Effective subdirect decompositions of regular semigroups

Semigroup Forum (2008) 77: 500–519DOI 10.1007/s00233-008-9114-0

R E S A R C H A RT I C L E

Effective subdirect decompositions of regularsemigroups

Miroslav Ciric · Žarko Popovic ·Stojan Bogdanovic

Received: 1 March 2007 / Accepted: 23 September 2008 / Published online: 6 November 2008© Springer Science+Business Media, LLC 2008

Abstract By an “effective” subdirect decomposition we mean a subdirect decom-position for which an effective construction of the corresponding family of factorcongruences is given. In this paper we construct several commuting pairs of factorcongruences which decompose any regular semigroup into a pullback product. Forregular and completely regular semigroups whose idempotents form subsemigroupsbelonging to certain varieties of bands, we give precise structural descriptions of thecomponents in these pullback products.

Keywords Subdirect product · Pullback product · Regular semigroup · Semilatticedecomposition · Orthodox semigroup · Orthogroup · Orthocryptogroup

1 Introduction

There are several ways of approaching subdirect decompositions of semigroups.In most cases they have been obtained from various semigroup theoretical con-structions, such as, for example, Hall-Yamada’s construction of orthodox semigroups

Communicated by Francis J. Pastijn.

Research supported by Ministry of Science, Republic of Serbia, Grant No. 144011.

M. Ciric (�)Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, P.O. Box 224,18000 Niš, Serbiae-mail: [email protected]

Ž. Popovic · S. BogdanovicFaculty of Economics, University of Niš, Trg Kralja Aleksandra 11, 18000 Niš, Serbia

Ž. Popovice-mail: [email protected]

S. Bogdanovice-mail: [email protected]

mailto:[email protected]



Effective subdirect decompositions of regular semigroups 501

(see [11]). In particular, Yamada’s construction of orthogroups has been widely usedin Petrich and Reilly’s book [16], in studying subdirect decompositions of regularorthogroups.

Another way is based on the famous Birkhoff representation theorem. Formulatedin terms of semigroups, it asserts that every semigroup can be represented as a subdi-rect product of subdirectly irreducible semigroups, and it can often reduce studyingthe structure of semigroups from a given class to studying subdirectly irreduciblemembers of this class.

In this paper we follow a third way of approaching subdirect decompositionswhich is based on another Birkhoff theorem proved in [1]. This theorem, stated interms of semigroups, says that a semigroup S is a subdirect product of a family ofsemigroups {Si}i∈I if and only if there exists a family {�i}i∈I of congruence relationson S such that

⋂

i∈I

�i = �S and S/�i∼= Si, for each i ∈ I. (1)

We call such a family {�i}i∈I a family of factor congruences, and by an “effective”subdirect decomposition we mean a subdirect decomposition for which an effectiveconstruction of the corresponding family of factor congruences is given.

The subdirect product is a quite general construction, and there are reasons toconsider various special cases. If {Si}i∈I is a family of semigroups having a commonhomomorphic image H , i.e., for each i ∈ I there exists a homomorphism ϕi of Si

onto H , then

S ={(ai)i∈I ∈

∏

i∈I

Si

∣∣∣aiϕi = ajϕj , for all i, j ∈ I

}

is a special subdirect product of a family of semigroups {Si}i∈I , called a pullbackproduct of the family of semigroups {Si}i∈I , with respect to the semigroup H andthe family of homomorphisms {ϕi}i∈I . Pullback products first appeared in universalalgebra, in a paper by Fuchs [8]. In semigroup theory, where they are known as spinedproducts, their intensive study was initiated by Kimura [13], who proved that a bandis regular if and only if it is pullback product of a left regular and a right regular band.Popularity of pullback products in semigroup theory is mainly a result of the fact thatpullback products are much easier to construct and preserve various properties oftheir factors that other subdirect products do not necessarily preserve. For example,they preserve complete regularity.

The first theorem which characterizes pullback products in terms of factor con-gruences was given by Fleischer [7]. Stated in terms of semigroups, it asserts thata semigroup S is a pullback product of semigroups P and Q with respect to theircommon homomorphic image H if and only if there exists a pair �, �′ of factor con-gruences on S such that � and �′ commute, S/� ∼= P , S/�′ ∼= Q, and S/� ◦ �′ ∼= H .Then we call � and �′ a commuting pair of factor congruences.

For pullback products with more than two factors, a similar theorem was provedby Wenzel [19], who showed that factor congruences which determine pullback prod-ucts must satisfy some conditions which can be viewed as a generalization of thoseappearing in the Chinese remainder theorem. Namely, if {�i}i∈I is a family of congru-ences on a semigroup S and � = ∨

i∈I �i , then {�i}i∈I is called absolutely permutable

502 M. Ciric et al.

if for any family {ai}i∈I ⊆ S satisfying (ai, aj ) ∈ �, for all i, j ∈ I , there exists a ∈ S

such that (ai, a) ∈ �i , for every i ∈ I . According to the mentioned Wenzel’s theorem,a semigroup S is a pullback product of a family of semigroups {Si}i∈I with respectto a semigroup H if and only if there exists an absolutely permutable family {�i}i∈I

of congruences on S satisfying (1), and S/� ∼= H , where � = ∨i∈I �i .

The main aim of this paper is to construct several commuting pairs of factor con-gruences which decompose any regular semigroup into a pullback product, and forregular and completely regular semigroups whose idempotents form subsemigroupsbelonging to certain varieties of bands, we give precise structural descriptions of thecomponents in these pullback products.

In Sect. 3 we study congruence relations �l , �r and �d , as well as congruenceopenings �0, �0, �0 and �0 of Green’s relations. Congruences �l and �r are knownas kernels of the right and left regular representation of a semigroup, respectively,and �d was already used in [14]. We prove that �l and �r , �l and �0, �r and�0, and �d and �0 form commuting pairs of factor congruences on any regularsemigroup. This gives four types of pullback product decompositions of a regularsemigroup, including the decomposition into a pullback product of a left reductiveand a right reductive regular semigroup with respect to a reductive regular semigroup,discovered by Kopamu [14]. More precise characterizations of the components inthese pullback products are given for orthodox semigroups whose idempotents forma left (right) quasi-normal band or a normal band. In the later case the well-knownresult by Yamada [20] is obtained.

However, for orthodox semigroups whose idempotents form bands more generalthan left or right quasi-normal bands, congruences �l , �r and �d do not give enoughinformation on structure of related factors. For that reason, in Sect. 4 we constructthree new types of congruences, denoted by �l , �r and �d , which will give betterresults. We prove that �l and �r , �l and �0, �r and �0, and �d and �0 form com-muting pairs of factor congruences on any regular semigroup, and this gives four newtypes of pullback product decompositions of a regular semigroup.

In Sects. 5 and 6 we show that congruences �l , �r and �d are exceptionally ef-ficient in the study of orthogroups, which results from the facts that they respect thegreatest semilattice decomposition of a semigroup, and an orthogroup and its semi-group of idempotents have the same greatest semilattice homomorphic image. The-orem 5.4 gives a pullback product representation of orthogroups, Theorem 6.2 givesa similar representation of left orthocryptogroups, whereas Theorem 6.4 gives fourpullback product representations of orthocryptogroups. The obtained results gener-alize results on pullback product representations of regular orthogroups presented inPetrich and Reilly’s book [16].

Let us note that various subdirect and pullback decompositions of orthodox semi-groups have been also studied in [10, 18, 21, 22] and [23].

2 Preliminaries

Throughout this paper, �X denotes the diagonal relation on a set X, composition oftwo binary relations α and β on X is denoted by α ◦ β , and we say that α and β

commute if α ◦ β = β ◦ α.


Let S be a semigroup. The set of all idempotents of S, if it is non-empty, is denotedby E(S). For a non-empty subset X of S, L(X), R(X) and J (X) denote respectivelythe left, right and two-sided ideal of S generated by X. If L(X) = R(X), then X iscalled a duo subset of S. Clearly, in this case L(X) = R(X) = J (X). It can be easilyverified that every ideal is a duo subset. Another natural example of a duo subset isthe set of all idempotents of a regular semigroup.

For Green’s relations on a semigroup S the usual notation �, �, �, � and � willbe used. Congruence openings of �, � and � are denoted by �0, �0 and �0, i.e.,

(a, b) ∈ �0 ⇔ (∀x ∈ S1) (xa, xb) ∈ �,

(a, b) ∈ �0 ⇔ (∀y ∈ S1) (ay, by) ∈ �,

(a, b) ∈ �0 ⇔ (∀x, y ∈ S1) (xay, xby) ∈ �.

By the well-known fact that every right congruence contained in � commutes withevery left congruence contained in � (see Lemma I.7.2 of [16]), we have that �0 ◦�0 = �0 ◦ �0, and we set �0 = �0 ◦ �0. It is very important to remember that �0

does not denote the congruence opening of �, which is in general larger than �0. Fora type of relations defined on any semigroup, if we sometimes need to emphasizethat we deal with a relation of type on a particular semigroup S, we write S orS instead of . For example, we write �S , �0

S , etc. The natural partial order ≤ on aband E is defined by e ≤ f ⇔ ef = f e = e, for any e, f ∈ E.

A semigroup S is regular if for any a ∈ S there exists x ∈ S such that a = axa.In this case, for any a ∈ S there exists x ∈ S satisfying a = axa and x = xax, whichis called an inverse of a. Such an inverse is not necessarily unique, and the set of allinverses of a ∈ S is denoted by V (a). If every a ∈ S has a unique inverse, then S iscalled an inverse semigroup.

A semigroup S is completely regular if every a ∈ S has an inverse x which com-mutes with a, i.e., ax = xa. Such an inverse is unique for every a ∈ S and is denotedby a−1. It is well-known that S is completely regular if and only if it is a union ofgroups, i.e., if each Green’s �-class is a subgroup of S, and for each a ∈ S, a−1 is aninverse of a in a subgroup of S, so it is called a group inverse of a. Also, for a ∈ S, bya0 we denote the identity of the subgroup of S containing a, i.e., a0 = aa−1 = a−1a.

In our study the lattice �() of varieties of bands will play an outstanding role, sowe recall some notions, notation and results concerning it. We will use the character-ization of the lattice �() given by J.A. Gerhard and M. Petrich in [9]. They definedinductively three systems of words as follows:

G2 = x2x1, H2 = x2, I2 = x2x1x2,

Gn = xnGn−1, Hn = xnGn−1xnHn−1, In = xnGn−1xnIn−1

(for n ≥ 3), and they showed that the lattice �() can be represented by the diagramgiven in Fig. 1. In this figure we also give notation of varieties in the lowest partof �() and names of bands belonging to them. Let us note that [u = v] denotesthe variety of bands determined by a semigroup identity u = v, i.e., this is a briefnotation for the semigroup variety [x2 = x, u = v]. For a word w, w denotes thedual of w, that is, the word obtained by reversing the order of the letters in w.

504 M. Ciric et al.

Fig. 1 The lattice �() of varieties of bands

We distinguish five columns of �(), each consisting of the following varieties:

First column: [Gn = In], n ≥ 2, and [Gn = Hn], n ≥ 3;Second column: [Gn = In]∧[Gn+1 = Hn+1], n ≥ 2, [Gn = Hn]∧[Gn = In], n ≥ 3,

and ��;Third column: [Gn = In] ∧ [Gn = In], n ≥ 2, [Gn = Hn] ∧ [Gn = Hn], n ≥ 3, �e

and �;Fourth column: [Gn+1 = Hn+1]∧ [Gn = In], n ≥ 2, [Gn = In]∧ [Gn = Hn], n ≥ 3,

and ��;Fifth column: [Gn = In], n ≥ 2, and [Gn = Hn], n ≥ 3.

We also distinguish left diagonals and right diagonals of �(), by which we meanthe following closed intervals of the lattice �():

Left diagonals: [�,��], [��,�e], [�, [G2 = I2]], [� , [G3 = H3]],[[Gn = In], [Gn+1 = In+1], n ≥ 2, and [[Gn = Hn], [Gn+1 = Hn+1], n ≥ 3,

Right diagonals: [�,��], [��,�e], [�, [G2 = I 2]], [� , [G3 = H 3]],[[Gn = In], [Gn+1 = In+1], n ≥ 2, and [[Gn = Hn], [Gn+1 = Hn+1], n ≥ 3.


A band belonging to a variety � ∈ �() is called a �-band.For any variety � ∈ �(), varieties ��0,��0 ,��0 ∈ �() are defined by:

��0 = H({S/�0 |S ∈ �}), ��0 = H({S/�0 |S ∈ �}),��0 = H({S/�0 |S ∈ �}),

where H(�) denotes the class of all homomorphic images of semigroups from aclass �.

It was proved in [4] that � �→ ��0 , � �→ ��0 and � �→ ��0 are opening operatorson the lattice �(), and the open elements with respect to the first two operators arerespectively startpoints of left and right diagonals of �().

We restate two results from [4] that will be used in the further work.

Theorem 2.1 [4] Let S be a band and � a variety of bands. Then the followingconditions are equivalent:

(i) S is a �-band;(ii) S is a subdirect product of a ��0 -band and a ��0 -band;

(iii) S is a pullback product of a ��0 -band and a ��0 -band with respect to a��0 -band.

Theorem 2.2 [4] Let � be a variety of a bands. Then � is the intersection point ofthe left diagonal starting in ��0 and the right diagonal starting in ��0 , and

� = ��0 ∨ ��0 and ��0 ∧ ��0 = ��0 .

This assertion is graphically demonstrated by Fig. 2, in (a) for a variety � in thethird column, in (b) for � and � in the second and fourth column, and in (c) for �and � in the first and fifth column.

Fig. 2 Operators � �→ ��0 , � �→ ��0 and � �→ ��0 on �()

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We complete this section stating two lemmas that will be used in the further work.The first of them is another formulation of the well-known Lallement’s lemma.

Lemma 2.1 Let S be a regular semigroup and let φ be a homomorphism of S onto asemigroup T . Then φ maps E(S) onto E(T ).

Lemma 2.2 Let S be an orthodox semigroup and E = E(S). Then

�E = �S ∩ (E × E), �E = �S ∩ (E × E),

�0E = �0

S ∩ (E × E) and �0E = �0

S ∩ (E × E).

Consequently,

E(S/�0S) ∼= E/�0

E and E(S/�0S) ∼= E/�0

E.

For undefined notions and notation we refer to the books by J.M. Howie [12] andM. Petrich and N.R. Reilly [16].

3 Subdirect decompositions of regular semigroups

Let relations �l , �r and �d on a semigroup S be defined by:

(a, b) ∈ �l ⇔ (∀x ∈ S) xa = xb,

(a, b) ∈ �r ⇔ (∀y ∈ S) ay = by,

(a, b) ∈ �d ⇔ (∀x, y ∈ S) xay = xby.

It is well-known that all of them are congruence relations on S, and both �l and�r are contained in �d . These congruence relations were introduced by A.H. Clif-ford and G.B. Preston [6] and B.M. Schein [17], and they were intensively used byS.J.L. Kopamu [14] in a study of subdirect decompositions and band compositions ofregular semigroups.

Let X be a non-empty subset of a semigroup S. Relations �l,X , �r,X and �d,X onS, introduced in [5], are defined by:

(a, b) ∈ �l,X ⇔ (∀x ∈ X) xa = xb,

(a, b) ∈ �r,X ⇔ (∀y ∈ X) ay = by,

(a, b) ∈ �d,X ⇔ (∀x, y ∈ X) xay = xby.

All of them are equivalence relations on S, �l,X is a right congruence, �r,X is a leftcongruence, and both �l,X and �r,X are contained in �d,X . Also, the following istrue:

Lemma 3.1 [5] Let X be a duo subset of a semigroup S. Then �l,X , �r,X and �d,X

are congruence relations on S.


Furthermore, if K = J (X), then

�l,X = �l,K, �r,X = �r,K and �d,X = �d,K .

Lemma 3.2 Let S be a regular semigroup and E = E(S). Then

�l = �l,E, �r = �r,E and �d = �d,E.

Proof This is an immediate consequence of Lemma 3.1 and the fact that regularityof S yields L(E) = R(E) = J (E) = S. �

The following theorem describes the main properties of congruences �l , �r

and �d .

Theorem 3.1 Let S be a regular semigroup. Then

(a) �l ◦ �r = �r ◦ �l = �d and �l ∩ �r = �S ;(b) �l ⊆ �0, �r ⊆ �0 and �d ⊆ �0;(c) �l ∩ �0 = �r ∩ �0 = �d ∩ �0 = �S ;(d) �l ◦ �0 = �0 ◦ �l , �r ◦ �0 = �0 ◦ �r and �d ◦ �0 = �0 ◦ �d .

Proof (a) Since �l and �r are contained in �d , we have that

�l ◦ �r ⊆ �d ◦ �d = �d and �r ◦ �l ⊆ �d ◦ �d = �d .

To prove the opposite inclusions consider a, b ∈ S such that (a, b) ∈ �d . Since S

is regular, we can take some a′ ∈ V (a) and b′ ∈ V (b). If c = ba′a, then for everyx, y ∈ S we have that

xa = xaa′a = xba′a = xc,

cy = ba′ay = bb′ba′ay = bb′aa′ay = bb′ay = bb′by = by,

which yields (a, c) ∈ �l and (c, b) ∈ �r , and hence, (a, b) ∈ �l ◦ �r . Similarly, ifd = aa′b, we obtain that (a, d) ∈ �r and (d, b) ∈ �l , so (a, b) ∈ �r ◦ �l . Therefore,we have proved that �l ◦ �r = �r ◦ �l = �d .

Further, consider a, b ∈ S such that (a, b) ∈ �l ∩ �r . Take an arbitrary a′ ∈ V (a)

and b′ ∈ V (b). Then (a, b) ∈ �l ∩ �r yields

a = aa′a = aa′b = aa′bb′b = aa′ab′b = ab′b = bb′b = b.

Therefore, we conclude that �l ∩ �r = �S .(b) Let a, b ∈ S such that (a, b) ∈ �l . If a′ ∈ V (a) and b′ ∈ V (b), then

a = (aa′)a = (aa′)b ∈ Sb and b = (bb′)b = (bb′)a ∈ Sa,

so (a, b) ∈ �. Thus, �l ⊆ �, and since �l is a congruence, then �l ⊆ �0. Analo-gously we prove that �r ⊆ �0. Finally,

�d = �l ◦ �r ⊆ �0 ◦ �0 = �0.

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(c) Let a, b ∈ S such that (a, b) ∈ �d ∩ �0. Then by (a, b) ∈ �0 ⊆ � we have thatb = au = va, for some u,v ∈ S1, and for an arbitrary a′ ∈ V (a), by (a, b) ∈ �d itfollows

a = aa′aa′a = aa′ba′a = aa′aua′a = aua′a = ba′a = vaa′a = va = b.

Thus, �d ∩ �0 = �S . Similarly we prove that �l ∩ �0 = �S and �r ∩ �0 = �S .(d) The equalities �l ◦ �0 = �0 ◦ �l and �r ◦ �0 = �0 ◦ �r are immediate con-

sequences of (b) and the well-known fact that every right congruence contained in �commutes with every left congruence contained in � (cf. Lemma I.7.2 of [16]).

Let a, b ∈ S such that (a, b) ∈ �d ◦ �0. Then there exists c ∈ S such that (a, c) ∈�d and (c, b) ∈ �0. Take an arbitrary a′ ∈ V (a) and set d = aa′ba′a. By (c, b) ∈�0 ⊆ � we have that b = cu = vc, for some u,v ∈ S1.

Now, for arbitrary s, t ∈ S1, by (a, c) ∈ �d and (c, b) ∈ �0 it follows that

sat = saa′aa′at = saa′ca′at � saa′ba′at = sdt,

so (a, d) ∈ �0. On the other hand, for any x, y ∈ S, by (a, c) ∈ �d it follows that

xdy = xaa′ba′ay = xaa′cua′ay = xaa′aua′ay = xaua′ay

= xcua′ay = xba′ay = xvca′ay = xvaa′ay = xvay = xvcy = xby,

so (d, b) ∈ �d . Therefore, (a, b) ∈ �0 ◦ �d , and we have that �d ◦ �0 ⊆ �0 ◦ �d .Similarly we prove the opposite inclusion �0 ◦ �d ⊆ �d ◦ �0, and finally, we havethat �d ◦ �0 = �0 ◦ �d . �

Therefore, �l and �r , �l and �0, �r and �0, and �d and �0 form commutingpairs of factor congruences on any regular semigroup, so we have the following:


(a) S is a pullback product of S/�l and S/�r with respect to S/�d ;(b) S is a pullback product of S/�l and S/�0 with respect to S/�l ◦ �0;(c) S is a pullback product of S/�r and S/�0 with respect to S/�r ◦ �0;(d) S is a pullback product of S/�d and S/�0 with respect to S/�d ◦ �0.

Proof This is an immediate consequence of Theorem 3.1. �

Theorem 3.2 raises a question on the structure of the factors appearing in it. Thefactors from (a) were described by Kopamu in [14]. Recall that a semigroup S isleft reductive if for every a, b ∈ S, by xa = xb for every x ∈ S it follows a = b, orequivalently, if �S

l = �S , and it is right reductive if for every a, b ∈ S, ay = by forevery y ∈ S implies a = b, that is, if �S

r = �S . If S is both left and right reductive,i.e., if �S

d = �S , then S is said to be reductive.Kopamu proved the following:

Theorem 3.3 Every regular semigroup is a pullback product of a left reductive anda right reductive regular semigroup with respect to a reductive regular semigroup.


However, the concepts of left, right and two sided reductivity are quite general,and there are reasons to search for some more precise structural descriptions of com-ponents in the considered pullback products. In the sequel we will show that idempo-tents can give a considerable amount of information on the structure of these compo-nents.

A semigroup S is orthodox if it is regular and E(S) is a subsemigroup of S, andif E(S) belongs to a given class (or rather a variety) � of bands, then S is called�-orthodox.

Lemma 3.3 A regular semigroup S is orthodox if and only if any one of the semi-groups S/�l , S/�r and S/�d is orthodox.

Proof If S is orthodox, then it is evident that every one of the semigroups S/�l , S/�r

and S/�d is orthodox. To prove the converse implication, it is enough to prove that forevery a ∈ S, (a, a2) ∈ �d implies a = a2. Indeed, if (a, a2) ∈ �d , then xay = xa2y,for all x, y ∈ S, and if we take an arbitrary a′ ∈ V (a) and if we set x = aa′ andy = a′a, then a = aa′aa′a = aa′a2a′a = a2.

This completes the proof of the lemma. �

Lemma 3.4 Let S be a regular semigroup. Then

(a) S is �� -orthodox if and only if S/�r is ��-orthodox;(b) S is �� -orthodox if and only if S/�l is ��-orthodox.

If S is orthodox, then

(c) S is �� -orthodox if and only if S/�0 is � -orthodox;(d) S is �� -orthodox if and only if S/�0 is � -orthodox.

Proof The assertions (a) and (b) follow by Lemmas 2.1 and 3.2 and Lemma 3.4 of[14], and (c) and (d) follow by Lemma 2.2 and Theorem 8 of [4]. �

Now we give the following structural characterization of �� -orthodox semi-groups.

Theorem 3.4 A semigroup S is an �� -orthodox semigroup if and only if it is apullback product of a ��-orthodox semigroup and an � -orthodox semigroupwith respect to an inverse semigroup.

Proof According to Theorem 3.2, S is a pullback product of semigroups S/�r andS/�0 with respect to S/�r ◦ �0, and by Lemma 3.4, S/�r is ��-orthodox andS/�0 is � -orthodox, whereas S/�r ◦ �0 is an inverse (�-orthodox) semigroup,as a common homomorphic image of S/�r and S/�0.

Conversely, let S be a pullback product of an ��-orthodox semigroup and an� -orthodox semigroup with respect to an inverse semigroup. Seeing that a pull-back product of regular semigroups with respect to an inverse semigroup is also reg-ular, we have that S is regular. On the other hand, E(S) is a subdirect product of aleft regular and a right normal band, so it is a right quasi normal band. �

510 M. Ciric et al.

A dual theorem can be stated for �� -orthodox semigroups. In the sequel wegive similar results for -orthodox semigroups.

Theorem 3.5 Let S be a regular semigroup. Then the following conditions are equiv-alent:

(i) S is an -orthodox semigroup;(ii) S/�l is an � -orthodox semigroup;

(iii) S/�r is an � -orthodox semigroup;(iv) S/�d is an inverse semigroup.

Proof It is well-known that E(S) is a normal band if and only if axyb = ayxb, forall a, x, y, b ∈ E(S), and by the definition of relations �l , �r and �d we have thatthis is equivalent to (xyb, yxb) ∈ �l , for all x, y, b ∈ E(S), to (axy, ayx) ∈ �r , forall a, x, y ∈ E(S), as well as to (xy, yx) ∈ �d , for all x, y ∈ E(S). By this we easilyconclude that the conditions (i)–(iv) are equivalent. �

As an immediate consequence of Theorems 3.2 and 3.5 (or Theorem 3.4 and itsdual) we obtain the well-known theorem proved by M. Yamada in [20].

Theorem 3.6 A semigroup S is an -orthodox semigroup if and only if it is apullback product of an � -orthodox semigroup and an � -orthodox semigroupwith respect to an inverse semigroup.

4 Subdirect decompositions respecting semilattice decompositions

For orthodox semigroups whose idempotents form bands more general than left orright quasi-normal bands, congruences �l , �r and �d do not give enough informa-tion on the structure of related factors. For that reason we define a new system ofcongruences that will give better results.

Let σ be the least semilattice congruence on a semigroup S, let Y be the greatestsemilattice homomorphic image of S, and let {Sα |α ∈ Y } be the collection of alldifferent σ -classes, i.e., the greatest semilattice decomposition of S. Moreover, forany α ∈ Y let (α] = {β ∈ Y |β ≤ α} and

S(α] =⋃

β∈(α]Sβ.

It is known that S(α] is a completely semiprime ideal of S, for every α ∈ Y .Define relations �l , �r and �d on S by:

�l =⋃

α∈Y

(Sα × Sα ∩ �l,S(α]

), �r =

⋃

α∈Y

(Sα × Sα ∩ �r,S(α]

),

�d =⋃

α∈Y

(Sα × Sα ∩ �d,S(α]

).


We can observe that similar definitions can be given starting from any semilatticecongruence, but here it is especially important to work with the least one.

Lemma 4.1 Let S be an arbitrary semigroup. Then �l , �r and �d are congruencerelations on S.

Proof Let (a, b) ∈ �d and c ∈ S. Then a, b ∈ Sα and (a, b) ∈ �d,S(α] , for some α ∈ Y .Assume that c ∈ Sβ , for some β ∈ Y . Then ca, cb, ac, bc ∈ Sαβ , and for all x, y ∈S(αβ] we have that x, y, xc, cy ∈ S(αβ] ⊆ S(α], so

x(ca)y = (xc)ay = (xc)by = x(cb)y and x(ac)y = xa(cy) = xb(cy) = x(bc)y,

and hence, (ca, cb), (ac, bc) ∈ �d,S(αβ] . Therefore, (ca, cb), (ac, bc) ∈ �d , and wehave proved that �d is a congruence relation.

Analogously we prove that �l and �r are congruence relations. �

The main properties of congruences �l , �r and �d are given by the followingtheorem.


(a) �l ◦ �r = �r ◦ �l = �d and �l ∩ �r = �S ;(b) �l ⊆ �0, �r ⊆ �0 and �d ⊆ �0;(c) �l ∩ �0 = �r ∩ �0 = �d ∩ �0 = �S ;(d) �l ◦ �0 = �0 ◦ �l , �r ◦ �0 = �0 ◦ �r and �d ◦ �0 = �0 ◦ �d .

Proof (a) It is evident that �l and �r are contained in �d , and as in the proof ofTheorem 3.1, we conclude that �l ◦ �r ⊆ �d and �r ◦ �l ⊆ �d .

Let (a, b) ∈ �d , i.e., a, b ∈ Sα and (a, b) ∈ �d,S(α] . According to Theorem 3.1, wehave that �d,S(α] = �l,S(α] ◦ �r,S(α] , i.e., there exists c ∈ S(α] such that (a, c) ∈ �l,S(α]and (c, b) ∈ �r,S(α] , and c can be chosen so that c = ba′a. By this it follows thatc ∈ Sα , so (a, c) ∈ �l and (c, b) ∈ �r . Hence, (a, b) ∈ �l ◦ �r , and we have provedthat �d ⊆ �l ◦ �r . Similarly we prove the inclusion �d ⊆ �r ◦ �l .

If (a, b) ∈ �l ∩ �r , then a, b ∈ Sα , for some α ∈ Y , and

(a, b) ∈ �l,S(α] ∩ �r,S(α] = �S(α] ,

so a = b. Thus, �l ∩ �r = �S .(b) Let (a, b) ∈ �l . Then a, b ∈ Sα and (a, b) ∈ �l,S(α] , for some α ∈ Y , so

(a, b) ∈ �l,S(α] ⊆ �S(α] ⊆ �S.

Thus, �l ⊆ �, and since �l is a congruence, then �l ⊆ �0. Analogously we provethat �r ⊆ �0 and �d ⊆ �0.

(c) If (a, b) ∈ �l ∩ �0, then a, b ∈ Sα and (a, b) ∈ �l,S(α] , for some α ∈ Y , and(a, b) ∈ �0

S(α] , so by Theorem 3.1 it follows that a = b. Thus, �l ∩ �0 = �S .

Analogously we prove that �r ∩ �0 = �S and �d ∩ �0 = �S .

512 M. Ciric et al.

(d) By the same arguments used in the proof of the corresponding part of Theo-rem 3.1, we obtain that �l ◦ �0 = �0 ◦ �l and �r ◦ �0 = �0 ◦ �r .

Let (a, b) ∈ �d ◦�0, i.e., let there exist c ∈ S such that (a, c) ∈ �d and (c, b) ∈ �0.Then a, c ∈ Sα and (a, c) ∈ �d,S(α] , for some α ∈ Y , and

(c, b) ∈ �0 ⊆ � ⊆ � ⊆ σ,

so b ∈ Sα and (c, b) ∈ �S ∩ (Sα × Sα) = �Sα , which implies that b = cu = vc, forsome u,v ∈ S1

α . Take an arbitrary a′ ∈ V (a) and set d = aa′ba′a. Then d ∈ Sα .Now, for arbitrary s, t ∈ S1 we have that saa′, a′at ∈ S(α], and by (a, c) ∈ �d,S(α]

and (c, b) ∈ �0 it follows that

sat = saa′aa′at = saa′ca′at � saa′ba′at = sdt,

so (a, d) ∈ �0. Furthermore, for any x, y ∈ S(α], by (a, c) ∈ �d,S(α] it follows that

xdy = xaa′ba′ay = xaa′cua′ay = xaa′aua′ay = xaua′ay

= xcua′ay = xba′ay = xvca′ay = xvaa′ay = xvay = xvcy = xby,

so (d, b) ∈ �d,S(α] , whence (d, b) ∈ �d . Therefore, (a, b) ∈ �0 ◦�d , and we have that�d ◦ �0 ⊆ �0 ◦ �d . Similarly we prove the opposite inclusion �0 ◦ �d ⊆ �d ◦ �0,and finally, we have that �d ◦ �0 = �0 ◦ �d . �

In view of Theorem 4.1, �l and �r , �l and �0, �r and �0, and �d and �0 formcommuting pairs of factor congruences on any regular semigroup, so we have thefollowing:


(a) S is a pullback product of S/�l and S/�r with respect to S/�d ;(b) S is a pullback product of S/�l and S/�0 with respect to S/�l ◦ �0;(c) S is a pullback product of S/�r and S/�0 with respect to S/�r ◦ �0;(d) S is a pullback product of S/�d and S/�0 with respect to S/�d ◦ �0.

Proof This is an immediate consequence of Theorem 4.1. �

5 Subdirect decompositions of orthogroups

Congruences �l , �r and �d are defined so that they respect the greatest semilat-tice decomposition of a semigroup. However, this fact can not be fully exploited ifwe work with an orthodox semigroup S and we want to derive information on thestructure of the factors S/�l , S/�r and S/�d from properties of its subsemigroup ofidempotents E(S), because an orthodox semigroup and its subsemigroup of idempo-tents do not necessarily have the same greatest semilattice homomorphic image.

In contrast to orthodox semigroups, an orthogroup and its subsemigroup of idem-potents have the same greatest semilattice homomorphic image, which enables con-gruences �l , �r and �d to be used in the study of orthogroups with full force.


Recall that a semigroup S is an orthogroup if it is completely regular and E(S)

is a subsemigroup of S. If, in addition, E(S) belongs to a given class (or rather avariety) � of bands, then S is called an �-orthogroup.

Lemma 5.1 A regular semigroup S is orthodox if and only if any one of the semi-groups S/�l , S/�r and S/�d is orthodox.

Proof This follows by Lemma 3.3. �

For an orthogroup S, the next theorem establishes relationships between propertiesof the bands E(S), E(S/�l) and E(S/�r ).

Theorem 5.1 Let S be an orthogroup.

(a) If [�1,�2] is a left diagonal of �(), then

E(S) ∈ �2 ⇔ E(S/�l) ∈ �1.

(b) If [�1,�2] is a right diagonal of �(), then

E(S) ∈ �2 ⇔ E(S/�r ) ∈ �1.

Proof (a) We distinguish six cases concerning a left diagonal [�1,�2].(a1) Let [�1,�2] = [[Gn = Hn], [Gn+1 = Hn+1]], for n ≥ 3.Further, let E(S) ∈ �2, let the letters x1, . . . , xn take arbitrary values a1, . . . , an ∈

E(S), respectively, let G∗n and H

∗n be the corresponding values of the words Gn

and Hn in E(S), let α1, . . . , αn ∈ Y such that a1 ∈ Sα1 , . . . , an ∈ Sαn , and set α =α1 · · ·αn.

It is clear that G∗n ,H

∗n ∈ Sα . Consider an arbitrary an+1 ∈ E(S(α]). Then an+1 ∈

E(Sβ), for some β ∈ (α], so an+1, an+1G∗n an+1 ∈ E(Sβ), and since E(Sβ) is a rec-

tangular band, then

an+1G∗nan+1 = an+1(an+1G

∗nan+1)an+1 = an+1.

Now we have that

an+1G∗n = an+1G

∗n an+1H

∗n = an+1H

∗n ,

so (G∗n ,H

∗n ) ∈ �l,E(S(α]) = �l,S(α] , and hence, (G

∗n ,H

∗n ) ∈ �l . By this it follows that

E(S/�l) ∈ [Gn = Hn].Conversely, let E(S/�l) ∈ [Gn = Hn]. Let the letters x1, . . . , xn, xn+1 take ar-

bitrary values a1, . . . , an, an+1 ∈ E(S), respectively, let G∗n and H

∗n denote the

corresponding values of words Gn and Hn in E(S), let α1, . . . , αn ∈ Y such thata1 ∈ Sα1, . . . , an ∈ Sαn , and set α = α1 · · ·αn.

Then G∗n ,H

∗n ∈ Sα and (G

∗n ,H

∗n ) ∈ �l,S(α] , and since an+1G

∗nan+1 ∈ S(α], we

have that

an+1G∗nan+1H

∗n = (an+1G

∗nan+1)H

∗n = (an+1G

∗nan+1)G

∗n

= (an+1G∗n )2 = an+1G

∗n .

514 M. Ciric et al.

Therefore, E(S) ∈ [xn+1Gn = xn+1Gnxn+1Hn] = [Gn+1 = Hn+1].(a2) The proof of the case [�1,�2] = [[Gn = In], [Gn+1 = In+1]], for n ≥ 2, is

analogous to the proof of (a1).(a3) Let [�1,�2] = [� , [G3 = H3]].First, assume that E(S) ∈ [G3 = H3] = [axy = axyay], and consider arbitrary

x, y, z ∈ E(S). Evidently, xyz, yxz ∈ Sα , for some α ∈ Y . Take any a ∈ E(S(α]), i.e.,a ∈ E(Sβ), for some β ∈ (α]. Then axyza, ayxza ∈ E(Sβ) and E(Sβ) is a rectangu-lar band, so axyza = a(axyza)a = a and ayxza = a(ayxza)a = a, and

axyz = axyzaz = az = ayxzaz = ayxz.

Therefore, (xyz, yxz) ∈ �l,E(S(α]) = �l,S(α] , i.e., (xyz, yxz) ∈ �l , and we concludethat E(S/�l) ∈ [xyz = yxz] = � .

Conversely, let E(S/�l) ∈ � = [xyz = yxz]. Let a, x, y ∈ E(S). Then wehave that xya, yxa, xay, axy ∈ Sα , for some α ∈ Y , and by (xya, yxa) ∈ �l,S(α] andaxy ∈ S(α] we obtain that

axya = (axy)(xya) = (axy)(yxa) = axyxa,

whence axyay = axyxay. On the other hand, by (xay, axy) ∈ �l,S(α] and axy ∈ S(α]it follows that

axyay = axyxay = (axy)(xay) = (axy)(axy) = axy.

Hence, E(S) ∈ [axy = axyay] = [G3 = H3].(a4–6) The case [�1,�2] = [�, [G2 = I2]], and the cases [�1,�2] = [��,�e]

and [�1,�2] = [�,��] can be proved similarly as the case (a3).(b) The proof of this part is dual to the proof of (a). �

Another theorem shows relationships between bands E(S/�d), E(S/�l) andE(S/�r ).

Theorem 5.2 Let S be an orthogroup.

(a) If � is an endpoint of some right diagonal of �(), then

E(S/�d) ∈ � ⇔ E(S/�l) ∈ �.

(b) If � is an endpoint of some left diagonal of �(), then

E(S/�d) ∈ � ⇔ E(S/�r ) ∈ �.

Proof (a) Let E(S/�d) ∈ �.Suppose that � = [Gn = Hn] = [Gn−1xn = Hn−1xnGn−1xn], for some n ≥ 3.

Let the letters x1, . . . , xn take arbitrary values a1, . . . , an ∈ E(S), respectively, letG∗

n−1 and H ∗n−1 be the corresponding values of the words Gn−1 and Hn−1 in E(S),

let α1, . . . , αn ∈ Y such that a1 ∈ Sα1, . . . , an ∈ Sαn , and set α = α1 · · ·αn. Clearly,G∗

n−1an,H∗n−1anG

∗n−1an ∈ Sα .


Consider now an arbitrary u ∈ S(α] and set v = G∗n−1an. Then by u,v ∈ S(α] and

(G∗n−1an,H

∗n−1anG

∗n−1an) ∈ �d,S(α] it follows that

uG∗n−1an = uG∗

n−1anv = uH ∗n−1anG

∗n−1anv = uH ∗

n−1anG∗n−1an,

and hence, (G∗n−1an,H

∗n−1anG

∗n−1an) ∈ �l,S(α] , i.e., (G∗

n−1an,H∗n−1anG

∗n−1an) ∈

�l . Therefore, E(S/�l) ∈ �.If � = [Gn = In], for some n ≥ 2, � = �e or � = ��, then one can verify in

a similar way that E(S/�l) ∈ �.Conversely, if E(S/�l) ∈ �, then clearly E(S/�d) ∈ �, since E(S/�d) is a ho-

momorphic image of E(S/�l).(b) The proof of this part is dual to the proof of (a). �

The previous two theorems can be gathered into one, as follows.

Theorem 5.3 Let S be an orthogroup and � a variety of bands. Then the followingconditions are equivalent:

(i) E(S) ∈ �;(ii) E(S/�l) ∈ ��0 and E(S/�r ) ∈ ��0 ;

(iii) E(S/�d) ∈ ��0 .

Proof By Theorem 2.2, the variety � is the intersection point of the left diagonal of�() starting in ��0 and the right diagonal of �() starting in ��0 , and

� = ��0 ∨ ��0 and ��0 ∧ ��0 = ��0 .

By this and by Theorems 5.1 and 5.2 we obtain that (i)⇔(ii) and (ii)⇔(iii). �

Now we are ready to prove the main theorem of this section that gives a pullbackproduct representation of an orthogroup.

Theorem 5.4 Let � be a variety of bands.A semigroup S is a �-orthogroup if and only if it is a pullback product of a

��0 -orthogroup and a ��0 -orthogroup with respect to a ��0 -orthogroup.

Proof Let S be a �-orthogroup. According to Theorem 4.2, S is a pullback prod-uct of semigroups S/�l and S/�r with respect to a semigroup S/�d , and by The-orem 5.3, S/�l is a ��0 -orthogroup, S/�r is a ��0 -orthogroup, and S/�d is a��0 -orthogroup.

Conversely, let S be a pullback product of a ��0 -orthogroup P and a ��0 -ortho-group Q with respect to a ��0 -orthogroup. Then S is also an orthogroup and E(S) isa subdirect product of a ��0 -band E(P ) and a ��0 -band E(Q), so by Theorem 2.1we obtain that E(S) ∈ �. Therefore, S is a �-orthogroup. �

516 M. Ciric et al.

6 Subdirect decompositions of orthocryptogroups

In the previous section only pullback product decompositions of orthogroups deter-mined by the pair �l and �r were considered. If we would use pairs �l and �0,or �r and �0, we would not get anything new. Namely, for a �-orthogroup S weknow that both S/�l and S/�0 are ��0 -orthogroups, and both S/�r and S/�0 are��0 -orthogroups, and in the general case we can not make any clear distinction be-tween properties of S/�l and S/�0, or between properties of S/�r and S/�0.

However, there is one significant difference between �l and �0, as well as be-tween �r and �0. On a completely regular semigroup S, any one of �l , �r and �d

may be a band congruence only if S is a band, whereas the classes of completely reg-ular semigroups on which �0 and �0 are band congruences are much larger. In thissection we consider orthogroups on which �0 or �0, or both, are band congruences.

A completely regular semigroup S is called left cryptic or a left cryptogroup if� is a left congruence on S. Dually we define a right cryptic semigroup or a rightcryptogroup.

The main properties of left cryptic semigroups are described by the followingtheorem.

Theorem 6.1 The following conditions on a completely regular semigroup S areequivalent:

(i) S is left cryptic;(ii) For any a, b ∈ S, Sab = Sab2;

(iii) � ⊆ �0;(iv) �0 is a band congruence;(v) S is a band of left groups.

Proof (i)⇔(ii) This is a known result proved in [16], as Proposition II.8.2. It is statedhere for the sake of completeness.

(i)⇒(iii) Let � be a left congruence and let (a, b) ∈ �. Then for any x ∈ S1 wehave that (xa, xb) ∈ � ⊆ �, whence (a, b) ∈ �0. Thus, � ⊆ �0.

(iii)⇒(iv) For any a ∈ S we have that (a, a2) ∈ �, and it is clear that (iii) im-plies (iv).

(iv)⇔(ii) This equivalence is obvious.(iv)⇔(v) By the proof of Theorem 4 [2], for any band congruence � on a semi-

group S, every �-class is a left simple semigroup if and only if � ⊆ �0, and seeingthat S is completely regular, we have that (iv) is equivalent to (v). �

Some more general results have been obtained in [2] and [3].A left cryptic orthogroup is called a left orthocryptogroup. If, in addition, its

subsemigroup of idempotents belongs to a given class or a variety � of bands, then itis called a left �-orthocryptogroup.

Dually we define right orthocryptogroups and right �-orthocryptogroups.The following theorem gives a pullback product representation of left orthocryp-

togroups.


Theorem 6.2 Let � be a variety of bands.Then S is a left �-orthocryptogroup if and only if S is a pullback product of a

��0 -band and a left ��0 -orthocryptogroup with respect to a ��0 -band.

Proof Let S be a left �-orthocryptogroup. By Theorem 4.2, S is a pullback prod-uct of S/�0 and S/�r with respect to S/�r ◦ �0. Since S is a left orthocryp-togroup, then S/�r is also a left orthocryptogroup, and by Theorem 5.3, it is a left��0 -orthocryptogroup. Also, in view of Theorem 6.1, S/�0 is a ��0 -band, and as acommon homomorphic image of S/�r and S/�0, S/�r ◦ �0 is a ��0 -band.

Conversely, let S be a pullback product of a ��0 -band and a left ��0 -ortho-cryptogroup. Seeing that both components in this product are left orthocryptogroups,then S is also a left orthocryptogroup. Moreover, E(S) is a ��0 -band and a��0 -band, so by Theorem 2.1 it follows that E(S) is a �-band. Consequently, S

is a left �-orthocryptogroup. �

A dual theorem can be stated for right �-orthocryptogroups.If a semigroup S is both left and right cryptic, i.e., if � is a congruence on S,

then S is said to be cryptic or a cryptogroup. As known, in this case � is a bandcongruence, so cryptogroups are just those semigroups which can be decomposedinto a band of groups.

A cryptogroup S in which E(S) is a subsemigroup of S is called an orthocryp-togroup. If, in addition, E(S) belongs to a given class or a variety � of bands, then S

is said to be a �-orthocryptogroup.In Theorem 4.1 we have proved that �l and �0, �r and �0, and �d and �0

commute, but we have not been able to say anything about their products.Here we show that on an orthocryptogroup all these products are equal to �0, and

we also show that the triple �d , �0 and �0 is absolutely permutable.

Theorem 6.3 Let S be an orthocryptogroup. Then

(a) �l ◦ �0 = �0 ◦ �l = �0;(b) �r ◦ �0 = �0 ◦ �r = �0;(c) �d ◦ � = � ◦ �d = �0;(d) �d ∨ �0 ∨ �0 = �0 and �d , �0 and �0 are absolutely permutable.

Proof (a) According to Theorem 4.1, it is enough to prove that �0 ⊆ �l ◦ �0.Let a, b ∈ S such that (a, b) ∈ �0. Then there is u ∈ S such that (a,u) ∈ �0 and

(u, b) ∈ �0, and by �0,�0 ⊆ � it follows that a,u, b ∈ Sα , for some α ∈ Y . If we setv = u0a, then also v ∈ Sα .

Consider any x ∈ S(α], i.e., x ∈ Sβ , for some β ∈ Y , β ≤ α. Since (a,u) ∈ �0

implies (a0, u0) ∈ �0, then (x0a0, x0u0) ∈ �, and hence, x0u0 = x0u0x0a0. Butx0, x0u0 ∈ E(Sβ) and E(Sβ) is a rectangular band, so we have that x0u0x0 = x0.Thus x0u0 = x0a0, and

xv = xx0u0a = xx0a0a = xa.

518 M. Ciric et al.

Therefore, (a, v) ∈ �l . On the other hand, the �0-class containing a and u is a leftgroup, whence (v,u) ∈ � = �0 ⊆ �0, and by (u, b) ∈ �0 we obtain (v, b) ∈ �0.Hence, �0 ⊆ �l ◦ �0, as required.

(b) This is dual to (a).It is convenient to prove (d) next.(d) It is obvious that �d ∨ �0 ∨ �0 = �0 ∨ �0 = �0.Consider a, b, c ∈ S which belong to the same �0-class D0 of S. In view of (a),

there exists u ∈ S such that (a,u) ∈ �l and (u, c) ∈ �0. Then u ∈ D0, and by (b),(u, v) ∈ �r and (v, b) ∈ �0, for some v ∈ S. Now

(a, v) ∈ �l ◦ �r = �d, (b, v) ∈ �0 and (c, v) ∈ �0 ◦ �r = �0,

and thus, �d , �0 and �0 are absolutely permutable.(c) This one obtains from the proof of (d), for b = c. �

Finally, we give four pullback product representations of orthocryptogroups.

Theorem 6.4 Let � be a variety of bands. Then the following conditions on a semi-group S are equivalent:

(i) S is a �-orthocryptogroup;(ii) S is a pullback product of a ��0 -orthocryptogroup and a �-band;

(iii) S is a pullback product of a ��0 -band, a ��0 -orthocryptogroup and a ��0 -band;

(iv) S is a pullback product of a ��0 -band and a ��0 -orthocryptogroup;(v) S is a pullback product of a ��0 -orthocryptogroup and a ��0 -band.

All these pullback products are with respect to a ��0 -band.

Proof Let S be a �-orthocryptogroup. Then S possesses all pullback product decom-positions listed in Theorem 4.2, and in addition to that, by (d) of Theorem 6.3, S is apullback product of S/�0, S/�d and S/�0 with respect to S/�0.

Next, by Theorem 5.3, S/�l , S/�r and S/�d are respectively a ��0 -, ��0 -, anda ��0 -orthogroup, whereas �0, �0 and � are band congruences, so S/�0 and S/�0

are respectively a ��0 - and a ��0 -band, and S/� is evidently a �-band. Therefore,all statements (ii)–(v) are true.

Conversely, let S possess any of the pullback product decompositions listed in(ii)–(v). Since all components in these pullback products are orthocryptogroups,then S is also an orthocryptogroup. Furthermore, E(S) is a subdirect product of a��0 - and a �-band, or a ��0 -, ��0 -, and a ��0 -band, or a ��0 and a ��0 -band, andby Theorem 2.1, in any of these cases E(S) is a �-band. Hence, if any of statements(ii)–(v) hold, S is a �-orthocryptogroup. �

There is also a characterization of an orthocryptogroup as a pullback product of aband and a Clifford semigroup (semilattice of groups), given by Petrich in [15] (seealso [16], Theorem II.8.5). The corresponding commuting pair of factor congruencesconsists of � and the least Clifford congruence on this orthocryptogroup (see [16],Proposition II.5.6).


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Effective subdirect decompositions of regular semigroups

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