# Congruences and group congruences on a semigroup

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DOI: 10.1007/s00233-012-9425-z

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- Gigoń, R.S. Semigroup Forum (2013) 86: 431. doi:10.1007/s00233-012-9425-z

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## Abstract

We show that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup *S* and the set of all group congruences on *S*. We describe also group congruences on *E*-inversive (*E*-)semigroups. In particular, we generalize the result of Meakin (J. Aust. Math. Soc. 13:259–266, 1972) concerning the description of the least group congruence on an orthodox semigroup, the result of Howie (Proc. Edinb. Math. Soc. 14:71–79, 1964) concerning the description of *ρ*∨*σ* in an inverse semigroup *S*, where *ρ* is a congruence and *σ* is the least group congruence on *S*, some results of Jones (Semigroup Forum 30:1–16, 1984) and some results contained in the book of Petrich (Inverse Semigroups, 1984). Also, one of the main aims of this paper is to study of group congruences on *E*-unitary semigroups. In particular, we prove that in any *E*-inversive semigroup, \(\mathcal{H}\cap\sigma\subseteq\kappa\), where *κ* is the least *E*-unitary congruence. This result is equivalent to the statement that in an arbitrary *E*-unitary *E*-inversive semigroup *S*, \(\mathcal{H}\cap\sigma= 1_{S}\).

### Keywords

Group congruence*E*-inversive semigroup

*E*-semigroupIdempotent-surjective semigroupEventually regular semigroupIdempotent pure congruenceIdempotent-separating congruence

*E*-unitary congruence

## 1 Introduction and preliminaries

Let *S* be an inverse semigroup with semilattice of idempotents *E*. Define an inverse subsemigroup *N* of *S* to be *normal* if it is *full* (i.e., *E*⊆*N*), *closed* (i.e., *Nω*=*N*, where *ω*:2^{S}→2^{S} is a *closure operator* given by *Aω*={*s*∈*S*:∃ *a*∈*A* [*as*∈*A*]} for all *A*⊆*S*), and *self-conjugate* (i.e., *s*^{−1}*Ns*⊆*N* for every *s*∈*S*). It follows from [11] (see p. 181) that there exists an inclusion-preserving bijection between the set of normal subsemigroups of *S* and the set of group congruences on *S*. In fact, the relation *ρ*_{N}={(*a*,*b*)∈*S*×*S*:*ab*^{−1}∈*N*} is a group congruence on *S* and \(\operatorname{ker}\rho_{N} = N\). These results were generalized in [9] and [16]. It is easy to see that (*a*,*b*)∈*ρ*_{N} if and only if *ax*,*bx*∈*N* for some *x*∈*S*.

The main purpose of the next section is a description of group congruences on a semigroup *S* in the terms of some special subsemigroups of *S*. Our description is simpler than that of Dubreil (see 10.2 [1]) and a little more general than the description of Gomes [6] (*nota bene* our proof is simpler). We apply this description to determine group congruences (in particular, the least group congruence) on some special classes of semigroups; namely: *E*-inversive (*E*-)semigroups (in particular, idempotent-surjective (*E*-)semigroups), eventually regular semigroups.

We divide this paper into seven sections. In Sect. 2 we describe group congruences on an arbitrary semigroup *S* in the terms of *normal* subsemigroups of *S* (see below for the definition). In Sect. 3 we investigate group congruences on *E*-inversive semigroups. In particular, we show that the least group congruence on an *E*-inversive semigroup exists (in general, this is false: see Example 1.2). In Sect. 4 and 5 we study group congruences on *E*-inversive *E*-semigroups and *E*-unitary *E*-inversive semigroups, respectively. Further, in Sect. 6 we use the results of Sect. 2 for an easy description of all group congruences on eventually regular semigroups (in terms of full, closed and self-conjugate subsemigroups) and we give some remarks on group congruences on inverse semigroups. Finally, in Sect. 7, some remarks on the hypercore of a semigroup are given (see [8]).

Let *S* be a semigroup. Denote by *Reg*(*S*) the set of *regular elements* of *S*, that is, *Reg*(*S*)={*a*∈*S*:*a*∈*aSa*} and by *V*(*a*) the set of *inverses* of *a*∈*S*, i.e., the set {*x*∈*S*:*a*=*axa*,*x*=*xax*}. Note that if *a*∈*S* is regular, say *a*=*axa* for some *x*∈*S*, then *xax*∈*V*(*a*). Also, *S* is called *regular* if *V*(*a*)≠∅ for every *a*∈*S*. Further, *S* is said to be *eventually regular* if every element *a* of *S* has a regular power. In such a case, by *r*(*a*) we shall mean the *regular index* of *a*, i.e., the least positive integer *n* for which *a*^{n}∈*Reg*(*S*).

Let *S* be a semigroup, *a*∈*S*. The set *W*(*a*)={*x*∈*S*:*x*=*xax*} is called the set of all *weak inverses* of *a* and so the elements of *W*(*a*) will be called *weak inverse elements* of *a*. A semigroup *S* is said to be *E-inversive* if for every *a*∈*S* there exists *x*∈*S* such that *ax*∈*E*_{S}, where *E*_{S} (or briefly *E*) is the set of idempotents of *S* (more generally, if *A*⊆*S*, then *E*_{A} denotes the set of idempotents of *A*). It is easy to see that a semigroup *S* is *E*-inversive if and only if *W*(*a*) is nonempty for all *a*∈*S*. Hence if *S* is *E*-inversive, then for every *a*∈*S* there is *x*∈*S* such that *ax*,*xa*∈*E*_{S} (see [19, 20]). Clearly, eventually regular semigroups are *E*-inversive. We remark that the class of eventually regular semigroups is very wide and contains the class of regular, group-bound (in particular, periodic, finite) semigroups. In [7] Hall observed that the set *Reg*(*S*) of a semigroup *S* with *E*_{S}≠∅ forms a regular subsemigroup of *S* if and only if the product of any two idempotents of *S* is regular. In that case, *S* is said to be an *R-semigroup*. Also, we say that *S* is an *E-semigroup* if *E*_{S} is a subsemigroup of *S*. Evidently, every *E*-semigroup is an *R*-semigroup. Finally, [eventually] regular *E*-semigroups are called [*eventually*] *orthodox*.

A generalization of the concept of eventually regular will also prove convenient. Define a semigroup *S* to be *idempotent-surjective* if whenever *ρ* is a congruence on *S* and *aρ* is an idempotent of *S*/*ρ*, then *aρ* contains some idempotent of *S*. It is well known that eventually regular semigroups are idempotent-surjective [2]. Further, we have the following known result [10] (we include a simple proof for completeness).

### Result 1.1

*Every idempotent*-*surjective semigroup**S**is**E*-*inversive*.

### Proof

Let *a*∈*S*. From the definition of a Rees congruence on *S* follows that the ideal *SaS* has at least one idempotent, that is, *xay*=*e*∈*E*_{S}, where *x*,*y*∈*S*. Hence *exaye*=*e*. Thus *yex*=(*yex*)*a*(*yex*), so *yex*∈*W*(*a*), as required. □

A subset *A* of *S* is said to be (respectively) *full*; *reflexive* and *dense* if *E*_{S}⊆*A*; ∀*a*,*b*∈*S* [*ab*∈*A*⟹*ba*∈*A*] and ∀*s*∈*S* ∃ *x*,*y*∈*S* [*sx*,*ys*∈*A*]. Also, define the *closure operator**ω* on *S* by *Aω*={*s*∈*S*:∃ *a*∈*A* [*as*∈*A*]} (*A*⊆*S*). We shall say that *A*⊆*S* is *closed* (in *S*) if *Aω*=*A*. Finally, a subsemigroup *N* of a semigroup *S* is *normal* if it is full, dense, reflexive and closed (if *N* is normal, then we shall write *N*◁*S*). Moreover, if a subsemigroup of *S* is full, dense and reflexive, then it is called *seminormal* [6].

By the *kernel*\(\operatorname{ker}\rho\) of a congruence *ρ* on a semigroup *S* we shall mean the set {*x*∈*S*:(*x*,*x*^{2})∈*ρ*}. Finally, denote by \(\mathcal{C}(S)\) the complete lattice of all congruences on a semigroup *S*.

### Example 1.2

Consider the semigroup of positive integers (ℕ,+) (with respect to addition). It is well known that every group congruence on ℕ is of the following form: *ρ*_{n}={(*k*,*l*)∈ℕ×ℕ:*n*|(*k*−*l*)} (*n*>0). Note that *E*_{ℕ}=∅, so a semigroup without idempotents possesses group congruences but ℕ has not least group congruence. Also, \(\operatorname{ker}\rho_{n} = n\rho_{n} = \{n, 2n, 3n, \ldots\}\).

## 2 Group congruences—general case

Let *S* be a semigroup, \(\rho \in \mathcal{C}(S)\). We say that *ρ* is a *group* congruence if *S*/*ρ* is a group. Denote by \(\mathcal{GC}(S)\) the set of group congruences on *S*. Clearly, if \(\rho \in \mathcal{GC}(S)\), then \(\operatorname{ker}\rho\) is the identity of the group *S*/*ρ*. Finally, by \(\mathcal{N}(S)\) we shall mean the set of all normal subsemigroups of *S*.

The following two lemmas are almost evident and we omit their easy proofs.

### Lemma 2.1

*Let**ρ**be a group congruence on a semigroup**S*. *Then*\(\operatorname{ker}\rho \lhd S\).

### Lemma 2.2

*Let**ρ*_{1},*ρ*_{2}*be group congruences on a semigroup**S*. *Then**ρ*_{1}⊂*ρ*_{2}*if and only if*\(\operatorname{ker}\rho_{1} \subset \operatorname{ker}\rho_{2}\).

### Lemma 2.3

*Let a subsemigroup**B**of a semigroup**S**be dense and reflexive*. *Then**ρ*_{1,B}=*ρ*_{2,B}=*ρ*_{3,B}=*ρ*_{4,B}.

### Proof

Let (*a*,*b*)∈*ρ*_{2,B}. Then *ax*=*yb* for some *x*,*y*∈*B*. Also, *as*∈*B* for some *s*∈*S*, since *B* is dense and so *sa*∈*B*, since *B* is reflexive. Hence *asy*∈*B* and so (*sy*)*a*∈*B*. It follows that (*sy*)*b*=*s*(*yb*)=*s*(*ax*)=(*sa*)*x*∈*B*. Thus (*sy*)*a*, (*sy*)*b*∈*B*. We have just shown that *ρ*_{2,B}⊂*ρ*_{3,B}.

Conversely, if *xa*,*xb*∈*B* for some *x*∈*S*, then *ax*,*bx*∈*B* (since *B* is reflexive), so *a*(*xb*)=(*ax*)*b*, where *ax*,*xb*∈*B*. Hence (*a*,*b*)∈*ρ*_{2,B}. Thus *ρ*_{2,B}=*ρ*_{3,B}.

Dually, *ρ*_{1,B}=*ρ*_{4,B}. Since *B* is reflexive, then *ρ*_{1,B}=*ρ*_{3,B}. □

If *B* is a dense, reflexive subsemigroup of *S*, then we denote the above four relations by *ρ*_{B}. We have the following theorem.

### Theorem 2.4

*Let**B**be a dense and reflexive subsemigroup of a semigroup**S*. *Then the relation**ρ*_{B}*is a group congruence on**S*. *Moreover*, \(B\subseteq B\omega = \operatorname{ker}\rho_{B}\). *If**B**is normal*, *then*\(B = \operatorname{ker}\rho_{B}\).

*Conversely*, *if**ρ**is a group congruence on**S*, *then there exists a normal subsemigroup**N**of**S**such that**ρ*=*ρ*_{N} (*in fact*, \(N = \operatorname{ker}\rho\)). *Thus there exists an inclusion*-*preserving bijection between the set of all normal subsemigroups of**S**and the set of all group congruences on**S*.

### Proof

Let *a*∈*S*. Since *B* is dense, then there exists *x*∈*S* such that *xa*∈*B*. Hence *ρ*_{B} is reflexive. Obviously, *ρ*_{B} is symmetric. Also, since *B* is a semigroup, then *ρ*_{B} is transitive. Consequently, *ρ*_{B} is an equivalence relation on *S*. Moreover, *ρ*_{B} is a left congruence on *S*. Indeed, let (*a*,*b*)∈*ρ*_{B},*c*∈*S*. Then *ax*,*bx*∈*B* and *zc*∈*B* for some *x*,*z*∈*S*, so *zcax*,*zcbx*∈*B*. It follows that (*ca*)(*xz*),(*cb*)(*xz*)∈*B*, since *B* is reflexive. Therefore (*ca*,*cb*)∈*ρ*_{B}. By symmetry, *ρ*_{B} is a right congruence on *S*. Finally, *S*/*ρ*_{B} is a group. Indeed, let *a*∈*S*,*b*∈*B* and *ax*,*xa*∈*B* for some *x*∈*S*. Then *bax*∈*B*. Hence *xa*,*x*(*ba*)∈*B*, so (*ba*,*a*)∈*ρ*_{B}. Since *B* is dense, then *S*/*ρ*_{B} is a group, as required.

Since *b*(*bb*)=(*bb*)*b* for every *b*∈*B*, then \(B \subset \operatorname{ker}\rho_{B}\). Also, \(B\omega = \operatorname{ker}\rho_{B}\). Indeed, let \(s \in \operatorname{ker}\rho_{B}\). Then (*s*,*b*)∈*ρ*_{B} for some *b*∈*B*. Hence *b*_{1}*s*=*bb*_{2} for some *b*_{1},*b*_{2}∈*B*. Thus *s*∈*Bω*, so \(\operatorname{ker}\rho_{B} \subset B\omega\). Conversely, let *s*∈*Bω*. Then *bs*∈*B* for some *b*∈*B*. Since *bb*∈*B*, then (*s*,*b*)∈*ρ*_{B}, so \(s \in \operatorname{ker}\rho_{B}\). Thus \(B\omega \subset \operatorname{ker}\rho_{B}\), as exactly required. Finally, if *B* is normal, then *B*=*Bω*. Hence \(B = \operatorname{ker}\rho_{B}\).

Conversely, let *ρ* be a group congruence on *S*. By Lemma 2.1, \(\operatorname{ker}\rho \lhd S\). Put \(\operatorname{ker}\rho = N\). Then by Lemma 2.2, *ρ*=*ρ*_{N}, since \(N = \operatorname{ker}\rho_{N} = \operatorname{ker}\rho\). It is now easy to see that the map \(\phi : \mathcal{N}(S) \to \mathcal{GC}(S)\), where *Nϕ*=*ρ*_{N} for every \(N \in \mathcal{N}(S)\), is an inclusion-preserving bijection between the set of all normal subsemigroups of *S* and the set of all group congruences on *S* (with the inverse \(\phi^{- 1} : \mathcal{GC}(S) \to \mathcal{N}(S)\), where \(\rho \phi^{- 1} = \operatorname{ker}\rho\) for all \(\rho \in \mathcal{GC}(S)\)). Note that *ϕ*^{−1} is an inclusion-preserving mapping, too. □

Since the first part of Theorem 2.4 is true for an arbitrary dense and reflexive subsemigroup of *S*, then we get the following corollary.

### Corollary 2.5

*Let**B**be a dense and reflexive subsemigroup of**S*. *Then**Bω*◁*S*.

### Example 2.6

*S*={

*a*,

*b*,

*c*,

*e*,

*f*} be the semigroup with the multiplication table given below: It is easy to see that

*E*is a dense and reflexive subsemigroup of

*S*but

*E*is not closed, since

*ea*∈

*E*and

*a*∉

*E*. Also,

*N*={

*a*,

*e*,

*f*} is normal. Indeed, the group congruence

*ρ*

_{E}has two

*ρ*

_{E}-classes:

*N*and {

*b*,

*c*}, since

*ae*,

*ee*,

*bb*,

*bc*∈

*E*and (

*e*,

*b*)∉

*ρ*

_{E}. Note also that \(E \subset \operatorname{ker}\rho_{N} = N, E \not= N\) and

*ρ*

_{E}=

*ρ*

_{N}. It follows that there is no a one-to-one correspondence between the set of all seminormal subsemigroups of

*S*and the set of all group congruences on

*S*.

### Remark 1

Obviously, every subgroup of a group is full and unitary but not every subgroup of a group is reflexive (for example: each two element subgroup of the group of all permutations of the six-element set *X* is not reflexive). It is well known that a subgroup *H* of a group *G* is normal if and only if the relation *ρ*_{H} is a congruence on *G*. We have a corresponding result:

*Let**A**be a closed subsemigroup of a semigroup**S. Then**A**is normal if and only if*\(\rho_{A} \in \mathcal{GC}(S)\).

Indeed, let \(\rho_{A} \in \mathcal{GC}(S)\). From *A*=*Aω* and the second paragraph of the proof of Theorem 2.4 we obtain that \(A = \operatorname{ker}\rho_{A}\). Thus *A*◁*S* (Lemma 2.1). The converse of the result follows from Theorem 2.4.

The set of all group congruences on a semigroup *S* (in general) does not form a lattice. Indeed, let (ℝ,+) be the semigroup of real positive numbers with respect to addition. Put *M*=ℕ and *N*={*x*,2*x*,3*x*,…}, where *x*∈ℝ∖ℚ. Then *M*,*N*◁*S* but *M*∩*N*=∅.

We generalize now the results of Howie [12], LaTorre [16] and Hanumantha Rao and Lakshmi [9].

### Theorem 2.7

*Let*

*B*

*be a seminormal subsemigroup of a semigroup*\(S, \rho \in \mathcal{C}(S)\).

*Then*:

- (i)
*ρ*∨*ρ*_{B}=*ρ*_{B}*ρρ*_{B}; - (ii)
\(\rho \lor \rho_{B} \in \mathcal{GC}(S)\);

- (iii)
(

*x*,*y*)∈*ρ*∨*ρ*_{B}*if and only if*(*ax*,*yb*)∈*ρ**for some**a*,*b*∈*B*.

### Proof

(i). Since *ρ*,*ρ*_{B}⊂*ρ*∨*ρ*_{B}, *ρ*_{B}*ρρ*_{B}⊂*ρ*∨*ρ*_{B}. Also, *ρ*_{B}*ρρ*_{B} is a reflexive, symmetric and compatible relation on *S*. We show that *ρ*_{B}*ρρ*_{B} is transitive. Then *ρ*∨*ρ*_{B}=*ρ*_{B}*ρρ*_{B}. Indeed, let (*r*,*s*),(*s*,*t*)∈*ρ*_{B}*ρρ*_{B}. Then (a) (*w*,*s*),(*s*,*x*)∈*ρ*_{B}; (b) (*y*,*w*),(*x*,*z*)∈*ρ*; (c) (*r*,*y*),(*z*,*t*)∈*ρ*_{B} for some *w*,*x*,*y*,*z*∈*S*. From (a) we obtain (*w*,*x*)∈*ρ*_{B}, so *aw*=*xb* for some *a*,*b*∈*B*. From (b) follows that (*aw*,*ay*),(*xb*,*zb*)∈*ρ*. Hence (*ay*,*zb*)∈*ρ*, since *aw*=*xb*. Finally, by (c), (*r*,*ay*),(*zb*,*t*)∈*ρ*_{B}, since \(B \subset \operatorname{ker}\rho_{B}\), so (*r*,*ay*)∈*ρ*_{B},(*ay*,*zb*)∈*ρ*,(*zb*,*t*)∈*ρ*_{B}. Thus (*r*,*t*)∈*ρ*_{B}*ρρ*_{B}, as required.

(ii). This is evident.

(iii). Let (*x*,*y*)∈*ρ*∨*ρ*_{B}. Then (*x*,*r*)∈*ρ*_{B},(*r*,*s*)∈*ρ* and (*s*,*y*)∈*ρ*_{B} for some *r*,*s*∈*S*. Hence *ax*=*rb*,*cs*=*yd* for some elements *a*,*b*,*c*,*d* of *B*. Therefore (*ca*)*x*=*c*(*ax*)=*c*(*rb*)=(*crb*)*ρ*(*csb*)=(*cs*)*b*=(*yd*)*b*=*y*(*db*), where *ca*,*db*∈*B*. Conversely, let (*ax*,*yb*)∈*ρ* for some *a*,*b*∈*B*. Since (*x*,*ax*),(*yb*,*y*)∈*ρ*_{B}, then (*x*,*y*)∈*ρ*_{B}*ρρ*_{B}=*ρ*∨*ρ*_{B} (by (i)). □

*A*be a nonempty subset of a semigroup \(S, \rho \in \mathcal{C}(S)\). Put

### Corollary 2.8

*Let**B**be a seminormal subsemigroup of a semigroup*\(S, \rho \in \mathcal{C}(S)\). *Then*\(\operatorname{ker}(\rho \lor \rho_{B}) = (B\rho)\omega\). *In particular*, (*Bρ*)*ω*◁*S*.

### Proof

Let \(x \in \operatorname{ker}(\rho \lor \rho_{B})\). Then there exists *b*∈*B* such that (*x*,*b*)∈*ρ*∨*ρ*_{B}, since \(B \subset \operatorname{ker}(\rho \lor \rho_{B})\). Hence (*ax*,*bc*)∈*ρ* for some *a*,*c*∈*B* (by Theorem 1.6(iii)). Thus *ax*∈*Bρ*. It follows that *x*∈(*Bρ*)*ω*. Conversely, if *x*∈(*Bρ*)*ω*, then *ax*∈*Bρ* for some *a*∈*Bρ*, so (*ax*,*b*),(*a*,*c*)∈*ρ* for some *b*,*c*∈*B*. It follows that (*cx*,*b*)∈*ρ*. Hence ((*cc*)*x*,*cb*)∈*ρ*. Thus (*x*,*c*)∈*ρ*∨*ρ*_{B}. Consequently, \(x \in\operatorname{ker}(\rho \lor \rho_{B})\). □

Also, by Theorem 1.6(i) and Proposition 2.3(ii) in [15] we obtain the following (see Corollary 3.2 [15]) corollary.

### Corollary 2.9

*Every group congruence on a semigroup**S**is dually right modular element of*\(\mathcal{C}(S)\).

### Corollary 2.10

*Let**B**be a seminormal subsemigroup of a semigroup*\(S, \rho \in \mathcal{C}(S)\). *Then**ρ*∨*ρ*_{B}=*S*×*S**if and only if* (*Bρ*)*ω*=*S*.

Let *B* be a seminormal subsemigroup of a semigroup \(S, \rho_{1}, \rho_{2} \in \mathcal{C}(S)\). Suppose that (*x*,*y*)∈(*ρ*_{1}∨*ρ*_{B})∩(*ρ*_{2}∨*ρ*_{B}). Then (*ax*)*ρ*_{2}(*yb*), where *a*,*b*∈*B*. Moreover, *ax*(*ρ*_{1}∨*ρ*_{B})*x*,*x*(*ρ*_{1}∨*ρ*_{B})*y*,*y*(*ρ*_{1}∨*ρ*_{B})*yb*, so *ax*(*ρ*_{1}∨*ρ*_{B})*yb*. Thus (*cax*,*ybd*)∈*ρ*_{1}, where *c*,*d*∈*B*. It follows that (*caxd*,*cybd*)∈*ρ*_{1}. Moreover, (*caxd*,*cybd*)∈*ρ*_{2}. Hence (*xd*,*cy*)∈(*ρ*_{1}∩*ρ*_{2})∨*ρ*_{B}. Thus (*x*,*y*)∈(*ρ*_{1}∩*ρ*_{2})∨*ρ*_{B}, since (*ρ*_{1}∩*ρ*_{2})∨*ρ*_{B} is a group congruence on *S* and \(c, d \in B \subset \operatorname{ker}((\rho_{1} \cap \rho_{2}) \lor \rho_{B})\). We have just shown that (*ρ*_{1}∨*ρ*_{B})∩(*ρ*_{2}∨*ρ*_{B})⊂(*ρ*_{1}∩*ρ*_{2})∨*ρ*_{B}. The converse inclusion is evident. Thus we may conclude that (*ρ*_{1}∨*ρ*_{B})∩(*ρ*_{2}∨*ρ*_{B})=(*ρ*_{1}∩*ρ*_{2})∨*ρ*_{B}.

We have the following theorem (see Theorem III.5.6 [21] and Theorem 4 [23]).

### Theorem 2.11

*Let*

*B*

*be a seminormal subsemigroup of a semigroup*

*S*.

*Then the mapping*\(\phi : \mathcal{C}(S) \to \mathcal{GC}(S)\),

*where*

*for every*\(\rho \in \mathcal{C}(S)\),

*is a*(

*lattice*)

*homomorphism of*\(\mathcal{C}(S)\)

*onto the*(

*modular*)

*lattice*[

*ρ*

_{B},

*S*×

*S*]

*of all group congruences on*

*S*

*containing*

*ρ*

_{B}.

### Proof

We have just proved that (*ρ*_{1}∩*ρ*_{2})*ϕ*=*ρ*_{1}*ϕ*∩*ρ*_{2}*ϕ* for all \(\rho_{1}, \rho_{2} \in \mathcal{C}(S)\). Clearly, (*ρ*_{1}∨*ρ*_{2})*ϕ*=*ρ*_{1}*ϕ*∨*ρ*_{2}*ϕ* for all \(\rho_{1}, \rho_{2} \in \mathcal{C}(S)\) and evidently *ϕ* is onto [*ρ*_{B},*S*×*S*]. □

We have the following corollary (see Theorem 4.5 [15]).

### Corollary 2.12

*Let**B**be a seminormal subsemigroup of a semigroup**S*. *Then**ρ*_{B}*distributes over meet*.

*S*be a semigroup,

*N*◁

*S*. Put

*S*/

*ρ*

_{N}by

*S*/

*N*. In particular, \(\mathcal{P}(S/N; \{N\})\) is the set of all subgroups of the group

*S*/

*N*. Remark that if \(A \in \mathcal{P}(S; N)\), then

*A*is full and dense.

The proofs of the following two propositions are standard and so we omit the proofs.

### Proposition 2.13

*Let**S**be a semigroup*, *N*◁*S*. *Then there exists an inclusion*-*preserving bijection**ϕ**between the set*\(\mathcal{P}(S; N)\)*and the set*\(\mathcal{P}(S/N; \{N\})\). *Moreover*, \(M \in \mathcal{P}(S; N)\)*and**M*◁*S**if and only if**Mϕ*◁*S*/*N*.

### Proposition 2.14

*Let*

*ϕ*

*be an epimorphism of a semigroup*

*S*

*onto a group*(

*G*,⋅,1).

*Then*:

- (i)
Ker(

*ϕ*)=*ϕϕ*^{−1}*is a group congruence on**S*; - (ii)
*N*={1}*ϕ*^{−1}◁*S*; - (iii)
Ker(

*ϕ*)=*ρ*_{N}.

*Conversely*, *if**N*◁*S*, *then**N**is the kernel of the canonical homomorphism of**S**onto**S*/*N*.

### Example 2.15

We now describe all normal subsemigroups of the bicyclic semigroup *S*=ℕ_{0}×ℕ_{0}, where (*k*,*l*)(*m*,*n*)=(*k*−*l*+max{*l*,*m*},*n*−*m*+max{*l*,*m*}). It is known that every (non-identical) homomorphic image of the bicyclic semigroup is a cyclic group. Also, it is almost evident that *E*_{S}={(0,0),(1,1),(2,2),…}◁*S* and (*k*,*l*)*ρ*_{E}(*m*,*n*) if and only if *k*+*n*=*l*+*m*, so *S*/*ρ*_{E}≅(ℤ,+). It follows that (*i*ℤ)*ϕ*^{−1}={(*m*,*n*)∈*S*:(*m*)_{i}=(*n*)_{i}} for every *i*∈ℕ. The conclusion is that every cyclic group is a homomorphic image of the bicyclic semigroup.

We have also the following well known proposition (from group theory).

### Proposition 2.16

*Let*

*S*

*be a semigroup*;

*M*,

*N*◁

*S*

*and*

*M*⊆

*N*.

*Then*:

- (i)
*M*◁*N*; - (ii)
*N*/*M*◁*S*/*M*; - (iii)
(

*S*/*M*)/(*N*/*M*)≅*S*/*N*.

Every full and closed subsemigroup *A* of an *E*-inversive semigroup *S* is itself *E*-inversive. Indeed, let *a*∈*A*. Then *ax*∈*E*_{S}=*E*_{A} for some *x*∈*S*, so *x*∈*Aω*=*A*. Consequently, there is *x*∈*A* such that *ax*∈*E*_{A}.

Finally, by way of contrast, we prove in the present section the following proposition which is valid for the class of all *E*-inversive semigroups.

### Proposition 2.17

*Let*

*S*

*be an*

*E*-

*inversive semigroup*,

*N*◁

*S*.

*Suppose also that a subsemigroup*

*M*

*of*

*S*

*is full and closed*.

*Then*:

- (i)
*M*∩*N*◁*M*; - (ii)
*N*◁(*MN*)*ω*; - (iii)
*M*/(*M*∩*N*)≅(*MN*)*ω*/*N*.

### Proof

(i). It is clear that *E*_{S}⊂*M*∩*N*, so *M*∩*N* is a full subsemigroup of *M*. Let *a*,*b*∈*M* be such that *ab*∈*M*∩*N*. Then *ba*∈*M* and *ba*∈*N* (since *N* is reflexive in *S*). Hence *ba*∈*M*∩*N*. Hence *M*∩*N* is reflexive in *M*. Further, if *x*∈(*M*∩*N*)*ω*, then *yx*∈*M*∩*N* for some *y*∈*M*∩*N*, so *x*∈*M*∩*N* (because *N* and *M* are closed). Since *M*∩*N* is full and closed, then it is *E*-inversive, so it is dense in *M*. Thus *M*∩*N*◁*M*.

(ii). We show that (*MN*)*ω* is a subsemigroup of *S*. Let *a*,*b*∈(*MN*)*ω*. Then *m*_{1}*n*_{1}*a*=*m*_{2}*n*_{2} for some *m*_{1},*m*_{2}∈*M*,*n*_{1},*n*_{2}∈*N*. Since *S* is *E*-inversive, then \(W(m_{1}) \not= \emptyset\). Hence *mm*_{1},*m*_{1}*m*∈*E*_{S}⊂*M* for some *m*∈*S*. Thus *m*∈*M* (since *M* is closed), (*mm*_{1})*n*_{1}*a*=(*mm*_{2})*n*_{2}. Therefore (*n*_{1}*a*,*mm*_{2})∈*ρ*_{N}, since *mm*_{1}∈*E*_{S}⊂*N*, so (*a*,*m*_{3})∈*ρ*_{N} (*m*_{3}∈*M*). Similarly, (*b*,*m*_{4})∈*ρ*_{N} for some *m*_{4}∈*M*. It follows that (*ab*,*m*_{5})∈*ρ*_{N}, where *m*_{5}∈*M*. Hence *n*_{3}*ab*=*m*_{5}*n*_{4} for some *n*_{3},*n*_{4}∈*N*. Thus (*m*_{5}*n*_{3})*ab*=(*m*_{5}*m*_{5})*n*_{4}. Consequently, *ab*∈(*MN*)*ω*. Furthermore, *N*⊂(*MN*)*ω*. Indeed, let *n*∈*N*. Then *n*_{1}*n*=*en*_{2} for some *e*∈*E*_{S},*n*_{1},*n*_{2}∈*N*. Hence we have (*en*_{1})*n*=*en*_{2}∈*MN*, so *n*∈(*MN*)*ω*. Consequently, *N*◁(*MN*)*ω* (since *N*◁*S*).

The proof of the condition (iii) is standard. □

## 3 Group congruences on an *E*-inversive semigroup

Note that if a semigroup *S* is *E*-inversive, then every full subsemigroup of *S* is dense (since *E*_{S} is dense), so a subsemigroup *A* of *S* is normal if and only if *A* is full, reflexive and closed. It follows that *S* has a least normal subsemigroup *U*. Thus the least group congruence on an arbitrary *E*-inversive semigroup exists. Denote it by *σ* or *σ*_{S}. Then *σ*=*ρ*_{U} and \(\operatorname{ker}\sigma = U\) (Theorem 2.4).

Firstly, we have the following proposition.

### Proposition 3.1

*Let**S**be an**E*-*inversive semigroup*. *Then*\(\mathcal{GC}(S) = [\sigma, S \times S]\). *Thus*\(\mathcal{GC}(S)\)*is a complete sublattice of*\(\mathcal{C}(S)\).

*Also*,

*ρ*

_{M}∨

*ρ*

_{N}=

*ρ*

_{M}

*ρ*

_{N}=

*ρ*

_{N}

*ρ*

_{M}

*for all*

*M*,

*N*◁

*S*.

*Hence the lattice*

*is modular*.

### Proof

The first part of the above proposition is clear. We show its second part. Let *a*(*ρ*_{M}*ρ*_{N})*b*. Then (*a*,*c*)∈*ρ*_{M},(*c*,*b*)∈*ρ*_{N}, where *c*∈*S*. Take any *x*∈*W*(*c*). Then *xc*,*cx*∈*E*_{S},(*cxa*)*ρ*_{N}(*bxa*),(*bxa*)*ρ*_{M}(*bxc*), so (*a*,*bxa*)∈*ρ*_{N},(*bxa*,*b*)∈*ρ*_{M}. Hence (*a*,*b*)∈*ρ*_{N}*ρ*_{M}. Therefore *ρ*_{M}*ρ*_{N}⊂*ρ*_{N}*ρ*_{M}. We may equally well show the opposite inclusion. Consequently, *ρ*_{M}∨*ρ*_{N}=*ρ*_{M}*ρ*_{N}=*ρ*_{N}*ρ*_{M}. In the light of Proposition I.8.5 [11], the lattice \((\mathcal{GC}(S), \subseteq, \cap, \circ)\) is modular. □

Let *M*,*N* be normal subsemigroups of a semigroup *S*. From Proposition 3.1 and Corollary 2.8 we obtain that \(\operatorname{ker}(\rho_{M} \rho_{N}) = \operatorname{ker}(\rho_{N} \rho_{M}) = (M\rho_{N})\omega = (N\rho_{M})\omega\). In fact, if *S* is *E*-inversive, then \(\operatorname{ker}(\rho_{M} \rho_{N}) = \operatorname{ker}(\rho_{N} \rho_{M}) = M\rho_{N} = N\rho_{M}\). Indeed, let \(x \in \operatorname{ker}(\rho_{M} \rho_{N})\). Then (*x*,*e*)∈*ρ*_{M}*ρ*_{N} for some *e*∈*E*_{S}. Hence (*x*,*n*)∈*ρ*_{M}, (*n*,*e*)∈*ρ*_{N}, where *n*∈*S* (in fact, \(n \in \operatorname{ker}\rho_{N} = N\)). Thus *x*∈*Nρ*_{M}. Conversely, if *x*∈*Nρ*_{M}, then (*x*,*n*)∈*ρ*_{M} for some *n*∈*N*. Hence (*x*,*n*)∈*ρ*_{M},(*n*,*e*)∈*ρ*_{N}, where *e*∈*E*_{S}. Thus (*x*,*e*)∈*ρ*_{M}*ρ*_{N}, that is, \(x \in \operatorname{ker}(\rho_{M} \rho_{N})\), so \(\operatorname{ker}(\rho_{M} \rho_{N}) = N\rho_{M}\). Similarly, \(\operatorname{ker}(\rho_{N} \rho_{M}) = M\rho_{N}\). This implies the required equalities. Also, \(\operatorname{ker}(\rho_{M} \rho_{N}) = (MN)\omega\). Indeed, let *x*∈*Mρ*_{N}. Then *n*_{1}*x*=*mn*_{2} for some *n*_{1},*n*_{2}∈*N*,*m*∈*M*. Hence (*mn*_{1})*x*∈*MN*. Thus *x*∈(*MN*)*ω*. We have proved that \(\operatorname{ker}(\rho_{M} \rho_{N}) \subset (MN)\omega\). Conversely, let *x*∈(*MN*)*ω*. Then *m*_{1}*n*_{1}*x*=*m*_{2}*n*_{2} for some *m*_{1},*m*_{2}∈*M*,*n*_{1},*n*_{2}∈*N*. Since *S* is *E*-inversive, then *mm*_{1}=*e*∈*E*_{S}⊂*M* for some *m*∈*S*. It follows that *m*∈*M* (since *M* is closed), so *en*_{1}*x*=*mm*_{2}*n*_{2}. Hence (*x*,*mm*_{2})∈*ρ*_{N}. Thus \(x \in M\rho_{N} = \operatorname{ker}(\rho_{M} \rho_{N})\), so \((MN)\omega \subset \operatorname{ker}(\rho_{M} \rho_{N})\), as exactly required.

In fact, we have just shown that in an arbitrary *E*-inversive semigroup *S*, *ρ*_{(MN)ω}=*ρ*_{M}*ρ*_{N}=*ρ*_{N}*ρ*_{M}=*ρ*_{(NM)ω} for all *M*,*N*◁*S*. Moreover, notice that \(\operatorname{ker}(\rho_{M} \cap \rho_{N}) = \operatorname{ker}\rho_{M} \cap \operatorname{ker}\rho_{N} = M \cap N\) (*M*,*N*◁*S*), so *ρ*_{M}∩*ρ*_{N}=*ρ*_{M∩N} for *M*,*N*◁*S*. Consequently, the lattice \((\mathcal{N}(S), \subseteq, \cap, \lor)\), where *M*∨*N*=(*MN*)*ω* for all *M*,*N*◁*S*, is isomorphic to the lattice \((\mathcal{GC}(S), \subseteq, \cap, \circ)\) (by the inclusion-preserving bijection *ϕ*, see the proof of Theorem 2.4). Note also that the lattice \((\mathcal{N}(S), \subseteq, \cap, \lor)\) is complete (since it has the greatest element *S* and the intersection of any nonempty family of normal subsemigroups of *S* is a normal subsemigroup of *S*).

For terminology and elementary facts about lattices the reader is referred to the book [21] (Sect. I.2). The following result will be useful (see Exercise I.2.15(iii) in [21]).

### Lemma 3.2

*Every lattice isomorphism of complete lattices is a complete lattice isomorphism*.

From the above consideration we obtain the following theorem.

### Theorem 3.3

*Let**S**be an**E*-*inversive semigroup*. *Then there exists a* (*lattice*) *isomorphism**ϕ**between the lattice*\((\mathcal{N}(S), \subseteq, \cap, \lor)\), *where**M*∨*N*=(*MN*)*ω**for all**M*,*N*◁*S*, *and the lattice*\((\mathcal{GC}(S), \subseteq, \cap, \circ)\). *In fact*, *ϕ**is defined by**Nϕ*=*ρ*_{N}*for every*\(N \in \mathcal{N}(S)\). *Moreover*, *ϕ**is a complete lattice isomorphism*.

Finally, we have the following proposition.

### Proposition 3.4

*Let**S**be an**E*-*inversive semigroup*, *N*◁*S*. *Then* (*a*,*b*)∈*ρ*_{N}*if and only if**ab*^{∗}∈*N**for some* (*all*) *b*^{∗}∈*W*(*b*).

### Proof

(⟹). Let *na*=*bm*, where *n*,*m*∈*N*, and *b*^{∗}∈*W*(*b*). Then *nab*^{∗}=*bmb*^{∗}. Since *b*^{∗}*bm*∈*N* and *N* is reflexive, then *nab*^{∗}∈*N*. Hence *ab*^{∗}∈*Nω*=*N*.

(⟸). Let *ab*^{∗}=*n*∈*N* for some *b*^{∗}∈*W*(*b*). Then *a*(*b*^{∗}*b*)=*nb*, so (*a*,*b*)∈*ρ*_{N} (by Lemma 2.3). □

## 4 Group congruences on an *E*-semigroup

First, we “generalize” some results from orthodox semigroups to *E*-semigroups (see Theorem VI.1.1 [11]).

### Proposition 4.1

*Let*

*S*

*be a semigroup*.

*The following conditions are equivalent*:

- (i)
*S**is an**E*-*semigroup*; - (ii)
∀

*a*,*b*∈*S*[*W*(*b*)*W*(*a*)⊆*W*(*ab*)].

*Moreover*,

*the condition*(i)

*implies the following condition*:

- (iii)
∀

*e*∈*E*_{S}[*W*(*e*)⊆*E*_{S}].

*If in addition**S**is an**R*-*semigroup*, *then the conditions* (i)*–*(iii) *are equivalent*.

### Proof

The proof is closely similar to the proof of Theorem VI.1.1 [11]. □

### Corollary 4.2

*Let*

*S*

*be an*

*E*-

*semigroup*.

*Then*:

- (i)
∀

*e*∈*E*_{S}[*W*(*e*),*V*(*e*)⊆*E*_{S}]; - (ii)
∀

*a*∈*S*,*a*^{∗}∈*W*(*a*),*e*∈*E*_{S}[*aea*^{∗},*a*^{∗}*ea*∈*E*_{S}]; - (iii)
∀

*a*∈*S*,*a*^{∗}∈*W*(*a*),*e*,*f*∈*E*_{S}[*ea*^{∗},*a*^{∗}*e*,*ea*^{∗}*f*∈*W*(*a*)].

### Proof

(i). This follows from Proposition 4.1.

(ii). This follows from the proof of Proposition VI.1.4 [11].

(iii). Let *a*∈*S*,*a*^{∗}∈*W*(*a*),*e*,*f*∈*E*_{S}. Since *e*∈*W*(*e*) and *f*∈*W*(*f*), then *ea*^{∗}∈*W*(*e*)*W*(*a*)⊆*W*(*ae*). Hence *ea*^{∗}=*ea*^{∗}*aeea*^{∗}=(*ea*^{∗})*a*(*ea*^{∗}). Therefore *ea*^{∗}∈*W*(*a*). Similarly, *a*^{∗}*e*∈*W*(*a*). Finally, *ea*^{∗}*f*∈*W*(*e*)*W*(*a*)*W*(*f*)⊆*W*(*fae*) and so *ea*^{∗}*f*=*ea*^{∗}*ffaeea*^{∗}*f*=(*ea*^{∗}*f*)*a*(*ea*^{∗}*f*). Hence *ea*^{∗}*f*∈*W*(*a*). □

### Proposition 4.3

*Let*

*S*

*be an*

*E*-

*inversive*

*E*-

*semigroup*.

*Then*

### Proof

Let (*a*,*b*)∈*ρ*_{2,E} and *a*^{∗}∈*W*(*a*). Then *ae*=*fb* for some *e*,*f*∈*E*. Moreover, *a*^{∗}*f*∈*W*(*a*) (Corollary 4.2(iii)), so (*a*^{∗}*f*)*a*,*a*(*a*^{∗}*f*)∈*E*. Further, *a*^{∗}*fb*=*a*^{∗}*ae*∈*E*. We have just shown that *xa*,*ax*,*xb*∈*E* for some *x*∈*S*. Thus *ρ*_{2,E}⊂*ρ*_{4,E}.

On the other hand, if *xa*,*xb*∈*E* for some *x*∈*S*, say *xa*=*e*,*xb*=*f*, then (*efx*)*a*(*efx*)=*ef*(*xa*)*efx*=*efx*, so *efx*∈*W*(*a*). Also, *fxbfx*=*f*(*xb*)*fx*=*fx*, i.e., *fx*∈*W*(*b*). Hence *efx*∈*W*(*b*) (Corollary 4.2(iii)). Thus \(W(a) \cap W(b) \not= \emptyset\). It follows that *ay*,*by*,*ya*,*yb*∈*E* for some *y*∈*S*. Dually, if *ax*,*bx*∈*E* for some *x*∈*S*, then *ay*,*by*,*ya*,*yb*∈*E* for some *y*∈*S*. Thus *ρ*_{4,E}=*ρ*_{1,E}. In fact, we get \(\rho_{4, E} = \rho_{1, E} = \{(a, b) \in S \times S: W(a) \cap W(b) \not= \emptyset \}\). Finally, if *x*∈*W*(*a*)∩*W*(*b*), then *a*(*xb*)=(*ax*)*b* and *xb*,*ax*∈*E*. Thus *ρ*_{2,E}=*ρ*_{4,E}=*ρ*_{1,E}. We may equally well show that *ρ*_{3,E}=*ρ*_{4,E}=*ρ*_{1,E}. Consequently, *ρ*_{1,E}=*ρ*_{2,E}=*ρ*_{3,E}=*ρ*_{4,E}. □

### Lemma 4.4

*Let*

*S*

*be an*

*E*-

*inversive*

*E*-

*semigroup*.

*Then*:

- (i)
∀

*a*∈*S*∃*e*,*f*∈*E*_{S}[*ea*,*af*∈*Reg*(*S*)]; - (ii)
\(\forall a \in S~\exists\, r \in \mathit{Reg}(S)~[W(a) \cap W(r) \not= \emptyset]\).

### Proof

Let *a*∈*S*,*x*∈*W*(*a*). Then (*ax*)*a*,*a*(*xa*)∈*Reg*(*S*), where *ax*,*xa*∈*E*_{S}, so (i) holds. Also, *r*=*axa*∈*Reg*(*S*) and *xrx*=*x*. Thus *x*∈*W*(*a*)∩*W*(*r*). □

Denote the above four relations from Proposition 4.3 by *ρ*_{E}. Recall that from the proof of Proposition 4.3 follows that \(\rho_{E} = \{(a, b) \in S \times S: W(a) \cap W(b) \not= \emptyset \}\).

### Theorem 4.5

*In any**E*-*inversive**E*-*semigroup*, *σ*=*ρ*_{E}. *Moreover*, \(\operatorname{ker}\sigma = E_{S}\omega\). *Thus**E*_{S}*ω*◁*S*.

### Proof

*ρ*

_{E}is an equivalence relation on

*S*. Let (

*a*,

*b*)∈

*ρ*

_{E},

*c*∈

*S*. Then

*x*∈

*W*(

*a*)∩

*W*(

*b*). Take any

*y*∈

*W*(

*c*). In the light of Proposition 4.1,

*ca*,

*cb*)∈

*ρ*

_{E}. Thus

*ρ*

_{E}is a left congruence on

*S*. We may equally well show that

*ρ*

_{E}is a right congruence on

*S*. Also, if

*e*,

*f*∈

*E*

_{S}, then

*ee*,

*ef*∈

*E*

_{S}. Consequently, (

*e*,

*f*)∈

*ρ*

_{E}for all

*e*,

*f*∈

*E*

_{S}. Lemma 4.4(ii) says that every

*ρ*

_{E}-class of

*S*contains a regular element. This implies that

*S*/

*ρ*

_{E}is a group.

*E*

_{S}

*ω*◁

*S*(Theorem 2.4). Finally,

*ρ*

_{E}⊆

*ρ*

_{N}for ever

*N*◁

*S*. Indeed,

*E*

_{S}⊆

*N*. Hence

*E*

_{S}

*ω*⊆

*Nω*=

*N*. Thus \(\rho_{E} = \rho_{E_{S}\omega} \subseteq \rho_{N}\) (Theorem 2.4). Consequently,

*σ*=

*ρ*

_{E}. □

### Corollary 4.6

*The least group congruence*

*σ*

*on an*

*E*-

*inversive*

*E*-

*semigroup is given by*

### Remark 2

Note that the condition “∃ *e*∈*E*_{S} [*eae*=*ebe*]” from the above corollary is equivalent to the apparently weaker condition “∃ *s*∈*S* [*sas*=*sbs*]”.

From Result 1.1 and Theorem 4.5 we obtain the following theorem.

### Theorem 4.7

*In any idempotent*-*surjective**E*-*semigroup*, *σ*=*ρ*_{E}.

*S*be a semigroup. A congruence

*ρ*on

*S*is called

*idempotent pure*if

*eρ*⊆

*E*

_{S}for every

*e*∈

*E*

_{S}. Note that if

*S*is idempotent-surjective, then

*ρ*is idempotent pure if and only if \(\operatorname{ker}\rho = E_{S}\). Let \(\mathcal{E}\) be an equivalence relation on

*S*induced by the partition: {

*E*

_{S},

*S*∖

*E*

_{S}}. Then \(\mathcal{E}^{\flat}\) (defined in [13], see p. 27) is the greatest idempotent pure congruence on

*S*. Put \(\tau = \mathcal{E}^{\flat}\). Then (see [13], p. 28)

*S*is

*E*-inversive, then

*τ*⊆

*σ*. Indeed, let (

*a*,

*b*)∈

*τ*and

*b*

^{∗}∈

*W*(

*b*). Then

*bb*

^{∗}∈

*E*

_{S},(

*ab*

^{∗},

*bb*

^{∗})∈

*τ*. Hence \(ab^{*} \in E_{S} \subseteq \operatorname{ker}\sigma\). In the light of Proposition 3.4, (

*a*,

*b*)∈

*σ*, as exactly required. In the following corollary we give an alternative proof of this fact.

### Corollary 4.8

*If**ρ**is a congruence on an idempotent*-*surjective**E*-*semigroup**S*, *then*\(\operatorname{ker}(\rho \lor \sigma) = (\operatorname{ker}\rho)\omega\). *In particular*, *τ*⊆*σ*.

### Proof

*τ*∨

*σ*=

*σ*. Thus

*τ*⊆

*σ*. □

*ρ*be a congruence on a semigroup

*S*. By the

*trace*\(\operatorname{tr}\rho\) of

*ρ*we shall mean the restriction of

*ρ*to

*E*

_{S}. Also, we say that

*ρ*is

*idempotent-separating*if \(\operatorname{tr}\rho = 1_{E_{S}}\). Edwards in [3] shows that if

*S*is an eventually regular semigroup, then the relation \(\theta = \{(\rho_{1}, \rho_{2}) \in \mathcal{C}(S) \times \mathcal{C}(S): \operatorname{tr}\rho_{1} = \operatorname{tr}\rho_{2}\}\) is a complete congruence on \(\mathcal{C}(S)\) and proves that every

*θ*-class

*ρθ*is a complete sublattice of \(\mathcal{C}(S)\) with the maximum element

*ρ*). Edwards generalizes some of these results for the class of all idempotent-surjective semigroups [4]. In fact, if

*S*is an arbitrary idempotent-surjective semigroup, then every

*θ*-class

*ρθ*is the interval [1(

*ρ*),

*μ*(

*ρ*)], where

*μ*is the maximum idempotent-separating congruence on

*S*(see [4] for more details).

It is easily seen that the class of idempotent-surjective semigroups is closed under homomorphic images [10]. Using the obvious terminology we show next that every homomorphism of idempotent-surjective *E*-semigroups can be factored into a homomorphism preserving the maximal group homomorphic images and an idempotent-separating homomorphism. Firstly, we have need the following lemma.

### Lemma 4.9

*Let**ρ**be a congruence on an idempotent*-*surjective**E*-*semigroup**S*, *a*,*b*∈*S*. *Then* (*aρ*,*bρ*)∈*σ**in**S*/*ρ**implies* (*a*,*b*)∈*σ**if and only if**ρ*⊆*σ*.

### Proof

The proof is closely similar to the proof of Lemma III.5.9 [21]. □

*S*be an idempotent-surjective

*E*-semigroup, \(\rho \in \mathcal{C}(S)\). Clearly, (

*a*,

*b*)∈

*σ*implies (

*aρ*,

*bρ*)∈

*σ*. In the light of Lemma 4.9, if

*ρ*⊆

*σ*, then (

*a*,

*b*)∈

*σ*if and only if (

*aρ*,

*bρ*)∈

*σ*. Hence

*S*/

*σ*≅(

*S*/

*ρ*)/

*σ*, that is,

*S*and

*S*/

*ρ*have isomorphic maximal group homomorphic images. In that case, we may say that

*ρ*

*preserves the maximal group homomorphic images*. Since for any congruence

*ρ*on

*S*we have 1(

*ρ*)⊆

*ρ*, then we obtain the following factorization:

The following proposition generalizes Proposition III.5.10 [21].

### Proposition 4.10

*Every homomorphism of idempotent*-*surjective**E*-*semigroups can be factored into a homomorphism preserving the maximal group homomorphic images and an idempotent*-*separating homomorphism*.

### Proof

Let *ρ* be any congruence on an idempotent-surjective *E*-semigroup *S*. Since *ρ*⊆*S*×*S*, then 1(*ρ*)⊆1(*S*×*S*). Clearly, *σ*∈[1(*S*×*S*),*S*×*S*] and so 1(*ρ*)⊆*σ*. It follows that the canonical epimorphism of *S* onto *S*/1(*ρ*) preserves the maximal group homomorphic images. Finally, an epimorphism *ϕ*:*S*/1(*ρ*)→*S*/*ρ* (defined by the obvious way) is idempotent-separating, since \(\operatorname{tr}\rho = \operatorname{tr}(1(\rho))\). The thesis of the proposition is a consequence of the above factorization. □

## 5 Group congruences on an *E*-unitary semigroup

A nonempty subset *A* of a semigroup *S* is called *left* [*right*] *unitary* if *as*∈*A* [*sa*∈*A*] implies *s*∈*A* for every *a*∈*A*,*s*∈*S*. Also, we say that *A* is *unitary* if it is both left and right unitary. Finally, a semigroup *S* with \(E_{S} \not= \emptyset\) is said to be *E*-*unitary* if *E*_{S} is unitary.

### Proposition 5.1

*Let*

*S*

*be a semigroup with*\(E_{S} \not= \emptyset\).

*The following conditions are equivalent*:

- (i)
*S**is**E*-*unitary*; - (ii)
*E*_{S}*is left unitary*; - (iii)
*E*_{S}*is right unitary*.

*Also*, *if**S**is an**E*-*unitary**E*-*inversive semigroup*, *then**S**is an**E*-*semigroup*.

### Proof

\(\mathrm{(i)} \Longrightarrow \mathrm{(ii)}\). This is trivial.

\(\mathrm{(ii)} \Longrightarrow \mathrm{(iii)}\). Let *s*∈*S*,*e*∈*E*_{S}. If *se*=*f*∈*E*_{S}, then *fsef*=*f* and so we get (*efs*)(*efs*)=*efs*, that is, *efs*∈*E*_{S}. Hence *fs*∈*E*_{S}. Thus *s*∈*E*_{S}.

\(\mathrm{(iii)} \Longrightarrow \mathrm{(i)}\). We may equally well show like above that *E*_{S} is left unitary. Thus the condition (i) holds.

Finally, let *S* be an *E*-unitary *E*-inversive semigroup. If *e*,*f*∈*E*_{S},*x*∈*W*(*ef*), then *xef*∈*E*_{S}. Hence *xef*,*x*∈*E*_{S}. Thus *ef*∈*E*_{S}. □

### Corollary 5.2

*Let*

*S*

*be an*

*E*-

*inversive semigroup*.

*Then the following conditions are equivalent*:

- (i)
*S**is**E*-*unitary*; - (ii)
\(\operatorname{ker}\sigma = E_{S}\);

- (iii)
*τ*=*σ*.

*In particular*, *if**S**is an**E*-*unitary**E*-*inversive semigroup*, *then**E*_{S}◁*S*.

### Proof

\(\mathrm{(i)} \Longrightarrow \mathrm{(ii)}\). In the light of Proposition 5.1 and Theorem 4.5, \(\operatorname{ker}\sigma = E_{S}\omega\). Also, *S* is left unitary, that is, *E*_{S} is closed. Thus \(\operatorname{ker}\sigma = E_{S}\).

\(\mathrm{(ii)} \Longrightarrow \mathrm{(iii)}\). We have mentioned above that *τ*⊆*σ*. On the other hand, the main assumption implies that *σ* is idempotent pure. Hence *σ*⊆*τ*. Thus *τ*=*σ*.

\(\mathrm{(iii)} \Longrightarrow \mathrm{(i)}\). Let *a*∈*S*,*e*,*f*∈*E*_{S}. If *ea*=*f*, then \(a \in \operatorname{ker}\sigma = \operatorname{ker}\tau = E_{S}\), that is, *E*_{S} is left unitary. In the light of Proposition 5.1, *S* is *E*-unitary. □

### Remark 3

Notice that if a semigroup is not *E*-inversive, then Corollary 5.2 is false. Indeed, let *F*_{X}^{1} be the free monoid on the set *X*. Then *F*_{X}^{1} is *E*-unitary but *τ* is induced by the partition {*F*_{X},{1}}. Thus *τ* is not a group congruence.

From Proposition 3.4 and Corollary 5.2 we obtain the following proposition.

### Proposition 5.3

*Let**S**be an**E*-*unitary**E*-*inversive semigroup*. *Then* (*a*,*b*)∈*σ**if and only if**ab*^{∗}∈*E*_{S}*for some* (*all*) *b*^{∗}∈*W*(*b*).

### Corollary 5.4

*Let**A**be an**E*-*inversive subsemigroup of an**E*-*unitary**E*-*inversive semigroup**S*. *Then**σ*_{A}=*σ*_{S}∩(*A*×*A*).

### Proof

Clearly, *σ*_{A}⊂*σ*_{S}∩(*A*×*A*). The converse follows from Proposition 5.3. □

In [14] Howie and Lallement showed that \(\sigma \cap \mathcal{H} = 1_{S}\), when *S* is an *E*-unitary regular semigroup. We prove a corresponding result.

### Theorem 5.5

*Let**S**be an**E*-*unitary**E*-*inversive semigroup*. *Then*\(\sigma \cap \mathcal{H} = 1_{S}\). *Moreover*, *if in addition**E*_{S}*forms a semilattice*, *then*\(\sigma \cap \mathcal{L} = \sigma \cap \mathcal{R} = 1_{S}\).

### Proof

*S*be an

*E*-unitary

*E*-inversive semigroup. Suppose also that

*E*

_{S}forms a semilattice. Then

*E*

_{S}is normal (Corollary 5.2), so if \((a, b) \in \sigma \cap \mathcal{L}\), then

*ax*=

*e*,

*bx*=

*f*∈

*E*

_{S}for some

*x*∈

*S*(see Proposition 5.3) and

*sa*=

*b*,

*tb*=

*a*for some

*s*,

*t*∈

*S*. Hence

*se*=

*sax*=

*bx*=

*f*∈

*E*

_{S},

*tf*=

*tbx*=

*ax*=

*e*∈

*E*

_{S}. Thus

*s*,

*t*∈

*E*

_{S}(since

*E*

_{S}is unitary), so since idempotents commute and

*ta*=

*tb*,

We may equally well show that \(\sigma \cap \mathcal{R} = 1_{S}\).

If *S* is *E*-unitary, then *E*_{S} is normal, too. Let \((a, b) \in \sigma \cap \mathcal{H}\). By the above proof and its dual we conclude that *a*=*eb*=*bf* and *b*=*ga*=*ah* for some *e*,*f*,*g*,*h*∈*E*_{S}. In the light of Proposition 2 in [18], *a*=*b*. □

### Remark 4

The assumption that *S* is an *E*-inversive semigroup is important. Indeed, let *S*=(ℝ_{0},+) be the semigroup of nonnegative real numbers with respect to addition. Then *S* is an *E*-unitary commutative semigroup. Put *M*=ℕ_{0} and *N*={0,*x*,2*x*,3*x*,…} (where *x*∈ℝ∖ℚ). Then *M*,*N*◁*S* but *M*∩*N*={0} is not normal, so *S* has no least group congruence.

The converse of Theorem 4.15 is not valid (in general). Indeed, let *S*=〈*x*〉, where *x*=(2 3 4 5 6 7 5) is a mapping of \(\mathcal{T}(\{1, 2, \ldots, 7\})\). Then *S*=*M*(4,3) is the monogenic semigroup with index 4 and period 3, say *S*={*x*,*x*^{2},…,*x*^{6}}. Also, the cyclic subgroup *K*_{x} of *S* with the unit *e* is equal {*x*^{4},*x*^{5},*x*^{6}=*e*}. Since *x*^{3}*e*=*x*^{7}*x*^{2}=*x*^{4}*x*^{2}=*e*, then *S* is not *E*-unitary. On the other hand, *σ* is induced by the partition: {{*x*,*x*^{4}},{*x*^{2},*x*^{5}},{*x*^{3},*e*}} and \(\mathcal{H}\) by the partition: {*K*_{x},{*x*},{*x*^{2}},{*x*^{3}}}. Thus \(\sigma \cap \mathcal{H} = 1_{S}\).

From Theorem 5.5 and Corollary 5.2 we have the following corollary.

### Corollary 5.6

*Let*

*S*

*be an*

*E*-

*unitary*

*E*-

*inversive semigroup*.

*Then*

*Moreover*,

*if in addition*

*E*

_{S}

*forms a semilattice*,

*then*

Recall that a congruence *ρ* on a semigroup *S* is *E*-*unitary* if *S*/*ρ* is *E*-unitary. In [5] the author described the least *E*-unitary congruence *κ* on an idempotent-surjective semigroup. Also, for every congruence *ρ* on an idempotent-surjective semigroup *S* there exists the least *E*-unitary congruence *κ*_{ρ} on *S* containing *ρ* [5].

Let *S* be an idempotent-surjective semigroup, *N*◁*S*. Define the relation \(\hat{\rho_{N}}\) on \(\mathcal{C}(S)\) by the following rule: \((\rho_{1}, \rho_{2}) \in \hat{\rho_{N}} \Leftrightarrow \rho_{1} \lor \rho_{N} = \rho_{2} \lor \rho_{N}\) (\(\rho_{1}, \rho_{2} \in \mathcal{C}(S)\)). Then \(\hat{\rho_{N}}\) is a congruence on \(\mathcal{C}(S)\), since \(\phi \phi^{- 1} = \hat{\rho_{N}}\) (see Theorem 2.11).

Also, we prove the following proposition.

### Proposition 5.7

*Let**S**be an idempotent*-*surjective semigroup*, \(N \lhd S, \rho \in \mathcal{C}(S)\). *Then the elements**ρ*,*κ*_{ρ},*ρ*∨*ρ*_{N}*are*\(\hat{\rho_{N}}\)-*equivalent and**ρ*⊆*κ*_{ρ}⊆*ρ*∨*ρ*_{N}. *Moreover*, *the element**ρ*∨*ρ*_{N}*is the largest in the*\(\hat{\rho_{N}}\)-*class*\(\rho \hat{\rho_{N}}\).

### Proof

Since *κ*_{ρ} is the least *E*-unitary congruence containing *ρ* and clearly *ρ*∨*ρ*_{N} is *E*-unitary, then *ρ*⊆*κ*_{ρ}⊆*ρ*∨*ρ*_{N}. Hence *ρ*∨*ρ*_{N}⊆*κ*_{ρ}∨*ρ*_{N}⊆*ρ*∨*ρ*_{N}. Therefore *ρ*∨*ρ*_{N}=*κ*_{ρ}∨*ρ*_{N}. Thus \((\rho, \kappa_{\rho}) \in \hat{\rho_{N}}\). Evidently, \((\rho, \rho \lor \rho_{N}) \in \hat{\rho_{N}}\). This implies the first part of the proposition. The second part is clear. □

### Remark 5

Recall from [22] that in the class of inverse semigroups not every \(\hat{\sigma}\)-class has a least element.

Finally, it is easy to see that the least *E*-unitary congruence *κ* on an arbitrary *E*-inversive semigroup exists, too. We show that \(\mathcal{H}\cap\sigma\subseteq\kappa\) in any *E*-inversive semigroup. Firstly, we have need the following useful proposition.

### Proposition 5.8

*Let**B**be the least seminormal subsemigroup of an**E*-*inversive semigroup**S*. *If**ϕ**is an epimorphism of**S**onto an**E*-*unitary semigroup**T*, *then**Bϕ*⊆*E*_{T}.

### Proof

Put *A*=(*E*_{T})*ϕ*^{−1}. Clearly, *A* is a full subsemigroup of *S*, so *A* is dense. Further, if *xy*∈*A*, then *E*_{T}∋(*xy*)*ϕ*=*xϕ*⋅*yϕ*=*yϕ*⋅*xϕ*=(*yx*)*ϕ* (since *E*_{T} is reflexive), so *yx*∈*A*. Hence *B*⊆*A*. Thus *Bϕ*⊆*Aϕ*⊆((*E*_{T})*ϕ*^{−1})*ϕ*⊆*E*_{T}. □

We may now prove the following equivalent theorem to Theorem 5.5.

### Theorem 5.9

*In any**E*-*inversive semigroup**S*, \(\mathcal{H}\cap\sigma\subseteq\kappa\). *If in addition**E*_{S}*forms a semilattice*, *then*\(\mathcal{L}\cap\sigma\subseteq\kappa\)*and*\(\mathcal{R}\cap\sigma\subseteq\kappa\).

### Proof

Indeed, *σ*=*ρ*_{B}, where *B* is the least seminormal subsemigroup of *S*. Let \((a, b)\in\mathcal{H}\cap\sigma\). Then clearly \((a\kappa, b\kappa)\in\mathcal{H}^{S/\kappa}\). Also, *ax*=*yb* for some *a*,*b*∈*B*. In the light of Proposition 5.8, (*aκ*)(*xκ*)=(*yκ*)(*bκ*), where *aκ*,*bκ*∈*E*_{S/κ}. Hence \((a\kappa, b\kappa)\in\mathcal{H}^{S/\kappa}\cap\sigma_{S/\kappa}= 1_{S/\kappa}\) (Theorem 5.5). Thus \(\mathcal{H}\cap\sigma\subseteq\kappa\), as required. □

## 6 Group congruences on an eventually regular semigroup

Group congruences on eventually regular semigroups were described in [9] by Hanumantha Rao and Lakshmi. In the paper [9] the following definition was introduced: a subset *A* of *S* is called *self-conjugate* if *x*^{r(x)−1}(*x*^{r(x)})^{∗}*Ax*⊆*A* and *xAx*^{r(x)−1}(*x*^{r(x)})^{∗}⊆*A* for all *x*∈*S*,(*x*^{r(x)})^{∗}∈*V*(*x*^{r(x)}). We say that *A* is *self-conjugate* if the former condition holds.

### Lemma 6.1

*Let**N**be a subsemigroup of an eventually regular semigroup**S*. *Then**N**is normal if and only if**N**is full*, *self*-*conjugate and closed*.

### Proof

Let *N* be normal, *x*∈*S*,(*x*^{r(x)})^{∗}∈*V*(*x*^{r(x)}). Then *N* is full and closed. Also, *x*^{r(x)}(*x*^{r(x)})^{∗}*N*⊆*EN*⊆*N*, so *x*^{r(x)−1}(*x*^{r(x)})^{∗}*Nx*⊆*N*, since *N* is reflexive.

Let *N* be full, self-conjugate and closed, *xy*∈*N*,(*x*^{r(x)})^{∗}∈*V*(*x*^{r(x)}). Then *x*^{r(x)−1}(*x*^{r(x)})^{∗}(*xy*)*x*∈*x*^{r(x)−1}(*x*^{r(x)})^{∗}*Nx*⊆*N*, i.e., (*x*^{r(x)−1}(*x*^{r(x)})^{∗}*x*)(*yx*)∈*N*, where *x*^{r(x)−1}(*x*^{r(x)})^{∗}*x*∈*E*_{S}⊆*N*. Hence *yx*∈*Nω*=*N*, so *N* is reflexive. Thus *N*◁*S*. □

### Lemma 6.2

*Let*

*S*

*be an eventually regular semigroup*,

*N*◁

*S*.

*Then*

### Proof

Let (*a*,*b*)∈*ρ*_{N} and (*b*^{r(b)})^{∗}∈*V*(*b*^{r(b)}). Then *na*=*bm* for some *n*,*m*∈*N*. Hence *nab*^{r(b)−1}(*b*^{r(b)})^{∗}=*bmb*^{r(b)−1}(*b*^{r(b)})^{∗}. Also, since *b*^{r(b)−1}(*b*^{r(b)})^{∗}*b*∈*E*_{S}, then *mb*^{r(b)−1}(*b*^{r(b)})^{∗}*b*∈*NE*_{S}⊆*N*, so *nab*^{r(b)−1}(*b*^{r(b)})^{∗}=*bmb*^{r(b)−1}(*b*^{r(b)})^{∗}∈*N*, since *N* is reflexive. Consequently, *ab*^{r(b)−1}(*b*^{r(b)})^{∗}∈*Nω*=*N*.

Conversely, let *a*,*b*∈*S*,(*b*^{r(b)})^{∗}∈*V*(*b*^{r(b)}) and *ab*^{r(b)−1}(*b*^{r(b)})^{∗}=*n*∈*N*. Then *a*(*b*^{r(b)−1}(*b*^{r(b)})^{∗}*b*)=*nb*, where *b*^{r(b)−1}(*b*^{r(b)})^{∗}*b*∈*E*_{S}⊆*N*. Hence (*a*,*b*)∈*ρ*_{N}. □

We have the following corollary (see Theorem 1 [9]).

### Corollary 6.3

*Let*

*S*

*be an eventually regular semigroup*,

*N*◁

*S*.

*Then*

*is a group congruence on*

*S*.

Finally, we give some remarks concerning group congruences on inverse semigroups. Firstly, consider the following result (see Exercise 7(ii) [11], p. 181).

### Statement 6.4

*An inverse subsemigroup**N**of an inverse semigroup**S**is normal if and only if* (*Nx*)*ω*=(*xN*)*ω**for every**x*∈*S*.

This result is false. Indeed, let *S* be a Clifford semigroup. Put \(N = \mathcal{Z}(S)\), where \(\mathcal{Z}(S) = \{s \in S: \forall a \in S~[sa = as]\}\). Clearly, *N* is a full subsemigroup of *S*. Also, *N* is self-conjugate. If the result is valid, then *N* is normal (since *Nx*=*xN* for every *x*∈*S*). Hence *ρ*_{N}=*S*×*S*=*ρ*_{S}, when *S*=*S*^{0}. It follows that every Clifford semigroup is commutative, a contradiction. Consequently, we conclude that the above result is false. Moreover, the assumptions of the result and the conditions: “*N* is full” and “*N* is self-conjugate” do not imply that (*Nx*)*ω*=(*xN*)*ω* for every *x*∈*S*.

It is clear that every subgroup of a group is full and closed. We prove now a correct version of the above statement.

### Proposition 6.5

*A full and closed inverse subsemigroup**N**of an inverse semigroup**S**is normal if and only if* (*Nx*)*ω*=(*xN*)*ω**for every**x*∈*S*.

### Proof

It is easy to see that if *N* is normal, then (*Nx*)*ω*=(*xN*)*ω* for every *x*∈*S*.

*Nx*)

*ω*=(

*xN*)

*ω*for every

*x*∈

*S*. It is easy to check that two relations

*ρ*

_{1}={(

*a*,

*b*)∈

*S*×

*S*:

*ab*

^{−1}∈

*N*} and

*ρ*

_{2}={(

*a*,

*b*)∈

*S*×

*S*:

*a*

^{−1}

*b*∈

*N*} are equivalences on

*S*and that

*xρ*

_{1}=(

*Nx*)

*ω*,

*xρ*

_{2}=(

*xN*)

*ω*for every

*x*∈

*S*. Also,

*ρ*

_{1}is right compatible and

*ρ*

_{2}is left compatible. Indeed, we show first that the equality (

*A*(

*Bω*))

*ω*=(

*AB*)

*ω*holds for all

*A*,

*B*⊆

*S*. Recall from [11] that

*natural partial order*on (an inverse semigroup)

*S*(i.e.,

*a*≤

*b*⇔∃

*e*∈

*E*

_{S}[

*a*=

*eb*]). Notice that ≤ is compatible. Let

*x*∈(

*A*(

*Bω*))

*ω*. Then

*ay*≤

*x*for some

*a*∈

*A*,

*y*∈

*Bω*(that is,

*b*≤

*y*for some

*b*∈

*B*). Hence

*ab*≤

*ay*≤

*x*. Thus

*x*∈(

*AB*)

*ω*. We have just proved that (

*A*(

*Bω*))

*ω*⊂(

*AB*)

*ω*. The opposite inclusion is clear. Let now (

*a*,

*b*)∈

*ρ*

_{2},

*c*∈

*S*. Then (

*aN*)

*ω*=(

*bN*)

*ω*and so (

*c*(

*aN*)

*ω*)

*ω*=(

*c*(

*bN*)

*ω*)

*ω*. Therefore (

*caN*)

*ω*=(

*cbN*)

*ω*. Thus

*ρ*

_{2}is a left congruence on

*S*. We may equally well show that

*ρ*

_{1}is a right congruence on

*S*. Since (

*Nx*)

*ω*=(

*xN*)

*ω*and

*xρ*

_{1}=(

*Nx*)

*ω*,

*xρ*

_{2}=(

*xN*)

*ω*for every

*x*∈

*S*, then

*ρ*

_{1}=

*ρ*

_{2}is a congruence on

*S*. Put for simplicity

*ρ*=

*ρ*

_{1}=

*ρ*

_{2}. Finally, if

*e*∈

*E*

_{S}, then

*E*

_{S}⊆

*N*=

*Nω*=(

*eN*)

*ω*. Hence

*ρ*is a group congruence on

*S*and \(\operatorname{ker}\rho = N\). Thus

*N*◁

*S*, as required. □

### Corollary 6.6

*A Clifford semigroup**S**is commutative if and only if*\(\mathcal{Z}(S)\)*is closed in**S* (*i*.*e*., *if and only if for every**s*∈*S**there exists*\(z \in \mathcal{Z}(S)\)*such that**z*≤*s*).

### Lemma 6.7

*Let**S**be a finite inverse semigroup with semilattice of idempotents**E*. *Then**Eω*=*S**if and only if**S**has zero*.

### Proof

It is clear that if *S* has zero, then *Eω*=*S*. Conversely, let *Eω*=*S*. Since *E* is finite, then *E* has the least idempotent with respect to the natural partial order, say 0. Let *s*∈*S*=*Eω*. Then *e*=*fs* and *e*=*sg* for some *e*,*f*,*g*∈*E* (see Proposition V.2.2 in [11]). Hence 0=0*s*=*s*0. Thus *S*=*S*^{0}, as required. □

By an analogy to groups we may introduce the concept of a *σ*-*simple inverse semigroup* in the class of finite inverse semigroups without 0. From Lemma 6.7 follows that every finite inverse semigroup *S* without zero has at least one non-universal group congruence, so *S* has exactly one non-universal group congruence if and only if *S*/*Eω* is a simple group. Hence we may say that a finite inverse semigroup *S* without zero is *σ*-*simple* if *S*/*Eω* is a simple group. This definition is equivalent to the following definition: *S* is *σ*-simple if *S* has exactly two normal subsemigroups, namely: *Eω* and *S*.

### Example 6.8

Let (*E*,≤) be a chain with the least element 0. Put *S*=*E*∪{*a*}, where *a*∉*E* and *aaa*=*a*. Assume also that *aa*=0. Hence *a*=*aaa*=0*a*=*a*0. It is easy to see that if a binary operation on *S* is associative, then *ea*=*ae*=*a* for every *e*∈*E*_{S}. For example, *ea*=*e*(0*a*)=(*e*0)*a*=0*a*=*a*. Conversely, it is straightforward to verify that such defined binary operation is associative. Thus *S* is a semigroup. Since *a*=*a*^{−1}, then *S* is an inverse semigroup. Finally, *E*=*Eω*, so *S*/*E*={*E*,{*a*}}.

## 7 The hypercore of a semigroup

In [8] Hall and Munn studied the hypercore of a semigroup. In this section we give some remarks on the hypercore of *E*-inversive *E*-semigroups and inverse semigroups.

Let *S* be a semigroup with \(E_{S} \not= \emptyset\). Denote by ℘_{S} the set of all subsemigroups *A* of *S* such that *A* has no cancellative congruences except the universal congruence. Note that {*e*}∈℘_{S} for every *e*∈*E*_{S}. Define the *hypercore*\(\operatorname{hyp}(S)\) of *S*, as follows: \(\operatorname{hyp}(S) = \langle \bigcup \{A: A \in \wp_{S} \} \rangle\) [8]. Furthermore, by the *core*\(\operatorname{core}(S)\) of an *E*-inversive semigroup *S* we shall mean \(\operatorname{ker}\sigma\).

In [8] the authors showed the following two results.

### Result 7.1

*Let*

*S*

*be an*

*E*-

*inversive semigroup*.

*Then*:

- (i)
\(\operatorname{hyp}(S) \in \wp_{S}\);

- (ii)
\(\operatorname{hyp}(S)\)

*is full and unitary*; - (iii)
\(\forall \rho \in \mathcal{GC}(S)~[\operatorname{hyp}(S) \subseteq \operatorname{ker}\rho]\).

### Result 7.2

*In any**E*-*inversive semigroup**S*, \(\operatorname{hyp}(S)\)*is the greatest**E*-*inversive subsemigroup of**S**with no non*-*universal group congruence*.

Let *U* be the least full unitary subsemigroup of an *E*-inversive semigroup *S*. Clearly, \(U \subseteq \operatorname{hyp}(S) \subseteq \operatorname{core}(S)\).

Finally, we have the following proposition.

### Proposition 7.3

*Let**S**be an**E*-*inversive**E*-*semigroup such that*\(1_{S} \notin \mathcal{GC}(S)\). *Then*\(U = \operatorname{hyp}(S) = \operatorname{core}(S) = E_{S}\omega\). *In particular*, *E*_{S}*ω**has no non*-*universal group congruence*.

*If in addition**S**is an inverse semigroup and**E*_{S}*ω**is finite*, *then**E*_{S}*ω**is an inverse semigroup with zero*. *In particular*, *every finite inverse semigroup**S* (*which is not a group*) *contains exactly one normal inverse subsemigroup with zero*.

### Proof

Let *S* be an *E*-inversive *E*-semigroup. Then \(\operatorname{core}(S) = E_{S}\omega\) (Theorem 4.5). Since *E*_{S}⊆*U* and *U* is closed, then *E*_{S}*ω*⊆*U*, so \(U = \operatorname{hyp}(S) = \operatorname{core}(S) = E_{S}\omega\). In the light of Result 7.2, *E*_{S}*ω* has no non-universal group congruence.

If *S* is an inverse semigroup, then obviously \(U = \operatorname{hyp}(S) =\operatorname{core}(S) = E_{S}\omega\) has no non-universal group congruence. Finally, if *E*_{S}*ω* is finite, then *E*_{S}*ω* has zero (Lemma 6.7). The rest of the proposition is now immediate. □

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