Semigroups with certain finiteness conditions and Chernikov groups
The main purpose of this short survey is to show how groups of special structure, which are accepted to be called Chernikov groups, appeared in the considerations of semigroups with certain finiteness conditions. A structure of groups with several such conditions has been described (they turned out...
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nasplib_isofts_kiev_ua-123456789-1522122025-02-09T14:37:09Z Semigroups with certain finiteness conditions and Chernikov groups Shevrin, L.N. The main purpose of this short survey is to show how groups of special structure, which are accepted to be called Chernikov groups, appeared in the considerations of semigroups with certain finiteness conditions. A structure of groups with several such conditions has been described (they turned out to be special types of Chernikov groups). Lastly, a question concerning some special type of Chernikov groups is recalled; this question was raised by the author more than 35 years ago, and it is still open. English version of the paper published in Russian in the book “Algebra and Linear Inequalities. To the Centennial of the Birthday of S. N. Chernikov”, Ekaterinburg, 2012. Supported by the Russian Foundation for Basic Research, grant 10-01-00524. 2012 Article Semigroups with certain finiteness conditions and Chernikov groups / L.N. Shevrin // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 2. — С. 299–306. — Бібліогр.: 6 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/152212 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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The main purpose of this short survey is to show how groups of special structure, which are accepted to be called Chernikov groups, appeared in the considerations of semigroups with certain finiteness conditions. A structure of groups with several such conditions has been described (they turned out to be special types of Chernikov groups). Lastly, a question concerning some special type of Chernikov groups is recalled; this question was raised by the author more than 35 years ago, and it is still open. |
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Shevrin, L.N. |
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Shevrin, L.N. Semigroups with certain finiteness conditions and Chernikov groups Algebra and Discrete Mathematics |
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Shevrin, L.N. |
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Shevrin, L.N. |
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Semigroups with certain finiteness conditions and Chernikov groups |
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Semigroups with certain finiteness conditions and Chernikov groups |
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Semigroups with certain finiteness conditions and Chernikov groups |
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Semigroups with certain finiteness conditions and Chernikov groups |
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Semigroups with certain finiteness conditions and Chernikov groups |
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semigroups with certain finiteness conditions and chernikov groups |
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Інститут прикладної математики і механіки НАН України |
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2012 |
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Semigroups with certain finiteness conditions and Chernikov groups / L.N. Shevrin // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 2. — С. 299–306. — Бібліогр.: 6 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT shevrinln semigroupswithcertainfinitenessconditionsandchernikovgroups |
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2025-11-26T22:58:25Z |
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1849895585416478720 |
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h.Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 13 (2012). Number 2. pp. 299 – 306
c© Journal “Algebra and Discrete Mathematics”
Semigroups with certain finiteness conditions
and Chernikov groups∗
L. N. Shevrin
Communicated by V. V. Kirichenko
Dedicated to the 100th birthday of Sergei Nikolaevich Chernikov
Abstract. The main purpose of this short survey is to show
how groups of special structure, which are accepted to be called
Chernikov groups, appeared in the considerations of semigroups
with certain finiteness conditions. A structure of groups with several
such conditions has been described (they turned out to be special
types of Chernikov groups). Lastly, a question concerning some
special type of Chernikov groups is recalled; this question was raised
by the author more than 35 years ago, and it is still open.
1. Introduction. One of the main lines in investigations of the
present writer was devoted to study of semigroups with finiteness con-
ditions. I recall that, given a class of algebraic systems, by a finiteness
condition is meant any property which is possessed by all finite sys-
tems of this class. In the works of the mentioned line, such conditions
were formulated in terms of subsemigroups or in terms of the lattice of
subsemigroups of a semigroup. When describing a structure of systems
with a non-trivial finiteness condition, one should clarify, so to say, a char-
acter and a degree of “deviations” from the property of being a finite
system. Applying to the semigroups having been considered, the revealed
∗English version of the paper published in Russian in the book “Algebra and Linear
Inequalities. To the Centennial of the Birthday of S. N. Chernikov”, Ekaterinburg, 2012.
Supported by the Russian Foundation for Basic Research, grant 10-01-00524.
Key words and phrases: finiteness condition, epigroup, finitely assembled semi-
group, Chernikov group.
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300 Semigroups and Chernikov groups
deviations almost always took place in their maximal subgroups. A key
general result obtained by the author was a result giving a complete re-
duction to the group case; it will be formulated below. Thereby a further
clearing the structure of semigroups under consideration or exact interre-
lations between classes of such semigroups required either references to
known results about groups or obtaining new group-theoretic results or,
if there was success neither in the former nor in the latter, raising open
questions concerning groups. All these three situations occurred in reality
in a massive of effected investigations. A detailed presentation of the
relevant rather rich material is given in Chapter IV of the monographs [1]
and [2] (the latter is not simply an English translation of the former but
its revised and enlarged version), and the interested reader is referred to
this chapter. The main purpose of this paper is to narrate briefly how
groups of special structure, which are accepted to be called Chernikov
groups (see a definition in Section 3 below), appeared in the considerations
mentioned. At the end of the paper, I want to recall a question concerning
some special type of Chernikov groups; this question was raised by the
author more than 35 years ago, and it is still open.
2. Reduction Theorem. It is assumed that the reader is acquainted
with more or less standard algebraic notions used in the paper. I give
only definitions of several notions which are not well-known. An element
of a semigroup is called a group element if it is contained in a subgroup
of the given semigroup. A semigroup is called an epigroup if for any of its
elements some power of this element is a group element. A semigroup is
called finitely assembled if it has finitely many non-group elements and
finitely many idempotents. Thereby a semigroup S is finitely assembled
if and only if it has finitely many maximal subgroups, say, G1, . . . , Gn
(as is known, maximal subgroups are mutually disjoint), and the set
S \ (G1 ∪ . . . ∪ Gn) is finite. If here all groups G1, . . . , Gn possess some
property θ (in other words, are θ-groups), then we say that S is finitely
assembled from θ-groups. Any finitely assembled semigroup is obviously
an epigroup.
By a basis of a semigroup S we mean an irreducible, i.e. minimal,
generating set of S.
We need the following requirements which may be satisfied for
a semigroup-theoretic property θ:
A) any subsemigroup of a θ-semigroup is a θ-semigroup;
B) any θ-semigroup has no unique infinite basis;
C) any semigroup covered by a finite set of θ-subsemigroups is itself
a θ-semigroup.
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L. N. Shevrin 301
A general result mentioned in Section 1 is formulated as follows.
Let θ be a finiteness condition satisfying the requirements A)–C) listed
above. An epigroup is a θ-semigroup if and only if it is finitely assembled
from θ-groups.
This statement (let us call it Reduction Theorem) gives, under highly
general requirements on a property θ, a description of θ-epigroups effect-
ing a complete reduction to groups. This reduction, in particular, can
be taken into account when constructing counter-examples for proving
the distinction between some finiteness conditions examined. Namely,
if, for some properties θ1 and θ2 embraced by Reduction Theorem, there
exists a θ1-semigroup S which is not a θ2-semigroup, then, according
to this theorem, S necessarily contains a subgroup which is a θ1-group
but not a θ2-group; therefore it is clear that a search of corresponding
counter-examples has to be done only for groups.
A general scheme given in Reduction Theorem embraces many concrete
finiteness conditions. And for each of them the fulfillment of requirement A)
is practically always trivial; requirement B) is often fulfilled even in
a stronger variant: any θ-semigroup has no infinite basis at all (and usually
it is easy to see); only verification of the fulfillment of requirement C),
as a rule, is not straightforward and sometimes turns out to be rather
non-trivial.
Note that semigroups with the conditions being studied are in most
cases automatically periodic, so, in concrete formulations for such cases,
there is no need to mention the requirement for the semigroup under
consideration to be an epigroup (since any periodic semigroup is an
epigroup). Furthermore, since any subsemigroup of a periodic group is
a subgroup, in formulations for these cases we may mention groups with
the same condition applying to subgroups. A typical example of such
a situation is presented by the minimal condition (for subsemigroups and,
respectively, for subgroups). In accordance with one of the common terms,
we shall call corresponding semigroups and groups artinian. Artinian
semigroups are obviously periodic, so a concrete version of Reduction
Theorem for them looks as follows.
A semigroup is artinian if and only if it is finitely assembled from
artinian groups.
3. On artinian groups. S. N. Chernikov in his group-theoretic
works devoted considerable attention to artinian groups. He revealed
a special type of such groups having a very distinct structure; the term
“Chernikov group” for such a group has become common for a long time
(Chernikov in his papers used the term “extremal group”). A group
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302 Semigroups and Chernikov groups
is called a Chernikov group if it is an extension of a direct product
of finitely many quasicyclic groups by a finite group. (For the sake of
accuracy, it is worth noting that with zero quantity of quasicyclic factors
in a Chernikov group we obtain merely a finite group.) Chernikov has
found, in particular, that any locally soluble artinian group is necessarily
a Chernikov group; the same takes place under a weaker condition, namely
the minimal condition for abelian subgroups. The interested reader can
find the proofs of the corresponding results in [3], Chapter 4, Section 2,
in [4], Section 24, or in [5], Section 3.4. Related results concerning groups
with one or another of the minimal conditions were obtained later by
some pupils of S. N. Chernikov. The most principal of them is a result
by V. P. Shunkov (1970) proving that a locally finite artinian group is
a Chernikov group (the same result was obtained also by O. H. Kegel and
B. A. F. Wehrfritz in the same year). Shunkov has established a stronger
result that a locally finite group with the minimal condition for abelian
subgroups is a Chernikov group; it gave the answer to one question posed
by Chernikov in 1959.
The last result increased the interest to an old question whether in gen-
eral artinian groups are exhausted by Chernikov groups; this question was
posed by Chernikov as long ago as in 1940. Counter-examples were con-
structed only at the junction of the 1970s and 1980s by A. Yu. Ol’shanskii
owing to the effective technique of geometric methods in combinatorial
group theory (see his monograph [6]). Among them infinite groups whose
proper subgroups have prime orders (and in the known examples of such
groups these orders either distinct or the same). In [1] and [2] we called
any such infinite group an Ol’shanskii group. The existence of Ol’shanskii
groups allowed to obtain answers to certain other questions which were
open for a long time.
4. On groups of finite breadth. Chernikov groups appeared also
when the author of the present paper considered a number of other
finiteness conditions which were not considered for groups before. The
first of such conditions is finiteness of breadth. The notion of breadth came
from lattice theory, and the initial definition is given in lattice-theoretic
terms as applied to the lattice of subsemigroups of a semigroup and the
lattice of subgroups of a group (as it was already noted, one may read
more widely in Chapter IV of [1] or [2]). However there is an equivalent
condition that can be formulated in terms of generating sets. Namely,
a semigroup [group] is of breadth b if and only if for any b + 1 of its
elements at least one belongs to the subsemigroup [subgroup] generated
by the others, and b is the least number with this property. The property
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L. N. Shevrin 303
of being a semigroup of finite breadth is embraced by the scheme given
in Reduction Theorem, and it easy to ascertain that semigroups and
groups of finite breadth are periodic. Therefore, according to this theorem,
a semigroup is of finite breadth if and only is it is finitely assembled from
groups of finite breadth.
Note that this reduction had been proved by the author separately
for the property of having finite breadth and for several other finiteness
conditions in the paper “Certain finiteness conditions in the theory of
semigroups” (Izv. AN SSSR. Ser. Matem., 1965) published before a general
reduction scheme was found, which took place considerably later. Before
that paper the author published the paper “Semigroups of finite breadth”
(In: “Theory of semigroups and its applications”, Saratov Univ., 1965),
where semigroups and groups satisfying this condition and some related
conditions were first studied.
What can one say about groups of finite breadth? Some general prop-
erties of such groups had been revealed in the paper “Semigroups of finite
breadth” mentioned. Among other properties, it was established there that
any Chernikov group has finite breadth, and for abelian groups (as well as
for groups of a certain wider class) finiteness of breadth is equivalent to
the property of being an artinian group. Examples of groups showing dif-
ference of these two properties were unknown, which impelled the author
to pose two questions about the fulfillment of possible implications con-
necting these finiteness conditions. These questions were formulated in the
paper “Semigroups of finite breadth” and were included in a list of seven
open questions about possible implications between several finiteness
conditions given in the paper “Certain finiteness conditions in the theory
of semigroups” mentioned above. They were included also in the first edi-
tion of “Kourovka Notebook (Unsolved problems of group theory)”, 1965;
these are questions 1.81 a) and b). Counter-examples giving negative
answers to them were constructed in the 1980s by Ol’shanskii’s pupils
G. S. Deryabina and S. V. Ivanov. Note that Ol’shanskii groups did not
give answers to these questions; indeed, any Ol’shanskii group is artinian
and has breadth 2. Soon Ivanov and V. N. Obraztsov (also Ol’shanskii’s
pupil) constructed examples of groups giving negative answers to a num-
ber of other questions about possible implications between the finiteness
conditions (and their combinations) considered by the author; see details
and references in Chapter IV of [1] or [2], an indication to the solution
of the first two question of the author is also given in [6].
Taking into account known properties of groups of finite breadth and
the result by Shunkov mention above, one can prove that a locally finite
group has finite breadth if and only if it is a Chernikov group. So for
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304 Semigroups and Chernikov groups
locally finite groups finiteness of breadth and the property of being an
artinian group are equivalent.
The author have also established that a locally finite group has at
most countable the subgroup lattice if and only if it is a Chernikov group.
In view of Reduction Theorem we obtain the following consequence
from the group-theoretic results given above. The following conditions for
a locally finite semigroup S are equivalent: (1) S is artinian, (2) S has
finite breadth, (3) S has at most countable the subsemigroup lattice, (4) S is
finitely assembled from Chernikov groups.
5. Three more lattice finiteness conditions. The first two of the
conditions considered below distinguish special types of Chernikov groups
in the class of all groups; the third one distinguishes a special type
of such groups in the class of locally finite groups. We start with necessary
definitions concerning partially ordered sets. We call a partially ordered set
(poset) narrow if in it any antichain is finite. If the powers of such a poset P
are bounded by a natural number, then P is said to be of finite width and
the greatest power of antichains is called the width of P . The dimension
of a poset 〈P,6〉 is the least cardinal number δ such that the relation 6 is
the set intersection of δ linear orders on P (or, equivalent, P is embedded
in a direct product of δ chains).
When considering semigroups [groups] with just the listed restrictions
on the subsemigroup [subgroup] lattice, let us arrange to transfer the pred-
icate in the corresponding term from the lattice mentioned to a semigroup
[group] itself. So, we shall speak about narrow semigroups and groups as
well as about semigroups and groups of finite width. Note that the term
“narrow” as applied to lattices, semigroups and groups was introduced
in [2], while in the monograph [1] and two preceding works, where narrow
semigroups and groups were considered (a supplement to the paper “Semi-
groups of finite breadth”, 1971, and the survey “Semigroups and their
subsemigroup lattices” written jointly with A. J. Ovsyannikov, 1983),
instead of the adjective “narrow” the prefix NF was used (which was
definitely less successful), i.e. the terms “NF -lattice”, “NF -semigroup”,
“NF -group” appeared there.
As to the property of having finite dimension, it is advisable to speak
not “semigroups of finite dimension” but “semigroups of finite lattice
dimension” (and the same for groups); the reason is that the first term
for groups and semigroups was used in literature in another sense, see
comments in Chapter IV of [1] or [2].
5.1. Narrow semigroups and groups. The property of being a nar-
row semigroup is embraced by the general reduction scheme described in
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L. N. Shevrin 305
Section 2, and it is easy to ascertain that narrow semigroups are periodic
and artinian. An exhaustive description of infinite narrow groups is given
by the following theorem.
An infinite group is a narrow group if and only if it is the direct
product of a finite group and finitely many quasicyclic groups for distinct
primes not dividing the order of the finite factor.
Note that in the proof of this theorem some results of Chernikov’s
paper “Infinite groups with finite layers” (Matem. Sb., 1948) are essentially
used.
As it is seen, narrow groups represent a special type of Chernikov
groups and, in particular, have finite breadth. In view of Reduction
Theorem we obtain that any narrow semigroup is a semigroup of finite
breadth. It is worth noting that this implication cannot be deduced
only from lattice-theoretic considerations; indeed, there exist examples
of narrow lattices which are not of finite breadth.
5.2. Semigroups and groups of finite width. From definitions it
directly follows that any semigroup of finite width is a narrow semigroup.
But the property of being a semigroup of finite width is not embraced
by our general reduction scheme; namely, requirement C) is not fulfilled
for this property (there exist counter-examples). However requirements A)
and B) are obviously fulfilled. Then, in view of the necessity part of Reduc-
tion Theorem (I did not distinguish it separately above), any semigroup
of finite width is finitely assembled from groups of finite width. This nec-
essary condition is not sufficient for the given property (counter-examples
exist), so the search of a necessary and sufficient condition is retained as
an unsolved problem.
An exhaustive description of infinite groups of finite width is given
by the following theorem based naturally on the theorem from Section 5.1.
An infinite group is a group of finite width if and only if it is the
direct product of a finite group and a quasicyclic group for some prime
not dividing the order of the finite factor.
5.3. Semigroups and groups of finite lattice dimension. It can
be proved that any lattice of finite dimension has finite breadth (which
is not greater than dimension), therefore any semigroup [group] of finite
lattice dimension is a semigroup [group] of finite breadth and, in particular,
periodic. The property of being a semigroup of finite lattice dimension
is embraced by our general reduction scheme, so, in view of Reduction
Theorem, a semigroup is of finite lattice dimension if and only if it is
finitely assembled from groups of finite lattice dimension.
The problem of describing arbitrary groups of finite lattice dimension
seems unrealistic; I may note in this connection that any Ol’shanskii group
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306 Semigroups and Chernikov groups
is of lattice dimension 2. However for locally finite groups, the problem
indicated, in all likelihood, can be solved in an exhaustive way. This claim
is explained by the following theorem together with a hypothesis attached
to it.
Any locally finite group of finite lattice dimension is an extension
of a direct product of finitely many quasicyclic groups for distinct primes
by a finite group.
Hypothesis: the converse statement is valid as well, i.e. any group
of the indicated structure is of finite lattice dimension.
The formulated result was obtained by the author in the mid 1970s
and reported (together with the hypothesis) in a special course “Periodic
semigroups” that was being delivered at Ural State University those years.
It was announced (again with the hypothesis) in “Abstracts of the XI
All-Union Symposium on Group Theory”, 1989, and its proof was first
published in [1]. A question about the validity of this hypothesis was in-
cluded in the 11th edition of “Kourovka Notebook”, 1990 (question 11.116)
as well as in the monographs [1] and [2] (question 13.10.4). It is still open.
References
[1] Shevrin L. N., Ovsyannikov A. J. Semigroups and Their Subsemigroup Lattices.
Sverdlovsk, Ural University Press, Part 1, 1990, Part 2, 1991 (in Russian).
[2] Shevrin L. N., Ovsyannikov A. J. Semigroups and Their Subsemigroup Lattices.
Dordrecht, Kluwer Academic Publishers, 1996.
[3] Chernikov S. N. Groups with Given Properties of System of Subgroups, Moscow,
Nauka, 1980 (in Russian).
[4] Kargapolov M. I., Merzlyakov Yu. I. Fundamentals of the Theory of Groups. 3rd
edition, Moscow, Nauka, 1982 (in Russian). English transl. of the 2nd edition:
New-York, Springer-Verlag, 1979.
[5] Robinson D. J. S. Finiteness Conditions and Generalized Soluble Groups. Parts 1, 2.
Berlin, Springer-Verlag, 1972.
[6] Ol’shanskii A. Yu. Geometry of Defining Relations in Groups. Moscow, Nauka,
1989 (in Russian). English transl.: Dordrecht, Kluwer Academic Publishers, 1991.
Contact information
Lev N. Shevrin Institute of Mathematics and Computer Science,
Ural Federal University, Lenina 51,
620083 Ekaterinburg, Russia
E-Mail: Lev.Shevrin@usu.ru
Received by the editors: 07.05.2012
and in final form 18.05.2012.
L. N. Shevrin
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