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

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Epigroups 357

3.43 Proposition

A. An epigroup S in which the set Gr S is a retract is decomposable into a semilattice of

rectangular bands of unipotent epigroups.

B. If an epigroup S is decomposable into a band of unipotent epigroups and Gr S is a

subsemigroup, then Gr S is a retract.

The premise in A is not a necessary condition, and the conclusion in B is not a suﬃ-

cient condition for an epigroup to have the group part as a retract. The ﬁrst assertion is

conﬁrmed by the semigroups LZ(n)forn>1, which, being completely regular semigroups,

trivially satisfy this condition but undecomposable into a band of unipotent epigroups (see

Theorem 3.31). The second assertion is conﬁrmed by the semigroup V which, as is easily

seen, does not satisfy the identity

xy = x · y but is decomposable even into a semilattice

of (three) unipotent semigroups. Thus the class of epigroups under discussion lies strictly

between the classes ﬁguring in the premise in A and the conclusion in B of Proposition 3.43.

In [104] the problem of ﬁnding a divisor characterization of epigroups of this class was noted.

Such a characterization has been found in [124].

3.44 Theorem For an epigroup S,thesetGr S is a retract if and only if there are no

semigroups B

, L

3,1

, R

3,1

, V among the epidivisors of S.

The work [124] also contains (without proofs) divisor characterizations of epigroups whose

group part is a quasiideal, or a left ideal, or an ideal, or a retract ideal. Recall that the set

Q ⊆ S is called a quasiideal if QS ∩ SQ ⊆ Q. The characterizations mentioned involve the

following semigroups supplementing our “cast of characters”:

C = a, e |e

= e, ae = ea = a, a

=0;

Y = a, e, f |e

= e, f

= f, ef = fe =0,ea= af = a;

P = a, e |e

= e, ea = a, ae =0,

and the semigroup

←−

P which is dual to P . These semigroups have a very simple structure; C,

P ,and

←−

P are three-element, Y is four-element.

3.45 Theorem For an epigroup S,thesetGr S is a quasiideal of S if and only if there are

no semigroups C, Y , V among the epidivisors of S.

3.46 Theorem For an epigroup S,thesetGr S is a left ideal of S if and only if there are

no semigroups C, P among the epidivisors of S.

Theorem 3.46 and its dual immediately imply the following consequence.

3.47 Corollary For an epigroup S,thesetGr S is an ideal of S if and only if there are no

semigroups C, P ,

←−

P among the epidivisors of S.

By combining this corollary and Theorem 3.44, we obtain a characterization of epigroups

whose group part is a retract ideal. Since the semigroup P is, as easily seen, isomorphic to a

subsemigroup in both B

and V , this characterization is reduced to the following statement.

358 L. N. Shevrin

3.48 Theorem For an epigroup S,thesetGr S is a retract ideal of S if and only if there

are no semigroups C, P ,

←−

P , L

3,1

, R

3,1

among the epidivisors of S.

Let us now turn to epigroups satisfying the identity

xy = x·y. To the property indicated by

Lemma 3.43, the following properties have obviously to be added: all the maximal subgroups

are abelian and the subsemigroup generated by the idempotents satisﬁes the identity

x = x,

which is equivalent to the identity x

= x. Thus we have the following fact.

3.49 Proposition An epigroup S satisfying the identity

xy = x · y is decomposable into

a semilattice of rectangular bands of nil-extensions of abelian groups, and the subsemigroup

E

 satisﬁes the identity x

= x.

That the converse is false is demonstrated by the following example.

3.50 Example Let S be the semigroup given by the presentation

S = a, f |a

= a

,fa= f

= f.

(In [104] this semigroup was denoted by L

3,2

; observe that the semigroup L

3,1

which occured

several times above is a homomorphic image of the semigroup L

3,2

.) Obviously S consists

of ﬁve elements a, a

, f , af , a

f, and it is a chain of the three-element left zero semigroup

{f,af, a

f} and the semigroup a = C

2,1

. Hence S trivially satisﬁes all the conditions of

Proposition 3.49 (in particular E

 = E

satisﬁes even the identity x

= x). However the

premise is not fulﬁlled, since

af = af = a

f = a · f.

Therefore one may pose the following problem (mentioned in [104]).

3.51 Problem Find a structural characterization (in particular, in terms of forbidden divi-

sors) of the epigroups obeying the identity

xy = x · y.

If we limit ourselves to archimedean epigroups, then, as was shown in [104], the converse

of Proposition 3.49 is true, i.e., we have

3.52 Proposition An archimedean epigroup S satisﬁes the identity

xy = x · y if and only

if S is a rectangular band of nil-extensions of abelian groups, and the subsemigroup E



satisﬁes the identity x

= x.

At last, let us turn to epigroups satisfying the identity

xy = y ·x. They are characterized

from several points of view by the following statement proved in [104, §6].

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

(1) S satisﬁes the identity

xy = y · x;

(2) S satisﬁes the identities (xy)

=(yx)

, (x

)

= x

;

(3) there are no semigroups B

, L

, R

, V among the epidivisors of S;

(4a) S is a semilattice of unipotent epigroups, and E

is a subsemigroup (which is then a

semilattice);

Epigroups 359

(4b) S is a semilattice of unipotent epigroups, and Gr S is a subsemigroup.

Recall that a unary semigroup is called involutory if its unary operation is an anti-auto-

morphism of order 2. Theorem 3.53 has the following immediate

3.54 Corollary An epigroup S is an involutory semigroup (with respect to the operation

x →

x) if and only if S is a semilattice of groups.

3.5 Unipotency classes and Green’s relations

We will denote by K the equivalence relation on an epigroup corresponding to the partition

of the given epigroup into its unipotency classes. In this subsection we are going to talk

about the interaction between the relation K and the relations D, L, R, H;amongthe

dual situations for L, R, we will, for deﬁniteness, consider L. The results presented in this

subsection are proved in [104, §7].

Note ﬁrst that in any epigroup K⊆

√

H and, in any semigroup,

√

H⊆

√

L⊆

√

D.An

example of non-trivial connections is provided by Corollary 3.36 of Theorem 3.35. By anal-

ogy with the condition K =

√

D there, we will be interested in the equalities K =

√

L and

K =

√

H, which deﬁne wider classes of epigroups. As in the case of D, these equalities

are equivalent, respectively, to the inclusions L⊆Kand H⊆K. The equalities K = L and

K = H are stronger conditions; they, along with the equality K = D, were considered in [46]

(and, for periodic semigroups, in [85]). Since, as is easy to prove, K⊆Dholdsinanepigroup

S if and only if S is a completely regular semigroup, a characterization of the corresponding

epigroups follows directly from Cliﬀord’s theorem on the decomposability of any completely

regular semigroup into a semilattice of completely simple semigroups: in an epigroup S,we

have K = D [K = L, K = H] if and only if S is a semilattice of groups [a semilattice of right

groups, a completely regular semigroup].

Let us turn to the condition L⊆K. The following is true.

3.55 Lemma An epigroup in which L⊆Kis a semilattice of right archimedean epigroups.

There arises the question: Does the property mentioned in the conclusion of Lemma 3.55

characterize the epigroups under consideration? A negative answer is provided by the follow-

ing example.

3.56 Example Let S be the semigroup deﬁned by the presentation

S = a, g |a

= a

= g, g

a = ag

= a, ga

= a

= aga.

It is convenient to put e = a

, i = g

. It follows directly from the deﬁning relations that e, i

are idempotents, and e is a right zero. Then eg, eg

, eg

are also right zeros. It is now easy

to see that S consists of 16 elements and has ﬁve torsion classes: K

= g, K

= {a, gag

,e},

= {ag

,gag,eg}, K

= {ag

,ga,eg

}, K

= {ag, gag

,eg

The semigroup S is a chain of two right archimedean semigroups: K

∪K

2 ∪K

360 L. N. Shevrin

and K

(the latter is simply a cyclic group of order 4). The L-classes and R-classes are

= R

= g,

= {e},L

= {eg},L

= {eg

},L

= {eg

= {a, ga},L

= {ag, gag},L

= {ag

,gag

},L

= {ag

,gag

= eg,R

= ag,R

= gag.

In particular, S has L-equivalent elements lying in diﬀerent torsion classes.

It is possible to exhibit a semigroup with the desired properties having fewer elements

(e.g., the semigroup in Example 3.61), but the one in Example 3.56 is needed for one of the

conclusions below. For this reason, we indicate its H-classes: H

= g and the remaining

H-classes are singletons; thus H⊆K.

The following lemma, together with the preceding one, “brackets” the class of epigroups

under discussion.

3.57 Lemma If S is a right archimedean epigroup or a semilattice of right archimedean

epigroups in which the archimedean components are decomposable into a band of unipotent

epigroups, then L⊆Kin S.

The disjunction in the premise of Lemma 3.57 is also not a characteristic property of epi-

groups in which L⊆K, as is demonstrated by the example of the semigroup obtained from

3,1

by adjoining an identity element: the realtion L in this semigroup coincides with the

equality relation, but it is not archimedean, and its archimedean component R

3,1

is indecom-

posable into a band of unipotent epigroups. We see that the class of epigroups in question

lies strictly between the classes appearing in the premise of Lemma 3.57 and the conclusion of

Lemma 3.55. How can one characterize it in the spirit of the earlier characterizations in this

section? There is a divisor characterization of this class. Indeed, it is closed under subepi-

groups (which is obvious) and homomorphic images (which is proved in [104, Lemma 28]).

Thus we may pose the following problem (already noted in [104]).

3.58 Problem Find a divisor characterization of epigroups in which L⊆K.

It is easy to see that within each of the varieties E

, a direct product of epigroups such that

L⊆Kalso has this property. Therefore, in view of what was said before the formulation of

Problem 3.58, the class of such epigroups is a subvariety of E

. An equational characterization

of this class is provided by the following statement.

3.59 Proposition The largest subvariety of E

consisting of epigroups in which L⊆Kis

deﬁned within E

by the identity ((xy)

=(y(xy)

Let us now turn to epigroups in which H⊆K. The class of such epigroups is considerably

wider than the preceding one and consists not only of semilattices of archimedean epigroups

or even of unipotently partitionable epigroups, as is shown by the example of the semigroup

,whereH coincides with the equality relation. The connection between this class and

those considered earlier in this section is demonstrated by the following statement which is

an analogue of Lemma 3.57 in a more general situation.

Epigroups 361

3.60 Proposition If S is an archimedean semigroup or a semilattice of rectangular bands

of unipotent epigroups, then H⊆Kin S.

Since under the condition H = K we obtain the class of all completely regular semigroups,

one might put the question: Does the inclusion H⊆Khold in any epigroup? A negative

answer is provided by the following example.

3.61 Example Let S be the semigroup deﬁned by the presentation

S = a, g |a

= a

= g, ga = ag

,ag

= a, ga

= a

.

It follows from the deﬁning relations that g

a = a and a

= aga, so all the deﬁning

relations of the semigroup in Example 3.56 are satisﬁed. Thus S is a homomorphic image of

that semigroup; S is obtained from it by a pairwise “fusing” of the following elements: ga

and ag

, a and gag

, ag and gag

, ag

and gag. This implies the “fusion” of the R-classes

and R

as well as the L-classes L

and L

, L

and L

.ThusS has the following

2-element H-classes: H

= {a, ga}, H

= {ag, gag}; the elements of each of them lie in

diﬀerent torsion classes, e.g., a ∈ K

2 , ga ∈ K

2 . Thus the inclusion H⊆Kis not fulﬁlled

in S.

In contrast to the situation for epigroups in which L⊆K, the problem of ﬁnding a divisor

characterization for epigroups such that H⊆Kcannot be posed. This is caused by the fact

that the class of these epigroups is not homomorphically closed; it follows from the arguments

in Examples 3.56 and 3.61: as was noted above, H⊆Kin the epigroup of Example 3.56, and

the epigroup of Example 3.61 is a homomorphic image of the former.

4 Finiteness conditions

4.0 Introductory remarks

Given a class of algebraic systems, by a ﬁniteness condition is meant any property which is

possessed by all ﬁnite systems of this class. Imposing ﬁniteness conditions is a classical ap-

proach in investigations of algebraic systems of diﬀerent kinds. For semigroups, a broad group

of such conditions deals with conditions formulated in terms of ideals or congruences of certain

types, in particular, in terms of the partially ordered sets of principal (one-sided or two-sided)

ideals. Some facts in this direction are presented in [13, §6.6]; see also works [26, 30, 31, 42]

as well as [45, §3.6]. For example, in [26] the interdependence of several minimal conditions

(including, by the way, the property of being an epigroup) has been clariﬁed. In this sec-

tion we shall focus attention on another group of conditions which are expressed basically in

terms of subsemigroups, especially in terms of the lattice of subsemigroups. They distinguish

some subclasses within the class of epigroups. A trivial ﬁniteness condition is the property

of being plainly a ﬁnite semigroup. To describe semigroups with some non-trivial ﬁniteness

conditions, we should clarify, so to speak, a character and a degree of “deviations” from the

property of being a ﬁnite semigroup. For semigroups considered in Subsections 4.1 and 4.2,

such deviations will always happen in the maximal subgroups only. Thus we have a complete

reduction to groups here. Information on groups with the conditions under consideration is

given in Subsection 4.3.

362 L. N. Shevrin

Note that the conditions in question are hereditary for subsemigroups (or subepigroups),

so, as in the situations examined in Section 3, the idea of “forbidden” objects can be used for

a characterization of the corresponding semigroups. These objects are now subsemigroups,

and the role of forbidden subsemigroups is played by semigroups having a unique inﬁnite

basis. The absence of such subsemigroups is a crucial factor in a general scheme (presented

in Subsection 4.1) embracing many concrete ﬁniteness conditions.

For more detailed information on the matters discussed in Subsections 4.1–4.3 (including

the proofs of many results), see [108, Chapter IV]. In Subsection 4.4 we will brieﬂy touch on

several ﬁniteness conditions for nilsemigroups, predominantly calling the reader’s attention

to some open problems.

4.1 Finitely assembled semigroups

AsemigroupS is called ﬁnitely assembled (f.a.)ifthesetsE

and S \ Gr S are ﬁnite. Any

f.a. semigroup is evidently an epigroup. The term “ﬁnitely assembled” is justiﬁed by the

observation that such a semigroup looks as if it is assembled from a ﬁnite family of (disjoint)

groups and a ﬁnite set of (non-group) elements. If in such a semigroup S all its maximal

subgroups belong to a ﬁxed class X,wesaythatS is f.a. from groups of X. The structure

of f.a. semigroups is fairly clariﬁed by the following description.

4.1 Proposition AsemigroupS is ﬁnitely assembled if and only if S has a ﬁnite series of

ideals in which each factor is either ﬁnite or is a rectangular band of ﬁnitely many inﬁnite

groups (this takes place for the kernel of S only) or is obtained from such a band by the

adjunction of a zero.

Note that this proposition in its non-trivial part is based on some statement relating to

a more general situation, namely, when a given epigroup has ﬁnitely many idempotents (this

is Proposition 11.1 in [108]).

A key role in the subsequent considerations of this subsection is played by certain prop-

erties formulated in terms of bases. By a basis of an algebraic system is meant an irreducible

(i.e., minimal) generating set of the given system. In order to distinguish bases of an epi-

group regarded as a (usual) semigroup or a unary semigroup, an epigroup basis of the given

epigroup will be called an epibasis. Recall that a semigroup of idempotents is often called a

band.

4.2 Proposition Any epigroup with ﬁnitely many idempotents and inﬁnitely many non-

group elements has a (unipotent) subsemigroup [subepigroup] with a unique inﬁnite basis

[epibasis].

4.3 Proposition Any epigroup with inﬁnitely many idempotents has a subsemigroup with a

unique inﬁnite basis. Such a subsemigroup may be taken to be either a nilpotent semigroup

or a band; so in both cases it is a subepigroup with a unique inﬁnite epibasis.

Remark that the inevitable question of whether it is possible to retain only a band in the

conclusion of Proposition 4.3 has a negative answer, which the following example shows.

4.4 Example Consider a semigroup T = t

= t

,i∈ N, and denote by T

the set of all

elements in T of the form t

···t

,wheren ≥ 3 and adjacent elements t

are distinct.

Epigroups 363

Evidently T

is an ideal of T . It is seen that the Rees quotient semigroup S = T/T

has

inﬁnitely many idempotents; however all the maximal subbands in S are two-element: each

of them consists of the zero T

and {t

} for some i.

Propositions 4.2 and 4.3 immediately imply the following

4.5 Theorem If an epigroup has no subsemigroups with a unique inﬁnite basis, then it is

ﬁnitely assembled.

The property in the premise of this theorem is not characteristic for ﬁnitely assembled

semigroups. A trivial counter-example is given by any group having a subsemigroup with a

unique inﬁnite basis, for example, by a non-cyclic free group (which, as is well known, contains

a free subsemigroup of countable rank). However, by considering the subepigroup versions

of Propositions 4.2 and 4.3, we obtain another corollary of them which gives a necessary and

suﬃcient criterion.

4.6 Theorem Finitely assembled semigroups are exactly the epigroups without subepigroups

with a unique inﬁnite epibasis.

For an abstract semigroup-theoretic property θ, a semigroup possessing this property will

be called a θ-semigroup. For quite a number of ﬁniteness conditions, it is possible to cover

by a general scheme the description of semigroups satisfying these conditions. This scheme is

based on Theorem 4.5, or Theorem 4.6. To expound it, we need the following requirements

which may be satisﬁed for a property θ:

(A) θ is hereditary for subsemigroups;

(B) any θ-semigroup has no unique inﬁnite basis;

semigroup.

4.7 Theorem Let θ be a ﬁniteness condition satisfying the requirements (A)–(C).Thena

semigroup is a θ-epigroup if and only if it is ﬁnitely assembled from θ-groups.

When considering epigroups, one may also deal with similar requirements concerning

subepigroups:



) θ is hereditary for subepigroups;



)anyθ-epigroup has no unique inﬁnite epibasis;



) any epigroup being the union of a ﬁnite family of θ-subepigroups is itself a θ-epigroup.

4.7



Theorem Let θ be a ﬁniteness condition satisfying the requirements (A



)–(C



) and, in

addition, such that an ideal extension of a θ-epigroup by a ﬁnite semigroup is a θ-epigroup.

Then a semigroup is a θ-epigroup if and only if it is ﬁnitely assembled from θ-groups.

These theorems give, under highly general requirements on a property θ, a description

of θ-epigroups eﬀecting a complete reduction to the case of groups. So further possible

advances in clariﬁcation of the structure of epigroups under consideration becomes entirely a

364 L. N. Shevrin

task of group theory. It concerns, in particular, the interrelation between diﬀerent conditions.

Indeed, let E(θ)andG(θ) denote the classes of all θ-epigroups and all θ-groups, respectively;

then we have the following immediate consequence of Theorems 4.7 and 4.7



4.8 Proposition Let θ

and θ

be two ﬁniteness conditions satisfying the requirements in

Theorem 4.7 or in Theorem 4.7



. Then the inclusion E(θ

) ⊆E(θ

) holds if (and obviously

only if ) G(θ

) ⊆G(θ

). This, in turn, implies similar conclusions for strict inclusion and

for the equality relation. Furthermore, if E(θ

) ⊂E(θ

),thenanyθ

-epigroup which is not a

-epigroup contains a subgroup which, being a θ

-group, is not a θ

-group.

The general scheme presented above has arisen as a result of several subsequent approx-

imations. It was ﬁrst proposed, for the case of periodic semigroups and in a slightly weaker

form, in [93]. A strengthened variant was exhibited in [94]. Later the author found an im-

proved variant, but, as before, for periodic semigroups only; it was announced in [105] (notice

that f.a. semigroups were called almost ﬁnite there). Finally, the latter was extended to epi-

groups in [106]. Theorems 4.5 and 4.7 were announced in [99], Theorems 4.6 and 4.7



were

stated in [100].

In Subsection 4.2 we consider a number of concrete ﬁniteness conditions covered by this

general scheme, namely, by Theorem 4.7. It should be noted that in most cases the semigroups

under consideration turn out to be periodic. Since a subsemigroup of a periodic semigroup

is automatically an epigroup, there is no need to apply Theorem 4.7



in such cases, and

in the formulation of the corresponding speciﬁcations of Theorem 4.7 the assumption that

a semigroup be a priori an epigroup may be omitted. Furthermore, since a subsemigroup

of a periodic group is in fact a subgroup, the corresponding properties of groups may then

be formulated in terms of subgroups. Information about groups with the conditions under

discussion will be given in Subsection 4.3.

4.2 Certain concrete ﬁniteness conditions

In the applications of Theorem 4.7 to concrete cases, one can easily verify requirements (A)–

Almost all of the conditions considered below are formulated in terms of the lattice Sub of all

subsemigroups of a semigroup S. (In order to have the right to speak of Sub S as a lattice,

one has to treat the empty set as a subsemigroup).

If Sub S satisﬁes the minimal [maximal] condition, we will call such a semigroup S a

Min-semigroup [Max-semigroup]. It is obvious that any Min-semigroup is periodic. For this

condition Theorem 4.7 turns into the following statement.

4.9 Theorem AsemigroupS is a Min-semigroup if and only if S is ﬁnitely assembled from

groups with the minimal condition for subgroups.

It is clear that the property of a semigroup of having only ﬁnite proper subsemigroups is

stronger than the property of being a Min-semigroup. Therefore Theorem 4.9 immediately

implies the following consequence in which a reduction to groups turns out to be simply

“dismembered”.

4.10 Corollary In a semigroup S, all the proper subsemigroups are ﬁnite if and only if S

is either a ﬁnite semigroup or an inﬁnite group whose proper subgroups are ﬁnite.

Epigroups 365

The assertion of this corollary for the case of commutative semigroups was proved in [35],

and it was observed there that an inﬁnite abelian group whose proper subgroups are ﬁnite is a

quasicyclic group. Recall that a group G is called quasicyclic if it is given by the presentation

G = a

,...,a

=1,a

n+1

= a

for all n ∈ N,

where p is a prime number. The characteristic property of quasicyclic groups is that the

lattice of subgroups is an inﬁnite chain (being in reality isomorphic to the chain N with

respect to the ordinary order).

In connection with Corollary 4.10, note that an interesting development of the theme

“semigroups whose proper subsemigroups have a lesser cardinality than that of the whole

semigroup” was accomplished in [48]. A semigroup with the property just mentioned was

called a J´onsson semigroup there, and the following statement was proved.

4.11 Theorem If the generalized continuum hypothesis is assumed, then any J´onsson semi-

group is a group.

Let us turn to Max-semigroups. Such a semigroup does not have to be periodic or even

an epigroup; an example is provided by the inﬁnite cyclic semigroup. Therefore in the

corresponding concrete version of Theorem 4.7 one cannot eliminate the assumption that a

semigroup under consideration be an epigroup. We obtain the following criterion.

4.12 Theorem An epigroup S is a Max-semigroup if and only if S is ﬁnitely assembled from

Max-groups.

As for Max-semigroups which are not epigroups, the situation is far from clear. One may

ﬁrst notice the following property which is weaker than ﬁnite assembleness.

4.13 Proposition Any Max-semigroup has ﬁnitely many idempotents.

Further information about Max-semigroups can be given for commutative semigroups.

Here we have a complete description. Recall that any commutative semigroup is decompos-

able into a semilattice of archimedean semigroups, so one may speak about the archimedean

components of such a semigroup.

4.14 Theorem For a commutative semigroup S, the following conditions are equivalent:

(1) S is a Max-semigroup;

(2) S has ﬁnitely many archimedean components each of which is an ideal extension of a

semigroup embeddable in an (abelian)Max-group by a ﬁnite nilpotent semigroup;

(3) S is embeddable in a (commutative) semigroup which is ﬁnitely assembled from Max-

groups.

Condition (2) here is a modiﬁed variant of the criterion formulated in [37], the criterion

given by condition (3) was found in [107]. Note that certain information about commutative

Max-semigroups was previously obtained in [47]. In particular, the following statement has

been proved there.

366 L. N. Shevrin

4.15 Proposition For an archimedean commutative semigroup S without idempotents, the

following conditions are equivalent:

(1) S is a Max-semigroup;

(2) S is ﬁnitely generated;

(3) S is embeddable in the direct product of the inﬁnite cyclic semigroup and a ﬁnite unipo-

tent semigroup.

A question concerning possible clariﬁcation of the structure of arbitrary Max-semigroups

was posed by the author in the late sixties and was formulated in [105, Question 50]. Con-

dition (3) of Theorem 4.14 suggests a plausible conjecture whose aﬃrmation would mean

obtaining a good reducing description of Max-semigroups. This conjecture is that the answer

to question (a) in the following problem is aﬃrmative. It seems advisable to consider also

questions (b) and (c) in this problem which are consequently weaker from the point of view

of the positive answers.

4.16 Problem

(a) Is any Max-semigroup embeddable in a semigroup which is ﬁnitely assembled from Max-

groups?

(b) Is any Max-semigroup embeddable in a ﬁnitely assembled semigroup?

These questions were ﬁrst formulated in [107], together with the following problem.

4.17 Problem Is any Max-semigroup that can be embedded in a group embeddable in a

Max-group?

In what follows we mention several more conditions covered by Theorem 4.7. There is no

need to reproduce the special formulations of the theorem for these conditions; remark only

that each of the conditions under consideration implies periodicity of a semigroup satisfying

this condition. We present mostly the deﬁnitions of these conditions and note some inter-

relations between them. As already remarked in Subsection 4.0, all relevant details can be

found in [108, Chapter IV].

AsemigroupS is said to have ﬁnite rank [ﬁnite breadth] r if any ﬁnitely generated

subsemigroup of S is generated by at most r elements [for any r + 1 elements of S,atleast

one belongs to the subsemigroup generated by the others], and r is the least number with

this property. A semigroup which is not a semigroup of ﬁnite rank [ﬁnite breadth] is called a

semigroup of inﬁnite rank [inﬁnite breadth]; similarly we shall use the word “inﬁnite” for to

other conditions which have a numerical meaning. The formulated deﬁnition of a semigroup

of ﬁnite breadth is equivalent to the deﬁnition given originally in terms of the subsemigroup

lattice; for a semigroup S, the property of having ﬁnite rank cannot be expressed in terms

of Sub S. Any semigroup of ﬁnite breadth is of ﬁnite rank which is not greater than the

breadth. The direct product of a countable set of cyclic groups which have distinct prime

orders provides an example of a semigroup of rank 1 and of inﬁnite breadth.