Kudryavtsev V.B., Rosenberg I.G. Structural Theory of Automata, Semigroups, and Universal Algebra

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Epigroups 347

epigroups is negative. Moreover, unipotently partitionable epigroups cannot be characterized

in any way in terms of forbidden epidivisors or Rees epidivisors. The fact is that the class of

such epigroups (even the ﬁnite ones) is not closed under Rees quotient semigroups. This is

demonstrated by the following example.

3.13 Example Let S = c, d |cdc = c, dcd = d, c

d = cd

= c

c = dc

= d

.Itis

easy to see that S has six elements and four torsion classes, all of which are subsemigroups:

{c, c

}, {d, d

}, {cd},and{dc}. The subset I = {c

} is an ideal of S,andS/I

∼

The largest homomorphically closed subclass of the class of unipotently partitionable

epigroups, and only this subclass, has the divisor characterization just mentioned; it can be

also transparently characterized from slightly diﬀerent natural points of view. This is done

in the following theorem proved in [104, §2]; the equivalence of conditions (1), (2a)–(2c),

(3a)–(3c) was already stated in [96].

3.14 Theorem The following conditions on an epigroup S are equivalent:

(1) all homomorphic images of S are unipotently partitionable;

(2a) in any homomorphic image S



with zero, the set Nil S



is a subsemigroup;

(2b) for any ideal I of S,theset

√

I is a subsemigroup;

(2c) in any Rees epidivisor H



of S,thesetNil H



is a subsemigroup;

(3a) there is no semigroup B

among the epidivisors of S;

(3b) there is no semigroup B

among the Rees epidivisors of S;

(3c) there is no subsemigroup B

in any homomorphic image of S;

(3d) for any ideal I of S, there are no subsemigroup B

in S/I;

(4) for any subepigroup H of S, Reg H ⊆Gr H;

(5) for any e, f, i ∈ E

, e |i and f |i imply ef |i or fe|i.

The fact that just epidivisors but not arbitrary divisors appear in conditions (3a), (3b)

is essential. For (3a) this is obvious: groups trivially satisfy the conditions of Theorem 3.14,

but any semigroup is a divisor of a free group of suitable cardinality. The following example

shows that also in condition (3b) epidivisors cannot be replaced by arbitrary divisors.

3.15 Example Let S be the Rees matrix semigroup M

G;2, 2;

−1

,whereG =

g is the inﬁnite cyclic group with identity element e.Putc =(1,g,1), d =(2,g,2),

Q = {(i, g

,λ) |i, λ ∈{1, 2},n>1}. It can be shown by a direct calculation that Q is an

ideal of the subsemigroup c, d, cdc = c, dcd = d, c

∈ Q, c, d\Q = {c, d, cd, dc},and

therefore c, d/Q

∼

.ThusB

occurs among the Rees divisors of S.ButS satisﬁes the

conditions of Theorem 3.14, which can be seen at once, for instance, from condition (2b),

since S, being a completely simple semigroup, contains no proper ideals.

348 L. N. Shevrin

3.3 Epigroups decomposable into a band of archimedean semigroups

The title of this subsection shows what will be the subject of our considerations here. In view

of Observation 2.2, any congruence class on an arbitrary epigroup that is a subsemigroup is

a subepigroup. Therefore, the components of any decomposition of an epigroup into a band

are subepigroups, so from now on we will speak about bands of epigroups with various

properties. Epigroups decomposable into a band of archimedean epigroups are characterized

from diﬀerent points of view by Theorem 3.16 which may be considered as the culmination of

this section and serves as a base for the rest of its material. To facilitate comprehension, as in

Theorem 3.14, we present a rather long list of equivalent conditions grouped together by type.

The reader will detect certain parallels between conditions on this list and the corresponding

conditions in Theorem 3.14 which, of course, was used to prove Theorem 3.16. The proof

of this theorem is given in [104, §3]. The equivalence of conditions (1a) and (4a) was stated

in [95] and later mentioned also in [23]; the equivalence of (1a), (2a)–(2c), and (3a)–(3c) was

stated in [97]; the equivalence of (1b), (6a) was ﬁrst established in [74]; the characterization

given by the ﬁrst of the identities in (8) was established in [82, §II.6]. We mention also

that one can ﬁnd in [23] several other conditions on a semigroup that are equivalent to

the condition that it be decomposable into a semilattice of archimedean epigroups; these

conditions were reproduced in [5, Chapter X], where the corresponding semigroups were

called GV -semigroups. Various results on semigroups decomposable into a semilattice of

archimedean semigroups are contained in the survey [8].

Notice that in the case of ﬁnite semigroups, the semigroups under discussion are rather

popular in certain investigations, where they are commonly presented by condition (4c); see,

for instance, [1, Chapter 8] which is devoted to the study of so-called implicit operations on

such semigroups and, as is noted at the beginning of this chapter, contains some of the main

results of the whole book.

Along with the semigroup B

, among the “cast of characters” in Theorem 3.16 is the

related 5-element semigroup A

. It is also well known both as the matrix semigroup

$

















&

and a completely 0-simple semigroup: it is isomorphic to the Rees matrix semigroup

{e};2, 2;



0 e

#

over the 0-group {e, 0}.AsB

, it can clearly be given by a presentation:

= c, d |c

=0,d

= d, cdc = c, dcd = d.

If ρ is a binary relation on a semigroup S,wedenoteby

√

ρ the binary relation on S deﬁned

by the following condition:

√

ρy ⇐⇒ there exist m, n such that x

ρy

3.16 Theorem The following conditions on an epigroup S are equivalent:

(1a) S is a band of archimedean epigroups;

Epigroups 349

(1b) S is a semilattice of archimedean epigroups;

(2a) in any homomorphic image S



with zero, the set Nil S



is an ideal;

(2b) for any ideal I of S,theset

√

I is an ideal;

(2c) in any Rees epidivisor H



of S,thesetNil H



is an ideal;

(3a) there are no semigroups A

, B

among the epidivisors of S;

(3b) there are no semigroups A

, B

among the Rees epidivisors of S;

(3c) there are no semigroups A

, B

among the subsemigroups of any homomorphic image

of S;

(3d) for any ideal I of S, there are no semigroups A

, B

among the subsemigroups of S/I;

(4a) Reg S =GrS;

(4b) for any subepigroup H of S, Reg H = H ∩ Reg S;

(4c) each regular D-class in S is a (automatically completely simple) subsemigroup;

(4d) each completely 0-simple principal factor of S is a completely regular semigroup, i.e.,

it contains no non-zero nil-elements;

(5a) for any e, f, i ∈ E

, i |e and f |e imply if |e;

(5b) for any a, b ∈ S and e ∈ E

, a |e and b |e imply ab |e;

(6a) for any a ∈ S and e ∈ E

, a |e implies a

|e;

(6b) for any a ∈ S, J(a) ⊆

J(a

);

(7a)

√

D is a transitive relation (and is therefore an equivalence relation);

(7b)

√

D is the smallest semilattice congruence;

(8) S satisﬁes either (hence both) of the identities

((xy)

(yx)

(xy)

)

=(xy)

, ((xy)

yx(xy)

)

=(xy)

We mention some direct consequences of Theorem 3.16 pertaining to varieties consisting of

semilattices of archimedean epigroups. Condition (8) of Theorem 3.16 provides an equational

characterization of such varieties. Denote by AS

the class of all epigroups in E

that are

decomposable into a semilattice of archimedean epigroups.

3.17 Lemma For any n the class AS

is a subvariety of E

Note that AS

= E

, but for n>1 we have the strict inclusions AS

⊂E

Structural characterizations of varieties of semilattices of archimedean epigroups are pro-

vided by Proposition 3.18 which is based on Theorem 3.16 in conjunction with Theorem 3.14.

As for the indicator characterization (3) here, the elimination of the semigroup A

(cf. con-

dition (3a) in Theorem 3.16) is explained by the well-known and easily veriﬁed fact that B

is a Rees divisor of the direct product A

× A

350 L. N. Shevrin

3.18 Proposition The following conditions for a variety V of epigroups are equivalent:

(1) V consists of semilattices of archimedean epigroups (i.e., in view of Observation 2.11

and Lemma 3.17, V is contained in some variety AS

);



) V consists of unipotently partitionable epigroups;

(2) in any epigroup S with zero belonging to V,thesetNil S is an ideal;



) in any epigroup S with zero belonging to V,thesetNil S is a subsemigroup;

(3) V does not contain B

;

(4) each regular epigroup in V is completely regular;



) each completely 0-simple epigroup in V is completely regular.

In the case of periodic varieties, the equivalence of (1), (1



), and (3) was established earlier

in [81] which also contains an equational characterization; the corresponding information was

reproduced in [109, Theorem 7.1]. As in [81], criterion (3) of Proposition 3.18 can be employed

to further study the varieties of epigroups. From this criterion and the fact (mentioned in [81])

that the variety var B

has the non-modular lattice of subvarieties we obtain:

3.19 Corollary A variety of epigroups whose lattice of subvarieties is modular consists of

semilattices of archimedean epigroups.

Another application of the same criterion leads to the following consequence.

3.20 Corollary The following conditions for a variety V of epigroups are equivalent:

(1) V is a variety of ﬁnite degree;



) each nilsemigroup in V is nilpotent;

(2) V consists of semilattices of nilpotent extensions of completely simple semigroups.

Here the equivalence of (1) and (1



) is guaranteed by the theorem in [88] on the nilpotency

of a nilsemigroup in which the nilpotency degrees of its nilpotent subsemigroups are uniformly

bounded. The characterization given by condition (2) extends to varieties of epigroups the

same condition found for periodic varieties in [81]. It would be interesting to extend the

other characterizations of the periodic varieties of ﬁnite degree obtained in [81] (see also [109,

Theorem 8.1]), as well as in [113] and [114], to the case of varieties of epigroups.

Let us return to Corollary 3.19. It can be considered as a starting point for a solution of

the following problem already mentioned in fact in [104].

3.21 Problem Describe the varieties of epigroups V such that the lattice L(V) is modular.

Among such varieties are all the completely regular varieties, since it is known that L(E

)is

modular (see [61,62,67]). Further, there is a description of semigroup varieties whose lattice

of subvarieties is modular, and, since these varieties are periodic (it follows, for instance,

from the result of [83]), they also belong to the family ﬁguring in Problem 3.21. Moreover,

in all likelihood, just the description mentioned can be extended to the case of varieties of

Epigroups 351

epigroups. Remark that this description has solved a long-known problem posed in [20].

The corresponding result had been stated in [121] and published with a proof “modulo the

nil case” in [120, 126]. The original version of the proof for the nil case was given only

in the dissertation [122]. A revised and improved version of the proof for this case was

given in passing in the course of the proofs of certain stronger results obtained in the recent

works [118, 119, 125].

In the remainder of this subsection we characterize epigroups of several well-known sub-

classes of the class of semilattices of archimedean epigroups. Speciﬁcally, we will mainly be

interested in epigroups that are decomposable into a semilattice of nil-extensions of rectan-

gular groups (Theorem 3.22), into a band of right archimedean epigroups (Theorem 3.28),

into a semilattice of right archimedean epigroups (Theorem 3.30), into a band of unipotent

epigroups (Theorem 3.31), into a semilattice of unipotent epigroups (Theorem 3.35), or into

a rectangular band of unipotent epigroups (Theorem 3.39). The proofs of all these theorems

are given in [104, §§4–6]. Notice that in a number of papers some of the types of epigroups

considered below have been distinguished within the class of semilattices of archimedean epi-

groups. We mention in this regard the papers [6, 7,22, 23, 46, 74, 75]; see also [5, Chapters IX

and X] as well as some sections in [9], and the bibliography therein.

It is easy to see that if a semigroup S is decomposable into a semilattice of archimedean

semigroups, then such a decomposition is unique, and it is precisely the greatest semilattice

decomposition of S. Its components will be called the archimedean components.Thenext

object we consider is the case when the kernels of the archimedean components of an epigroup

with the property under consideration are rectangular groups. Recall that a rectangular group

is a completely simple semigroup in which the set of all idempotents is a subsemigroup, or,

equivalently, a direct product of a group, a left zero semigroup, and a right zero semigroup.

Among the “cast of characters” in Theorem 3.22, along with the semigroups A

, B

,arethe

semigroups M

, n ∈ N, deﬁned by the presentations

= e, f |e

=(efe)

= e, f

=(fef)

= f.

The semigroup M

is isomorphic to the Rees matrix semigroup M[C

1,n

;2, 2; (

εε

εγ

)], where ε

is the identity element and γ is a generator of the group C

1,n

3.22 Theorem The following conditions on an epigroup S are equivalent:

(1) S is a semilattice of nil-extensions of rectangular groups;

(2a) S satisﬁes the identity (xy)

(yx)

(xy)

=(xy)

;

(2b) S satisﬁes the identities ((xy)

(yx)

(xy)

)

=(xy)

, x

)

=(x

)

;

(3) there are no semigroups A

, B

, M

for any prime p among the epidivisors of S.

Note that condition (2a) of Theorem 3.22 is related to the equivalent (but not formulated

in terms of identities) condition Q

of [74], and condition (2b) appeared in [82, §II.6]. Note

also that for an element a of an epigroup, the equality aa

= a

holds if and only if a

has ﬁnite order and period 1, i.e., a

= a

n+1

for some natural number n.Inviewofthis

remark and condition (8) of Theorem 3.16, we see that condition (2b) of Theorem 3.22 can

be transformed into the following criterion (see [5, §2 of Chapter X]):

352 L. N. Shevrin

3.23 Corollary AsemigroupS is a semilattice of nil-extensions of rectangular groups if and

only if S is a semilattice of archimedean epigroups, and, for any e, f ∈ E

, the element ef

has ﬁnite order and period 1.

Corollary 3.23 directly implies:

3.24 Corollary If S is an epigroup that is decomposable into a semilattice of archimedean

epigroups, and Gr S is a subsemigroup, then E

isasubsemigroupif (and obviously only if )

in each archimedean component, the set of idempotents is a subsemigroup.

We will state two corollaries pertaining to the special case of completely regular semi-

groups. Recall that a completely regular semigroup S in which E

is a subsemigroup is called

an orthogroup. Corollary 3.24, applied to completely regular semigroups, yields the following

long-known fact (see [65, Corollary IV.3.2 and Proposition IV.3.7]):

3.25 Corollary A semigroup is an orthogroup if and only if it is a semilattice of rectangular

groups.

Condition (3) of Theorem 3.22 and Corollary 3.25 give the following criterion:

3.26 Corollary A completely regular semigroup is an orthogroup if and only if there is no

semigroup M

for any prime p among its epidivisors.

It is impossible to remove from Corollary 3.24 the requirement that Gr S be a subsemi-

group, i.e., the condition that the set of idempotents is a subsemigroup distinguishes a proper

subclass of the class of epigroups described by Theorem 3.22. This subclass is characterized

by Proposition 3.27 (which in turn overlaps Corollary 3.26). In this proposition, there appears

yet another “character”—the semigroup V deﬁned by the presentation

V = e, f |e

= e, f

= f, fe =0.

It is obvious that V consists of four elements: V = {e, f, ef, 0}.

3.27 Proposition The following conditions on an epigroup S are equivalent:

(1) S is a semilattice of archimedean epigroups, and E

is a subsemigroup;

(2) S satisﬁes the identities (xy)

(yx)

(xy)

=(xy)

, (x

)

= x

;

(3) there are no semigroups B

, V , M

for any prime p among the epidivisors of S.

Remark that in the original formulation of this proposition in [104] there was an extra

forbidden epidivisor A

which may in reality be omitted; indeed, it is easy to see that the

semigroup V occurs among the subsemigroups of A

, so a “ban” of V implies the same for

. A similar remark should be made regarding the formulation of Theorem 3.53 below.

In the problem of characterizing semigroups that are decomposable into a band of right

[left] archimedean epigroups, we will consider, for deﬁniteness, the ﬁrst of the dual situations.

Epigroups 353

The solution of this problem for this case is given by Theorem 3.28. We must add to the

“cast of characters” the following semigroups:

= e, f |ef = e, fe = f,

3,1

= a, f |a

f = a

,fa= f

= f,

LZ(n)=g, f |g

n+1

= g, g

f = fg = f

= f.

Obviously L

is merely a two-element left zero semigroup. The semigroup L

3,1

consists of four

elements and is an ideal extension of a three-element left zero semigroup by the semigroup

2,1

. The semigroup LZ(n) consists of 2n elements and is a chain of two semigroups: an

n-element left zero semigroup and the cyclic group C

1,n

3.28 Theorem The following conditions on an epigroup S are equivalent:

(1) S is a band of right archimedean epigroups;

(2) S satisﬁes the identity (x

)

(xy)

=(xy)

;

(3) there are no semigroups A

, B

, L

3,1

, LZ(n) for all n>1(it suﬃces to consider n

being prime) among the epidivisors of S.

The equivalence of conditions (1) and (3) of Theorem 3.28 was announced in [98]. For

the particular case of completely regular semigroups, we have the following consequence.

3.29 Corollary A completely regular semigroup is a band of right groups if and only if there

is no semigroup LZ(p) for any prime p among its epidivisors.

We will say that an idempotent e of a semigroup S switches divisors from right to left

[from left to right] if, for any x ∈ S, e ∈ Sx implies e ∈ xS [e ∈ xS implies e ∈ Sx].

3.30 Theorem The following conditions on an epigroup S are equivalent:

(1) S is a semilattice of right archimedean epigroups;

(2a) S satisﬁes the identity (xy)

(yx)

=(yx)

;

(2b) S satisﬁes the identity x

(yx)

=(yx)

;

(3) there are no semigroups A

, B

, L

among the epidivisors of S;

(4) each regular D-class of S is a right group;

(5) each idempotent of S switches divisors from right to left;

(6) S satisﬁes any of the conditions of Theorem 3.16 and does not contain L

as a sub-

semigroup.

Note that condition (5) is related to the equivalent condition Q

in [74], and the identity

in (2b), as well as the identity in condition (2) of Theorem 3.35 below, appeared in [82, §II.6].

We denote by R

, R

3,1

, RZ(n) the semigroups dual to L

, L

3,1

, LZ(n), respectively.

354 L. N. Shevrin

3.31 Theorem The following conditions on an epigroup S are equivalent:

(1) S isabandofunipotentepigroups;

(2) S satisﬁes the identity (xy)

=(x

)

;

(3) there are no semigroups A

, B

, L

3,1

, R

3,1

, LZ(n), RZ(n) for any n>1(it suﬃces to

consider n being prime) among the epidivisors of S.

Remark that the implication (1) ⇒ (2) here is obvious; the rest follows from Theorem 3.28

and its dual, taking into account that, for an epigroup, the property of being unipotent is

equivalent to the property of being bi-archimedean (see Proposition 3.6). For completely

regular semigroups we obtain the following divisor criterion complementing other long-known

criteria for decomposability into a band of groups (see, for example, [13, Vol. I, p. 129]).

3.32 Corollary A completely regular semigroup is a band of groups if and only if there are

no semigroups LZ(n) and RZ(n) for any n>1 among its epidivisors.

In turn, Corollary 3.32 automatically implies the following indicator characterization

found in [76].

3.33 Corollary A variety of completely regular semigroups consists of bands of groups if

and only if it does not contain the semigroups LZ(n) and RZ(n) for any n>1.

Here is one more direct consequence of Theorem 3.31; for the case of periodic semigroups

this result appeared in [71].

3.34 Corollary Any epigroup with two idempotents is a band of unipotent epigroups.

The next theorem follows automatically from Theorem 3.30 and its dual.

3.35 Theorem The following conditions on an epigroup S are equivalent:

(1) S is a semilattice of unipotent epigroups;

(2) S satisﬁes the identity (xy)

=(yx)

;

(3) there are no semigroups A

, B

, L

, R

among the epidivisors of S;

(4) each regular D-class of S is a group;

(5) each idempotent of S switches divisors from left to right and from right to left;

(6) S satisﬁes any of the conditions of Theorem 3.16 and does not contain L

or R

subsemigroups.

We limit ourselves to three examples of well-known facts that follow directly from The-

orem 3.35. Recall that a semigroup S is called weakly commutative if ab∩bSa = ∅ for all

a, b ∈ S.

3.36 Corollary The list of equivalent conditions in Theorem 3.35 can be augmented by the

following ones:

Epigroups 355

(7) in S,theequalityK =

√

D holds;

(8) S is weakly commutative.

Note that the equality K =

√

D is equivalent to the inclusion D⊆K. In this form condi-

tion (7) occured earlier. The equivalence of (1), (7), and (8) applied to periodic semigroups

was established in [50], taking into account [85]; the corresponding results were extended to

epigroups in [46].

3.37 Corollary Any epigroup with central idempotents (in particular, any commutative epi-

group) is decomposable into a semilattice of unipotent epigroups.

Periodic semigroups with central idempotents were already considered in [40,41]. Almost

all of the results of these two papers can be extended to epigroups. In particular, for any

semilattice A and any family of unipotent epigroups U

(α ∈ A), there exists an epigroup

that is decomposable into a semilattice of these epigroups; it can be realized by a strong

semilattice structure, which extends Cliﬀord’s construction describing semilattices of groups

(for the latter, see, for instance, [49, Section A10]).

3.38 Corollary AsemigroupS is a chain of unipotent epigroups if and only if S is an

epigroup in which E

is a chain.

Remark that in [5, p. 142], this result was given in a somewhat weaker form, where

the property of being an epigroup was replaced by the property of being decomposable into

a semilattice of archimedean epigroups (in the terminology of [5], the property of being a

GV -semigroup).

3.39 Theorem The following conditions on an epigroup S are equivalent:

(1) S is a rectangular band of unipotent epigroups;

(2a) S satisﬁes the identity (xyx)

= x

;

(2b) S satisﬁes the identities (x

)

= x

, (xy)

=(x

)

;

(3) there are no two-element semilattice or semigroups L

3,1

, R

3,1

among the epidivisors of

(4a) S is archimedean, and Gr S is a retract of S;

(4b) S is a subdirect product of a completely simple semigroup and a nilsemigroup.

The equivalence of conditions (1) and (4b) was announced in [95], and one of the results

in [22] established the equivalence of (1) and (4a) (it also appeared in [23] and in [5, §2of

Chapter IX]; condition (4a) was given in the cited papers in the following equivalent form:

S is a retract extension of a completely simple semigroup by a nilsemigroup).

356 L. N. Shevrin

3.4 Epigroups whose group part is a subsemigroup

It is natural to consider epigroups in which the operation x → x is an endomorphism or

an anti-endomorphism. Such epigroups are distinguished by the identities

xy = x · y and

xy = y · x, respectively. In either case, the epigroup in question satisﬁes the identity

xy = x · y. (3.1)

Since an element a of an epigroup S belongs to GrS if and only if a =

a, the identity (3.1)

implies, in turn, that the set Gr S is a subsemigroup of S. In this subsection we pay attention

to all four of the mentioned restrictions on an epigroup.

We start with the last restriction deﬁning the largest of the corresponding classes of

epigroups. Note that several particular types of epigroups satisfying this condition occurred

in Subsections 3.1 and 3.3. We give now an indicator characterization of the epigroups under

consideration in terms of forbidden epidivisors. This result has been obtained in [124]. To

formulate it, we need to augment our “cast of characters” by a new family of individual

semigroups.

The following construction was presented in [78]. Let G be a group. Deﬁne a multipli-

cation on the disjoint union R(G)ofG

with the cartesian square G × G by preserving the

multiplication on G

and letting, for all g ∈ G,[h, j], [k, ] ∈ G × G,

[h, j] · g =[h, jg],g· [h, j]=[hg

−1

,j], [h, j] · 0=0· [h, j]=0,

[h, j] · [k, ]=



[h, ]ifj = k,

0otherwise.

Then R(G) becomes a semigroup being obviously an epigroup. Recall that, as in Subsec-

tion 2.1, we denote by C

1,∞

and C

1,n

the inﬁnite cyclic group and the cyclic group of order

n, respectively. The semigroup V was introduced in Subsection 3.3.

3.40 Theorem For an epigroup S,thesetGr S isasubsemigroupifandonlyifthereare

no semigroups V , R(C

1,∞

), R(C

1,p

) for any prime p among the epidivisors of S.

The role of the semigroup V in general is clariﬁed by the following fact established also

in the paper [124].

3.41 Proposition The semigroup V occurs among the epidivisors of an epigroup S if (and

obviously only if ) there exist e, f ∈ E

such that ef /∈ Gr S.

Let us turn to epigroups satisfying the identity (3.1). First of all, they are characterized

by the following simple property, observed in [104, §1].

3.42 Lemma In an epigroup S, the mapping x →

x is an endomorphism if and only if the

set Gr S is a retract. In this case, the indicated mapping is a unique retraction of S onto

Gr S.

Part A of the next proposition clariﬁes a structure of such epigroups and shows that they

form a subclass of the class of epigroups considered in Theorem 3.16, part B gives a suﬃcient

condition for the property under discussion.