Kudryavtsev V.B., Rosenberg I.G. Structural Theory of Automata, Semigroups, and Universal Algebra
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Epigroups 367
AsemigroupS is said to be narrow [of finite length ] if all antichains in the lattice Sub S
are finite [all chains in Sub S are finite, their lengths are bounded by ,and is the least
number with this property]. A simple lattice-theoretic observation shows that a semigroup
of finite length has finite breadth which is not greater than the length. A quasicyclic group
provides an example of a semigroup of breadth 1 and infinite length. Any narrow semigroup
is a Min-semigroup (it is easy to show) and has finite breadth (it follows from Theorems 4.26
and 4.22, taking into account Proposition 4.3).
A semigroup is said to be of finite lattice dimension d if Sub S has dimension d,which
means that SubS is embeddable in the direct product of d chains, and d is the least number
with this property. Any semigroup of finite lattice dimension has finite breadth which is not
greater than this dimension (this is a manifestation of a non-trivial lattice-theoretic fact).
The direct product of two quasicyclic p-groups for the same p provides an example of a
semigroup of breadth 2 which is not narrow and has infinite lattice dimension.
There are some other conditions of the same character. We mention only one of them
here; not being included in our general scheme (namely, requirement (C) is not satisfied),
it is closely connected with the conditions considered above. We will say that a semigroup
S is of finite width w if Sub S has finite width w, which means that the cardinalities of all
antichains in Sub S are bounded by w,andw is the least number with this property. Any
semigroup of finite width is obviously narrow. The direct product of two quasicyclic groups
for distinct primes provides an example of a narrow semigroup of infinite width.
4.18 Theorem (a) Any semigroup of finite width is finitely assembled from groups of finite
width. (b) A finitely assembled semigroup S in which at most one of the maximal subgroups
is an infinite group of finite width, and the others are finite, is a semigroup of finite width.
The necessary [sufficient] condition given by statement (a) [resp., (b)] is not sufficient
[necessary]; the counter-examples are given in [108]. The gap between these two conditions
seems to be rather narrow. This augments the interest in the following problem which was
actually noted in [90] and formulated explicitly in [106, Part 1].
4.19 Problem Find a necessary and sufficient condition describing the structure of semi-
groups of finite width.
4.3 The case of groups
First of all, note that, for the subgroup lattice of a group, the minimal condition implies
periodicity of the given group; the same takes place for finiteness of breadth, finiteness of
dimension, the property of being narrow (and thereby for finiteness of length or width).
Hence, for groups satisfying these conditions, there is no difference between subsemigroups
and subgroups.
We start our considerations here by mentioning that there was the long-standing problem
of whether any group of finite length is in fact finite. This problem was posed by I. Kaplansky
in the forties, see [4, Problem 43]. Groups providing a negative solution have been constructed
in [55–57], see also [58, Chapter 9]. In these groups every proper subgroup is of prime order,
so the subgroup lattice of such a group has the following diagram.
368 L. N. Shevrin
Here the prime orders of proper subgroups can be either distinct or the same. In the latter
case one obtains the answer to a question which was known as the Tarski’s problem: Does
there exist an infinite group whose proper subgroups have the same prime order? The groups
constructed in the works mentioned above gave the solution of other problems open for a long
time (see, for instance, comments at the beginning of §27 in [58], or comments on Question Γ4
in [33, §1]). The proofs of such results are based on fruitful geometric methods to study groups
presented by defining relations, see [58].
Let us turn to the other conditions considered in Subsection 4.2. We will use the notation
Min [resp., Max] for the minimal [maximal] condition for subsemigroups, Rn [resp., Br, Ln] for
finiteness of rank [breadth, length], and Per for periodicity. We have the following implication,
observed above: Ln =⇒ Min & Max & Br, Br =⇒ Rn =⇒ Per. There were open questions
concerning more precise interrelations between these conditions. Several such questions were
formulated in [89] (see also [105, Questions 51–55], and [33], where all the questions from [89]
are reproduced). In [33] the examination of the interdependence of the listed conditions was
completed: the questions have been solved negatively, even in their strongest form. The main
result of [33] is the following theorem.
4.20 Theorem (a) The conjunction Min & Max & Br does not imply Ln. (b) The conjunc-
tion Min & Max & Rn does not imply Br. (c) The conjunction Min & Max does not imply Rn.
(d) The conjunction Min & Br does not imply Max. (e) The conjunction Max & Br does not
imply Min.
Previously, an example of a group constructed in [14] showed that Min does not imply
Br. In [33] there are some related results as well. For instance, the problem discussed
in [105, Question 56] was settled. That question was about the interrelation between Min
and the property of having an at most countable subgroup lattice for periodic groups. It
turned out that both implications are not true for the conditions in question, see Theorems 4
and 5 in [33] (the latter is taken from [54]). In contrast to the situation for arbitrary periodic
groups, these two conditions as well as Min and Br are equivalent for locally finite groups,
see Theorem 4.22 below. Recall that a group G is called locally finite (l.f.) if every finitely
generated subgroup of G is finite.
A special type of l.f. groups is represented by so-called Chernikov groups. A group is said
to be a Chernikov group if it is an extension of a direct product of finitely many quasicyclic
groups by a finite group. It is well known that Chernikov groups satisfy the minimal condition
for subgroups. One of the crucial results in this area is given by the following theorem proved
independently in [38] and [111] (for a proof, see also [39]).
4.21 Theorem A locally finite group satisfies the minimal condition for abelian subgroups
if and only if it is a Chernikov group.
Taking into account this theorem as well as some properties of groups of finite breadth
established in [90], we obtain the equivalence of conditions (1), (2), and (4) of the next
theorem. As to condition (3), it can be added in view of Theorem 13.9 of [108].
Epigroups 369
4.22 Theorem For a locally finite group G, the following conditions are equivalent:
(1) G is a Min-group;
(2) G is a Br-group;
(3) the subgroup lattice of G is at most countable;
(4) G is a Chernikov group.
Any Max-group is obviously Noetherian, i.e., satisfies the maximal condition for sub-
groups, but the former condition is stronger than the latter, even in the case of abelian
groups, as the following statement shows.
4.23 Proposition An infinite abelian group is a Max-group if and only if it is the direct
product of the infinite cyclic group and a finite group.
This statement can be generalized as follows, where the situation is completely transpar-
ent. Recall that, for some property θ, a group is said to be a finite extension of a θ-group,or
θ-by-finite, if it has a normal subgroup of finite index which is a θ-group.
4.24 Theorem Any soluble Max-group is necessarily cyclic-by-finite. Conversely, any cy-
clic-by-finite and, more generally, any Max-by-finite group is a Max-group.
The assertions of this theorem for the case of soluble groups were proved in [107]. As to
the converse assertion in the general case, it was noted in [107] as an open question. This
question was reproduced also in [106, Part 1], and in [34], and it was soon settled in [59],
that has been reflected in [108, Lemma 13.5.5].
The existence of the groups mentioned in Theorem 4.20 gives no hope of obtaining a
satisfactory description of periodic Max-groups. As to non-periodic Max-groups, apparently
there is also no hope of obtaining a transparent description: the types of such groups are
too diverse, as the following statements show. These statements were proved in [34], see
Theorems 1 and 2 there.
4.25 Theorem There are continuously many non-isomorphic 2-generated torsion-free groups
in each of which every maximal subsemigroup is a cyclic group, and distinct maximal sub-
groups intersect trivially. For any sufficiently big prime p (e.g., p>10
80
) there are con-
tinuously many non-isomorphic 2-generated non-periodic groups G such that the identity
x
p
(x
p
y
p
)
p
=(x
p
y
p
)
p
x
p
holds in G; every maximal subsemigroup of G is a cyclic group either
of infinite order or of order p, and distinct maximal subgroups of G intersect trivially.
The structure of infinite narrow groups and infinite groups of finite width is described by
the following statements proved in [92] and [90], respectively. One can see that such groups
are automatically locally finite.
4.26 Theorem An infinite group is narrow [of finite width] if and only if it is the direct prod-
uct of a finite group and finitely many quasicyclic groups with distinct primes [a quasicyclic
p-group for some prime p] not dividing the order of the finite factor.
The next theorem was stated in [101].
370 L. N. Shevrin
4.27 Theorem A locally finite group of finite lattice dimension is a finite extension of the
direct product of finitely many quasicyclic groups for distinct primes.
It seems quite plausible that the converse to this statement is true as well. But there is
no proof by now, therefore one has to formulate the following
4.28 Problem Does any group which is a finite extension of the direct product of finitely
many quasicyclic groups for distinct primes have finite lattice dimension?
4.4 On nilsemigroups
Being of independent interest, along with groups, as one of the most important types of
unipotent epigroups, nilsemigroups occur in various semigroup-theoretic investigations. In
particular, they appear as certain components of the structure of some epigroups examined;
see relevant statements in Section 3. Remark that nilsemigroups could serve as the subject
matter of a separate survey, and the author hopes to prepare such a survey in the future. Here
we restrict our consideration only to information relating to some long-standing problems.
Note first that all the conditions considered in Subsection 4.2, as applied to nilsemigroups,
are reduced to the property of simply being a finite semigroup. This may be regarded as
an immediate consequence of the general result given by Theorem 4.7. But what actually
happens is that the proof of this result (in essence, of Propositions 4.1 and 4.2) uses in one
of the key steps the following fact about nilsemigroups established in [87].
4.29 Theorem Any nilsemigroup whose nilpotent subsemigroups are finite is itself finite.
Applications of this fact are based on the observation that, for a nilpotent semigroup T
(which is not a singleton), the set T \T
2
is its unique basis, and T is finite if and only if this
basis is finite. Hence for many properties which are hereditary for subsemigroups, obvious
arguments reduce questions discussed for nilsemigroups to their nilpotent subsemigroups.
It is easy to ascertain that a finite nilsemigroup is in fact nilpotent. Therefore the prop-
erty of being a nilpotent semigroup may be regarded as the strongest non-trivial finiteness
condition for the class Ni. There are diverse weaker such conditions; we will consider only
a few of them.
A semigroup is called locally nilpotent (l.n.) if each of its finitely generated subsemigroups
is nilpotent. Remark that if we speak about a nilpotent subsemigroup H of some semigroup S,
then the zero of H does not have to be the zero of S, but it is obvious that any l.n. semigroup
is unipotent and so is a nilsemigroup. Since a finitely generated nilpotent semigroup is finite,
l.n. semigroups are precisely locally finite nilsemigroups. There exists an infinite 2-generated
[3-generated] semigroup satisfying the identity x
3
=0[x
2
= 0]. Such nilsemigroups which
are not l.n. can be easily constructed by using the famous so-called Thue-Morse words (see,
for instance, [49, Section A3], or [45, §2.6]).
A proper subclass of the class of l.n. semigroups is formed by semigroups with the idealizer
condition. Recall that the idealizer of a subsemigroup T in a semigroup S is the greatest
subsemigroup in S containing T as an ideal. A semigroup S is said to satisfy the idealizer
condition,ortobeanI-semigroup, if any proper subsemigroup of S is distinct from its
idealizer. A well ordered series {0} = A
0
⊂ A
1
⊂···⊂A
α
⊂ A
α+1
⊂···⊂A
β
= S of ideals
of a semigroup S with zero (where, as usual, A
α
=
γ<α
A
γ
for every limit ordinal α)is
Epigroups 371
called an ascending annihilator series (a.a.s.)ofS if SA
α+1
∪A
α+1
S ⊆ A
α
for every α<β.
It is easy to see that every semigroup with an a.a.s. is an I-semigroup. In [86] it was shown
that every I-semigroup is l.n. However, it is unknown whether the converse is true, so the
following problem is still open.
4.30 Problem Does the idealizer condition imply the existence of an ascending annihilator
series in a given semigroup?
This problem arose in the early sixties and was explicitly formulated by the author later
in [91, Problem 1.50a] in the following equivalent form: Does every I-semigroup have the
non-trivial annihilator? Recall that the annihilator of a semigroup S with zero is the set
{a ∈ S |ax = xa =0foreachx ∈ S}. The equivalence of these two formulations can
be easily derived from the fact that a homomorphic image of an I-semigroup is again an
I-semigroup.
Another open problem concerns semigroups all of whose proper subsemigroups are nilpo-
tent. We call such a semigroup almost nilpotent.IfH is a proper nilpotent subsemigroup
of a semigroup S, and the zero of H is the zero of S,thenH is distinct from its idealizer.
Therefore every almost nilpotent nilsemigroup is an I-semigroup. As is shown in [88], almost
nilpotent semigroups not being nilsemigroups are exhausted by 2-element bands and cyclic
groups of prime order. For the nil case, we have the following problem.
4.31 Problem Is any almost nilpotent nilsemigroup in fact nilpotent?
This problem was first noted in [86] and, after that, in [88]. It was formulated later
in [91, Problem 1.52], and from that time this intriguing problem attracted the attention of
a number of researchers. There are results which give an affirmative answer under different
additional restrictions on a semigroup; they concern Problem 4.30 as well. We mention
particularly the paper [28], where, among other results, certain connections of Problems 4.30
and 4.31 with yet another problem relating to I-semigroups (namely, Problem 1.51a from [91])
was established. A survey of the results concerning the problems under discussion was given
in [103]. As was noted in the work mentioned, these problems are connected: namely, a
counter-example to Problem 4.31 will be at the same time a counter-example to Problem 4.30.
In [103] some properties are pointed out which have to be possessed by an almost nilpotent
but non-nilpotent nilsemigroup if such a semigroup exists. These properties show that a
corresponding example should be rather freakish. An approach to produce such an example
was also suggested in [103]; moreover, even a candidate for a counter-example was proposed
there. However, in the plan of a hypothetical proof, a concrete place was indicated where the
construction proposed might lead to a collapse. Such a collapse was, indeed, detected later
in [112]. So, Problem 4.31, being open by now for more than 40 years, is waiting for new
attempts to attack it.
The following diagram of inclusions shows the interrelations between the above-mentioned
classes of semigroups.
372 L. N. Shevrin
.
Continuous lines correspond to strict inclusions, dotted ones mean that we have two unsolved
problems here.
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