On \((E,\widetilde{\mathcal {H}}_E)\)abundant semigroups and their subclasses
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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 inverses1 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 nonregular 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 Eample 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.
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 
 (1)
Completely regular semigroups are exactly union of groups, and semilattices of completely simple semigroups;
 (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)
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 nonregular semigroups and led to the definition of new classes of semigroups (adequate semigroups [12], abundant semigroups [13], ample semigroups [32], and their onesided versions).
The origin of the relations \({\widetilde{\mathcal {K}}}\) lies in the thesis of ElQallali [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 ElQallali [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 \((\forall e\in E) \widetilde{L}_E[e]=Se\).

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 Usuperabundant 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 Usuperabundant semigroups, or weakly Usuperabundant 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 (\(E\subseteq Z(S)\)).
Classes of nonregular 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 Esemiabundant (weakly left Eabundant)  \((\forall a \in S, \exists e\in E)\,a\widetilde{\mathcal {L}}_Ee\)  \((E,\widetilde{\mathcal {L}}_E)\)abundant 
Weakly Esuperabundant  \((\forall a \in S, \exists e\in E)\,a\widetilde{\mathcal {H}}_Ee\)  \((E,\widetilde{\mathcal {H}}_E)\)abundant 
Weakly Esuperabundant 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 Esimple  
\((\forall a \in S, \exists e\in E)\; a\widetilde{\mathcal {H}}_Ee\,(\forall a,b,c,d\in S)\; a\widetilde{\mathcal {H}}_Eb, c\widetilde{\mathcal {H}}_Ed \Rightarrow ac\widetilde{\mathcal {H}}_Ebd\,(\forall a\in S, \forall e\in E)\; ae=ea\)  EClifford 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 Esemiabundant semigroups, but also weakly left Eabundant 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 Esemigroups, 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 Esemigroups 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 Eample semigroups), with a special emphasis on central idempotents (Clifford restriction semigroups). We then get an analog of Clifford’s decomposition Theorem, and a “Ptheorem” 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 Esimple 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 Esimple 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
 (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)
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 nonempty 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
 (I)If \(\alpha \beta \gamma =\delta \) then$$\begin{aligned} \displaystyle a_{\delta }p_{[\delta ,a]\left\langle b,\left\langle c,\delta \right\rangle \right\rangle } b_{\delta } p_{[\delta ,b]\langle c,\delta \rangle }c_{\delta }=a_{\delta }p_{[\delta ,a]\langle b,\delta \rangle } b_{\delta } p_{\left[ \left[ \delta ,a \right] ,b\right] \langle c,\delta \rangle }c_{\delta } \end{aligned}$$
 (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$$\begin{aligned} a\circ b=\left( \left\langle a, \left\langle b,\alpha \beta \right\rangle \right\rangle , a_{\alpha \beta }p_{[\alpha \beta ,a]\langle b,\alpha \beta \rangle } b_{\alpha \beta }, \left[ \left[ \alpha \beta ,a\right] ,b\right] \right) \end{aligned}$$  (III)If \(\gamma \le \alpha \beta , i\in I_{\gamma }, \lambda \in \Lambda _{\gamma }\) then$$\begin{aligned} \left( \left\langle a, \left\langle b,i\right\rangle \right\rangle , a_{\gamma }p_{[\gamma ,a]\langle b,\gamma \rangle } b_{\gamma }, \left[ \left[ \gamma ,a\right] ,b\right] \right) =\left( \langle a\circ b, i\rangle , (a\circ b)_{\gamma }, [\lambda , a\circ b]\right) \end{aligned}$$
 (IV)(i) If \(a=(i,x,\lambda ), b=(j,y,\lambda )\in S_{\alpha }\) then for all \(e\in E\) and \(d\in S_{\beta }\)
 (a)
\(a\circ e=a\Leftrightarrow b\circ e=b\),
 (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 \(d\in S_{\beta }\) (a)
\(e\circ a=a\Leftrightarrow e\circ b=b\),
 (b)
\(e\circ (d\circ a)=d\circ a\Leftrightarrow e\circ (d\circ b)=d\circ b\).
 (a)
We will also need some technical results (see [50] and references therein), listed as follows:
Lemma 2.5
 (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)
For any \(e,f\in E\), \(e\widetilde{\mathcal {D}}_Ef\) if and only if exists \(h\in E, e\mathcal {L}h\mathcal {R}f\).
 (1)
For all \(e\in E\), \(\widetilde{\mathcal {H}}_E(e)\) is a monoid.
 (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)
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
 (1)
The relations \(\widetilde{\mathcal {L}}_E\) and \(\widetilde{\mathcal {R}}_E\) are right and left congruences;
 (2)
The relation \(\widetilde{\mathcal {D}}_E\) is a semilattice congruence;
 (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. f, g) 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 Esimple semigroups. Obviously, they are also the completely \((E,\widetilde{\mathcal {H}}_E)\)abundant, \(\widetilde{\mathcal {J}}_E\)simple semigroups of Theorems 2.2, 2.3 and 2.4 by Lemma 2.1.
We then improve the previous decomposition and composition theorems.
Theorem 2.7
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
 (i)
If \(\alpha \ge \beta , i\in I_{\beta }, \lambda \in \Lambda _{\beta }\) then \(p_{\lambda \langle a,i\rangle } a_{\beta } p_{[\beta ,a]i}=p_{\lambda \langle a,\beta \rangle } a_{\beta } p_{[\lambda ,a]i}\)
 (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$$\begin{aligned} a\circ b=\left( \left\langle a, \left\langle b,\alpha \beta \right\rangle \right\rangle , a_{\alpha \beta }p_{[\alpha \beta ,a]\langle b,\alpha \beta \rangle } b_{\alpha \beta }, \left[ \left[ \alpha \beta ,a\right] ,b\right] \right) \end{aligned}$$  (iii)If \(\gamma \le \alpha \beta , i\in I_{\gamma }, \lambda \in \Lambda _{\gamma }\) then$$\begin{aligned} \left( \left\langle a, \left\langle b,i\right\rangle \right\rangle , a_{\gamma }p_{[\gamma ,a]\langle b,\gamma \rangle } b_{\gamma }, \left[ \left[ \gamma ,a\right] ,b\right] \right) =\left( \langle a\circ b, i\rangle , (a\circ b)_{\gamma }, [\lambda , a\circ b]\right) \end{aligned}$$
 (iv)
If \(\gamma \le \beta , e\in E_{\beta }\) then \(e_{\gamma }=p^{1}_{[\gamma ,e]\langle e,\gamma \rangle }\)
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 Esimple 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
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 e, f, g 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
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
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
 (1)
If \(a,b\in eSe\) and \(f\in E\cap \widetilde{\mathcal {H}}_E(ab)\) then \(f\omega e\);
 (2)
\(I_e\) is an ideal of eSe;
 (3)
\(\widetilde{\mathcal {H}}_E(e)=eSe\backslash I_e\);
 (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
 (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)
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)
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)
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
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 \(x=\pi _e(x)=\pi _f(y)=y\). \(\square \)
 (1)
Every element \(a\in S\) has a power in a subgroup of S;
 (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.
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
 \(\Rightarrow \)

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
 (1)
S is \((E,\widetilde{\mathcal {H}}_E)\)abundant and \(\widetilde{\mathcal {D}}_E\)simple (completely Esimple);
 (2)
S is completely \((E,\widetilde{\mathcal {H}}_E)\)abundant and \(\widetilde{\mathcal {J}}_E\)simple;
 (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 Esimple semigroups.
Corollary 3.10
Let S be a completely Esimple 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 Esimple 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 Esimple 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 Esimple semigroups. We start with a lemma.
Lemma 3.11
Let S be a completely Esimple 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
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 Esimple.
Conversely, let S be completely Esimple. 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).
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
 (1)
\(ab\in \widetilde{\mathcal {R}}_E(a)\cap \widetilde{\mathcal {L}}_E(b)\);
 (2)
\(aba\widetilde{\mathcal {H}}_Ea\);
 (3)
\(\widetilde{\mathcal {H}}_E\) is a congruence.
Proof
 (1)
As \(ea=a\) then \(aS=eaS\subseteq eS=\widetilde{\mathcal {R}}_E(a)\), and dually \(Sb\subseteq \widetilde{\mathcal {L}}_E(b)\).
 (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)
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 Esimple 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 Esimple 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
 (1)
S is a completely Esimple semigroup;
 (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)
S is the disjoint union of the local submonoids \(eSe, e\in E(J)\) for a completely simple semigroup J.
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 nonregular 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
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 quasiidentities 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
Proof
For the implication, we apply the quasiidentities 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 quasiidentities 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
Proof
 \(A\Rightarrow B\)
 \(B\Rightarrow C\)

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, \(x^+=x^+x^{++}=x^+x^+\). Thenby Eq. (9), and dually. \(\square \)$$\begin{aligned} (xy^+)^+y^+=(xy^+)^+(xy^+y^+)^+y^{++}=(xy^+y^+)^+=(xy^+)^+ \end{aligned}$$
Among these semigroups, \((\widetilde{\mathcal {L}}_E,\widetilde{\mathcal {R}}_E)\)congruent semigroups need a priori additional quasiidentities.
Lemma 4.4
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 quasiidentity is dual.
Conversely, assume that S is \((S^+,\widetilde{\mathcal {H}}_{S^+})\)abundant and satisfies these quasiidentities. 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
Proof
For the implication, we apply the quasiidentities of Lemma 4.4 to \((yx)^+(yx)=(yx)\) and \((xy)(xy)^+=(xy)\). For the converse we prove that these identities imply the quasiidentities 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
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
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 Esimple semigroups.
Proposition 4.9
Proof
Assume that S is completely Esimple. 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
 (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)
\(\mathcal {C}S^+\mathcal {A}=\mathcal {V}(1,2,3,4,10,11)=\mathcal {V}(1,2,9,12,13)\) is the subvariety of unary completely \((S^+,\widetilde{\mathcal {H}}_{S^+})\)abundant semigroups;
 (3)
\(S^+ \mathcal {C}\mathcal {G}=\mathcal {V}(1,2,3,4,14)\) is the subvariety of unary completely \((S^+,\widetilde{\mathcal {H}}_{S^+})\)abundant, \(\widetilde{\mathcal {H}}_{S^+}\)congruent semigroups (\(S^+\)cryptogroups);
 (4)
\(\mathcal {C}S^+\mathcal {S}=\mathcal {V}(1,2,15,16,17,18)=\mathcal {V}(1,2,9,15,16)\) is the subvariety of unary completely \(S^+\)simple semigroups;
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
 (1)
Every \(\widetilde{\mathcal {R}}_E\)class \(\widetilde{\mathcal {R}}_E(a)\) contains a (necessarily unique) idempotent, that we denote by \(a^+\);
 (2)
The relation \(\widetilde{\mathcal {R}}_E\) is a left congruence;
 (3)
The left ample condition \((\forall a\in S, \forall e\in E) ae=(ae)^+a \) is satisfied.
Weakly right Eample semigroups are defined dually. Weakly Eample semigroups are those semigroups both left and right weakly Eample.
It is well known that left restriction semigroups are precisely weakly left Eample semigroups, see for instance [26] Theorem 4.13. Precisely, a left restriction semigroup is weakly left \(S^+\)ample and a weakly Eample 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
 (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)
\((S,.,\,^+)\) is a right restriction semigroup and its set of projection \(S^+=\{x^+\, x\in S\}\) is a semilattice of central idempotents (in S);
 (3)
\((S,.,\,^+,\,^+)\) is a restriction semigroup and its set of projection \(S^+=\{x^+\, x\in S\}\) is a semilattice of central idempotents (in S).
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 selfdual, \((S,., \,^+)\) 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. \(\square \)
 (1)
Elements of \(S^+\) are central idempotents;
 (2)
Every \(\widetilde{\mathcal {H}}_{S^+}\)class \(\widetilde{\mathcal {H}}_{S^+}(a)\) contains a (necessarily unique) idempotent \(a^+\);
 (3)
The relation \(\widetilde{\mathcal {H}}_{S^+}\) is a congruence.
Proposition 5.2
 (1)
Elements of E are central idempotents;
 (2)
Every \(\widetilde{\mathcal {H}}_E\)class \(\widetilde{\mathcal {H}}_E(a)\) contains a (necessarily unique) idempotent, that we denote by \(a^+\);
 (3)
The relation \(\widetilde{\mathcal {H}}_E\) is a congruence.
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 Eample). 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
 (1)
S is a left restriction semigroup with \(a\mapsto a^+\) a retraction from S onto \(S^+\);
 (2)
S is a left restriction semigroup with \((xy)^+=x^+y^+\);
 (3)
S is a left restriction semigroup with \(S^+=\{x^+\, x\in S\}\) semilattice of central idempotents;
 (4)
S satisfies the identities \(x^+x=x\), \(x^+y=yx^+\) and \((xy)^{++}=x^+y^+\);
 (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. ThenDually, \(x^+y^+=y^+(xy)^+\). 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 \(S^+\).\(\square \)$$\begin{aligned} x^+y^+=(xy^+)^+= ((xy)^+x)^+= (xy)^+x^+. \end{aligned}$$
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 Eample semigroups with central idempotents). By Corollary 2.9, if S is a EClifford 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). EClifford restriction semigroups also admit such a characterization, using the generalizations of inverse semigroups we have seen: Esemiadequate semigroups and Ehresmann semigroups.
Corollary 5.5
 (1)
S is a EClifford restriction semigroup;
 (2)
S is \((E,\widetilde{\mathcal {H}}_E)\)abundant and Ehresmann;
 (3)
S is completely \((E,\widetilde{\mathcal {H}}_E)\)abundant and Esemiadequate.
 (4)
S is completely \((E,\widetilde{\mathcal {H}}_E)\)abundant and E is a semilattice.
 (5)
S is completely \((E,\widetilde{\mathcal {H}}_E)\)abundant and idempotents of E commute.
Proof
 \((1)\Rightarrow (2)\)

Assume that S is a EClifford 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 Eample semigroup with central idempotents, hence a EClifford 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 EClifford 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 EClifford 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 Esemiadequate 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
 (1)
S is a EClifford restriction semigroup;
 (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)
S is a semilattice E of monoids \(M_e, e\in E\), with identities e;
 (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 EClifford 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 EClifford restriction semigroup. \(\square \)
This theorem translates in the language of unary semigroups as follows:
Corollary 5.9
 (1)
\((S,.,\,^+)\) is a Clifford restriction semigroup;
 (2)
\((S,.,\,^+)\) is a semilattice of restriction monoids and \(a\mapsto a^+\) is an homomorphism;
 (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 ax, af, ex, ef, aex, aef 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].
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 onetoone 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 onetoone. By Theorem 5.8, S is a EClifford 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 onetoone, \(a=b\) 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 Erestriction 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 onetoone. \(\square \)
6 Application: Tregular semigroups
The aim of this section is to develop a concept close to unitregularity 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 Tregular (resp. Tdominated) if each element is Tregular (resp. Tdominated).
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
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 nonorthodox 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 unitregular 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 unitregular 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 Tdominated elements, when \(T\subseteq H(E(S))\).
Lemma 6.4
Let \(a\in S\) and \(T\subseteq H(E(S))\). Then a is Tdominated 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 Tregular semigroups. For \(E\subseteq E(S)\), we note as before \(H(E)=\bigcup _{e\in E} \mathcal {H}(e)\).
Theorem 6.5
 (1)
S is completely Esimple and H(E)dominated;
 (2)
S is completely Esimple and H(E)regular;
 (3)
S is isomorphic to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) over a unitregular monoid M with sandwich matrix with values in the group of units;
 (4)
There exists a completely simple subsemigroup J of S, S is Jdominated 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 Esimple, 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 unitregular.
 \((3)\Rightarrow (4)\)

Let S be isomorphic by \(\phi \) to a Rees matrix semigroup \(\mathcal {M}(M,I,\Lambda ,P)\) over a unitregular 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 Jdominated. It is completely Esimple by Theorem 2.3.
 \((4)\Rightarrow (1)\)

By Corollary 3.16, S is a regular completely Esimple 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
 (1)
There exists \(E\subseteq E(S)\), S is a completely \((E, \widetilde{\mathcal {H}}_E)\)abundant, H(E)dominated semigroup;
 (2)
There exists \(E\subseteq E(S)\), S is a completely \((E, \widetilde{\mathcal {H}}_E)\)abundant, H(E)regular semigroup;
 (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 unitregular monoids \(M_{\alpha }\) with sandwich matrices with values in the group of units;
 (4)
There exists a completely regular subsemigroup R of S, S is Rdominated 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 Tregular. 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 Tregular. 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 Tregular 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
 (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)
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)
There exists a subset \(E\subseteq E(S)\) such that S is a EClifford 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)
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)
The semigroup S is a strong semilattice Y of factorisable monoids \(F_{\alpha }\), \(\alpha \in Y\);
 (6)
There exists a regular subsemigroup T of S such that \(E(T)\subseteq Z(S)\), S is least Tregular, and the relation \(\sigma ^+=\{(a,b)\in S\, a^+=b^+\}\) is a congruence;
 (7)
There exists a Clifford subsemigroup T of S such that S is least Tregular and \(\widetilde{\mathcal {L}}_{E(T)}\) and \(\widetilde{\mathcal {R}}_{E(T)}\) are right and left congruences.
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 EClifford 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 unitregular.
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 Tregular. 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 Tregular.
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 \(h^\#\in \mathcal {H}(x^+)\). \(\square \)
References
 1.Billhardt, B., Giraldes, E., MarquesSmith, P., Martins, P.: Associate inverse subsemigroups of regular semigroups. Semigroup Forum 79, 101–118 (2009)MathSciNetCrossRefMATHGoogle Scholar
 2.Blyth, T.S., McFadden, R.: Unit orthodox semigroups. Glasg. Math. J. 24, 39–42 (1983)MathSciNetCrossRefMATHGoogle Scholar
 3.Blyth, T.S., Giraldes, E., MarquesSmith, P.: Associate subgroups of orthodox semigroups. Glasg. Math J. 36, 163–171 (1994)CrossRefMATHGoogle Scholar
 4.Chen, Y., He, Y., Shum, K.P.: Projectively condensed semigroups, generalized completely regular semigroups and projective orthomonoids. Acta Math. Hungar. 119(3), 281–305 (2008)MathSciNetCrossRefMATHGoogle Scholar
 5.Clifford, A.H.: Semigroups admitting relative inverses. Ann. Math. 42(4), 1037–1049 (1941)MathSciNetCrossRefMATHGoogle Scholar
 6.de Barros, C.M.: Sur les catégories ordonnées régulières. Cah. Topol. Géom. Différ. Catég. 11, 23–55 (1969)MATHGoogle Scholar
 7.Drazin, M.P.: A partial order in completely regular semigroups. J. Algebra 98(2), 362–374 (1986)MathSciNetCrossRefMATHGoogle Scholar
 8.Du, L., Guo, Y.Q., Shum, K.P.: Some remarks on \((l)\)Green’s relations and strongly RPP semigroups. Acta Math. Sci. Ser. B Engl. Ed. 31(4), 1591–1599 (2011)MathSciNetMATHGoogle Scholar
 9.ElQallali, A.: Structure theory for abundant and related semigroups. Ph.D. thesis, University of York (1980)Google Scholar
 10.Fountain, J.B.: A class of right PP monoids. Q. J. Math. Oxf. (2) 28, 285–305 (1977)MathSciNetCrossRefMATHGoogle Scholar
 11.Fountain, J.B.: Right PP monoids with central idempotents. Semigroup Forum 13, 229–237 (1977)MathSciNetCrossRefMATHGoogle Scholar
 12.Fountain, J.B.: Adequate semigroups. Proc. Edinb. Math. Soc. (2) 22(2), 113–125 (1979)MathSciNetCrossRefMATHGoogle Scholar
 13.Fountain, J.B.: Abundant semigroups. Proc. Lond. Math. Soc. (3) 44(1), 103–129 (1982)MathSciNetCrossRefMATHGoogle Scholar
 14.Fountain, J.B., Lawson, M.V.: The translational hull of an adequate semigroup. Semigroup Forum 32(1), 79–86 (1985)MathSciNetCrossRefMATHGoogle Scholar
 15.Fountain, J.B., Petrich, M.: Completely \(0\)simple semigroups of quotients. J. Algebra 101, 365–402 (1986)MathSciNetCrossRefMATHGoogle Scholar
 16.Fountain, J.B., Gomes, G.M.S., Gould, V.: A Munn type representation for a class of \(E\)semiadequate semigroups. J. Algebra 218(2), 693–714 (1999)MathSciNetCrossRefMATHGoogle Scholar
 17.Gomes, G.M.S., Gould, V.: Proper weakly left ample semigroups. Int. J. Algebra Comput. 9, 72–139 (1999)MathSciNetCrossRefMATHGoogle Scholar
 18.Gould, V.: Straight left orders. Stud. Sci. Math. Hung. 30, 355–373 (1995)MathSciNetMATHGoogle Scholar
 19.Gould, V.: Semigroups of left quotients: existence, straightness and locality. J. Algebra 267(2), 514–541 (2003)MathSciNetCrossRefMATHGoogle Scholar
 20.Gould, V., Szendrei, M.B.: Proper restriction semigroups, semidirect products and Wproducts. Acta Math. Hung. 141(1–2), 36–57 (2013)MathSciNetCrossRefMATHGoogle Scholar
 21.Green, J.A.: On the structure of semigroups. Ann. Math. 54(2), 163–172 (1951)MathSciNetCrossRefMATHGoogle Scholar
 22.Grillet, P.A.: Commutative Semigroups. Advances in Mathematics, vol. 2. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
 23.Guo, X., Guo, Y., Shum, K.P.: Left abundant semigroups. Commun. Algebra 32(6), 2061–2085 (2004)MathSciNetCrossRefMATHGoogle Scholar
 24.Hartwig, R.: How to partially order regular elements. Math. Jpn. 25, 1–13 (1980)MathSciNetMATHGoogle Scholar
 25.Hickey, J.B.: A class of regular semigroups with regularitypreserving elements. Semigroup Forum 81(1), 145–161 (2010)MathSciNetCrossRefMATHGoogle Scholar
 26.Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2(1), 21–57 (2009)MathSciNetMATHGoogle Scholar
 27.Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs. New Series, 12. The Clarendon Press, Oxford University Press, New York (1995)Google Scholar
 28.Jackson, M., Stokes, T.: An invitation to Csemigroups. Semigroup Forum 62(2), 279–310 (2001)MathSciNetCrossRefMATHGoogle Scholar
 29.Jones, P.R.: On lattices of varieties of restriction semigroups. Semigroup Forum 86(2), 337–361 (2013)MathSciNetCrossRefMATHGoogle Scholar
 30.Kilp, M.: Commutative monoids all of whose principal ideals are projective. Semigroup Forum 6, 334–339 (1973)MathSciNetCrossRefMATHGoogle Scholar
 31.Lallement, G., Petrich, M.: A generalization of the Rees theorem in semigroups. Acta Sci. Math. (Szeged) 30, 113–132 (1969)MathSciNetMATHGoogle Scholar
 32.Lawson, M.V.: The structure of type A semigroups. Q. J. Math. Oxf. Ser. (2) 37(147), 279–298 (1986)MathSciNetCrossRefMATHGoogle Scholar
 33.Lawson, M.V.: Abundant Rees matrix semigroups. J. Aust. Math. Soc. Ser. A 42(1), 132–142 (1987)MathSciNetCrossRefMATHGoogle Scholar
 34.Lawson, M.V.: Rees matrix semigroups. Proc. Edinb. Math. Soc. (2) 33(1), 23–37 (1990)MathSciNetCrossRefMATHGoogle Scholar
 35.Lawson, M.V.: Semigroups and ordered categories I: the reduced case. J. Algebra 141, 422–462 (1991)MathSciNetCrossRefMATHGoogle Scholar
 36.Liber, A.E.: On the theory of generalized groups. Dokl. Akad. Nauk. SSSR 97, 25–28 (1954). (Russian)MathSciNetGoogle Scholar
 37.Lopez Jr., A.M.: The maximal right quotient semigroup of a strong semilattice of semigroups. Pac. J. Math. 71(2), 477–485 (1977)MathSciNetCrossRefMATHGoogle Scholar
 38.Lyapin, E.S.: Semigroups. Translations of Mathematical Monographs, vol. 3. American Mathematical Society, Providence, RI (1963)MATHGoogle Scholar
 39.Ma, S.Y., Ren, X.M., Yuan, Y.: On completely \(\widetilde{\cal{J}}\)simple semigroups. Acta Math. Sin. (Chin. Ser.) 54(4), 643–650 (2011)MATHGoogle Scholar
 40.Miller, D.D., Clifford, A.H.: Regular \(\cal{D}\)classes in semigroups. Trans. Am. Math. Soc. 82(1), 270–280 (1956)MathSciNetMATHGoogle Scholar
 41.Mitsch, H.: A natural partial order on semigroups. Proc. Am. Math. Soc. 97(3), 384–388 (1986)MathSciNetCrossRefMATHGoogle Scholar
 42.Nambooripad, K.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)MathSciNetCrossRefMATHGoogle Scholar
 43.Petrich, M.: Lectures in Semigroups. Wiley, New York (1977)MATHGoogle Scholar
 44.Petrich, M.: A structure theorem for completely regular semigroups. Proc. Am. Math. Soc. 99(4), 617–622 (1987)MathSciNetCrossRefMATHGoogle Scholar
 45.Petrich, M.: On weakly ample semigroups. J. Aust. Math. Soc. 97, 404–417 (2014)MathSciNetCrossRefMATHGoogle Scholar
 46.Ren, X.M., Shum, K.P.: On superabundant semigroups whose set of idempotents forms a subsemigroup. Algebra Colloq. 14(2), 215–228 (2007)MathSciNetCrossRefMATHGoogle Scholar
 47.Ren, X.M., Shum, K.P., Guo, Y.Q.: A generalized Clifford theorem of semigroups. Sci. China A 53, 1097–1101 (2010)MathSciNetCrossRefMATHGoogle Scholar
 48.Sutov, E.G.: Potential divisibility of elements in semigroups. Leningr. Gosud. Ped. Inst. Uc. Zap. 166, 105–119 (1958). (Russian)MathSciNetGoogle Scholar
 49.Wang, Y., Ren, X.M., Ma, S.Y.: The translational hull of superabundant semigroups with semilattice of idempotents. Sci. Magna 2(4), 75–80 (2006)MathSciNetMATHGoogle Scholar
 50.Wang, Y.: Beyond regular semigroups. Ph.D. Thesis, University of York (2012)Google Scholar
 51.Wang, Y.: Beyond regular semigroups. Semigroup Forum 92(2), 414–448 (2016)MathSciNetCrossRefMATHGoogle Scholar
 52.Yuan, Y., Gong, C., Ma, S.Y.: The structure of \(U\)superabundant semigroups and the translational hull of completely \(\widetilde{\cal{J}}\)simple semigroups. Adv. Math. (China) 1, 35–47 (2014)MathSciNetMATHGoogle Scholar