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

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Epigroups 337

The semigroup D

n,k

is embeddable in a cyclic epigroup (and then, for example, in C

nk,∞

)

if and only if k>0, and this is the case if and only if the idempotent in D

n,k

is a unique

element of ﬁnite order. When k>0, D

n,k

is a duplicate extension of the inﬁnite cyclic group

b with attachment b

−k

;whenk =1,wehavesimplyD

n,1

= C

n,∞

As to epigroups having a two-element epigroup generating set, it makes no sense to ﬁnd

their classiﬁcation, because, as in the case of groups and semigroups, the following is true.

2.4 Theorem Any countable epigroup is embeddable in a 2-generated epigroup.

Similarly to known results concerning arbitrary countable semigroups [44, 60] and ﬁnite

semigroups [32], the following is true as well.

2.5 Theorem Any countable epigroup is embeddable in an epigroup generated by three idem-

potents.

Theorems 2.4 and 2.5 were announced in [70]; their proofs are still unpublished (I must

note with sorrow that the ﬁrst author of [70] died several years ago). Analogous results (with

a number of reﬁnements concerning periods or indices of elements) have been obtained earlier

in [69] for periodic semigroups. Note that the technique developed in [69] made it possible in

passing to reprove all the mentioned results of [32,44, 60] as well as the results of the pioneer

works [19, 25, 53] relating to embeddability of countable or ﬁnite semigroups in 2-generated

ones.

In some considerations concerning epigroups, it is useful to apply the following fact.

2.6 Lemma The subsemigroup E

 of an epigroup S is in fact a subepigroup.

This statement was apparently ﬁrst proved (among some other facts about epigroups) in

the unpublished work [115]. In published form it appeared in particular in [16, Corollary 1],

and in [63, Theorem 2.2]; its very short proof uses a trick from [21] showing that the pseudo-

inverse of a product of k idempotents can be expressed as a product of k +1 idempotents (see

also [27, Theorem 1.4.18]). The main line of examination in [63] deals with the set E

of an

epigroup S ∈E

,andE

is treated there as a special type of a universal algebra endowed with

operations t

, i ≥ 2, where t

is the i-ary operation deﬁned by the rule t

,...,e

···e

)

. This algebra is called the idempotent algebra of an epigroup S,andsome

properties of such algebras are examined in [63]. Note in this connection that there is another

type of (partial) algebraic systems which can be deﬁned on the set of all idempotents of a

semigroup, so-called biordered sets. Such biordered sets were characterized abstractly in [52].

In [16] the corresponding characterizations were given for epigroups properly, for periodic

semigroups, for ﬁnite semigroups, and, ﬁrst of all, for eventually regular semigroups. Recall

that a semigroup S is said to be an eventually regular [17] if S =

√

Reg S.Anyepigroup

is clearly eventually regular. Some conditions under which an eventually regular semigroup

turns out to be an epigroup or even a periodic semigroup are given in [18]. In particular, an

eventually regular semigroup S with ﬁnitely many regular D-classes is an epigroup if (and

obviously only if) S has no bicyclic subsemigroup.

Let us call a semigroup bi-unary if it is further endowed with two unary operations.

We may view inverse epigroups as bi-unary semigroups, having in mind the operations of

pseudo-inversion x →

x and taking the inverse element x → x

−1

. There arises a number of

natural questions pertaining to the study of such algebraic systems. Leaving them out of this

338 L. N. Shevrin

article, we mention only that Corollary 3.54 below presents a “degenerate” case of bi-unary

inverse epigroups—where the two signature unary operations coincide. In the general case,

the connection between these operations is provided by the following fact observed in [104, §7].

2.7 Proposition In any inverse epigroup, the operations of pseudo-inversion and taking the

inverse element are permutable, i.e., the identity (

−1

= x

−1

holds.

2.2 Identities of epigroups

In speaking of identities of epigroups, we have in mind the signature of unary semigroups,

i.e., for an identity u = v,inthetermsu and v both multiplication and pseudo-inversion may

appear (as well as the operation a → a

derivative from them: a

= aa). One can observe

the simpliest identities holding in any epigroup, which follow directly from the corresponding

equalities in (2.2):

xx = xx, x(x)

= x.

We mention further simple consequences from the deﬁnitions:

x = x, x = x, xx =(x)

Here are less trivial examples of the identities that hold in any epigroup:

= x

· x

= x

· y

, x

· y

=(x

)

Can one describe all the identities which hold in any epigroup? We will denote the class

of all epigroups by E.ForaclassX of algebraic systems of one type, the equational theory

eq X of this class is the set of all the identities which hold in any system from X.Abasis

of eq X,orabasis of identities of X, is a subset of eq X such that every identity from eq X

can be derived from identities of this subset. The question above may be reformulated, in

particular, as the problem of ﬁnding some basis of eq E. Another way to answer it is to

establish whether the theory eq E is decidable.

The same problem is interesting also for the class E

ﬁn

of all ﬁnite semigroups treated

as unary semigroups with pseudo-inversion. Observe that E

ﬁn

is a pseudovariety, i.e., it is

closed under taking subsystems, homomorphic images and ﬁnite direct products. Note in

this connection that the majority of important pseudovarieties of ﬁnite semigroups can be

speciﬁed in the class of all ﬁnite semigroups just by epigroup identities; one may see many

conﬁrmations of this thesis in, for example, [1].

An answer to the questions discussed in the previous paragraph is given by the following

theorem.

2.8 Theorem The equational theories of the classes E and E

ﬁn

coincide and have a basis

consisting of the identities x(yz)=(xy)z, x(

yx)=(xy)x, x(x)

= x, x

x = x, xx = xx,

=(x)

,wherep runs over the set of all primes. The theory eq E =eqE

ﬁn

is decidable.

Along with the classes E and E

ﬁn

, among important classes of epigroups are classes consist-

ing of epigroups S such that Gr S = E

, i.e., all the maximal subgroups of S are one-element.

Following widely-used terminology, we call the epigroups with this property combinatorial.

Remark that, as applied to ﬁnite semigroups, it is common to call such semigroups aperiodic.

Epigroups 339

I shall not discuss to what extent this term is apt, but in the general case it is deﬁnitely

unacceptable; indeed, combinatorial epigroups are obviously periodic, so, if they were called

aperiodic, we would have a collision between terminology and logic. However, taking into

account the traditional notation used for the class of ﬁnite semigroups with the property

under consideration, we will denote the class of all combinatorial epigroups by A and the

class of all ﬁnite combinatorial semigroups by A

ﬁn

2.9 Theorem The equational theories of the classes A and A

ﬁn

coincide and have a basis

consisting of the identities x(yz)=(xy)z, x(

yx)=(xy)x, xx = x, x = x, x

= x,wherep

runs over the set of all primes. The theory eq A =eqA

ﬁn

is decidable.

Theorems 2.8 and 2.9 were given in [129] together with a description of the main ideas of

their proofs. (Unfortunately, a full presentation of this material by the author is impossible

because of his tragic death in Black Sea in 2000). Remark that the list of identities which

appeared in the original formulation of Theorem 2.9 in [129] contains an extra identity

x·x = x

which can be derived from the others; indeed, from the third identity it follows that

x·x = x,

and then, in view of the fourth identity,

x · x = x · x = x = x.

Note in the conclusion of this subsection that many important types of epigroups can be

characterized in terms of identities; see various examples in Section 3.

2.3 On varieties of epigroups

The concept of an epigroup identity leads naturally to the concept of a variety of epigroups

treated as unary semigroups. For a class of algebraic systems X of one type, we denote by

L(X) the lattice of all varieties contained in X.TheclassE of all epigroups is not a variety

(in other words, L(E) has no greatest element) because it is not closed under direct products

(of inﬁnite families). Indeed, if we take the family of cyclic epigroups a

 whose indices are

unbounded, then the direct product



a

 is not an epigroup since, for example, no power

of the element (...,a

,...) belongs to a subgroup.

We denote by E

the class of all epigroups of index at most n. The equalities (2.2), along

with the remark that the smallest n with the property indicated in the third of the equalities

is ind(a), establish

2.10 Proposition The class E

is a variety of unary semigroups. It is deﬁned by the iden-

tities

(xy)z = x(yz),x

x = xx, xx

= x, x

n+1

x = x

In the lattice L(E), there is a naturally distinguished chain

⊂E

⊂···⊂E

⊂···

which, ﬁguratively speaking, can be regarded as the “spine” of this lattice. The reason is the

following property.

2.11 Observation For any variety V of epigroups, there exists n such that V⊆E

Indeed, a class of epigroups that is not contained in any of the varieties E

obviously

contains cyclic epigroups whose indices are unbounded, so one may refer to the argument

given at the end of the ﬁrst paragraph of this subsection.

340 L. N. Shevrin

In view of Observation 2.11, to any variety V of epigroups there corresponds a smallest n

such that V⊆E

. This number, ind V, which is equal to the maximal index of the epigroups

in V is naturally called the index of the variety V. (We should mention here that the term

“variety of ﬁnite index” has already been used in the theory of semigroups in a diﬀerent

sense, namely for varieties in which the nilpotency degrees of the nilpotent semigroups are

uniformly bounded. The corresponding concept can be appropriately extended to varieties

of epigroups and, to avoid a collision of terminology, we will call a variety with the latter

property a variety of ﬁnite degree. A certain characterization of such varieties is given below

in Corollary 3.20 of Proposition 3.18.)

The variety E

is nothing but the variety of all completely regular semigroups (treated as

unary semigroups). We will call any variety of such semigroups a completely regular (c.r.)

variety;soE

is the largest c.r. variety. Note that c.r. varieties are examined in several

chapters of [68]. Along with c.r. varieties, among the examined varieties of epigroups are

all periodic varieties, i.e., varieties consisting of periodic semigroups (a lot of such varieties

appeared as a matter of fact in diverse investigations devoted to semigroup varieties). Indeed,

in any periodic variety, an identity of the form x

= x

n+m

holds, and the pseudo-inversion

operation in this case can be expressed by a formula in the semigroup signature: for instance,

x = x

nm−1

. Conversely, any class of epigroups that is a semigroup variety obviously consists

of periodic semigroups. Since any non-periodic epigroup contains among its subepigroups the

inﬁnite cyclic group, and the latter generates the variety of all abelian groups, this variety is

the smallest non-periodic variety of epigroups.

Any variety of epigroups that contains E

will be called an over c.r. variety.Itisclear

that in studying varieties of epigroups we encounter three possible types of situations: (1) we

generalize facts previously established for c.r. or periodic varieties; (2) we ﬁnd properties of

varieties having signiﬁcant new consequences for c.r. and/or periodic varieties; (3) we consider

speciﬁc properties of over c.r. varieties that are not manifested (or are trivially manifested)

in c.r. and periodic varieties. Among the epigroup varieties that we consider in Section 3 are

both over c.r. varieties and proper subvarieties of E

, and we encounter the ﬁrst two of the

situations just mentioned.

As for possible investigation of epigroup varieties in general, one can, of course, proceed

in several directions, and it would be natural to pay attention to all of the main aspects:

equational, structural, algorithmic, and so on; see, for example, the introductions in the

survey articles [109,110]. Not attempting to list in detail the concrete problems and questions

that arise, we limit ourselves to a few remarks.

Turning to the lattice L(E), we can easily see that it has the same atoms as the lattice

of semigroup varieties, and for any n the upper cone E

∆

= {X ∈ L(E) |X ⊇E

} has a single

atom, namely E

∨ var C

n+1,1

, which is contained in any variety in E

∆

\{E

}.Oneofthe

standard properties examined for lattices of varieties (or lattices of related classes such as,

for instance, pseudovarieties) is the so-called the cover property. By deﬁnition, a lattice L

has this property if any element x ∈ L which is not the greatest element of L has a cover in

L, i.e., an element y such that x<yand there is no element of L lying strictly between x

and y. A brief survey of the main results and problems connected with the cover property in

the lattice of semigroup varieties is given in [123] whose central result exhibits an example of

a semigroup pseudovariety which has no cover in the lattice of all semigroup pseudovarieties.

A part of the paper cited is devoted to epigroup varieties. Namely, the following fact has

been established there.

Epigroups 341

2.12 Proposition The lattice L(E) has the cover property.

A more signiﬁcant result of [123] in this direction is concerned with a problem about the

cover property for the lattices L(E

); this problem was noted in [104]. To formulate the result

mentioned, denote by EA

the class of all epigroups from E

whose idempotent generated

subepigroups (see Lemma 2.6) are combinatorial. It is observed in [123] that EA

is a variety

of epigroups.

2.13 Theorem For any n>1 the variety EA

has no cover in the lattice L(E

As for the case n = 1, the corresponding problem was ﬁrst posed by the present writer

more that 25 years ago (see [91, Problem 2.62b]) and still remains open; we repeat it here.

2.14 Problem Does the lattice L(E

) of all completely regular varieties have the cover prop-

erty?

Another open problem pertaining to the lattice of subvarieties of a variety of epigroups

will be formulated in Subsection 3.3.

Among known results related to the lattice L(E), one may mention also the main result

of [3] which indicates certain intervals in L(E) such that no variety within any interval of the

type considered has a ﬁnite basis for its identities. The precise formulation of this statement

would require a number of additional deﬁnitions and special notations, so we omit it here.

In Section 3 we give, in particular, some characterizations of varieties consisting of epi-

groups which have certain “nice” decompositions.

There are many questions about epigroup varieties which are waiting to be examined.

Quite a number of such questions are brieﬂy discussed at the end of §1 in [104], so we will

mention only part of them here. In all likelihood, none of the lattices L(E

) has coatoms,

i.e. elements covered by the greatest element; it would be interesting to verify this conjec-

ture. The intervals [E

, E

n+1

] of the lattice L(E) deserve special attention. What are their

lattice properties (for example, the order types of maximal chains and the cardinalities of

antichains)? What are the interactions between these intervals (for example, when m<n,

is [E

, E

m+1

] embeddable into [E

, E

n+1

], and is it true that [E

, E

m+1

]and[E

, E

n+1

]arenot

isomorphic)?

We denote by Ni the class of all nilsemigroups, and we put Ni

= Ni ∩E

.With

each variety V of epigroups we can associate its “nil part” V∩Ni, which is a subvariety

of the variety Ni

ind V

, and its “completely regular part” V∩E

. It would be interesting to

examine diﬀerent aspects of the interrelation between a variety and its nil and completely

regular parts in certain situations; some speciﬁcations of this idea are proposed in [104, §1],

and the interested reader is referred to the work mentioned

It should be noted that the translator of the original Russian version of this work into English was not so

accurate and made many mistakes in choosing appropriate words in the translation. By the way, it began with

the title of the work: the original “To the theory of epigroups” has turned into “On the theory of epigroups”.

There are some blunders; for example, “given” was translated as “long” (Russian equivalents of these words,

written in Roman letters, are “dannyj” and “dlinnyj”, respectively, so only a very thoughtless translator could

mess up the words and in doing so write an expression making no sense). There are several terminological

slips; for example, “complete preimage” is written instead of the correct “inverse image” as well as the preﬁx

“super” instead of the correct “over” in the situation considered. There is plainly carelessness as well; for

example, “presents” is written instead of the correct “presence”, and so on, and the like.

I will allow myself to give two more similar examples taken from English versions of the surveys [109] and

342 L. N. Shevrin

Let us mention, ﬁnally, some questions that can be posed about free epigroups in various

varieties, including E

. Among such questions is that of the basis rank of E

for n>1(is

it equal to 2?) and the related question on the embeddability of the E

-free epigroup of

countable rank in an E

-free epigroup of ﬁnite rank. (For n = 1 the answers are known: it is

easy to show that the free completely regular semigroup of countable rank is not embeddable

in any ﬁnitely generated completely regular semigroup; also, as is shown in [64], the basis rank

of E

is inﬁnite.) It is natural to ask about the decidability of the word problem in E

-free

epigroups (for n = 1 the decidability was established in [24,36,117]), about a description, for

n>1, of Green’s relations and, in particular, the maximal subgroups, and about a description

of the partially ordered set of idempotents. By the way, one may regard the E

-free epigroups

as looking to some extent like the free Burnside semigroups, i.e., the free semigroups of the

variety deﬁned by the identity x

= x

n+m

for some m. Both certain structural properties

of such semigroups and the corresponding algorithmic aspects were considered in [43], and

perhaps some ideas of that work could be applied to the examination of the E

-free epigroups.

3 Certain decomp ositions

3.0 Introductory remarks

A common trend in a structural theory of the algebraic systems under examination is to ﬁnd

conditions under which these systems can be constructed in some way from more speciﬁc

ones, in particular, those having a more simple, or more rigid, or more clear structure. In

our case, the role of such “building blocks” will be played by completely simple semigroups

(including certain particular types of them, up to groups) as well as nilsemigroups (in par-

ticular, nilpotent semigroups). As to the ways of “assemblage”, we will deal mostly with

ideal extensions and decompositions into bands (including certain particular types of such

constructions, for instance, the semilattice decompositions).

Recall that a semigroup is said to be a band of subsemigroups S

, α ∈ A,ifS

form a

partition of S,andforanyα, β ∈ A,thereexistsγ ∈ A such that S

⊆ S

. This means, in

other words, that these subsemigroups (the components of the band under consideration) are

the classes of some congruence, ρ,say,onS, and the quotient semigroup S/ρ is a semigroup

of idempotents. If S/ρ is commutative, and hence is a semilattice, then S is said to be a

semilattice of semigroups S

; if the semilattice S/ρ is a chain, we say that S is a chain of the

corresponding semigroups.IfS/ρ is rectangular, i.e., it satisﬁes the identity xyx = x,thenS

is a rectangular band of the corresponding semigroups. A particular case of a rectangular band

is given by a left [right] bands of semigroups; in this case, for any two components S

, S

of a

given band, the inclusion S

⊆ S

] holds, that is all the components are right

[left] ideals. A decomposition of a semigroup into a semilattice of some subsemigroups is called

a semilattice decomposition. Similarly, we obtain the notion of a rectangular, left and right

decomposition as well. A rectangular decomposition is also called a matrix decomposition.

[110]. In the former, one of the egregious errors uses “arbitrary” instead of “derivative” (Russian equivalents

are “proizvol’nyj” and “proizvodnyj”, respectively, so there is no need to comment what were tranlator’s

attention and understanding). In the latter, “manifold” is used instead of “variety” (Russian equivalents of

these words are the same, namely, “mnogoobrazie”, so the translator of [110] was not aware of the subject

of this survey). The reader of these translations should be vigilant in order to be able to detect probable

incorrectness not due to the authors but introduced by the translator.

Epigroups 343

Both of these terms are justiﬁed by the fact that the components of a matrix decomposition

can be equipped with double indices running over some sets I and Λ such that the rule

iλ

jµ

⊆ S

iµ

,i,j∈ I, λ,µ ∈ Λ,

is fulﬁlled. If all the components of a given band belong to a certain class of semigroups X,

then we say that the corresponding semigroup is a band (or decomposable into a band ) of

semigroups from X. For instance, we may consider bands of groups, bands of commutative

semigroups, and the like. Recall in this connection that completely simple semigroups are

precisely rectangular bands of groups. A semigroup is a left [right] group if it is a left [right]

band of groups; as is known, there are several equivalent properties characterizing these

semigroups, see, for instance, [49, Section A5].

Diﬀerent decompositions into bands play an important role in the structural theory of

semigroups. Here it is particularly worthwhile to distinguish semilattice and matrix decom-

positions, not only because of their clearness but also due to the following fact: any band of

semigroups forming a family {S

} is a semilattice of rectangular bands of semigroups from

}; that is, the components S

can be grouped in subfamilies such that the union of the

components belonging to each subfamily is a rectangular band of these components, while

the initial semigroup is a semilattice of the unions mentioned. Notice that various questions

relating to semilattice and matrix decompositions of semigroups are extensively examined

in [65] and [66]; semilattice decompositions were studied in detail in [74], and the note [73]

is devoted to a certain description of the greatest semilattice decomposition of an epigroup.

Notice also that the topic “Bands of semigroups” is surveyed with due fullness in [102, §2.3]

and, somewhat more brieﬂy, in [49, Section A2]. A comprehensive survey of investigations

relating to various kinds of decompositions is given in [11].

One of the distinctive features of the approach carried out in Subsections 3.2–3.5 is the

primary interest in so-called indicator characterizations, i.e., characterizations in terms of

“forbidden” objects; the latter ones in our case are mainly epidivisors. In these divisor

characterizations, there appear mostly ﬁnite semigroups having an extremely clear structure.

This makes such characterizations very transparent and enables us to obtain, in most cases

as direct corollaries, many other criteria for the decomposability of epigroups (or, earlier, of

periodic semigroups). Moreover, divisor criteria immediately yield indicator characterizations

of the varieties consisting of the corresponding epigroups. These criteria were subsequently

applied to the theory of varieties as well; see, for example, [79–81, 114, 127].

3.1 Archimedean epigroups

AsemigroupS is called archimedean [left archimedean, right archimedean] if for any a, b ∈ S

there exists an n such that b |a

[resp., a

∈ Sb, a

∈ bS]. A semigroup is bi-archimedean if

it is left and right archimedean. A semigroup is unipotent if it has a unique idempotent. An

ideal extension of an arbitrary semigroup by a nilsemigroup [nilpotent semigroup] is called

a nil-extension [nilpotent extension]. Extending to epigroups the corresponding semigroup

terminology (see, for example, [110]), we will call an epigroup identity u = v heterotypical if

one of the words u, v contains a letter which does not occur in the other.

3.1 Proposition The following conditions on a semigroup S are equivalent:

(1) S is an archimedean epigroup;

344 L. N. Shevrin

(2) S is an epigroup in which E

is an antichain;

(3) S is an nil-extension of a completely simple semigroup;

(4) S is an epigroup satisfying some heterotypical identity;

(5) S is an epigroup with the identity (x

)

= x

;

(6) S is an epigroup with the identity (x

)

= x

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

(1) V consists of archimedean semigroups;

(2) one (hence each) of the identities appearing in Proposition 3.1 is satisﬁed in V;

(3a) V contains no two-element semilattice;

(3b) each regular epigroup in V is completely simple.

The equivalence of conditions (4)–(6) in Proposition 3.1 is an epigroup modiﬁcation of

a long-known property of periodic semigroups (see [10]). The equivalence of conditions (1)

and (3) was noted in [74]. Condition (3) leads to the following consequence.

3.3 Observation In an archimedean epigroup S,thesetGr S is the kernel of S,theGreen’s

relation D coincides with the Rees congruence corresponding to the ideal Gr S,andtherela-

tions L and R (hence also H) are congruences.

The one-sided version of Proposition 3.1 is given by the following.

3.4 Proposition The following conditions on a semigroup S are equivalent:

(1) S is a right [left] archimedean epigroup;

(2) S is an epigroup in which E

is a right [left] zero semigroup;

(3) S is an nil-extension of a right [left] group;

(4) S is an epigroup satisfying the identity x

= y

= x

Lemma 1.5, together with some additional elementary arguments, implies the following

statement distinguishing a speciﬁc subclass of the class of archimedean epigroups.

3.5 Proposition An epigroup S is a subdirect product of a completely regular semigroup and

a nilsemigroup if and only if Gr S is a retract ideal.

The theme of the retractability of the group part of an epigroup will again be heard

at the end of Subsection 3.3 and in Subsection 3.4. In particular, Theorem 3.48 gives a

characterization of epigroups appearing in Proposition 3.5 in terms of “forbidden epidivisors”.

Two extreme types of unipotent epigroups are groups and nilsemigroups. It turns out that

an arbitrary unipotent epigroup can be built of these polar types. This is seen in Proposi-

tion 3.6 obtained plainly by combining both the left and the right versions of Proposition 3.4,

taking into account Proposition 3.5.

Epigroups 345

3.6 Proposition The following conditions on a semigroup S are equivalent:

(1) S is a unipotent epigroup;

(2) S is a bi-archimedean epigroup;

(3) S is a nil-extension of a group;

(4) S is a subdirect product of a group and a nilsemigroup;

(5) S is an epigroup with the identity x

= y

3.2 Conditions under which unipotency classes are subsemigroups

One of the main “characters” in this section is the ﬁve-element Brandt semigroup B

.Itis

well known as a matrix semigroup

$

















&

Being a completely 0-simple inverse semigroup, it has a clear representation as the Rees

matrix semigroup M

[{e};2, 2; (

e 0

)] over the 0-group {e, 0}. When examining structural

properties of semigroups, it is convenient to deﬁne them by a presentation, also well known:

= c, d |c

= d

=0,cdc= c, dcd = d.

(To reduce the number of deﬁning relations, we at once employ a zero element in them.)

We will not make a distinction between the terms “the semigroup B

” and “a semigroup

isomorphic to B

”; an analogous convention will be applied to other concrete semigroups

that appear below. An ideal extension of an arbitrary semigroup by B

will be called, for

brevity, a B

-extension.

The semigroup B

is frequently encountered in various semigroup-theoretic investigations.

In particular, it plays a special role in our considerations. First of all, it provides an example

of a semigroup with the smallest number of elements that has a unipotency class (namely,

) that is not a subsemigroup. But it turns out (and it is rather surprising) that its speciﬁc

presence is a characteristic feature for any epigroup with this property. Namely, the following

statement (proved in [104, §2]) is true.

3.7 Proposition A unipotency class K

in an epigroup is not a subsemigroup if and only if

the subepigroup K

 contains a subsemigroup that is a B

-extension of a unipotent epigroup.

Any epigroup is a union of its unipotent (for instance, cyclic) subepigroups. If it has

a partition into unipotent subepigroups, this means that such a partition is unique and

its components are precisely the unipotency classes; in this case an epigroup will be called

unipotently partitionable. In the following theorem which provides a criterion for an epigroup

to be unipotently partitionable, the “if” part is delivered by Proposition 3.7, and the “only

if” part follows directly from the fact that a B

-extension of a unipotent epigroup with

idempotent e contains two elements of K

whose product is an idempotent diﬀerent from e.

3.8 Theorem An epigroup is unipotently partitionable if and only if it contains no subsemi-

group that is a B

-extension of a unipotent epigroup.

346 L. N. Shevrin

The triviality of verifying the properties of B

connected with the behavior of its elements

enables us to obtain as direct corollaries of this criterion (actually, of Proposition 3.7) all con-

ditions found before (pertaining mainly to periodic semigroups) for a unipotency class to be a

subsemigroup, and, in particular, conditions for an epigroup to be unipotently partitionable.

We limit ourselves here to only three corollaries. The assertion of the ﬁrst one was given

in [71] (and in a weaker form in [128]). The assertion of the second corollary in the case of

periodic semigroups was also mentioned in [71]. The third corollary, in view of condition (2)

of Proposition 3.1, follows automatically from the second; it was proved (as one of the main

results) in [22], and the corresponding result for periodic semigroups was given in [84].

3.9 Corollary A torsion class K of a periodic semigroup is a subsemigroup if and only if

for any a, b ∈ K there exist natural numbers p, q such that b

∈ab.

3.10 Corollary If an idempotent e of an epigroup S is such that the set of all upper bounds

for e in E

is a chain (in particular, if e is a maximal idempotent in E

),thenK

is a

subsemigroup.

3.11 Corollary Any archimedean epigroup is unipotently partitionable.

The latter corollary signiﬁcantly increases the supply of unipotently partitionable epi-

groups: among such epigroups are all the types of epigroups considered in Subsections 3.3

and 3.4. As for the varieties consisting of unipotently partitionable epigroups, see Proposi-

tion 3.18 below. It follows from Corollary 3.10 that in any epigroup with a ﬁnite number of

idempotents there are unipotency classes that are subsemigroups. On the other hand, there

exist epigroups that can be called antipodes of unipotently partitionable epigroups; in such

an epigroup none of the unipotency classes is a subsemigroup. An example of a periodic

semigroup with this property is given in [72].

If a unipotency class K

is not a subsemigroup, then there is nothing we can say about

the structure of the semigroup K

 (besides the properties indicated in the Corollary of

Lemma 1.3). The point is that any epigroup can be embedded in an epigroup generated as

a semigroup by a single unipotency class. More precisely, the following fact is true (see [104,

§7]).

3.12 Proposition Any semigroup S can be embedded in a semigroup T such that:

(a) T has a zero and it is generated by nil-elements of order 2;

(b) if S is an epigroup [periodic semigroup ],thenT is also an epigroup [periodic semigroup ];

n or n +1.

Let us return to Theorem 3.8. We now call attention to the fact that the existence

among the subsemigroups of a given semigroup S of a B

-extension of a unipotent epigroup

implies the presence of B

among the (Rees) epidivisors of S. Indeed, it is obvious that

an ideal extension of an epigroup by an epigroup is an epigroup, so a B

-extension of a

unipotent epigroup is an epigroup. The answer to the question of whether, in general, the

absence of B

among the epidivisors is a characteristic feature of unipotently partitionable