y^+=(xy^+)^+= ((xy)^+x)^+= (xy)^+x^+. \end{aligned}$$
Dually, \(x^+y^+=y^+(xy)^+\). It follows that \(x^+y^+(xy)^+=(xy)^+x^+y^+=x^+y^+\) and \(x^+y^+ \omega (xy)^+\). Also \(x^+y^+(xy)=(xy)^+x^+(xy)=(xy)^+(xy)=xy\) and by Equation (LC), \(x^+y^+(xy)^+=(xy)^+\). Finally \((xy)^+=x^+y^+\) and \(a\mapsto a^+\) is a homomorphism. By (LC), \(x^+=(x^+x)^+=x^+x^+\) and then by (LC) again, \(x^{++}=(x^+x^+)^+=x^+x^{++}=x^+\), and \(\,^+\) fixes elements of \(S^+\).\(\square \)

Example 5.4

Centrality of \(S^+\) is not sufficient. Let \(S=\{e,f,g=ef=fe\}\) be the semilattice of Example 3.1 and pose \(e^+=e\), \(f^+=f=g^+\). Then \((S,.,\,^+)\) is a unary semigroup with \(S^+\) a set of central idempotents (but not a semilattice). As \(fe^+=g\ne (fe)^+f=f\) then Eq. (LC) is not satisfied. Also \((e^+g)^+=f\ne e^+g^+=g\) and Eq. (LA) is not satisfied.

By analogy with Clifford semigroups, whose idempotents are central, which are completely regular (every \(\mathcal {H}\)-class contains an idempotent) and cryptic (\(\mathcal {H}\) is a congruence), we call the unary semigroups of Theorem 5.3 Clifford restriction semigroups. As those semigroups are defined by a set of identities, they form a variety of algebras of type (2, 1) and as such, are stable under direct product, homomorphic images and subalgebras. If we want to look at these semigroups as plain semigroups with a distinguished set of idempotents (as in Proposition 5.2), we will preferably call them E -Clifford restriction semigroups (instead of weakly E-ample semigroups with central idempotents). By Corollary 2.9, if S is a E-Clifford restriction semigroup, then \(H(E)=\bigcup _{e\in E}\mathcal {H}(e)\) is a Clifford subsemigroup of S.

It is known that Clifford semigroups may be simply defined as completely regular and inverse semigroups, that is completely regular semigroups whose set of idempotents commute (In this case E(S) is a semilattice). E-Clifford restriction semigroups also admit such a characterization, using the generalizations of inverse semigroups we have seen: E-semiadequate semigroups and Ehresmann semigroups.

Corollary 5.5

Let S be a semigroup, and \(E\subseteq E(S)\) be a set of idempotents. Then the following statements are equivalent:
  1. (1)

    S is a E-Clifford restriction semigroup;

     
  2. (2)

    S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and Ehresmann;

     
  3. (3)

    S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and E-semiadequate.

     
  4. (4)

    S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and E is a semilattice.

     
  5. (5)

    S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and idempotents of E commute.

     

Proof

\((1)\Rightarrow (2)\)

Assume that S is a E-Clifford restriction semigroup. Then by Proposition 5.2 S is \((E,\widetilde{\mathcal {H}}_E)\)-abundant and \(\widetilde{\mathcal {H}}_E\)-congruent. By Corollary 4.8, \(\widetilde{\mathcal {L}}_E\) is a right congruence and \(\widetilde{\mathcal {R}}_E\) is a left congruence, and S is an Ehresmann semigroup.

\((2)\Rightarrow (3)\)

Straightforward.

\((3)\Rightarrow (4)\)

Straightforward.

\((4)\Rightarrow (5)\)

Straightforward.

\((5)\Rightarrow (1)\)

Assume S is completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant and idempotents of E commute. Let \(a\in S\) and \(e\in E\), and let \(a^+\) be the only idempotent in \(E\cap \widetilde{\mathcal {H}}_E(a)\). As elements of E commute, then \(a^+e=ea^+\) hence \(ae=aa^+e=aea^+\). As \(ae\widetilde{\mathcal {H}}_E (ae)^+\) then \((ae)^+a^+=(ae)^+\). Also, by Proposition 4.2, \((ae)^+=(ae)^+e\) and \((ea)^+=e(ea)^+\). As by right congruence, \((ae)^+\widetilde{\mathcal {L}}_E ae \widetilde{\mathcal {L}}_E a^+e=ea^+\) and \((ae)^+ (ae)^+=(ae)^+\) then \(ea^+(ae)^+=ea^+\). It follows that \((ae)^{+}=ea^+\) and E is stable by multiplication, hence a semilattice. Also \(eae=ea^+ae=(ae)^+ae=ae\). Dually \(eae=ea\) and idempotents of E are central. Finally, we prove that the ample condition is satsfied. As \((ae)^{+}=ea^+\) then \((ae)^+ a= (ea^+) a=ea=ae\). Finally S is a weakly E-ample semigroup with central idempotents, hence a E-Clifford restriction semigroup. \(\square \)

Some results of the first sections apply to these semigroups. We consider first the subdirect product decomposition. Consider a monoid M (with or without 0), and \(M^0=M\dot{\bigcup }\{0\}\) the monoid M with an extra zero added. There are two distinct unary operations that make it a Clifford restriction semigroup, the operation \(a\mapsto a^*=1\) for all \(a\in M^0\), and the operation \(a\mapsto a^*=1\) for all \(a\in M\) and \(0^*=0\). It is this second operation we consider here. For any direct product P of monoids and monoids with a zero added, \(P=(\Pi _{i\in I} M_i)(\Pi _{j\in J} M_j^0)\), we thus define a product map \(a\mapsto a^*\) on P by \((a^*)_i=1_i, i\in I\), \((a^*)_j=0_j, j\in J\) if \(a_j=0_j\) and \((a^*)_j=1_j, j\in J\) otherwise. Obviously, \(a\mapsto a^*\) is a retraction of P onto its image \(E=P^*\). We call \((P,.,\,^*)\) a direct product of restriction monoids and restriction monoids with a zero added. By Theorem 5.3 and Birkoff’s HSP theorem applied to direct product of left restriction semigroups, P is a Clifford restriction semigroup.

Corollary 5.6

Let \((S,.,\,^+)\) be a subdirect product of restriction monoids and restriction monoids with a zero added. Then it is a Clifford restriction semigroup. Conversely, any Clifford restriction semigroup is such a subdirect product.

Proof

Let \((S,.,\,^+)\) be such a subdirect product. By Birkoff’s HSP theorem, it is a Clifford restriction semigroup as a subalgebra of direct products of Clifford restriction semigroups. For the converse, let \((S,.,^+)\) be a Clifford restriction semigroup. Then as a plain semigroup it is a E-Clifford restriction semigroup with \(E=S^+\subseteq Z(S)\). By Corollary 3.7, (S, .) is then a subdirect product of the factors \(\widetilde{\mathcal {H}}_E(e)=Se\), \(e\in E\) primitive (within E) and \(\widetilde{\mathcal {H}}_E^0(e)=Se/\left( \bigcup _{ f\in E, f<e} Sf\right) \), \(e\in E\) not primitive, which are monoids and monoids with a zero added. Let I be the set of primitive idempotents and J the set of non primitive idempotents of E (we thus have \(1_e=e\) for \(e\in I\cup J=E\)). Pose \(P=(\Pi _{e\in I} \widetilde{\mathcal {H}}_E(e))(\Pi _{e\in J} \widetilde{\mathcal {H}}_E^0(e))\). Then we can endow P with the previous retraction \(a\mapsto a^*\) to form a Clifford restriction semigroup. As (S, .) is completely abundant, then for all \(e\in E\) the \(\widetilde{\mathcal {H}}_E\)-class \(\widetilde{\mathcal {H}}_E(e)=Se\backslash \left( \bigcup _{ f\in E, f<e} Sf\right) \) is a monoid and the two operations \(\,^+\) and \(\,^*\) coincide on S. Finally, \((S,.,\,^+)\) is a subdirect product of \((P,.,\,^*)\). \(\square \)

Next example shows the interest of working with unary semigroups instead of plain semigroups.

Example 5.7

Consider once again Example 3.3. Then \(E=\{1>e>0\}\) and 0 is the only primitive idempotent of E. S is a subdirect product of \(P=\widetilde{\mathcal {H}}_E^{0_1}(1)\times \widetilde{\mathcal {H}}_E^{0_e}(e)\times \widetilde{\mathcal {H}}_E(0)\) with \(\widetilde{\mathcal {H}}_E(1)=S\) \(\widetilde{\mathcal {H}}_E(e)=\{e,0\}\) and \(\widetilde{\mathcal {H}}_E(0)=\{0\}\), but S is not completely \(\widetilde{\mathcal {H}}_E\)-abundant. If we endow P with the previous operation \(a\mapsto a^*\), we get that \((P,.,\,^*)\) is a Clifford restriction semigroup. But \(a\hookrightarrow (a,0_e,0)\) in P and \((a,0_e,0)^*=(1,0_e,0)\) that is not the image of an element of S (the image of 1 is (1, e, 0)). The restriction of \(\,^*\) to S is not defined.

As E-Clifford restriction semigroups are clearly completely \((E,\widetilde{\mathcal {H}}_E)\)-abundant semigroups, we can use the decomposition induced by Theorem 2.7. By a strong semilattice of monoids, we mean a strong semilattice Y of semigroups \(S_{\alpha }\) with identity \(1_{\alpha }\) such that the semigroup homomorphisms \(\phi _{\alpha ,\beta }, \alpha \ge \beta \) are monoid homomorphisms (they send identities to identities, \(1_{\alpha }\phi _{\alpha ,\beta }=1_{\beta }\) for all \(\alpha \ge \beta \)).

Some equivalences in Theorems 5.3 and 5.8 appear in a different form in [29], in the study of the lattice of varieties of restriction semigroups, and in [16] in the study of E-semiadequate semigroups. Also, we can check easily that a semilattice Y of unipotent monoids automatically satisfies that \(1_{\alpha }1_{\beta }=1_{\alpha \beta } (\forall \alpha ,\beta \in Y)\), and that \(E(S)\sim Y\) in this case. Hence we get an equivalence between E(S)-Clifford restriction semigroups and (strong) semilattices of unipotent monoids. This result, and those of Theorem 5.3 in the special case \(E=E(S)\), were actually proved by Petrich in the context of weakly (left) ample semigroup (Theorem 3.10 and Theorem 4.3 in [45]), that is (left) restriction semigroups with \(E=E(S)\).

Theorem 5.8

The following statements are equivalent:
  1. (1)

    S is a E-Clifford restriction semigroup;

     
  2. (2)

    S is a semilattice Y of monoids \(M_{\alpha }, \alpha \in Y\) with \(1_{\alpha }1_{\beta }=1_{\alpha \beta } (\forall \alpha ,\beta \in Y)\), and \(E=\{1_{\alpha }|\, \alpha \in Y\}\);

     
  3. (3)

    S is a semilattice E of monoids \(M_e, e\in E\), with identities e;

     
  4. (4)

    S is a strong semilattice Y of monoids \(M_{\alpha }, \alpha \in Y\), and \(E=\{1_{\alpha }|\, \alpha \in Y\}\).

     

Proof

\((1)\Rightarrow (2)\)

Let S be a E-Clifford restriction semigroup. By Theorem 2.7, S is a semilattice Y of completely \(E_{\alpha }\)-simple semigroups \(M_{\alpha }\), with Y the set of \(\widetilde{\mathcal {J}}_E=\widetilde{\mathcal {D}}_E\)-classes of S. As idempotents of \(E_{\alpha }\) commute with each \(M_{\alpha }\) and are primitive, then each \(M_{\alpha }\) contains a unique idempotent, whence it is a monoid, and S is a semilattice of monoids \(M_{\alpha }, \alpha \in Y\) with \(Y\sim E\sim \{1_{\alpha }|\, \alpha \in Y\}\). The set of identities thus satisfy \(1_{\alpha }1_{\beta }=1_{\alpha \beta } (\forall \alpha ,\beta \in Y)\).

\((2)\Rightarrow (3)\)

The application \(Y\rightarrow E\) that maps \(\alpha \) to \(1_{\alpha }\) is an isomorphism by hypothesis.

\((3)\Rightarrow (4)\)

Let S be a semilattice E of monoids \(M_{e}\), \(e\in E\), with e identity of the monoid. Then for all \(e,f\in E\), \(ef\in E\) is the identity of \(M_{ef}\) by construction. For all \(e\ge f\), it makes sense to define a map \(\phi _{e,f}:M_{e}\rightarrow M_{f}\) by \(x\phi _{e,f}=xf\). These are monoid homomorphisms by assumption, and they define S as a strong semilattice of monoids.

\((4)\Rightarrow (1)\)

Let S be a strong semilattice Y of monoids \(M_{\alpha }\), \(\alpha \in Y\), with homomorphisms \(\phi _{\alpha ,\beta } (\forall \alpha \ge \beta )\). Let E be the set of identities \(1_{\alpha }\) of the \(M_{\alpha }, \alpha \in Y\). Let \(a\in M_{\alpha }, f=1_{\beta }, \alpha ,\beta \in Y\). Then \(af=\phi _{\alpha ,\alpha \beta }(a)\phi _{\beta ,\alpha \beta }(f)=\phi _{\alpha ,\alpha \beta }(a)1_{\alpha \beta }=phi_{\alpha ,\alpha \beta }(a)\) since the homomorphisms send identities to identities. Dually \(fa=\phi _{\alpha ,\alpha \beta }(a)\) and the elements of E are central. As E is also a semigroup of idempotents, then E is a semilattice of central idempotents (and \(\widetilde{\mathcal {L}}_E=\widetilde{\mathcal {R}}_E=\widetilde{\mathcal {H}}_E\)).

Let \(a\in M_{\alpha }, b\in M_{\beta }\) such that \(a \widetilde{\mathcal {L}}_E b\). As \(a1_{\alpha }=a\) then \(b 1_{\alpha }=b\). As \(b=b1_{\alpha }\in M_{\beta }\cap M_{\beta \alpha }\) and the monoids are disjoint then \(\beta \alpha =\beta \). Symmetrically, \(\alpha \beta =\alpha \) and finally \(\alpha =\beta \). Conversely, let \(a,b\in M_{\alpha }\) and \(1_{\beta }\in E\) such that \(a1_{\beta }=a\). Then as above \(\alpha \beta =\alpha \). As \(b1_{\beta }=b\phi _{\alpha ,\alpha \beta } 1_{\beta }\phi _{\beta ,\alpha \beta }\), then \(b1_{\beta }=b\phi _{\alpha ,\alpha } 1_{\beta }\phi _{\beta ,\alpha }=b1_{\alpha }=b\) and \(a \widetilde{\mathcal {L}}_E b\). Finally, \(a \widetilde{\mathcal {L}}_E b\) if and only if they belong to the same monoid, which is clearly a congruence. Finally let \(a\in M_{\alpha }\). Then \(a\widetilde{\mathcal {H}}_E 1_{\alpha }\) where \(1_{\alpha }\) is the (unique) idempotent in the \(\widetilde{\mathcal {H}}_E\)-class \(\widetilde{\mathcal {H}}_E(a)\). By Proposition 5.2, S is a E-Clifford restriction semigroup. \(\square \)

This theorem translates in the language of unary semigroups as follows:

Corollary 5.9

The following statements are equivalent:
  1. (1)

    \((S,.,\,^+)\) is a Clifford restriction semigroup;

     
  2. (2)

    \((S,.,\,^+)\) is a semilattice of restriction monoids and \(a\mapsto a^+\) is an homomorphism;

     
  3. (3)

    \((S,.,\,^+)\) is a strong semilattice of restriction monoids.

     

Example 5.10

A semilattice of restriction monoids may not be a Clifford restriction semigroup. Consider the restriction semilattice \(S=\{e=e^+, f=f^+=g^+, g=ef=fe\}\) of Example 5.4. Then \(\sigma ^{+}\) is a congruence and each \(\sigma ^+\)-class is a restriction monoid (\(\sigma ^+(e)=\{e=e^+\}\), \(\sigma ^+(f)=\{f=f^+=g^+, g\}\)). But \(e^+f^+=g\ne (ef)^+=f\) and it is not a Clifford restriction semigroup.

Example 5.11

In this direction, a famous theorem is Theorem 1 of Fountain [11], which states the equivalence between right PP semigroups with central idempotents and semilattices of left cancellative monoids. We give a proof of this equivalence below using E(S)-Clifford restriction semigroups. Let S be a \(\mathcal {L}^*\)-abundant semigroup (right PP semigroup) with central idempotents. As \(\mathcal {L}^*\subseteq \widetilde{\mathcal {L}}\) and idempotents are central (hence \(\widetilde{\mathcal {L}}=\widetilde{\mathcal {R}}\)), it is \(\widetilde{\mathcal {H}}\)-abundant. Let \(a,b\in S, e,f\in E(S)\) such that \(a\widetilde{\mathcal {H}} e\) and \(b\widetilde{\mathcal {H}} f\). Then by unicity of idempotents in a \(\widetilde{\mathcal {H}}\)-class, \(a\mathcal {L}^* e\), \(b\mathcal {L}^* f\) and by right congruence, \(ab\mathcal {L}^*eb\) and \(be \mathcal {L}^* fe\). By centrality of idempotents \(ab \mathcal {L}^*eb=be \mathcal {L}^* fe=ef\), and \(ab \widetilde{\mathcal {H}} ef\), whence \(\widetilde{\mathcal {H}}\) is a congruence. It follows that S is a E(S)-Clifford restriction semigroup, hence a semilattice of unipotent monoids. It is straightforward to see that each \(\widetilde{\mathcal {H}}\)-class is a left cancellative monoid. Conversely, let S be a semilattice Y of semigroups \(S_{\alpha }\), such that each \(S_{\alpha }\) is a left cancellative monoid (in particular each \(S_{\alpha }\) is unipotent). Then S is a E(S)-Clifford restriction semigroup, and the semigroups \(S_{\alpha }\) are the \(\widetilde{\mathcal {H}}\)-classes. Let \(a\in S\), and e be the idempotent in \(\widetilde{\mathcal {H}}(a)\). As \(ae=a\) then \((\forall x,y\in S) ex=ey\implies ax=ay\). Conversely, let \(x\in S, y\in S^1\) such that \(ax=ay\), and let f be the idempotent in \(\widetilde{\mathcal {H}}(x)\). As \(\widetilde{\mathcal {H}}\) is a congruence then axafexefaexaef belong to the same \(\widetilde{\mathcal {H}}\)-class \(\widetilde{\mathcal {H}}(ef)\), and since \(ax=ay\), then \(ay, ey, aey\in \widetilde{\mathcal {H}}(ef)\). As idempotents are central then \(ax=ay\) implies \((aef)(xef)=(aef)(yef)\) and as each \(\widetilde{\mathcal {H}}\)-class is left cancellative, then \(xef=yef\), hence \((ex)(ef)=(ey)(ef)\). Once again by cancellation in the monoid \(\widetilde{\mathcal {H}}(ef)\), \(ex=ey\). This shows that \(a\mathcal {L}^* e\), and the semigroup S is right PP with central idempotents. We recover Theorem 1 of Fountain [11].

Finally, we provide a last characterization for proper Clifford restriction semigroups (see for instance [20]). Let S be a weakly E-ample semigroup. We denote the least congruence identifying the projections E by \(\sigma _E\). We have \(a\sigma _E b\Leftrightarrow \exists e\in E, ea=eb\Leftrightarrow \exists f\in E, af=bf\). The semigroup is proper if \(\widetilde{\mathcal {R}}_E\cap \sigma _E=\iota \) and \(\widetilde{\mathcal {L}}_E\cap \sigma _E=\iota \) (for instance, an inverse semigroup is proper if and only if it is E-unitary, [27] proposition 5.9.1). In this case \(s\theta =(s^+, s\sigma _E)\) is one-to-one from S to \(E\times S\sigma _E\) (but not a morphism in general). We note that \(S\sigma _E\) is a monoid, and that to each element \(m=s\sigma _E\) is associated an order ideal of E
$$\begin{aligned} {I}(m)=\left\{ e\in E|\, \exists a\in m, a^+=e\right\} . \end{aligned}$$
Moreover, these order ideals satisfy the following relation:
$$\begin{aligned} (\forall t,s\in S) {I}(t\sigma _E)\bigcap {I}(s\sigma _E)\subseteq {I}(ts\sigma _E). \end{aligned}$$
Also \((\forall e\in E){I}(e\sigma _E)=E.\) Pose \(\mathcal {M}(S\sigma _E, E, {I})=\{(e,m)\in E\times S\sigma _E|\, e\in {I}(m)\}\). It is a subsemigroup of \(E\times S\sigma _E\), and we can endow it with a unary operation \((e,m)^+=(e,1)\). By definition of the order ideals, \(\theta \) maps S onto \(\mathcal {M}(S\sigma _E, E, {I})\). Finally, assume that S is a E-Clifford restriction semigroup, with \(E=S^+=\{x^+|\, x\in S\}\). Then the map \(\theta : S\rightarrow \mathcal {M}(S\sigma _E, E, {I})\) is an isomorphism of unary semigroups. Conversely, let E be a semilattice and M be a monoid. Consider OrdI(E) the set of order ideals of E. Then \((OrdI(E),\subseteq )\) is a preorder (actually a poset). We preorder M by Green’s preorder \(\le _{\mathcal {J}}\), \(x\le _{\mathcal {J}} y\Leftrightarrow \exists t,s\in M, x=tys\). Let \({I}: (M,\le _{\mathcal {J}}) \rightarrow (OrdI(E),\subseteq )\) be a non-decreasing function (functor of the preorders) that sends 1 to E. Then \(\mathcal {M}(M, E,{I})=\{(e,m)\in E\times M|\, e\in {I}(m)\}\) endowed with the binary operation \((e,m)(f,n)=(ef,mn)\) and the unary operation \((e,m)^+=(e,1)\) is a proper Clifford restriction semigroup (subsemigroup of \(E\times M\)). Finally, we have shown that:

Theorem 5.12

A unary semigroup \((S,.,\,^+)\) is a proper Clifford restriction semigroup if and only if it is isomorphic to a (unary) semigroup \(\mathcal {M}(M, E,{I})\), for a given monoid M, semilattice E and preorder functor \({I}:(M,\le _{\mathcal {J}}) \rightarrow (OrdI(E),\subseteq )\) that sends 1 (identity of M) to E. In this case \(S^+\sim E\).

As we are also interested in the semilattice decomposition, we can translate the previous results:

Corollary 5.13

A semigroup S is a strong semilattice Y of monoids \(M_{\alpha }\), \(\alpha \in Y\), whose homomorphisms \(\phi _{\alpha ,\beta }:M_{\alpha }\rightarrow M_{\beta } (\alpha \ge \beta )\) are one-to-one if and only if it is isomorphic to a semigroup \(\mathcal {M}(M, Y,{I})\), for a given monoid M and preorder functor \({I}:(M,\le _{\mathcal {J}}) \rightarrow (OrdI(Y),\subseteq )\) that sends 1 (identity of M) to Y.

Proof

Assume S is a strong semilattice Y of monoids \(M_{\alpha }\), \(\alpha \in Y\), whose homomorphisms \(\phi _{\alpha ,\beta }:M_{\alpha }\rightarrow M_{\beta } (\alpha \ge \beta )\) are one-to-one. By Theorem 5.8, S is a E-Clifford restriction semigroup, with E the set of identities \(1_{\alpha }\) of the \(M_{\alpha }, \alpha \in Y\), or equivalently \((S,.,\,^+)\) is a Clifford restriction semigroup with \(a^+=1_{\alpha }\) for all \(\alpha \in Y\) and \(a\in M_{\alpha }\) (and \(Y\sim E\sim S^+\)). Let \((a,b)\in \widetilde{\mathcal {R}}_E\cap \sigma _E\). Then as shown previously, a and b belong to the same monoid \(M_{\alpha }\), and \(e=1_{\beta }\) satisfies \(ea=eb\). As \(ea=1_{\beta }\phi _{\beta ,\alpha \beta }a\phi _{\alpha ,\alpha \beta } = 1_{\alpha \beta }a\phi _{\alpha ,\alpha \beta }=a\phi _{\alpha ,\alpha \beta }\) and symmetrically \(eb=b\phi _{\alpha ,\alpha \beta }\) then \(a\phi _{\alpha ,\alpha \beta }=b\phi _{\alpha ,\alpha \beta }\) and as the homomorphisms are one-to-one, \(a=b\) and the semigroup S is proper. By Theorem 5.12, \((S,.,\,^+)\) is isomorphic to a semigroup \(\mathcal {M}(M, E,{I})\), for a given monoid M, semilattice E and preorder functor \({I}:(M,\le _{\mathcal {J}}) \rightarrow (OrdI(E),\subseteq )\) that sends 1 (identity of M) to Y, and \(Y\sim E\).

Conversely, let S be isomorphic to \(\mathcal {M}(M, Y,{I})\), for a given monoid M, semilattice Y and preorder functor \({I}:(M,\le _{\mathcal {J}}) \rightarrow (OrdI(Y),\subseteq )\) that sends 1 (identity of M) to Y. By Theorem 5.12, it is a proper E-restriction semigroup with \(Y\sim E\) and by Theorem 5.8 and its proof it is a strong semilattice Y of monoids \(M_{\alpha }\), \(\alpha \in Y\), whose homomorphisms \(\phi _{\alpha ,\beta }:M_{\alpha }\rightarrow M_{\beta } (\alpha \ge \beta )\) are defined by \(a\phi _{\alpha ,\beta }=a 1_{\beta }\). Let \(a,b\in M_{\alpha }\) such that \(a 1_{\beta }=b 1_{\beta }\). Then \((a,b)\in \widetilde{\mathcal {R}}_E\cap \sigma _E\) and since S is proper, \(a=b\) and the homomorphisms are one-to-one. \(\square \)

6 Application: T-regular semigroups

The aim of this section is to develop a concept close to unit-regularity using maximal subgroups of a semigroup instead of solely the group of units. This idea appears (in a different context) notably in the work of Fountain, Petrich, Gould and others on orders on semigroup (equivalently semigroups of quotients) [15, 18]. As noted by Gould [19]: “Their aim was to develop concepts that reflect the equal importance of all subgroups of a semigroup, not only the group of units, which of course may not even exist.”. It is then tempting to replace directly the group of units by the union of the maximal subgroups, that is replace units by local units (group invertible elements). However, doing uniquely this may not be sufficient, notably to get structure theorems. Indeed, a crucial property of units in a monoid is that they are majorants for the preorder \(\le _{\mathcal {H}}\) and maximal for the \(\omega \) preorder. Indeed, for any \(a\in S\) and \(u\in S^{-1}\), \(a=au^{-1} u=uu^{-1} a\) and a\(\le _{\mathcal {H}} u\). Also, if \(u\omega a\), \(u\in S^{-1}\), then \(\exists e,f\in E(S), u=ea=af.\) It follows that \(eu=u\) hence \(e=euu^{-1}=1\), \(a=u\) and \(f=1\). As second feature is that the identity is a central idempotent. Recall that a regular semigroup with central (resp. commuting) idempotents is a Clifford (resp. inverse) semigroup.

We start with a general definition.

Definition 6.1

Let S be a regular semigroup, T a subset of S. An element \(a\in S\) is T -regular (resp. T -dominated) if it admits an associate (resp. a majorant for the natural partial order) \(x\in T\). S is T-regular (resp. T-dominated) if each element is T-regular (resp. T-dominated).

As explained in the introduction of the section, we will be interested in subsets T of group invertible elements, \(T\subseteq H(E(S))\), stable by group inversion (\(T^\#\subseteq T\)).

Next lemma expresses the link between regularity and domination, for x a group (completely regular) element.

Lemma 6.2

Let \(a\in S,\; x\in H(E(S))\). Then
$$\begin{aligned} a\omega x \Longleftrightarrow ax^{^\#}a=a, \; a\le _{\mathcal {H}} x. \end{aligned}$$

Proof

Let \(a\in S, x\in H(E(S))\) such that \(a\omega x\). Then exists \(e,f\in E(S)\), \(a=ex=x f\). It follows that \(a\le _{\mathcal {H}} x\) and \(ax^\#a=ex x^\#x f=exf=af=a\).

Conversely, let \(x\in H(E(S))\) such that \(ax^\#a=a\) and \(a\le _{\mathcal {H}} x\). Then by cancellation properties \(a=(ax^\#)x=x(x^\#a)\), with \(ax,xa\in E(S)\), and \(a\omega x\). \(\square \)

Example 6.3

To see that the converse does not hold, consider the Rees matrix semigroup \(S=\mathcal {M}^{0}\left( G,\{1,2\},\{1,2\},\left( \begin{array}{cc} 1 &{} 1\\ 1 &{} 0 \end{array}\right) \right) \) with \(G=\{e\}\) (it is the smallest non-orthodox regular semigroup). Then \(a=(2,e,2)\) has a unique associate \(x=(1,e,1)\in E(S)\), but x is not a majorant of a for the natural partial order. It follows that a is H(E(S))-regular but not H(E(S))-dominated. Observe that all the other elements are idempotents and the whole semigroup S is H(E(S))-regular (but not H(E(S))-dominated).

It is well known that unit-regular elements of a monoid S can be characterized as elements of the form \(a=eu\) (resp. \(a=ue\)) with \(e\in E(S)\) and \(u\in S^{-1}\). In particular, a unit-regular monoid S satisfies \(S=E(S)\mathcal {H}(1)(=\mathcal {H}(1)E(S))\), and is therefore also called factorisable monoid.

An analog characterization is valid for T-dominated elements, when \(T\subseteq H(E(S))\).

Lemma 6.4

Let \(a\in S\) and \(T\subseteq H(E(S))\). Then a is T-dominated if and only if \(a=eh\), with \(e\in E(S), h\in T\) and \(e\le _{\mathcal {H}} h\) if and only if \(a=kf\), with \(f\in E(S), k\in T\) and \(f\le _{\mathcal {H}} k\).

Moreover, any two idempotents \(e,e'\) in the first decomposition are \(\mathcal {R}\)-related to a.

Proof

Let a be dominated by \(x\in T\). Then exists \(e,f\in E(S)\), \(a=ex=x f\). Conversely, assume \(a=eh\) with \(e\in E(S), h\in T\) and \(e\le _{\mathcal {H}} h\). Then \(hh^\#e=e\) by cancellation properties and \((h^\#e h)(h^\#e h)=(h^\#e h)\). It follows that \(a=eh=h(h^\#eh)\) with e and \(h^\#e h\) idempotents and \(a\omega h\). The second equivalence is dual.

Finally, assume that \(a=eh\), with \(e\in E(S), h\in T\) and \(e\le _{\mathcal {H}} h\). Then \(a\le _{\mathcal {R}} e\). But also \(e=ehh^\#\) by cancellation properties, hence \(e=ah^\#\) and \(e\le _{\mathcal {R}} a\). Finally \(e\mathcal {R}a\). It follows that two idempotent \(e,e'\) satisfy \(e\mathcal {R}a\mathcal {R}e'\). \(\square \)

We now apply the results of the previous sections to the case of T-regular semigroups. For \(E\subseteq E(S)\), we note as before \(H(E)=\bigcup _{e\in E} \mathcal {H}(e)\).

Theorem 6.5

Let S be a semigroup. Then the following statements are equivalent:
  1. (1)

    S is completely E-simple and H(E)-dominated;

     
  2. (2)

    S is completely E-simple and H(E)-regular;

     
  3. (3)

    S is isomorphic to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) over a unit-regular monoid M with sandwich matrix with values in the group of units;

     
  4. (4)

    There exists a completely simple subsemigroup J of S, S is J-dominated and the local submonoids \(eSe, e\in J\) are disjoint.

     

Proof

\((1)\Rightarrow (2)\)

This is the content of Lemma 6.2.

\((2)\Rightarrow (3)\)

Assume S is a completely E-simple, H(E)-regular semigroup. Then by Theorem 2.3 it is isomorphic to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) over a monoid M with sandwich matrix with values in the group of units. Let \(m\in M\) and pose \(a=(i,m,\lambda )\) (for an arbitrary choice of \(i\in I\) and \(\lambda \in \Lambda \)). As a is H(E)-regular, there exists an element \(x=(j,n,\mu )\in H(E)\) such that \(axa=a\). Let \(e\in E\) such that \(x\in \mathcal {H}(e)\). Then \(e=(j,p_{\mu j}^{-1},\mu )\). As \(xx^\#=x^\#x=e\), then n is invertible. As \(axa=(i, xp_{\lambda j} n p_{\mu i} x,\lambda )\) then \(x=xp_{\lambda j} n p_{\mu i} x\), and as \(p_{\lambda j} n p_{\mu i}\) is a product of units, it is a unit and m is unit-regular.

\((3)\Rightarrow (4)\)

Let S be isomorphic by \(\phi \) to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) over a unit-regular monoid M with sandwich matrix with values in the group of units G of M, and pose \(J=I\times G\times \lambda \). let \(a\in S\). Then \(a\phi =(i,m,\lambda )\). As M is unit regular, there exists a unit \(g\in G\) such that \(mgm=m\). Pose \(x=(i, p_{\lambda i}^{-1} g p_{\lambda i}^{-1},\lambda )\). Then \(x\phi ^{-1}\) belongs to J, and direct calculations show that \(a\omega x^\#= (i, g^{-1} ,\lambda )\phi ^{-1}\). It follows that S is J-dominated. It is completely E-simple by Theorem 2.3.

\((4)\Rightarrow (1)\)

By Corollary 3.16, S is a regular completely E-simple semigroup with \(E=Max=E(J)\) and \(J\subseteq RP(S)=H(E)\). \(\square \)

Theorem 6.5 extends directly to completely \((E, \widetilde{\mathcal {H}}_E)\)-abundant semigroups.

Theorem 6.6

Let S be a semigroup. Then the following statements are equivalent:
  1. (1)

    There exists \(E\subseteq E(S)\), S is a completely \((E, \widetilde{\mathcal {H}}_E)\)-abundant, H(E)-dominated semigroup;

     
  2. (2)

    There exists \(E\subseteq E(S)\), S is a completely \((E, \widetilde{\mathcal {H}}_E)\)-abundant, H(E)-regular semigroup;

     
  3. (3)

    The semigroup S is isomorphic to a semilattice Y with property \((P_Y)\) of Rees matrix semigroups \(\mathcal {M}(M_{\alpha },I_{\alpha },\Lambda _{\alpha },P_{\alpha })\) over unit-regular monoids \(M_{\alpha }\) with sandwich matrices with values in the group of units;

     
  4. (4)

    There exists a completely regular subsemigroup R of S, S is R-dominated and completely \((E(R), \widetilde{\mathcal {H}}_E(R))\)-abundant.

     

In Billhart et al. [1], the authors define the notion of an associate inverse semigroup, in relation with the natural partial order, where, by an associate inverse subsemigroup of a regular semigroup S, they mean a subsemigroup T of S containing a least associate of each \(x\in S\) for the natural partial order. Such a semigroup T is necessarily inverse. We propose here a slightly different notion. We say that S is least T -regular if for any element \(a\in S\), the set of idempotents \(\{e\in E(S)\cap T|\, a\in a\left( \mathcal {H}(e)\cap T\right) a\}\) admits a least element (with respect to the natural partial order on \(E(S)\cap T\)). We denote this least element by \(a^+\), and the equivalence relation \(a^+=b^+\) by \(a \sigma ^+ b\). Note that for T a regular subsemigroup of S, \(E=E(S)\cap T=E(T)\) and \(\mathcal {H}(e)\cap T=\mathcal {H}^T(e)\) is the maximal subgroup of T containing e. Also, if T is a regular subsemigroup of S such that \(E=E(T)\subseteq Z(S)\), then T is a Clifford semigroup by [27] theorem 4.2.1, and E a semilattice.

Lemma 6.7

Let S be a semigroup and T be a Clifford subsemigroup of S such that S is least T-regular. For any \(a\in S\), \(a \widetilde{\mathcal {H}}_{E(T)} a^+\). In particular S is \(\left( E(T),\widetilde{\mathcal {H}}_{E(T)}\right) \)-abundant.

Proof

Let \(a\in S\) with associate \(x\in \mathcal {H}^T(a^+)\) and let \(e\in E(T), ae=a\). Then \(a(ex)a=a\). As T is a completely regular semigroup, ex admits a group inverse \((ex)^\#\). As \(a^+\) is the least element of \(\{f\in E(T)|\, a\in a \mathcal {H}^T(f) a\}\) then \(a^+(ex)(ex)^\#=(ex)(ex)^\#a^+=a^+\). It follows that \(ea^+=a^+\), and as T is also inverse, \(a^+e=a^+\). Finally \(a\widetilde{\mathcal {L}}_{E(T)} a^+\). By duality, \(a\widetilde{\mathcal {R}}_{E(T)} a^+\) hence finally \(a\widetilde{\mathcal {H}}_{E(T)} a^+\). \(\square \)

Corollary 6.8

Let S be a semigroup and T be a Clifford subsemigroup of S such that S is least T-regular. If \(\widetilde{\mathcal {L}}_{E(T)}\) and \(\widetilde{\mathcal {R}}_{E(T)}\) are right and left congruences, then S is a E(T)-Clifford restriction semigroup (or equivalently, \((S,.,^+)\) is a Clifford restriction semigroup).

Proof

By Lemma 6.7, S is \(\left( E(T),\widetilde{\mathcal {H}}_{E(T)}\right) \)-abundant. It is \(\left( \widetilde{\mathcal {L}}_{E(T)},\widetilde{\mathcal {R}}_{E(T)}\right) \) by hypothesis and E(T) is a semilattice as T is inverse. By Corollary 5.5 S is a Clifford restriction semigroup. \(\square \)

Corollary 6.9

Let S be a least T-regular semigroup, with T a regular subsemigroup of S such that \(E=E(T)\subseteq Z(S)\). If \(\sigma ^+=\{(a,b)\in S|\, a^+=b^+\}\) is a congruence, then S is a E(T)-Clifford restriction semigroup (or equivalently, \((S,.,^+)\) is a Clifford restriction semigroup).

Proof

As T is a regular subsemigroup of S such that \(E(T)\subseteq Z(S)\), then T is a Clifford semigroup by [27] theorem 4.2.1, and E(T) a semilattice. By Lemma 6.7, S is \(\left( E,\widetilde{\mathcal {H}}_{E(T)}\right) \)-abundant. As \(\widetilde{\mathcal {H}}_{E(T)}=\sigma ^+\) is a congruence, we conclude by Proposition 5.2. \(\square \)

Combining these results with the previous characterizations of E(T)-Clifford restriction semigroups we get:

Theorem 6.10

Let S be a semigroup. The following statements are equivalent:
  1. (1)

    There exists a unary operation \(^+\) on S such that \((S,.,^+)\) is a Clifford restriction semigroup, and \((\forall x\in S, \exists y\in S) \; x\omega y \text { and } y\in \mathcal {H}({x^{+}})\).

     
  2. (2)

    There exists a unary operation \(^+\) on S such that \((S,.,^+)\) is a Clifford restriction semigroup, and \((\forall x\in S) \; x\in x\mathcal {H}(x^{+}) x\).

     
  3. (3)

    There exists a subset \(E\subseteq E(S)\) such that S is a E-Clifford restriction semigroup, and \((\forall x\in S) \; x\in x\mathcal {H}(e) x\) with \(e\in E\cap \widetilde{\mathcal {H}}_E(x)\).

     
  4. (4)

    The semigroup S is a semilattice Y of factorisable monoids \(F_{\alpha }\), \(\alpha \in Y\), with \(1_{\alpha }1_{\beta }=1_{\alpha \beta } (\forall \alpha ,\beta \in Y)\);

     
  5. (5)

    The semigroup S is a strong semilattice Y of factorisable monoids \(F_{\alpha }\), \(\alpha \in Y\);

     
  6. (6)

    There exists a regular subsemigroup T of S such that \(E(T)\subseteq Z(S)\), S is least T-regular, and the relation \(\sigma ^+=\{(a,b)\in S|\, a^+=b^+\}\) is a congruence;

     
  7. (7)

    There exists a Clifford subsemigroup T of S such that S is least T-regular and \(\widetilde{\mathcal {L}}_{E(T)}\) and \(\widetilde{\mathcal {R}}_{E(T)}\) are right and left congruences.

     
In this case, we have the equalities \(E=E(T)=S^+\) and \(\widetilde{\mathcal {L}}_{E(T)}=\widetilde{\mathcal {R}}_{E(T)} =\widetilde{\mathcal {H}}_{E(T)}=\sigma ^+\), and \(\sigma ^+\) is a semilattice congruence.

Proof

\((1)\Rightarrow (2)\)

This is the content of Lemma 6.2.

\((2)\Rightarrow (3)\)

We pose \(S^+=E\).

\((3)\Rightarrow (4)\)

Let S be a E-Clifford restriction semigroup such that \((\forall x\in S), \; x\in x\mathcal {H}(e) x\) with \(e\in E\cap \widetilde{\mathcal {H}}_E(x)\). By Theorem 5.8, S is a semilattice of the \(\widetilde{\mathcal {H}}_{E}\)-classes, that are monoids.

For any \(e\in E\), pose \(F_e=\widetilde{\mathcal {H}}_{E}(e)\), the \(\widetilde{\mathcal {H}}_{E}\)-class of e. We have to show that \(F_e\) is a factorisable monoid, or equivalently, that \(F_e\) is unit regular. Let \(a\in F_e\). Then \(e\in E\cap \widetilde{\mathcal {H}}_E(a)\). Let \(b\in \mathcal {H}(e), aba=a\). As \(\mathcal {H}\subseteq \mathcal {R}\subseteq \widetilde{\mathcal {R}}_E\) then \(b,b^\#\in F_e\) and b is a unit, whence a is unit-regular.

Also by Theorem 5.8 the product of the identities \(e\in F_e\) and \(f\in F_f\) is ef, identity of \(F_{ef}\).

\((4)\Rightarrow (5)\)

Let S be a semilattice Y of factorisable monoids \(F_{\alpha }\), \(\alpha \in Y\), with \(1_{\alpha }1_{\beta }=1_{\alpha \beta } (\forall \alpha ,\beta \in Y)\). For all \(a\alpha \ge \beta \), it makes sense to define a map \(\phi _{\alpha ,\beta }:F_{\alpha }\rightarrow F_{\beta }\) by \(x\phi _{\alpha ,\beta }=x1_{\beta }\). These are monoid homomorphisms by assumption, and they define S as a strong semilattice of factorisable monoids.

\((5)\Rightarrow (6)\)

Let S be a strong semilattice Y of factorisable monoids \(F_{\alpha }\), \(\alpha \in Y\), with homomorphisms \(\phi _{\alpha ,\beta } (\forall \alpha \ge \beta )\). For any \(\alpha \in Y\), each \(F_{\alpha }\) is decomposed as the product \(E_{\alpha }G_{\alpha }\), with \(E_{\alpha }\) the idempotents of \(F_{\alpha }\) and \(G_{\alpha }\) its group of units.

Let T be the set \(\displaystyle \bigcup _{\alpha \in Y} G_{\alpha }\). As the homomorphisms \(\phi _{\alpha ,\beta }\) are monoid homomorphisms, T is a semigroup (and by that, a Clifford semigroup). E(T) is thus the set of identities \(1_{\alpha }\) of the \(G_{\alpha }, \alpha \in Y\), and \(E(T)\subseteq Z(S)\).

Let \(a\in F_{\alpha }\). Then \(a=e_{\alpha }g_{\alpha }=ag_{\alpha }^\#a\), and S is T-regular. Let \(g_{\beta }\in T\), \(ag_{\beta }a=a\). Then \(\alpha \beta =\alpha \beta \alpha =\alpha \), and \(g_{\alpha }g_{\alpha }^\#\le g_{\beta }g_{\beta }^\#\). It follows that S is least T-regular.

Let \(a\in F_{\alpha }, b\in F_{\beta }\) such that \(a \widetilde{\mathcal {L}}_{E(T)} b\). As \(a1_{\alpha }=a\) then \(b 1_{\alpha }=b\). As \(b1_{\alpha }\in F_{\beta \alpha }\) and the monoids are disjoint then \(\beta \alpha =\beta \). Symmetrically, \(\alpha \beta =\alpha \) and finally \(\alpha =\beta \). Conversely, let \(a,b\in F_{\alpha }\) and \(t=1_{\beta }\in E(T)\) such that \(at=a\). Then \(\alpha \beta =\alpha \). As \(b1_{\beta }=b\phi _{\alpha ,\alpha \beta } 1_{\beta }\phi _{\beta ,\alpha \beta }\), then \(b1_{\beta }=b\phi _{\alpha ,\alpha } 1_{\beta }\phi _{\beta ,\alpha }=b1_{\alpha }=b\) and \(a \widetilde{\mathcal {L}}_E b\). Finally, \(a \widetilde{\mathcal {L}}_{E(T)} b\) if and only if they belong to the same monoid, which is clearly a congruence relation.

\((6)\Rightarrow (7)\)

As \(E(T)\subseteq Z(S)\), then T is a Clifford semigroup and \(\widetilde{\mathcal {L}}_{E(T)}=\widetilde{\mathcal {R}}_{E(T)}=\widetilde{\mathcal {H}}_{E(T)}=\sigma ^+\) is a congruence.

\((7)\Rightarrow (1)\)

By Corollary 6.8, \((S,.,^+)\) is a Clifford restriction semigroup and in particular, \(S^+=E(T)\subseteq Z(S)\). Let \(x\in S\). Then exists \(h\in \mathcal {H}^T(x^+)\subseteq \mathcal {H}(x^+)\) such that \(xhx=x\). Then \(x=xhh^\#hx=hh^\#xhx=h^\#hx=xhh^\#\) with \(hx,xh\in E(S)\), hence \(x\omega h^\#\) with \(h^\#\in \mathcal {H}(x^+)\). \(\square \)

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Université Paris NanterreNanterreFrance

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