Kudryavtsev V.B., Rosenberg I.G. Structural Theory of Automata, Semigroups, and Universal Algebra
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326 R. McKenzie and J. Snow
Thus G
A
p
(nk) is at least as great as the number of systems m
i
i<d
of non-negative integers
satisfying
i<d
m
i
≤ n, i.e., we have
G
A
p
(nk) ≥
n + d(k)
n
.
In particular, G
A
(k
2
) ≥
k+d(k)
k
. For any fixed positive integer M ,wehaved(k) ≥ M for
large k, and for such k, it follows that G
A
(k
2
) ≥
k+M
k
. Since this is a polynomial of degree
M in k,thenG
A
(k) cannot be bounded for all k by a polynomial of degree <M/2. This ends
our proof that if A is not directly representable then G
A
is not bounded by any polynomial
function. 2
15 Problems
15.1 Problem Is there an algorithm to determine, given a finite ring R with unit, whether
the variety
RM of left unitary R-modules is decidable? F. Point, M. Prest [40] and M. Prest
[41] provide a starting point for the exploration of what is known about this problem.
15.2 Problem Is there an algorithm to determine, given a finite algebra F of finite type
such that HSP(F) has modular congruence lattices, whether HSP(F)(orSP(F)) is finitely
axiomatizable? By a famous theorem of K. Baker [2], every finitely generated congruence
distributive variety of finite type is finitely axiomatizable. R. McKenzie [36] proved that
there is no algorithm to determine if HSP(F) is finitely axiomatizable where F ranges over
all finite algebras with one binary operation.
15.3 Problem Prove or disprove that for every finite set A and every operation m =
m(x, y, z)overA that satisfies Maltsev’s equations m(x, x, y) ≈ y ≈ m(y, x,x), there are
only countably many clones of operations on A containing m. A.A.Bulatov,P.M.Idziak
[3] has results on this problem.
15.4 Problem Is it true that every congruence modular variety of finite type that is residu-
ally finite has a finite residual bound? K. Kearnes, R. Willard [29] proved that this implication
holds for congruence distributive varieties of finite type, and more generally for congruence
meet semi-distributive varieties of finite type.
15.5 Problem Characterize the locally finite congruence modular varieties V that possess
first-order definable principal congruences—i.e., there is a first-order formula θ(x, y, u, v)so
that for all A ∈Vand {a, b, c, d}⊆A, A |= θ(a, b, c, d) if and only if a, b lies in the
congruence of A generated by c, d.
References
[1] G.E.Andrews,The Theory of Partitions, Encyclopedia Math. Appl. 2, Addison-Wesley,
Reading, MA, 1976.
[2] K. Baker, Finite equational bases for finite algebras in congruence-distributive equational
class, Advances in Math. 24 (1977), 207–243.
Congruence modular varieties 327
[3] A. A. Bulatov and P. M. Idziak, Counting Maltsev clones on small sets, Discrete Math.
268 (1–3) (2003), 59–80.
[4] S. Burris and R. McKenzie, Decidability and Boolean Representations, Memoirs Amer.
Math. Soc. 32 (246), 1981.
[5] G. Cz´edli and E. Horv´ath, All congruence identities implying modularity have Maltsev
conditions, preprint.
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[7] A. Day, A characterization of modularity for congruence lattices of algebras, Canadian
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371–403.
[10] I. Fleischer, A note on subdirect products, Acta Math. Acad. Sci. Hungar. 6 (1955),
463–465.
[11] R. Freese and R. McKenzie, Residually small varieties with modular congruence lattices,
Trans. Amer. Math. Soc. 264 (1981), 419–430.
[12] R. Freese and R. McKenzie, Commutator Theory for Congruence Modular Varieties,
London Mathematical Society Lecture Note Series 125, Cambridge Univ. Press, 1987.
[13] H.-P. Gumm, Uber die losungermenge von gleichungssystemen uber allgemeinen alge-
bren, Math Z. 162 (1978), 51–62.
[14] H.-P. Gumm, Algebras in permutable varieties: geometrical properties of affine algebras,
Algebra Universalis 9 (1) (1979), 8–34.
[15] H.-P. Gumm, An easy way to the commutator in modular varieties, Proc. Amer. Math.
Soc. 80 (1980), 393–397.
[16] H.-P. Gumm, Congruence modularity is permutability composed with distributivity,
Archiv der Math. (Basel) 36 (1981), 569–576.
[17] H.-P. Gumm, Geometrical Methods in Congruence Modular Varieties, Memoirs Amer.
Math. Soc. 36 (286), 1983.
[18] H.-P. Gumm and C. Herrmann, Algebras in modular varieties: Baer refinements, can-
cellation and isotopy, Houston J. Math. 5 (1979), 503–523.
[19] J. Hagemann and C. Herrmann, A concrete ideal multiplication for algebraic systems
and its relation to congruence distributivity, Arch. Math. Basel 32 (1979), 234–245.
328 R. McKenzie and J. Snow
[20] C. Herrmann, Affine algebras in congruence modular varieties, Acta Sci. Math. (Szeged)
41 (1979), 119–125.
[21] D. Hobby and R. McKenzie, The Structure of Finite Algebras,Amer.Math.Soc.Con-
temporary Mathematics Series 76, 1988 [revised edition 1996].
[22] P. M. Idziak, A characterization of finitely decidable congruence modular varieties,
Trans. Amer. Math. Soc. 349 (1997), 903–934.
[23] P. M. Idziak and R. McKenzie, Varieties with polynomially many models, Fundamenta
Mathematicae 170 (2001), 53–68.
[24] P. M. Idziak and R. McKenzie, and M. Valeriote, The structure of locally finite varieties
with polynomially many models, preprint.
[25] B. J´onsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967),
110–121.
[26] K. A. Kearnes, On the relationship between AP, RS and CEP, Proc.Amer.Math.Soc.
105 (4) (1989), 827–839.
[27] K. A. Kearnes and R. McKenzie, Commutator theory for relatively modular quasivari-
eties, Trans. Amer. Math. Soc. 331 (1992), 465–502.
[28] K. A. Kearnes and A. Szendrei, The relationship between two commutators, Inter. J.
Algebra. Comput. 8 (1998), 497–531.
[29] K. A. Kearnes and R. Willard, Residually finite, congruence meet semi-distributive vari-
eties of finite type have a finite residual bound, Proc. Amer. Math. Soc. 127 (10) (1999),
2841–2850.
[30] H. Lakser and W. Taylor and S. T. Tschantz, A new proof of Gumm’s theorem, Algebra
Universalis 20 (1985), 115–122.
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[32] R. McKenzie, Narrowness implies uniformity, Algebra Universalis 15 (1982), 67–85.
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salis 24 (1987), 224–250.
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Birkhauser, Progress in Mathematics 79, 1989.
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1–28.
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(1996), 49–104.
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Congruence modular varieties 329
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classes of algebras, Proc.Amer.Math.Soc.14 (1963), 105–109.
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[40] F. Point and M. Prest, Decidabity for theories of modules, J. London Math. Soc. 38 (2)
(1988), 193–206.
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bridge Univ. Press, 1988.
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1 (1971), 265–266.
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New York, 1970.
Epigroups
Lev N. SHEVRIN
∗
Department of Mathematics and Mechanics
Ural State University
Lenina 51, 620083 Ekaterinburg
Russia
Dedicated to Walter Douglas Munn on the occasion of his 75th birthday
Abstract
An epigroup is a semigroup in which some power of any element lies in a subgroup
of the given semigroup. The class of epigroups includes a number of important classes of
semigroups. Not only some particular types of epigroups but epigroups as such can serve
as the subject matter of a substantial theory. In this survey the following topics concerning
epigroups are considered: epigroups as unary semigroups, certain “nice” decompositions,
finiteness conditions.
AsemigroupS is called an epigroup if for any element x of S some power of x lies in
some subgroup of S. The class of epigroups is very large: it contains all periodic semigroups
and, in particular, all finite semigroups, all completely regular semigroups, all completely
0-simple semigroups. It also contains some important concrete semigroups; for example, the
semigroup of all matrices over a division ring is an epigroup. The study of completely 0-simple,
completely regular, and periodic semigroups was begun at the very outset of the development
of the theory of semigroups; we mention in this regard the pioneer works [12, 77, 84]. In
the course of semigroup-theoretic investigations during several decades, some properties of
epigroups in general were revealed via the study of particular types of epigroups. The first
work where epigroups were considered per se was the paper [51] in which these semigroups
were called pseudo-invertible. This paper was, in its turn, influenced by the paper [15] in
which the notion of the pseudo-inverse of an element had been introduced. Epigroups were
considered later (as one of the objects or the main object of attention) in several dozens
papers by different authors and under different names: quasi-completely regular, group-bound,
quasiperiodic and some others. The term “epigroup” was suggested by the present writer in
the late eighties and apparently has become fairly common by now. Being shorter and, what
is especially essential, being expressed by a noun, such a term is more flexible and definitely
more appropriate for a concept which pretends to be the object of a certain theory.
In this survey we present quite a number of basic facts about epigroups. Its contents are
revealed by the titles of the sections and subsections. Notice that a very brief presentation
∗
The work was partially supported by the President Program of a support of the Leading Scientific Schools
of the Russian Federation (grant 2227.2003.01) and by the Ministry of Education of the Russian Federation
(grants E02–1.0–143 and 04.01.059).
331
V. B. Kudryavtsev and I. G. Rosenberg (eds.),
Structural Theory of Automata, Semigroups, and Universal Algebra, 331–380.
© 2005 Springer. Printed in the Netherlands.
332 L. N. Shevrin
of these topics was given by the author earlier in [49, Sections A6 and A7], and, somewhat
more extensively, in [102, §6].
I thank Boris Vernikov for technical assistance in preparing the text of this article.
1 Preliminaries
All standard semigroup-theoretic information can be found, for example, in Chapter A of the
handbook [49]. We will recall some definitions and notation as needed. Let S be a semigroup.
An element a ∈ S is called a group element if a belongs to some subgroup of S.IfS has
a zero, then an element a ∈ S is called a nil-element if a
n
=0forsomen. We will denote
by E
S
or simply by E the set of all idempotents of S,byGrS the set of all group elements
of S (the group part of S), and by Reg S the set of all regular elements of S. Obviously
E
S
⊆ Gr S ⊆ Reg S. We will often view E as a partially ordered set with respect to the
natural order: e ≤ f ⇐⇒ ef = fe = e. We say that an element a of a semigroup S divides
an element b ∈ S,andwewritea |b,ifb belongs to the principal ideal J(a). If S has a zero,
we will denote by Nil S the set of all nil-elements of S; S is a nilsemigroup if Nil S = S.A
semigroup S with zero 0 is called nilpotent if there exists n such that x
1
x
2
···x
n
=0forany
x
1
,x
2
,...,x
n
∈ S; the smallest n with this property is called the degree of nilpotency of S.
We will denote by G
e
the maximal subgroup of S having the idempotent e as its identity
element. Thus Gr S =
e∈E
S
G
e
. The equality Gr S = S just defines the completely regular
semigroups. The set of all natural numbers is denoted by N.
For an arbitrary subset M of a semigroup S we put
√
M = {x ∈ S |x
n
∈ M for some n ∈ N}.
Thus the fact that S is an epigroup can be expressed by the equality S =
√
Gr S.ThatS is a
periodic semigroup can be expressed by the equality S =
√
E
S
; as is known, this is equivalent
to the property that, for any a ∈ S, the cyclic subsemigroup a generated by a is finite. If
S has a zero, then Nil S =
@
{0}.
For e ∈ E we put
K
e
=
@
G
e
.
By definition, for any element a of an epigroup S there exists e ∈ E
S
such that a ∈ K
e
.
That the idempotent e is defined here uniquely is a consequence of the following property
already mentioned in [51]: if a
m
∈ G
e
,thena
n
∈ G
e
for any n>m. So one may imagine that
elements of an epigroup S are pulled by idempotents to the group part of S, and each element
is “attached” to a single idempotent. This shows that the set of all idempotents plays the
role of a certain framework of an epigroup. The property noted means that any epigroup S
is partitioned into the subsets K
e
called unipotency classes. The partition into unipotency
classes gives, so to speak, a rough structure of an arbitrary epigroup. A unipotency class of
an epigroup does not have to be a subsemigroup; to find out when this takes place (and even
when all the unipotency classes are subsemigroups) is the goal of Subsection 3.2.
The following well-known statement (see [51] and also [13, Theorem 2.55]) is one of the
first general results concerning epigroups.
1.1 Proposition A semigroup is a [0-]simple epigroup if and only if it is completely [0-]
simple.
Epigroups 333
A semigroup is called stable if, for any of its J-classes, the sets of L-classes and R-classes
contained in it are antichains with respect to the natural order (L
a
≤ L
b
means L(a) ⊆ L(b);
R
a
≤ R
b
means R(a) ⊆ R(b)); there are some properties equivalent to the one indicated in this
definition. It is easy to show that any epigroup is a stable semigroup; this was done explicitly
in [26]. The following statement points out some useful properties of stable semigroups, and
hence of epigroups.
1.2 Proposition In a stable semigroup: (a) Green’s relations J and D coincide; (b) no
subsemigroup is bicyclic; (c) the idempotents belonging to the same D-class form an antichain.
Assertion (a) here is presented, for instance, in [13, Theorem 6.45]; Assertion (b) appears
explicitly, for instance, in [2]; Assertion (c) follows easily from (b) (without additional as-
sumptions about the semigroup in question), as can be seen, for example, from the argument
in the proof of Theorem 2.54 in [13].
The idempotent of the unipotency class to which an element a belongs will be denoted
by e
a
. The following properties were mentioned in [51].
1.3 Lemma (a) ae
a
= e
a
a; (b) ae
a
∈ G
e
a
.
Lemma 1.3 easily implies the following assertions.
1.4 Corollary (a) The group G
e
is the kernel (i.e., the smallest ideal) of the subsemigroup
K
e
generated by K
e
. (b) An idempotent e is the smallest element in the partially ordered
set E ∩K
e
.
The assertions in Lemma 1.3 and its Corollary pertain to the situation involving an
arbitrary semigroup with idempotents. In the case of epigroups, one can state somewhat
strengthened assertions by replacing the subsemigroup K
e
by the subepigroup K
e
gen-
erated by K
e
(see Subsection 2.1).
Recall that a retract of a semigroup S is a subsemigroup T for which there exists a
homomorphism ϕ : S → T which is the identity mapping on T ; such a homomorphism ϕ is
called a retraction.Aretract ideal is an ideal that is a retract. The following fact is well
known.
1.5 Lemma If T is a retract ideal of a semigroup S,thenS is a subdirect product of T and
the quotient semigroup S/T .
The concepts of the index and period of an element of finite order in a semigroup can
be extended to the case of elements of infinite order. The period of an element of infinite
order will be assumed to be infinite and will be denoted by the symbol ∞.Theindex of an
arbitrary element a of a semigroup S is defined as follows: if a∩Gr S = ∅, it is the smallest
natural number n such that a
n
∈ Gr S;ifa∩Gr S = ∅, it is 0. The index of an element a
will be denoted by ind(a). The elements of index 1 are precisely the group elements. As in
the case of elements of finite order, the pair (index, period ) will be called the type of a given
element. Thus an element of infinite order has type (n, ∞), where n is a non-negative integer.
In epigroups, and only in epigroups, all elements have indices that are natural numbers. If the
indices of all elements of an epigroup S are bounded by a natural number, we say that S is an
epigroup of finite index,anditsindex is defined to be max{ind(a) |a ∈ S}; the index of S is
denoted by ind S. The epigroups of index 1 are precisely the completely regular semigroups.
334 L. N. Shevrin
2 Epigroups as unary semigroups
2.1 Background
A unary semigroup is a semigroup with an additional unary operation. Note that important
representatives of such algebras are completely regular and inverse semigroups; see the corre-
sponding treatments, for instance, in [49, Sections A10 and A11]. Epigroups can be treated
as unary semigroups as well. To introduce the corresponding unary operation, recall that,
according to Assertion (b) of Lemma 1.3, for any element a of an epigroup we have ae
a
∈ G
e
a
.
Hence one may consider the element
a =(ae
a
)
−1
, (2.1)
where the right-hand side is the inverse in the group G
e
a
. This element is called the pseudo-
inverse of a, and the operation a →
a is just the stated one. The following equalities hold:
a
a = aa, aa
2
= a, a
n+1
a = a
n
for some n. (2.2)
The equalities (2.2) have been used in [15] for the original definition of pseudo-inverses. The
conditions (2.1) and (2.2) are equivalent in the sense that if a and
a are some elements of a
semigroup such that the equalities (2.2) hold, then a
n
is a group element, and if a
n
∈ G
e
,
then e = e
a
, and it is easy to ascertain that a ∈ G
e
and condition (2.1) holds.
The definition of the pseudo-inverse of an element a of non-zero index by (2.1) is more
convenient than by (2.2). Indeed, then we do not need to establish the uniqueness of
a for a
given a, the equalities (2.2) (where n =ind(a)) follow directly from (2.1), and we can write
them in the following form, which more clearly illuminates their “structural” meaning:
a
a = aa = e
a
,e
a
a = a, a
n
e
a
= a
n
. (2.2
)
But the original form of the equalities (2.2) is still valuable, particularly in connection with
defining epigroups by identities (see Subsection 2.2).
The idea of viewing epigroups as unary semigroups has been proposed in [104], and this
approach was shown in three ways there: in formulating statements solving some problems, in
posing some problems, in applying techniques. A number of results of this work are presented
below in this section and, especially, in Section 3.
In a particular case when an epigroup S is in fact completely regular, the operation a →
a
turns into the usual unary operation a → a
−1
of such semigroups, i.e., a is none other than
the inverse of a in the maximal subgroup containing a. Along with the basic unary operation
a →
a it is useful to consider the derivative operation a → e
a
. The latter is linked with the
former by the formula e
a
= aa. It is customary to denote e
a
by a
ω
in the theory of finite
semigroups, and by a
0
in the theory of completely regular semigroups. For the general case
of epigroups, we choose the notation a
ω
here.
Repeated application of pseudo-inversion will be denoted by an additional bar. It follows
from (2.1) that
a = a if and only if a is a group element. The same equality explains the
identities
x = x and x = x
2
x which were already mentioned in [15].
A subepigroup of an epigroup is a subset that is closed under both the operations of
multiplication and pseudo-inversion. It is easy to see that the subepigroups of an epigroup
are precisely the subsemigroups that are themselves epigroups. The simplest example of an
Epigroups 335
epigroup having subsemigroups which are not subepigroups is given by the infinite cyclic
group (the subepigroups of a group are obviously none other than its subgroups).
Note the following two elementary properties of the principal character; the latter is a
special case of a rudimentary general algebraic fact.
2.1 Observation A homomorphic image of an epigroup is an epigroup, and any semigroup
homomorphism ϕ of an epigroup S onto some epigroup is automatically an epigroup homo-
morphism, i.e., ϕ(
a)=ϕ(a) for any a ∈ S.
2.2 Observation Under a homomorphism of an epigroup, the inverse image of a subepigroup
(in particular, the inverse image of an idempotent) is a subepigroup.
A homomorphic image of a subsemigroup of a given semigroup is called a divisor (or
factor) of the semigroup. A divisor is called a Rees divisor if it is the Rees quotient semigroup
of some subsemigroup modulo an ideal. A divisor obtained from a subepigroup of an epigroup
will be called, for brevity, an epidivisor. Observation 2.1 shows that any epidivisor of an
epigroup is itself an epigroup. Since an ideal of an epigroup is obviously a subepigroup,
among the epidivisors of an epigroup are all its principal factors.
The subepigroup generated by a subset A of a given epigroup will be denoted by A,
while the single angular brackets are used to denote the subsemigroup generated by a set.
(We could also suggest the alternative notation epA, but the double angular brackets seem
to contrast better with the subsemigroup notation.) If S is an epigroup and S = A [resp.,
S = A], then A will be called an epigroup [semigroup] generating set of S.Itisworth
noting that if A is an epigroup generating set, and
A = {a |a ∈ A}, then the set A ∪ A
does not have to be a semigroup generating set of the given epigroup. Moreover, a finitely
generated epigroup (even if it is completely regular) may have no finite semigroup generating
sets: an example is the free completely regular semigroup of rank 2 (see [116]
1
). We can say,
however, that if S is an epigroup in which the mapping a →
a is an endomorphism or an
anti-endomorphism, then S = A obviously implies S = A ∪
A. Epigroups with these
properties of the pseudo-inversion operation will be considered in Subsection 3.4.
It is natural to call an epigroup having a one-element epigroup generating set mono-
genic or, in accordance with traditional group-theoretic and semigroup-theoretic terminology,
cyclic. We choose the latter. Consider an arbitrary cyclic epigroup C = a.Letind(a)=n.
The equalities (2.2
) show that Gr C is the cyclic group generated by a,GrC = G
e
a
, C is
a cyclic group if n = 1 (and only in this case), and C \ Gr C = {a,...,a
n−1
} if n>1.
In any event, C = K
e
a
, C is commutative, and Gr C is its kernel. We see that the finite
cyclic epigroups are precisely the finite cyclic semigroups whose structure and classification
are textbook facts. Our aim now is to classify infinite cyclic epigroups. To do this, we need
the construction described below.
Suppose T is an arbitrary semigroup, H a subsemigroup, and I an ideal of H.Wetake
asetF of cardinality equal to that of H \ I and disjoint from T ; the image of any element
1
Note, in order to avoid a misunderstanding, that in the work cited, as well as in a number of other
works of different authors, completely regular semigroups were called Clifford (another variant is Cliffordian)
semigroups. This good term had been proposed not later than the mid sixties (refer to the book [29]);
being shorter, it seemed to be rather appropriate for the notion which became the subject of a rich theory.
However, later many authors began to use the term “Clifford semigroup” only for the particular case of inverse
completely regular semigroups, i.e., semilattices of groups. Somewhat more detailed terminological remarks
on this matter are given in [108, p. 23].
336 L. N. Shevrin
x ∈ F under a fixed bijection ψ : F → H \ I will be denoted by x
. We define an operation ◦
on the union S = T ∪ F as follows:
x ◦ y =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
xy if x, y ∈ T,
xy
if x ∈ T, y ∈ F,
x
y if x ∈ F, y ∈ T,
x
y
if x, y ∈ F and x
y
∈ I,
ψ
−1
(x
y
)ifx, y ∈ F and x
y
∈ H \ I.
It is easy to verify that ◦ is associative; hence S becomes a semigroup that obviously contains
T as an ideal, and the quotient semigroup S/T is isomorphic to the Rees factor H/I.Thus
S is an ideal extension of T . Figuratively speaking, S is obtained from T by “attaching” a
copy of subsemigroup H so that the copy of I is identified with I and no other identifications
are made. We will call S an extension of T by the Rees factor H/I,oraduplicate extension
with attachment H and base I. A duplicate extension of the semigroup T is uniquely defined
up to isomorphism by the attachment H and base I; we will denote it by D(T,H,I). It is
easy to see that a duplicate extension is a retract extension, i.e., T is a retract of D(T,H,I).
In the degenerate case when I = H,wehavesimplyD(T,H,H)=T .
The following classification was given in [104, §1].
2.3 Proposition The infinite cyclic epigroups are exhausted by all possible duplicate ex-
tensions of the infinite cyclic group with attachment consisting of the positive powers of its
generator. Two such epigroups are isomorphic if and only if they have the same index.
The infinite cyclic epigroup of index n will be denoted by C
n,∞
. This notation is consistent
with the notation C
n,m
for the finite cyclic semigroup of type (n, m). It is also natural to
call (n, ∞)thetype of the epigroup C
n,∞
. Thus the family of all cyclic epigroups is {C
n,π
},
where n runs over the set N,andπ runs over the set N ∪{∞}. The epigroup C
1,π
is nothing
but the cyclic group of order π. A homomorphic image of the epigroup C
n,∞
is, of course, a
cyclic epigroup of index at most n; conversely, it follows easily from the description given by
Proposition 2.3 that any epigroup C
m,π
with m ≤ n is a homomorphic image of C
n,∞
.
The infinite cyclic epigroup C
n,∞
= a is generated as a semigroup by two elements: a
and b =
a. It can be defined in the class of all semigroups by the presentation
C
n,∞
= a, b |ab = ba, ab
2
= b, a
n+1
b = a
n
.
The semigroup C
n,∞
is an ideal extension of the infinite cyclic group by a cyclic nilpotent
semigroup of order n. The converse is obviously false, but the extensions of the indicated
type admit an exhaustive classification: if we ignore the degenerate case n =1,theyforma
two-parameter family {D
n,k
}, n>1, k ≥ 0, where the semigroup D
n,k
is defined for k>0
by the presentation
D
n,k
= a, b |ab = ba, ab
k+1
= b, a
n+1
b
k
= a
n
,
and for k = 0 by the presentation
D
n,0
= a, b, c |ab = ba = b, ac = ca = c, bc = cb = a
n
.