# Rectangular group congruences on a semigroup

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00233-012-9426-y

- Cite this article as:
- Gigoń, R.S. Semigroup Forum (2013) 87: 120. doi:10.1007/s00233-012-9426-y

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## Abstract

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup *S* is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when *S* is *E*-inversive. Finally, we prove that every rectangular group congruence on an *E*-inversive semigroup is uniquely determined by its kernel and trace.

### Keywords

Rectangular group congruenceGroup congruenceMatrix congruence## 1 Introduction and preliminaries

A groupoid *S* is called a *left* [*right*] *zero semigroup* if it satisfies the identity *xy*=*x* [*xy*=*y*]. Further, by a *rectangular band* we shall mean the direct product of a left zero and a right zero semigroup. Moreover, a semigroup *S* is said to be a *rectangular group* if it is isomorphic to the direct product *G*×*M* of a group *G* and a rectangular band *M*.

Let \(\mathcal{C}\) be a class of semigroups. We say that a congruence *ρ* on a semigroup *S* is a \(\mathcal{C}\)*-congruence* if \(S/\rho\in\mathcal{C}\). For example, if \(\mathcal{C}\) is the class of groups, then *ρ* is called a *group* congruence on *S* if *S*/*ρ* is a group. In way of an exception, a congruence *ρ* on a semigroup *S* is said to be a *matrix* congruence if *S*/*ρ* is a rectangular band. Note that every left [right] zero semigroup is a rectangular band, so every left [right] zero congruence on *S* is a matrix congruence. Also, clearly the least matrix congruence on any semigroup *S* exists. Denote it by *ψ*. Furthermore, every group congruence and every matrix congruence is a rectangular group congruence. Hence we say that a rectangular group congruence is *proper* if it is neither a group nor a matrix congruence. We first give necessary and sufficient conditions on a semigroup *S* in order that it will have a proper rectangular group congruence. Furthermore, we show that every rectangular group congruence on *S* is the intersection of a group congruence and a matrix congruence. In addition, if *S* is *E*-inversive, then this expression is unique. Moreover, we prove that each rectangular group congruence on an *E*-inversive semigroup is uniquely determined by its kernel and trace. Before we start our study, we recall some definitions.

Let *S* be a semigroup and *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} [7, 8]. Further, by *Reg*(*S*) we shall mean the set of *regular elements* of *S* (an element *a* of *S* is called *regular* if *a*∈*aSa*) and by *V*(*a*)={*x*∈*S*:*a*=*axa*,*x*=*xax*} the set of all *inverse elements* of *a*. It is well known that an element *a* of *S* is regular if and only if *V*(*a*)≠∅, so a semigroup *S* is regular if and only if *V*(*a*)≠∅ for every *a*∈*S*. Finally, a regular semigroup *S* is said to be *orthodox* if *E*_{S} forms a subsemigroup of *S*.

The following result seems to belong to folklore.

### Result 1.1

*The following conditions concerning a semigroup*

*S*

*are equivalent*:

- (i)
*S**is a rectangular band*; - (ii)
*S**is nonwhere commutative*,*i*.*e*., ∀*a*,*b*∈*S*[*ab*=*ba*⟹*a*=*b*]; - (iii)
∀

*a*,*b*∈*S*[*aba*=*a*]; - (iv)
∀

*a*,*b*,*c*∈*S*[*a*^{2}=*a*,*abc*=*ac*].

*A*of a semigroup

*S*is called

*left*[

*right*]

*dense*if the condition

*ab*∈

*A*implies that

*a*∈

*A*[

*b*∈

*A*] for all

*a*,

*b*∈

*S*. Further,

*A*is said to be

*quasi dense*if the following two conditions hold:

- (i)
∀

*a*∈*S*[*a*∈*A*⇔*a*^{2}∈*A*]; - (ii)
∀

*a*,*b*∈*S*[*ab*∈*A*⇔*aSb*⊆*A*].

*A*is a

*quasi ideal*of

*S*if

*AS*∩

*SA*⊆

*A*. For the connections between left [right] zero, matrix congruences on a semigroup

*S*and left dense right [right dense left] ideals, quasi dense subsemigroups of

*S*(respectively), the reader is referred to [9]. We note only some results of [9]. Firstly, denote by

*X*the set of all left dense right ideals of a semigroup

*S*and all right dense left ideals of a semigroup

*S*with the empty set included and

*S*excluded. Let 2

^{X}be a family of all subsets of

*X*and \(\mathcal{MC}(S)\) be the set of all matrix congruences on

*S*. Define the map \(\phi: 2^{X} \to\mathcal{MC}(S)\) by \(\mathcal{X} \phi= \rho _{\mathcal{X}}\) (\(\mathcal{X} \in2^{X}\)), where

### Result 1.2

(Theorem 5 [9])

*The map**ϕ**is antitone* (*i*.*e*., \(\mathcal{X} \subseteq \mathcal{Y} \Longrightarrow\rho_{\mathcal{Y}} \subseteq\rho_{\mathcal {X}}\)) *and maps* 2^{X}*onto*\(\mathcal{MC}(S)\).

### Result 1.3

(Corollary to Theorem 5 [9])

*The relation**ρ*_{X}*is the least matrix congruence on a semigroup**S*. *Moreover*, *we may replace* (*in the present result*) *the set**X**by the set**Y**of all quasi dense subsemigroups of**S*.

### Result 1.4

(A part of Proposition 4, Theorem 9 [9])

*The following conditions concerning a congruence*

*ρ*

*on a semigroup*

*S*

*are equivalent*:

- (i)
*ρ**is a matrix congruence on**S*; - (ii)
*every**ρ*-*class of**S**is a quasi dense subsemigroup of**S*; - (iii)
*every**ρ*-*class of**S**is a quasi ideal of**S*.

*Conversely*, *a subsemigroup**A**of**S**is quasi dense*, *when**A**is a matrix of some**ψ*-*classes of**S*. *Thus**A**is quasi dense if and only if**A**is a**ρ*-*class of some matrix congruence**ρ**on**S*.

### Result 1.5

(Theorem 14 [9])

*Let**S**be a matrix of semigroups**S*_{ıλ}, *where**ı*∈*I*, *λ*∈*Λ*, *such that every**S*_{ıλ}*has an identity element**e*_{ıλ}*and the set**M* (*say*) *of elements**e*_{ıλ} (*ı*∈*I*, *λ*∈*Λ*) *forms a subsemigroup of**S*. *Then**M**is a rectangular band*. *Further*, *S*_{ıλ}≅*S*_{ȷμ}*for all**ı*,*ȷ*∈*I*,*λ*,*μ*∈*Λ**and if we suppose that* 1∈*I*,*Λ*, *then**S**is isomorphic to the direct product**M*×*S*_{11}*of a rectangular band**M**and a semigroup**S*_{11}. *Moreover*, *the semigroups**S*_{ıλ}*are precisely the**ψ*-*classes of**S**and**S*_{ıλ}=*e*_{ıλ}*S*_{ıλ}*e*_{ıλ}=*e*_{ıλ}*Se*_{ıλ}*for all**ı*∈*I*,*λ*∈*Λ*.

Notice that if *S* is a rectangular group (that is, *S*≅*M*×*G*, where *M* is a rectangular band and *G* is a group), then we shall write rather *S*=*M*×*G* than *S*≅*M*×*G*. The following theorem is known but for example: Masat considered in [5, 6] a regular semigroup *S* such that *E*_{S} forms a rectangular band, and he did not know that *S* is a rectangular group, and so we include a simple proof for the completeness. Green’s relations on a semigroup *S* are denoted by \(\mathcal{L}^{S}\), \(\mathcal{R}^{S}\), \(\mathcal{H}^{S}\), \(\mathcal{D}^{S}\) and \(\mathcal{J}^{S}\). For undefined terms the reader is referred to the books [3, 4].

### Theorem 1.6

*The following conditions concerning a semigroup*

*S*

*are equivalent*:

- (i)
*S**is a rectangular group*; - (ii)
*S**is completely simple and orthodox*; - (iii)
*S**is completely regular and satisfies the identity*:*x*^{−1}*yy*^{−1}*x*=*x*^{−1}*x*; - (iv)
*S**is regular and**E*_{S}*forms a rectangular band*.

*Consequently*, *if**S**is a rectangular group*, *then*\(S \cong E_{S} \times \mathcal{H}_{e} = E_{S} \times eSe\)*for some* (*all*) *e*∈*E*_{S}.

### Proof

\(\mathrm{(ii)} \Longrightarrow\mathrm{(i)}\). If *S* is completely simple, then *S* is a matrix of groups \(\mathcal{H}_{e}\) (*e*∈*E*_{S}), since Lemma III.2.4 [4] implies that \(\mathcal{H}\) is a matrix congruence on *S*, so \(\mathcal{H} = \psi\). Clearly, every \(\mathcal {H}_{e}\) has an identity element *e*. Also, \((efe, e) \in\mathcal{H}\) for all *e*,*f*∈*E*_{S}, that is, *e*=*efe* for all *e*,*f*∈*E*_{S} (since *efe*∈*E*_{S}). Hence *E*_{S} is a rectangular band. Thus \(S \cong E_{S} \times \mathcal{H}_{e}\) for some (all) *e*∈*E*_{S} (by Result 1.5).

*S*=

*M*×

*G*, where

*M*is a rectangular band and

*G*is a group (with an identity 1). Define the mapping

^{−1}:

*S*→

*S*by (

*a*,

*g*)

^{−1}=(

*a*,

*g*

^{−1}) for all (

*a*,

*g*)∈

*S*. One can easily verify that (

*S*,⋅,

^{−1}) is completely regular, that is, (

*x*

^{−1})

^{−1}=

*x*,

*xx*

^{−1}=

*x*

^{−1}

*x*and

*xx*

^{−1}

*x*=

*x*for every

*x*∈

*S*. Further, suppose that

*x*=(

*a*,

*g*),

*y*=(

*b*,

*h*)∈

*S*. Then

*e*∈

*E*

_{S}. Since

*ee*

^{−1}=

*e*

^{−1}

*e*,(

*e*

^{−1})

^{−1}=

*e*, then

*e*

^{−1}

*ff*

^{−1}

*e*=

*e*

^{−1}

*e*, i.e.,

*efe*=

*e*for all

*e*,

*f*∈

*E*

_{S}. Thus

*E*

_{S}is a rectangular band. Consequently, the condition (iv) holds.

\(\mathrm{(iv)} \Longrightarrow\mathrm{(ii)}\). Clearly, each idempotent of *S* is primitive and \(\mathcal{D}^{E_{S}} = E_{S} \times E_{S}\). Since *S* is regular, then every element of *S* is \(\mathcal{D}\)-related with some of its idempotent. It follows that \(\mathcal{D}^{S} = \mathcal{J}^{S} = S \times S\). Thus *S* is completely simple and orthodox. □

By the *trace*\(\operatorname{tr}\rho\) of a relation *ρ* on a semigroup *S* we shall mean the restriction of *ρ* to the set *E*_{S}.

The following result will be useful in the proof of Theorems 2.2(iii), 2.5, 2.6.

### Result 1.7

(Corollary 2.7 [1])

*If**ρ**is a matrix congruence on an**E*-*inversive semigroup**S*, *then every**ρ*-*class of**S**is**E*-*inversive*. *Also*, *every matrix congruence on an**E*-*inversive semigroup is uniquely determined by its trace*.

Further, some preliminaries about group congruences on a semigroup *S* are needed. A subset *A* of *S* is called (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 said to be *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*.

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*}.

The following two results follow directly from the definition of the group.

### Lemma 1.8

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

### Lemma 1.9

*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 1.10

[2]

*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}.

If *B* is a seminormal subsemigroup of *S*, then we denote the above four relations by *ρ*_{B}.

The following theorem says that there exists an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup *S* and the set of all group congruences on *S*.

### Theorem 1.11

[2]

*Let**B**be a seminormal 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*.

Finally, the following remark will be useful.

### Remark 1

Denote by *σ* the least group congruence on a semigroup (if it exists). One can easily seen that if *S* is an *E*-inversive semigroup (and so *E*_{S} is dense), then there exists the least normal subsemigroup of *S*. In the light of the above theorem, every *E*-inversive semigroup possesses a least group congruence.

## 2 Rectangular group congruences

The following theorem gives necessary and sufficient conditions on a semigroup *S* in order that it has a proper rectangular group congruence. (Notice that a normal subsemigroup *N* of *S* is called *proper* if *N*≠*S*.)

### Theorem 2.1

*Let*

*S*

*be a semigroup*.

*The following conditions are equivalent*:

- (i)
*there exists a proper rectangular group congruence on**S*; - (ii)
*S**is a disjoint union of two or more quasi dense subsemigroups of**S**and contains a proper normal subsemigroup of**S*; - (iii)
*there exists a non*-*universal group and a non*-*universal matrix congruence on**S*.

### Proof

\(\mathrm{(i)} \Longrightarrow\mathrm{(ii)}\). Let *ρ* be a proper rectangular group congruence on *S*, say *S*/*ρ* is equal *M*×*G*, where *M* is a rectangular band, *G* is a group (with identity 1). Note that *M*≅*E*_{M×G}={(*m*,1):*m*∈*M*}. Further, for all *m*∈*M*, define *Q*_{m} to be the preimage of {*m*}×*G* by the canonical epimorphism *ρ*^{♮} from *S* onto *S*/*ρ*. It follows easily from Result 1.1(iv) that {*m*}×*G* is a quasi dense subsemigroup of *M*×*G*. Thus the preimage of {*m*}×*G* by *ρ*^{♮} is also such a subsemigroup of *S* (by *M*≅*E*_{M×G}). Since *ρ* is not a group congruence, then |*M*|>1, and so *S* has a proper matrix congruence (Result 1.3). Hence *S* is a disjoint union of two or more quasi dense subsemigroups of *S*, see Result 1.4 (notice that *S*=⋃{*Q*_{m}:*m*∈*M*}, where the union is disjoint, and this decomposition of *S* induced, by the first part of Result 1.4, a matrix congruence on *S*). Let \(N = \operatorname{ker}\rho\). Then *a*∈*N* if and only if *aρ*∈*E*_{M×G}, that is, if and only if *aρ*∈*M*×{1}. Clearly, *N* is a full subsemigroup of *S*. Also, *M*×{1} is reflexive in *M*×*G*, so *N* is reflexive in *S*. Furthermore, *N* is dense, since *S*/*ρ* is *E*-inversive. Finally, *N* is closed in *S*, since *M*×{1} is closed in *M*×*G*. Consequently, *N* is a normal subsemigroup of *S* and since *S*/*ρ* is not a rectangular band, then *N* is proper.

\(\mathrm{(ii)} \Longrightarrow\mathrm{(iii)}\). This follows from Result 1.4 and Theorem 1.11.

\(\mathrm{(iii)} \Longrightarrow\mathrm{(i)}\). Let *ρ*_{1} be a non-universal matrix congruence on *S* and *ρ*_{2} be a non-universal group congruence on *S*. We show that *ρ*=*ρ*_{1}∩*ρ*_{2} is a proper rectangular group congruence on a semigroup *S*. Indeed, let *a*∈*S*. Since *S*/*ρ*_{2} is regular, then (*axa*,*a*)∈*ρ*_{2} for some *x*∈*S*, so (*axa*,*a*)∈*ρ*. Therefore *S*/*ρ* is regular. Clearly, *xρ*_{2}, where \(x \in \operatorname{ker}\rho_{2}\) is an identity of the group *S*/*ρ*_{2} and so (*xyx*,*x*)∈*ρ*_{2} for all \(x, y \in \operatorname{ker}\rho_{2}\). Hence (*xyx*,*x*)∈*ρ* for all \(x, y \in \operatorname{ker}\rho _{2}\). On the other hand, if *xρ*∈*E*_{S/ρ}, then \(x \in \operatorname{ker}\rho_{2}\). It follows that *E*_{S/ρ} forms a rectangular band, therefore, *S*/*ρ* is a rectangular group (Theorem 1.6(iv)). Finally, suppose by way of contradiction that *ρ* is a matrix congruence on *S*, that is, (*aba*,*a*)∈*ρ* for all *a*,*b*∈*S*. Then (*aba*,*a*)∈*ρ*_{2} for all *a*,*b*∈*S*. Hence *S*/*ρ*_{2} must be a trivial group. Thus *ρ*_{2}=*S*×*S*, a contradiction from the assumption that *ρ*_{2} is a non-universal congruence on *S*. Similarly, if *ρ* is a group congruence, then *ρ*_{1} is a group congruence (since *ρ*⊆*ρ*_{1}), so *S*/*ρ*_{1} must be trivial. Hence *ρ*_{1} is the universal relation, but this is impossible. Consequently, *ρ* is a proper rectangular group congruence on *S*. □

We have just seen that the intersection of a group congruence on a semigroup *S* and a matrix congruence on *S* is a rectangular group congruence on *S*. Conversely, the part (i) of the following theorem (together with Theorem 2.1) implies that any rectangular group congruence on *S* can be expressed in this way.

### Theorem 2.2

*Let*

*ρ*

*be a rectangular group congruence on a semigroup*

*S*(

*and so*

*S*/

*ρ*=

*M*×

*G*,

*where*

*M*

*is a rectangular band*,

*G is a group*).

*Also*,

*let*

*Q*

_{m}(

*m*∈

*M*)

*be the preimage of*{

*m*}×

*G*

*by the canonical epimorphism*

*ρ*

^{♮}

*from*

*S*

*onto*

*S*/

*ρ*,

*and put*

*N*={

*s*∈

*S*:

*sρ*∈

*E*

_{M×G}}.

*Moreover*,

*denote by*

*υ*

*the matrix congruence on S*,

*induced by the partition*{

*Q*

_{m}:

*m*∈

*M*}

*of*

*S*(

*see the proof of “*\(\mathrm{(i)} \Longrightarrow\mathrm{(ii)}\)

*” in Theorem*2.1).

*Then*:

- (i)
*ρ*=*υ*∩*ρ*_{N}; - (ii)
*S*/*ρ*≅*S*/*υ*×*S*/*ρ*_{N}.

*If in addition*

*S*

*is*

*E*-

*inversive*,

*then*:

- (iii)
∀

*m*∈*M*[*N*∩*Q*_{m}◁*Q*_{m}]; - (iv)
∀

*m*∈*M*[\(S/\rho_{N} \cong Q_{m}/\rho_{(N \, \cap\: Q_{m})}\)].

### Proof

Firstly, notice that *N* is the preimage of *M*×{1_{G}} by *ρ*^{♮}. Secondly, every *Q*_{m} is a quasi dense subsemigroup of *S* (see Result 1.4). Also, if *S* is *E*-inversive, then each *Q*_{m} is an *E*-inversive subsemigroup of *S* (Result 1.7).

(i). Let (*a*,*b*)∈*ρ* and *aρ*=(*m*,*g*), where (*m*,*g*)∈*M*×*G*. Take *x*=(*m*,*g*^{−1}), where *g*^{−1} is a group inverse of *g* in *G*. Then clearly *xa*,*xb*∈*N* and so (*a*,*b*)∈*ρ*_{N} (see Remark 1). Also, *aρ*=(*m*,*g*)=*bρ*∈{*m*}×*G*. Hence *a*,*b*∈*Q*_{m}, so (*a*,*b*)∈*υ*. Thus (*a*,*b*)∈*υ*∩*ρ*_{N}. Consequently, *ρ*⊂*υ*∩*ρ*_{N}. Conversely, let (*a*,*b*)∈*υ*∩*ρ*_{N}. Then *aρ*,*bρ*∈{*m*}×*G* (*aρ*=(*m*,*g*_{1}), *bρ*=(*m*,*g*_{2})), *xa*,*xb*∈*N* for some *m*∈*M* and *x*∈*Q*_{n}, where *n*∈*M* (say *xρ*=(*n*,*g*)), and so (*xa*)*ρ*=(*nm*,*gg*_{1}). On the other hand, (*xa*)*ρ*∈*M*×{1_{G}}. Hence *g*_{1}=*g*^{−1}. We may equally well show that *g*_{2}=*g*^{−1}. Thus (*a*,*b*)∈*ρ*. Consequently, *ρ*=*υ*∩*ρ*_{N}, as exactly required.

*ρ*=

*υ*∩

*ρ*

_{N}by (i). Define the mapping

*ϕ*:

*S*/

*ρ*→

*S*/

*υ*×

*S*/

*ρ*

_{N}by (

*aρ*)

*ϕ*=(

*aυ*,

*aρ*

_{N}) (

*a*∈

*S*). Clearly,

*ϕ*is a monomorphism. We show that

*ϕ*is surjective. Let (

*aυ*,

*bρ*

_{N})∈

*S*/

*υ*×

*S*/

*ρ*

_{N}. Then

*a*∈

*Q*

_{m}, where

*m*∈

*M*. Take any element

*n*∈

*N*∩

*Q*

_{m}. Then

(iii). Let *m*∈*M*. Put *N*_{m}=*N*∩*Q*_{m}. Evidently, *N*_{m} is a full, reflexive and closed subsemigroup of *Q*_{m} (even if *S* is an arbitrary semigroup). By Result 1.7, *N*_{m} is dense in *Q*_{m}. Thus *N*_{m}◁*Q*_{m}.

(iv). Let *m*∈*M*. Define the map \(\phi: Q_{m}/\rho_{N_{m}}\to S/\rho _{N}\) by \((a\rho_{N_{m}})\phi= a\rho_{N}\) (*a*∈*Q*_{m}). Clearly, *ϕ* is a well-defined homomorphism. Furthermore, if *a*∈*S* and *n*∈*N*_{m}⊆*N*, then *nan*∈*Q*_{m} and \(((nan)\rho_{N_{m}})\phi= (nan)\rho_{N} = a\rho_{N}\). Thus *ϕ* is surjective. Finally, we show that *ϕ* is injective. Let \(a, b \in Q_{m}, (a\rho_{N_{m}})\phi= (b\rho _{N_{m}})\phi\). Then (*a*,*b*)∈*ρ*_{N}, so *ax*, *bx*∈*N* for some *x*∈*S*. Hence for every *n*∈*N*_{m}⊆*N*, *anx*, *bnx*∈*N*. Thus *n*(*anx*)*n*∈*N*∩(*Q*_{m}*NQ*_{m})⊆*N*∩*Q*_{m}=*N*_{m} and similarly: *n*(*bnx*)*n*∈*N*_{m}. Since *na*, *nb*, *nxn*∈*Q*_{m}, then \((na, nb) \in\rho_{N_{m}}\), and so \((a, b) \in\rho_{N_{m}}\), because \(n \in N_{m} = \operatorname{ker}\rho_{N_{m}}\). □

### Corollary 2.3

*If the least group congruence exists on a semigroup**S*, *then the relation**ψ*∩*σ**is the least rectangular group congruence on**S*. *In particular*, *in any**E*-*inversive semigroup*, *ψ*∩*σ**is the least rectangular group congruence*.

### Remark 2

If *S* is not *E*-inversive, then the least rectangular group congruence on a semigroup *S* may not exist. Indeed, consider the additive semigroup of non-negative integers ℕ. 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). Further, since ℕ has identity, then the least matrix congruence on ℕ is the universal relation, so any rectangular group congruence on ℕ is a group congruence (Theorem 2.2(i)). Consequently, ℕ has no least rectangular group congruence.

Let \(\mathcal{C}\) be a class of semigroups which is closed under homomorphic images. Note that if the least \(\mathcal {C}\)-congruence \(\rho_{\mathcal{C}}\) on a semigroup *S* exists, then the interval [\(\rho_{\mathcal{C}}\), *S*×*S*] consists of all \(\mathcal{C}\)-congruences on *S* and it is a complete sublattice of the complete lattice \(\mathcal{C}(S)\) of congruences on *S*. Evidently, the class of all groups [rectangular bands] is closed under homomorphic images. Using Theorem 1.6(iv) one can prove without difficulty that the class of all rectangular groups has this property. Denote by *θ* the least rectangular group congruence on an *E*-inversive semigroup. In particular, the intervals [*ψ*, *S*×*S*], [*σ*, *S*×*S*], [*θ*, *S*×*S*] consist of all matrix, group, rectangular group congruences on an *E*-inversive semigroup *S*, respectively, and they are complete sublattices of \(\mathcal{C}(S)\). Denote them by \(\mathcal{MC}(S)\), \(\mathcal{GC}(S)\), \(\mathcal{RGC}(S)\), respectively. Clearly, the direct product \(\mathcal{MC}(S) \times\mathcal{GC}(S)\) is a complete sublattice of \(\mathcal{C}(S) \times\mathcal{C}(S)\) (see [3, p. 37]).

For terminology and elementary facts about lattices the reader is referred to the book [10, Sect. I.2]. The following simple result will be useful (see Lemma I.2.8 and Exercise I.2.15(iii) in [10]).

### Result 2.4

*If**ϕ**is an order isomorphism of a lattice**L**onto a lattice**M*, *then**ϕ**is a lattice isomorphism*. *Moreover*, *every lattice isomorphism of complete lattices is a complete lattice isomorphism*.

We show that each rectangular group congruence on an *E*-inversive semigroup can be expressed as the unique intersection of a group and a matrix congruence.

### Theorem 2.5

*Every rectangular group congruence on an**E*-*inversive semigroup**S**is of the form**υ*∩*ρ*_{N}, *where**υ**is a matrix congruence on**S*, *N*◁*S*, *and this expression is unique*.

*Moreover*,

*there exists an inclusion*-

*preserving bijection*

*ϕ*

*between the complete lattice*\(\mathcal{MC}(S) \times\mathcal{GC}(S)\)

*and the complete lattice*\(\mathcal{RGC}(S)\).

*In fact*,

*ϕ*

*is defined by*:

*for every*\((\upsilon, \rho_{N}) \in\mathcal{MC}(S) \times\mathcal{GC}(S)\).

*Furthermore*,

*ϕ*

^{ −1}

*is an inclusion*-

*preserving bijection*(

*by the proof of Theorem*2.2(i)),

*so*

*ϕ*

*is an order isomorphism of the complete lattice*\(\mathcal{MC}(S) \times\mathcal{GC}(S)\)

*onto the complete lattice*\(\mathcal{RGC}(S)\).

*Consequently*,

*ϕ*

*is a complete lattice isomorphism between the lattices*\(\mathcal{MC}(S) \times\mathcal{GC}(S)\)

*and*\(\mathcal{RGC}(S)\),

*respectively*.

### Proof

Let *ρ* be a rectangular group congruence on *S*. Then *ρ* is the intersection of some matrix and some group congruence on *S* (Theorem 2.2(i)). Next, suppose that \(\rho= \upsilon_{1} \cap \rho_{N_{1}} = \upsilon_{2} \cap \rho_{N_{2}}\), where *υ*_{i} is a matrix congruence on *S*, *N*_{i}◁*S* (*i*=1,2). Let (*a*,*b*)∈*υ*_{1}. Since *υ*_{1}∩*υ*_{2} is a matrix congruence on *S*, then there are idempotents *e*, *f* of *S* such that \((a, e) \in\upsilon_{1} \cap\upsilon_{2}, (e, f) \in\rho_{N_{1}}, (f, b) \in \upsilon_{1} \cap\upsilon_{2}\) (Result 1.7), so \((e, f) \in\upsilon_{1} \cap\rho_{N_{1}} = \upsilon_{2} \cap\rho_{N_{2}} \subseteq\upsilon_{2}\). Hence (*a*,*b*)∈*υ*_{2}, i.e., *υ*_{1}⊂*υ*_{2}. We may equally well show the opposite inclusion. Put *υ*_{1}=*υ*_{2}=*υ*, so that \(\rho= \upsilon\cap \rho_{N_{1}} = \upsilon\cap \rho_{N_{2}}\). If \((a, b) \in \rho_{N_{1}}\), then \((aab, abb) \in\upsilon\cap\rho_{N_{1}} \subset\rho _{N_{2}}\), so \((a, b) \in\rho_{N_{2}}\) (by cancellation). Hence \(\rho _{N_{1}} \subset\rho_{N_{2}}\). By symmetry, \(\rho_{N_{2}} \subset\rho _{N_{1}}\). Thus \(\rho_{N_{1}} = \rho_{N_{2}}\), as exactly required.

The second part of the theorem follows directly from the above considerations and Result 2.4. □

Finally, from Result 1.7 we obtain the following theorem.

### Theorem 2.6

*Every rectangular group congruence on an**E*-*inversive semigroup**S**is uniquely determined by its kernel and trace*.

### Proof

*ρ*

_{1}, \(\rho_{2} \in\mathcal{GC}(S), \upsilon_{1}\), \(\upsilon_{2} \in\mathcal{MC}(S)\) be such that \(\operatorname{ker}(\upsilon_{1} \cap\rho _{1}) \subset \operatorname{ker}(\upsilon_{2} \cap\rho_{2})\) and \(\operatorname{tr}(\upsilon_{1} \cap\rho _{1}) \subset \operatorname{tr}(\upsilon_{2} \cap\rho_{2})\). Then

*ρ*

_{1}⊂

*ρ*

_{2}. Similarly, we obtain that \(\operatorname{tr}\upsilon_{1} \subset \operatorname{tr}\upsilon_{2}\). Hence

*υ*

_{1}⊂

*υ*

_{2}(this follows from the proof of Result 1.7, see [1]). Thus

*υ*

_{1}∩

*ρ*

_{1}⊂

*υ*

_{2}∩

*ρ*

_{2}. This implies the thesis of the theorem. □

### Open Access

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