On a semitopological polycyclic monoid
We study algebraic structure of the λ-polycyclic monoid Pλ and its topologizations. We show that the λ-polycyclic monoid for an infinite cardinal λ≥2 has similar algebraic properties so has the polycyclic monoid Pn with finitely many n≥2 generators. In particular we prove that for every infinite car...
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irk-123456789-1552552019-06-17T01:26:58Z On a semitopological polycyclic monoid Bardyla, S. Gutik, O. We study algebraic structure of the λ-polycyclic monoid Pλ and its topologizations. We show that the λ-polycyclic monoid for an infinite cardinal λ≥2 has similar algebraic properties so has the polycyclic monoid Pn with finitely many n≥2 generators. In particular we prove that for every infinite cardinal λ the polycyclic monoid Pλ is a congruence-free combinatorial 0-bisimple 0-E-unitary inverse semigroup. Also we show that every non-zero element x is an isolated point in (Pλ,τ) for every Hausdorff topology τ on Pλ, such that (Pλ,τ) is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on Pλ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies τ on Pλ such that (Pλ,τ) is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal λ≥2 any continuous homomorphism from a topological semigroup Pλ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains Pλ as a dense subsemigroup. 2016 Article On a semitopological polycyclic monoid / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 163-183. — Бібліогр.: 38 назв. — англ. 1726-3255 2010 MSC:Primary 22A15, 20M18. Secondary 20M05, 22A26, 54A10, 54D30,54D35, 54D45, 54H11. http://dspace.nbuv.gov.ua/handle/123456789/155255 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We study algebraic structure of the λ-polycyclic monoid Pλ and its topologizations. We show that the λ-polycyclic monoid for an infinite cardinal λ≥2 has similar algebraic properties so has the polycyclic monoid Pn with finitely many n≥2 generators. In particular we prove that for every infinite cardinal λ the polycyclic monoid Pλ is a congruence-free combinatorial 0-bisimple 0-E-unitary inverse semigroup. Also we show that every non-zero element x is an isolated point in (Pλ,τ) for every Hausdorff topology τ on Pλ, such that (Pλ,τ) is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on Pλ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies τ on Pλ such that (Pλ,τ) is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal λ≥2 any continuous homomorphism from a topological semigroup Pλ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains Pλ as a dense subsemigroup. |
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Bardyla, S. Gutik, O. On a semitopological polycyclic monoid Algebra and Discrete Mathematics |
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Bardyla, S. Gutik, O. |
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Bardyla, S. |
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On a semitopological polycyclic monoid |
| title_short |
On a semitopological polycyclic monoid |
| title_full |
On a semitopological polycyclic monoid |
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On a semitopological polycyclic monoid |
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On a semitopological polycyclic monoid |
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on a semitopological polycyclic monoid |
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Інститут прикладної математики і механіки НАН України |
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2016 |
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On a semitopological polycyclic monoid / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 163-183. — Бібліогр.: 38 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bardylas onasemitopologicalpolycyclicmonoid AT gutiko onasemitopologicalpolycyclicmonoid |
| first_indexed |
2025-07-14T07:19:01Z |
| last_indexed |
2025-07-14T07:19:01Z |
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1837606613924446208 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 21 (2016). Number 2, pp. 163–183
© Journal “Algebra and Discrete Mathematics”
On a semitopological polycyclic monoid
Serhii Bardyla and Oleg Gutik
Communicated by M. Ya. Komarnytskyj
Abstract. We study algebraic structure of the λ-polycyclic
monoid Pλ and its topologizations. We show that the λ-polycyclic
monoid for an infinite cardinal λ > 2 has similar algebraic properties
so has the polycyclic monoid Pn with finitely many n > 2 generators.
In particular we prove that for every infinite cardinal λ the polycyclic
monoid Pλ is a congruence-free combinatorial 0-bisimple 0-E-unitary
inverse semigroup. Also we show that every non-zero element x
is an isolated point in (Pλ, τ) for every Hausdorff topology τ on
Pλ, such that (Pλ, τ) is a semitopological semigroup, and every
locally compact Hausdorff semigroup topology on Pλ is discrete.
The last statement extends results of the paper [33] obtaining for
topological inverse graph semigroups. We describe all feebly compact
topologies τ on Pλ such that (Pλ, τ) is a semitopological semigroup
and its Bohr compactification as a topological semigroup. We prove
that for every cardinal λ > 2 any continuous homomorphism from
a topological semigroup Pλ into an arbitrary countably compact
topological semigroup is annihilating and there exists no a Hausdorff
feebly compact topological semigroup which contains Pλ as a dense
subsemigroup.
2010 MSC: Primary 22A15, 20M18. Secondary 20M05, 22A26, 54A10, 54D30,
54D35, 54D45, 54H11.
Key words and phrases: inverse semigroup, bicyclic monoid, polycyclic monoid,
free monoid, semigroup of matrix units, topological semigroup, semitopological semi-
group, Bohr compactification, embedding, locally compact, countably compact, feebly
compact.
164 On a semitopological polycyclic monoid
1. Introduction and preliminaries
In this paper all topological spaces will be assumed to be Hausdorff.
We shall follow the terminology of [8, 11, 14, 32]. If A is a subset of a
topological space X, then we denote the closure of the set A in X by
clX(A). By ω we denote the first infinite cardinal.
A semigroup S is called an inverse semigroup if every a in S possesses
an unique inverse, i.e. if there exists an unique element a−1 in S such that
aa−1a = a and a−1aa−1 = a−1.
A map which associates to any element of an inverse semigroup its inverse
is called the inversion.
A band is a semigroup of idempotents. If S is a semigroup, then we
shall denote the subset of all idempotents in S by E(S). If S is an inverse
semigroup, then E(S) is closed under multiplication. The semigroup
operation on S determines the following partial order 6 on E(S): e 6 f
if and only if ef = fe = e. This order is called the natural partial order
on E(S). A semilattice is a commutative semigroup of idempotents. A
semilattice E is called linearly ordered or a chain if its natural order is
a linear order. A maximal chain of a semilattice E is a chain which is
properly contained in no other chain of E. The Axiom of Choice implies
the existence of maximal chains in any partially ordered set. According
to [36, Definition II.5.12] chain L is called ω-chain if L is isomorphic to
{0, −1, −2, −3, . . .} with the usual order 6. Let E be a semilattice and
e ∈ E. We denote ↓e = {f ∈ E | f 6 e} and ↑e = {f ∈ E | e 6 f}.
If S is a semigroup, then we shall denote by R, L, J, D and H the
Green relations on S (see [16] or [11, Section 2.1]):
aRb if and only if aS1 = bS1;
aLb if and only if S1a = S1b;
aJb if and only if S1aS1 = S1bS1;
D = L◦R = R◦L;
H = L ∩ R.
A semigroup S is said to be:
• simple if S has no proper two-sided ideals, which is equivalent to
J = S × S in S;
• 0-simple if S has a zero and S contains no proper two-sided ideals
distinct from the zero;
• bisimple if S contains a unique D-class, i.e., D = S × S in S;
S. Bardyla, O. Gutik 165
• 0-bisimple if S has a zero and S contains two D-classes: {0} and
S \ {0};
• congruence-free if S has only identity and universal congruences.
An inverse semigroup S is said to be
• combinatorial if H is the equality relation on S;
• E-unitary if for any idempotents e, f ∈ S the equality ex = f
implies that x ∈ E(S);
• 0-E-unitary if S has a zero and for any non-zero idempotents
e, f ∈ S the equality ex = f implies that x ∈ E(S).
The bicyclic monoid C(p, q) is the semigroup with the identity 1
generated by two elements p and q subjected only to the condition pq = 1.
The distinct elements of C(p, q) are exhibited in the following useful array
1 p p2 p3 · · ·
q qp qp2 qp3 · · ·
q2 q2p q2p2 q2p3 · · ·
q3 q3p q3p2 q3p3 · · ·
...
...
...
...
. . .
and the semigroup operation on C(p, q) is determined as follows:
qkpl · qmpn = qk+m−min{l,m}pl+n−min{l,m}.
It is well known that the bicyclic monoid C(p, q) is a bisimple (and
hence simple) combinatorial E-unitary inverse semigroup and every non-
trivial congruence on C(p, q) is a group congruence [11]. Also the nice
Andersen Theorem states that a simple semigroup S with an idempotent
is completely simple if and only if S does not contains an isomorphic copy
of the bicyclic semigroup (see [1] and [11, Theorem 2.54]).
Let λ be a non-zero cardinal. On the set Bλ = (λ × λ) ∪ {0}, where
0 /∈ λ × λ, we define the semigroup operation “ · ” as follows
(a, b) · (c, d) =
{
(a, d), if b = c;
0, if b 6= c,
and (a, b) · 0 = 0 · (a, b) = 0 · 0 = 0 for a, b, c, d ∈ λ. The semigroup Bλ is
called the semigroup of λ×λ-matrix units (see [11]).
In 1970 Nivat and Perrot proposed the following generalization of the
bicyclic monoid (see [35] and [32, Section 9.3]). For a non-zero cardinal
λ, the polycyclic monoid Pλ on λ generators is the semigroup with zero
166 On a semitopological polycyclic monoid
given by the presentation:
Pλ =
〈
{pi}i∈λ ,
{
p−1
i
}
i∈λ
| pip
−1
i = 1, pip
−1
j = 0 for i 6= j
〉
.
It is obvious that in the case when λ = 1 the semigroup P1 is isomorphic
to the bicyclic semigroup with adjoined zero. For every finite non-zero car-
dinal λ = n the polycyclic monoid Pn is a congruence free, combinatorial,
0-bisimple, 0-E-unitary inverse semigroup (see [32, Section 9.3]).
We recall that a topological space X is said to be:
• compact if each open cover of X has a finite subcover;
• countably compact if each open countable cover of X has a finite
subcover;
• countably compact at a subset A ⊆ X if every infinite subset B ⊆ A
has an accumulation point x in X;
• countably pracompact if there exists a dense subset A in X such
that X is countably compact at A;
• feebly compact if each locally finite open cover of X is finite.
According to Theorem 3.10.22 of [14], a Tychonoff topological space X
is feebly compact if and only if each continuous real-valued function on
X is bounded, i.e., X is pseudocompact. Also, a Hausdorff topological
space X is feebly compact if and only if every locally finite family of
non-empty open subsets of X is finite. Every compact space is countably
compact, every countably compact space is countably pracompact, and
every countably pracompact space is feebly compact (see [3] and [14]).
A topological (inverse) semigroup is a Hausdorff topological space
together with a continuous semigroup operation (and an inversion, respec-
tively). Obviously, the inversion defined on a topological inverse semigroup
is a homeomorphism. If S is a semigroup (an inverse semigroup) and τ is
a topology on S such that (S, τ) is a topological (inverse) semigroup, then
we shall call τ a (inverse) semigroup topology on S. A semitopological
semigroup is a Hausdorff topological space together with a separately
continuous semigroup operation.
The bicyclic semigroup admits only the discrete semigroup topology
and if a topological semigroup S contains it as a dense subsemigroup then
C(p, q) is an open subset of S [13]. Bertman and West in [7] extended this
result for the case of semitopological semigroups. Stable and Γ-compact
topological semigroups do not contain the bicyclic semigroup [2, 30].
The problem of an embedding of the bicyclic monoid into compact-like
topological semigroups discussed in [5, 6, 27]. In [13] Eberhart and Selden
proved that if the bicyclic monoid C(p, q) is a dense subsemigroup of a
S. Bardyla, O. Gutik 167
topological monoid S and I = S \ C(p, q) 6= ∅ then I is a two-sided ideal
of the semigroup S. Also, there they described the closure the bicyclic
monoid C(p, q) in a locally compact topological inverse semigroup. The
closure of the bicyclic monoid in a countably compact (pseudocompact)
topological semigroups was studied in [6].
In [15] Fihel and Gutik showed that any Hausdorff topology τ on the
extended bicyclic semigroup CZ such that ( CZ, τ) is a semitopological
semigroup is discrete. Also in [15] studied a closure of the extended bicyclic
semigroup CZ in a topological semigroup.
For any Hausdorff topology τ on an infinite semigroup of λ×λ-matrix
units Bλ such that (Bλ, τ) is a semitopological semigroup every non-zero
element of Bλ is an isolated point of (Bλ, τ) [22]. Also in [22] was proved
that on any infinite semigroup of λ×λ-matrix units Bλ there exists a
unique feebly compact topology τA such that (Bλ, τA) is a semitopological
semigroup and moreover this topology τA is compact. A closure of an
infinite semigroup of λ×λ-matrix units in semitopological and topological
semigroups and its embeddings into compact-like semigroups were studied
in [18,22,23].
Semigroup topologizations and closures of inverse semigroups of mono-
tone co-finite partial bijections of some linearly ordered infinite sets,
inverse semigroups of almost identity partial bijections and inverse semi-
groups of partial bijections of a bounded finite rank studied in [9, 10,17,
20,23–25,28,29].
To every directed graph E one can associate a graph inverse semigroup
G(E), where elements roughly correspond to possible paths in E. These
semigroups generalize polycyclic monoids. In [33] the authors investigated
topologies that turn G(E) into a topological semigroup. For instance,
they showed that in any such topology that is Hausdorff, G(E) \ {0} must
be discrete for any directed graph E. On the other hand, G(E) need not
be discrete in a Hausdorff semigroup topology, and for certain graphs E,
G(E) admits a T1 semigroup topology in which G(E) \ {0} is not discrete.
In [33] the authors also described the algebraic structure and possible
cardinality of the closure of G(E) in larger topological semigroups.
In this paper we show that the λ-polycyclic monoid for in infinite car-
dinal λ > 2 has similar algebraic properties so has the polycyclic monoid
Pn with finitely many n > 2 generators. In particular we prove that for
every infinite cardinal λ the polycyclic monoid Pλ is a congruence-free,
combinatorial, 0-bisimple, 0-E-unitary inverse semigroup. Also we show
that every non-zero element x is an isolated point in (Pλ, τ) for every
Hausdorff topology on Pλ, such that Pλ is a semitopological semigroup,
168 On a semitopological polycyclic monoid
and every locally compact Hausdorff semigroup topology on Pλ is dis-
crete. The last statement extends results of the paper [33] obtaining for
topological inverse graph semigroups. We describe all feebly compact
topologies τ on Pλ such that (Pλ, τ) is a semitopological semigroup and
its Bohr compactification as a topological semigroup. We prove that for
every cardinal λ > 2 any continuous homomorphism from a topological
semigroup Pλ into an arbitrary countably compact topological semigroup
is annihilating and there exists no a Hausdorff feebly compact topological
semigroup which contains Pλ as a dense subsemigroup.
2. Algebraic properties of the λ-polycyclic monoid for an
infinite cardinal λ
In this section we assume that λ is an infinite cardinal.
We repeat the thinking and arguments from [32, Section 9.3].
We shall give a representation for the polycyclic monoid Pλ by means
of partial bijections on the free monoid Mλ over the cardinal λ. Put
A = {xi : i ∈ λ}. Then the free monoid Mλ over the cardinal λ is
isomorphic to the free monoid Mλ over the set A. Next we define for
every i ∈ λ the partial map α : Mλ → Mλ by the formula (u)αi = xiu
and put that Mλ is the domain and xiMλ is the range of αi. Then for
every i ∈ λ we may regard so defined partial map as an element of the
symmetric inverse monoid I(Mλ) on the set Mλ. Denote by Iλ the inverse
submonoid of I(Mλ) generated by the set {αi : i ∈ λ}. We observe that
αiα
−1
i is the identity partial map on Mλ for each i ∈ λ and whereas if
i 6= j then αiα
−1
j is the empty partial map on the set Mλ, i, j ∈ λ. Define
the map h : Pλ → Iλ by the formula (pi)h = αi and (p−1
i )h = α−1
i , i ∈ λ.
Then by Proposition 2.3.5 of [32], Iλ is a homomorphic image of Pλ and
by Proposition 9.3.1 from [32] the map h : Pλ → Iλ is an isomorphism.
Since the band of the semigroup Iλ consists of partial identity maps, the
identifying the semilattice of idempotents of Iλ with the free monoid M0
λ
with adjoined zero admits the following partial order on M0
λ:
u 6 v if and only if v is a prefix of u for u, v ∈ M0
λ,
and 0 6 u for every u ∈ M0
λ.
(1)
This partial order admits the following semilattice operation on M0
λ:
u ∗ v = v ∗ u =
{
u, if v is a prefix of u;
0, otherwise,
S. Bardyla, O. Gutik 169
and 0 ∗ u = u ∗ 0 = 0 ∗ 0 = 0 for arbitrary words u, v ∈ M0
λ.
Remark 2.1. We observe that for an arbitrary non-zero cardinal λ the
set M0
λ \ {0} with the dual partial order to (1) is order isomorphic to the
λ-ary tree Tλ with the countable height.
Hence, we proved the following proposition.
Proposition 2.2. For every infinite cardinal λ the semigroup Pλ is
isomorphic to the inverse semigroup Iλ and the semilattice E(Pλ) is
isomorphic to (M0
λ, ∗).
Let n be any positive integer and i1, . . . , in ∈ λ. We put
P λ
n 〈i1, . . . , in〉
=
〈
pi1
, . . . , pin
, p−1
i1
, . . . , p−1
in
| pik
p−1
ik
= 1, pik
p−1
il
= 0 for ik 6= il
〉
.
The statement of the following lemma is trivial.
Lemma 2.3. Let λ be an infinite cardinal and n be an arbitrary positive
integer. Then P λ
n 〈i1, . . . , in〉 is a submonoid of the polycyclic monoid Pλ
such that P λ
n 〈i1, . . . , in〉 is isomorphic to Pn for arbitrary i1, . . . , in ∈ λ.
Our above representation of the polycyclic monoid Pλ by means of
partial bijections on the free monoid Mλ over the cardinal λ implies the
following lemma.
Lemma 2.4. Let λ be an infinite cardinal. Then for any elements
x1,. . ., xk ∈Pλ there exist i1,. . ., in ∈λ such that x1, . . . , xk ∈P λ
n 〈i1, . . . , in〉.
Theorem 2.5. For every infinite cardinal λ the polycyclic monoid Pλ is a
congruence-free combinatorial 0-bisimple 0-E-unitary inverse semigroup.
Proof. By Proposition 2.2 the semigroup Pλ is inverse.
First we show that the semigroup Pλ is 0-bisimple. Then by the
Munn Lemma (see [34, Lemma 1.1] and [32, Proposition 3.2.5]) it is
sufficient to show that for any two non-zero idempotents e, f ∈ Pλ there
exists x ∈ Pλ such that xx−1 = e and x−1x = f . Fix arbitrary two
non-zero idempotents e, f ∈ Pλ. By Lemma 2.4 there exist i1, . . . , in ∈ λ
such that e, f ∈ P λ
n 〈i1, . . . , in〉. Lemma 2.3, Theorem 9.3.4 of [32] and
Proposition 3.2.5 of [32] imply that there exists x ∈ P λ
n 〈i1, . . . , in〉 ⊂ Pλ
such that xx−1 = e and x−1x = f . Hence the semigroup Pλ is 0-bisimple.
The above representation of the polycyclic monoid Pλ by means of
partial bijections on the free monoid Mλ over the cardinal λ implies that
170 On a semitopological polycyclic monoid
the H-class in Pλ which contains the unity is a singleton. Then since
the polycyclic monoid Pλ is 0-bisimple Theorem 2.20 of [11] implies that
every non-zero H-class in Pλ is a singleton. It is obvious that H-class in
Pλ which contains zero is a singleton. This implies that the polycyclic
monoid Pλ is combinatorial.
Suppose to the contrary that the monoid Pλ is not 0-E-unitary. Then
there exist a non-idempotent element x ∈ Pλ and non-zero idempotents
e, f ∈ Pλ such that xe = f . By Lemma 2.4 there exist i1, . . . , in ∈ λ
such that x, e, f ∈ P λ
n 〈i1, . . . , in〉. Hence the monoid P λ
n 〈i1, . . . , in〉 is not
0-E-unitary, which contradicts Lemma 2.3 and Theorem 9.3.4 of [32].
The obtained contradiction implies that the polycyclic monoid Pλ is a
0-E-unitary inverse semigroup.
Suppose the contrary that there exists a congruence C on the polycyclic
monoid Pλ which is distinct from the identity and the universal congruence
on Pλ. Then there exist distinct x, y ∈ Pλ such that xCy. By Lemma 2.4
there exist i1, . . . , in ∈ λ such that x, y ∈ P λ
n 〈i1, . . . , in〉. By Lemma 2.3
and Theorem 9.3.4 of [32], since the polycyclic monoid Pn is congruence-
free we have that the unity and zero of the polycyclic monoid Pλ are C-
equivalent and hence all elements of Pλ are C-equivalent. This contradicts
our assumption. The obtained contradiction implies that the polycyclic
monoid Pλ is a congruence-free semigroup.
From now for an arbitrary cardinal λ > 2 we shall call the semigroup
Pλ the λ-polycyclic monoid.
Fix an arbitrary cardinal λ > 2 and two distinct elements a, b ∈ λ.
We consider the following subset A = {bia : i = 0, 1, 2, 3, . . .} of the free
monoid Mλ. The definition of the above defined partial order 6 on M0
λ
implies that two arbitrary distinct elements of the set A are incomparable
in (M0
λ,6). Let B(bia) be a subsemigroup of Iλ generated by the subset
{
α ∈ Iλ : dom α = biaMλ and ran α = bjaMλ for some i, j ∈ ω
}
of the semigroup Iλ. Since two arbitrary distinct elements of the set A
are incomparable in the partially ordered set (M0
λ,6) the semigroup
operation of Iλ implies that the following conditions hold:
(i) αβ is a non-zero element of the semigroup Iλ if and only if ran α =
dom β;
(ii) αβ = 0 in Iλ if and only if ran α 6= dom β;
(iii) if αβ 6= 0 in Iλ then dom(αβ) = dom α and ran(αβ) = ran β;
(iv) B(bia) is an inverse subsemigroup of Iλ,
S. Bardyla, O. Gutik 171
for arbitrary α, β ∈ B(bia).
Now, if we identify ω with the set of all non-negative inte-
gers {0, 1, 2, 3, 4, . . .}, then simple verifications show that the map
h : B(bia) → Bω defined in the following way:
(a) if α 6= 0, dom α = biaMλ and ran α = bjaMλ, then (α)h = (i, j),
for i, j ∈ {0, 1, 2, 3, 4, . . .};
(b) (0)h = 0,
is a semigroup isomorphism.
Hence we proved the following proposition.
Proposition 2.6. For every cardinal λ > 2 the λ-polycyclic monoid Pλ
contains an isomorphic copy of the semigroup of ω×ω-matrix units Bω.
Proposition 2.7. For every non-zero cardinal λ and any α, β ∈ Pλ \{0},
both sets {χ ∈ Pλ : α · χ = β} and {χ ∈ Pλ : χ · α = β} are finite.
Proof. We show that the set {χ ∈ Pλ : α · χ = β} is finite. The proof in
other case is similar.
It is obvious that
{χ ∈ Pλ : α · χ = β} ⊆
{
χ ∈ Pλ : α−1 · α · χ = α−1 · β
}
.
Then the definition of the semigroup Iλ implies there exist words u, v ∈ Mλ
such that the partial map α−1 · β is the map from uMλ onto vMλ defined
by the formula (ux)(α−1 · β) = vx for any x ∈ Mλ. Since α−1 · α is
an identity partial map of Mλ we get that the partial map α−1 · β is
a restriction of the partial map χ on the set dom(α−1 · α). Hence by
the definition of the semigroup Iλ there exists words u1, v1 ∈ Mλ such
that u1 is a prefix of u, v1 is a prefix of v and χ is the map from u1Mλ
onto v1Mλ defined by the formula (u1x)(α−1 · β) = v1x for any x ∈ Mλ.
Now, since every word of free monoid Mλ has finitely many prefixes we
conclude that the set
{
χ ∈ Pλ : α−1 · α · χ = α−1 · β
}
is finite, and hence
so is {χ ∈ Pλ : α · χ = β}.
Later we need the following lemma.
Lemma 2.8. Let λ be any cardinal > 2. Then an element x of the λ-
polycyclic monoid Pλ is R-equivalent to the identity 1 of Pλ if and only
if x = pi1
. . . pin
for some generators pi1
, . . . , pin
∈ {pi}i∈λ.
Proof. We observe that the definition of the R-relation implies that xR1
if and only if xx−1 = 1 (see [32, Section 3.2]).
172 On a semitopological polycyclic monoid
(⇒) Suppose that an element x of Pλ has a form x = pi1
. . . pin
. Then
the definition of the λ-polycyclic monoid Pλ implies that
xx−1 = (pi1
. . . pin
) (pi1
. . . pin
)−1 = pi1
. . . pin
p−1
in
. . . p−1
i1
= 1,
and hence xR1.
(⇐) Suppose that some element x of the λ-polycyclic monoid Pλ is
R-equivalent to the identity 1 of Pλ. Then the definition of the semigroup
Pλ implies that there exist finitely many pi1
, . . . , pin
∈ {pi}i∈λ such that
x is an element of the submonoid P λ
n 〈i1, . . . , in〉 of Pλ, which is generated
by elements pi1
, . . . , pin
, i.e.,
P λ
n 〈i1, . . . , in〉
=
〈
pi1
, . . . , pin
, p−1
i1
, . . . , p−1
in
: pik
p−1
ik
= 1, pik
p−1
il
= 0 for ik 6= il
〉
.
Proposition 9.3.1 of [32] implies that the element x is equal to the unique
string of the form u−1v, where u and v are strings of the free monoid
M{pi1
,...,pin
} over the set {pi1
, . . . , pin
}. Next we shall show that u is the
empty string of M{pi1
,...,pin
}. Suppose that u = a1 . . . ak and v = b1 . . . bl,
for some a1, . . . , ak, b1, . . . , bl ∈ {pi1
, . . . , pin
} and u is not the empty-
string of M{pi1
,...,pin
}. Then the definition of the λ-polycyclic monoid Pλ
implies that
xx−1 =
(
u−1v
) (
u−1v
)−1
= u−1vv−1u
= (a1 . . . ak)−1 (b1 . . . bl) (b1 . . . bl)
−1 (a1 . . . ak)
= a−1
k . . . a−1
1 b1 . . . blb
−1
l . . . b−1
1 a1 . . . ak
. . .
= a−1
k . . . a−1
1 1a1 . . . ak
= a−1
k . . . a−1
1 a1 . . . ak 6= 1,
which contradicts the assumption that xR1. The obtained contradiction
implies that the element x has the form x = pi1
. . . pin
for some generators
pi1
, . . . , pin
from the set {pi}i∈λ.
3. On semigroup topologizations
of the λ-polycyclic monoid
In [13] Eberhart and Selden proved that if τ is a Hausdorff topology
on the bicyclic monoid C(p, q) such that ( C(p, q), τ) is a topological
S. Bardyla, O. Gutik 173
semigroup then τ is discrete. In [7] Bertman and West extended this results
for the case when ( C(p, q), τ) is a Hausdorff semitopological semigroup.
In [33] there proved that for any positive integer n > 1 every non-zero
element in a Hausdorff topological n-polycyclic monoid Pn is an isolated
point. The following proposition generalizes the above results.
Proposition 3.1. Let λ be any cardinal > 2 and τ be any Hausdorff
topology on Pλ, such that Pλ is a semitopological semigroup. Then every
non-zero element x is an isolated point in (Pλ, τ).
Proof. We observe that the λ-polycyclic monoid Pλ is a 0-bisimple semi-
group, and hence is a 0-simple semigroup. Then the continuity of right and
left translations in (Pλ, τ) and Proposition 2.7 imply that it is complete
to show that there exists an non-zero element x of Pλ such that x is an
isolated point in the topological space (Pλ, τ).
Suppose to the contrary that the unit 1 of the λ-polycyclic monoid
Pλ is a non-isolated point of the topological space (Pλ, τ). Then every
open neighbourhood U(1) of 1 in (Pλ, τ) is infinite subset.
Fix a singleton word x in the free monoid Mλ. Let ε be an idempotent
of the λ-polycyclic monoid Pλ which corresponds to the identity partial
map of xMλ. Since left and right translation on the idempotent ε are
retractions of the topological space (Pλ, τ) the Hausdorffness of (Pλ, τ)
implies that εPλ and Pλε are closed subsets of the topological space
(Pλ, τ), and hence so is the set εPλ ∪ Pλε. The separate continuity of the
semigroup operation and Hausdorffness of (Pλ, τ) imply that for every
open neighbourhood U(ε) 6∋ 0 of the point ε in (Pλ, τ) there exists an
open neighbourhood U(1) of the unit 1 in (Pλ, τ) such that
U(1) ⊆ Pλ \ (εPλ ∪ Pλε), ε · U(1) ⊆ U(ε) and U(1) · ε ⊆ U(ε).
We observe that the idempotent ε is maximal in Pλ \{1}. Hence any other
idempotent ι ∈ Pλ \ (εPλ ∪ Pλε) is incomparable with ε. Since the set
U(1) is infinite there exists an element α ∈ U(1) such that either α · α−1
or α−1 · α is an incomparable idempotent with ε. Then we get that either
ε · α = ε · (α · α−1 · α) = (ε · α · α−1) · α = 0 · α = 0 ∈ U(ε)
or
α · ε = (α · α−1 · α) · ε = α · (α−1 · α · ε) = α · 0 = 0 ∈ U(ε).
The obtained contradiction implies that the unit 1 is an isolated point of
the topological space (Pλ, τ), which completes the proof of our proposition.
174 On a semitopological polycyclic monoid
A topological space X is called collectionwise normal if X is T1-space
and for every discrete family {Fα}α∈J of closed subsets of X there exists
a discrete family {Sα}α∈J of open subsets of X such that Fα ⊆ Sα for
every α ∈J [14].
Proposition 3.2. Every Hausdorff topological space X with a unique
non-isoloated point is collectionwise normal.
Proof. Suppose that a is a non-isolated point of X. Fix an arbitrary
discrete family {Fα}α∈J of closed subsets of the topological space X.
Then there exists an open neighbourhood U(a) of the point a in X which
intersects at most one element of the family {Fα}α∈J. In the case when
U(a) ∩ Fα = ∅ for every α ∈ J we put Sα = Fα for all α ∈ J. If
U(a) ∩ Fα0
6= ∅ for some α0 ∈J we put Sα0
= U(a) ∪ Fα0
and Sα = Fα
for all α ∈J \ {α0}. Then {Sα}α∈J is a discrete family of open subsets
of X such that Fα ⊆ Sα for every α ∈J.
Propositions 3.1 and 3.2 imply the following corollary.
Corollary 3.3. Let λ be any cardinal > 2 and τ be any Hausdorff topology
on Pλ, such that Pλ is a semitopological semigroup. Then the topological
space (Pλ, τ) is collectionwise normal.
In [33] there proved that for arbitrary finite cardinal > 2 every Haus-
dorff locally compact topology τ on Pλ such that (Pλ, τ) is a topological
semigroup, is discrete. The following proposition extends this result for
any infinite cardinal λ.
Proposition 3.4. Let λ be an infinite cardinal and τ be a locally compact
Hausdorff topology on Pλ such that (Pλ, τ) is a topological semigroup. Then
τ is discrete.
Proof. Suppose to the contrary that there exist a Hausdorff locally com-
pact non-discrete semigroup topology τ on Pλ. Then by Proposition 3.1
every non-zero element the semigroup Pλ is an isolated point in (Pλ, τ).
This implies that for any compact open neighbourhoods U(0) and V (0)
of zero 0 in (Pλ, τ) the set U(0) \ V (0) is finite. Hence zero 0 of Pλ is an
accumulation point of any infinite subset of an arbitrary open compact
neighbourhood U(0) of zero in (Pλ, τ).
Put R1 is the R-class of the semigroup Pλ which contains the identity
1 of Pλ. Then only one of the following conditions holds:
S. Bardyla, O. Gutik 175
(1) there exists a compact open neighbourhood U(0) of zero 0 in (Pλ, τ)
such that U(0) ∩ R1 = ∅;
(2) U(0) ∩ R1 is an infinite set for every compact open neighbourhood
U(0) of zero 0 in (Pλ, τ).
Suppose that case (1) holds. For arbitrary x ∈ R1 we put
R[x] =
{
a ∈ R1 : x−1a ∈ U(0)
}
.
Next we shall show that the set R[x] is finite for any x ∈ R1. Suppose
to the contrary that R[x] is infinite for some x ∈ R1. Then Lemma 2.8
implies that x−1a is non-zero element of Pλ for every a ∈ R[x], and hence
by Proposition 2.7,
B =
{
x−1a : a ∈ R[x]
}
is an infinite subset of the neighbourhood U(0). Therefore, the above
arguments imply that 0 ∈ clPλ
(B). Now, the continuity of the semigroup
operation in (Pλ, τ) implies that
0 = x · 0 ∈ x · clPλ
(B) ⊆ clPλ
(x · B).
Then Lemma 2.8 implies that xx−1 = 1 for any x ∈ R1 and hence we
have that
x · B =
{
xx−1a : a ∈ R[x]
}
= {a : a ∈ R[x]} = R[x] ⊆ R1.
This implies that every open neighbourhood U(0) of zero 0 in (Pλ, τ)
contains infinitely many elements from the class R1, which contradicts
our assumption.
Suppose that case (2) holds. Then the set {0} is a compact minimal
ideal of the topological semigroup (Pλ, τ). Now, by Lemma 1 of [31]
(also see [8, Vol. 1, Lemma 3,12]) for every open neighbourhood W (0)
of zero 0 in (Pλ, τ) there exists an open neighbourhood O(0) of zero 0
in (Pλ, τ) such that O(0) ⊆ W (0) and O(0) is an ideal of clPλ
(O(0)),
i.e., O(0) · clPλ
(O(0)) ∪ clPλ
(O(0)) · O(0) ⊆ O(0). But by Proposition 3.1
all non-zero elements of Pλ are isolated points in (Pλ, τ), and hence
we have that clPλ
(O(0)) = O(0). This implies that O(0) is an open-
and-closed subsemigroup of the topological semigroup (Pλ, τ). Therefore,
the topological λ-polycyclic monoid (Pλ, τ) has a base B(0) at zero
0 which consists of open-and-closed subsemigroups of (Pλ, τ). Fix an
arbitrary S ∈ B(0). Then our assumption implies that there exists
x ∈ S ∩ R1. Since x ∈ R1, Lemma 2.8 implies that xx−1 = 1. Without
176 On a semitopological polycyclic monoid
loss of generality we may assume that x−1x 6= 1, because S is a proper
ideal of Pλ. Put B(x) =
〈
x, x−1
〉
. Then Lemma 1.31 of [11] implies that
B(x) is isomorphic to the bicyclic monoid, and since by Proposition 3.1 all
non-zero elements of Pλ are isolated points in (Pλ, τ), B0(x) = B(x) ⊔ {0}
is a closed subsemigroup of the topological semigroup (Pλ, τ), and hence
by Corollary 3.3.10 of [14], B0(x) with the induced topology τB from
(Pλ, τ) is a Hausdorff locally compact topological semigroup. Also, the
above presented arguments imply that 〈x〉∪{0} with the induced topology
from (Pλ, τ) is a compact topological semigroup, which is contained in
B0(x) as a subsemigroup. But by Corollary 1 from [19], (B0(x), τB) is
the discrete space, which contains a compact infinite subspace 〈x〉 ∪ {0}.
Hence case (2) does not hold.
The presented above arguments imply that there exists no non-
discrete Hausdorff locally compact semigroup topology on the λ-polycyclic
monoid Pλ.
The following example shows that the statements of Proposition 3.4
does not extend in the case when (Pλ, τ) is a semitopological semigroup
with continuous inversion. Moreover there exists a compact Hausdorff
topology τA-c on Pλ such that (Pλ, τA-c) is semitopological inverse semi-
group with continuous inversion.
Example 3.5. Let λ is any cardinal > 2. Put τA-c is the topology of the
one-point Alexandroff compactification of the discrete space Pλ \ {0} with
the narrow {0}, where 0 is the zero of the λ-polycyclic monoid Pλ. Since
Pλ \ {0} is a discrete open subspace of (Pλ, τA-c), it is complete to show
that the semigroup operation is separately continuous in (Pλ, τA-c) in the
following two cases:
x · 0 and 0 · x,
where x is an arbitrary non-zero element of the semigroup Pλ. Fix an
arbitrary open neighbourhood UA(0) of the zero in (Pλ, τA-c) such that
A = Pλ \ UA(0) is a finite subset of Pλ. By Proposition 2.7,
RA
x = {a ∈ Pλ : x · a ∈ A} and LA
x = {a ∈ Pλ : a · x ∈ A}
are finite not necessary non-empty subsets of the semigroup Pλ. Put
URA
x
(0) = Pλ \ RA
x , ULA
x
(0) = Pλ \ LA
x and UA−1 = Pλ \ {a : a−1 ∈ A}.
Then we get that
x · URA
x
(0) ⊆ UA(0), ULA
x
(0) · x ⊆ UA(0) and (UA−1)−1 ⊆ UA(0),
S. Bardyla, O. Gutik 177
and hence the semigroup operation is separately continuous and the
inversion is continuous in (Pλ, τA-c).
Proposition 3.6. Let λ is any cardinal > 2 and τ be a Hausdorff topology
on Pλ such that (Pλ, τ) is a semitopological semigroup. Then the following
conditions are equivalent:
(i) τ = τA-c;
(ii) (Pλ, τ) is a compact semitopological semigroup;
(iii) (Pλ, τ) is a feebly compact semitopological semigroup.
Proof. Implications (i) ⇒ (ii) and (ii) ⇒ (iii) are trivial and implication
(ii) ⇒ (i) follows from Proposition 3.1.
(iii) ⇒ (ii) Suppose there exists a feebly compact Hausdorff topology
τ on Pλ such that (Pλ, τ) is a non-compact semitopological semigroup.
Then there exists an open cover {Uα}α∈J which does not contain a
finite subcover. Let Uα0
be an arbitrary element of the family {Uα}α∈J
which contains zero 0 of the semigroup Pλ. Then Pλ \ Uα0
= AUα0
is an
infinite subset of Pλ. By Proposition 3.1, {Uα0
} ∪
{
{x} : x ∈ AUα0
}
is an
infinite locally finite family of open subset of the topological space (Pλ, τ),
which contradicts that the space (Pλ, τ) is feebly compact. The obtained
contradiction implies the requested implication.
It is well known that the closure clS(T ) of an arbitrary subsemigroup
T in a semitopological semigroup S again is a subsemigroup of S (see [37,
Proposition I.1.8(ii)]). The following proposition describes the structure
of a narrow of the λ-polycyclic monoid Pλ in a semitopological semigroup.
Proposition 3.7. Let λ is any cardinal > 2, S be a Hausdorff semitopo-
logical semigroup and Pλ is a dense subsemigroup of S. Then S \ Pλ ∪ {0}
is a closed ideal of S.
Proof. First we observe by Proposition I.1.8(iii) from [37] the zero 0 of
the λ-polycyclic monoid Pλ is a zero of the semitopological semigroup S.
Hence the statement of the proposition is trivial when S \ Pλ = ∅.
Assume that S \ Pλ 6= ∅. Put I = S \ Pλ ∪ {0}. By Theorem 3.3.9 of
[14], I is a closed subspace of S. Suppose to the contrary that I is not an
ideal of S. If I ·S * I then there exist x ∈ I\{0} and y ∈ Pλ\{0} such that
x · y = z ∈ Pλ \ {0}. By Theorem 3.3.9 of [14], y and z are isolated points
of the topological space S. Then the separate continuity of the semigroup
operation in S implies that there exists an open neighbourhood U(x) of the
point x in S such that U(x) ·{y} = {z}. Then we get that |U(x)∩Pλ| > ω
178 On a semitopological polycyclic monoid
which contradicts Proposition 2.7. The obtained contradiction implies the
inclusion I · S ⊆ I. The proof of the inclusion S · I ⊆ I is similar.
Now we shall show that I · I ⊆ I. Suppose to the contrary that there
exist x, y ∈ I \ {0} such that x · y = z ∈ Pλ \ {0}. By Theorem 3.3.9 of
[14], z is an isolated point of the topological space S. Then the separate
continuity of the semigroup operation in S implies that there exists an
open neighbourhood U(x) of the point x in S such that U(x) · {y} = {z}.
Since |U(x) ∩ Pλ| > ω there exists a ∈ Pλ \ {0} such that a · y ∈ a · I * I
which contradicts the above part of our proof. The obtained contradiction
implies the statement of the proposition.
4. Embeddings of the λ-polycyclic monoid into compact-
like topological semigroups
By Theorem 5 of [23] the semigroup of ω×ω-matrix units does not
embed into any countably compact topological semigroup. Then by Propo-
sition 2.6 we have that for every cardinal λ > 2 the λ-polycyclic monoid
Pλ does not embed into any countably compact topological semigroup
too.
A homomorphism h from a semigroup S into a semigroup T is called
annihilating if there exists c ∈ T such that (s)h = c for all s ∈ S. By
Theorem 6 of [23] every continuous homomorphism from the semigroup
of ω×ω-matrix units into an arbitrary countably compact topological
semigroup is annihilating. Then since by Theorem 2.5 the semigroup Pλ
is congruence-free Theorem 6 of [23] and Theorem 2.5 imply the following
corollary.
Corollary 4.1. For every cardinal λ > 2 any continuous homomorphism
from a topological semigroup Pλ into an arbitrary countably compact
topological semigroup is annihilating.
Proposition 4.2. For every cardinal λ > 2 any continuous homomor-
phism from a topological semigroup Pλ into a topological semigroup S
such that S × S is a Tychonoff pseudocompact space is annihilating, and
hence S does not contain the λ-polycyclic monoid Pλ.
Proof. First we shall show that S does not contain the λ-polycyclic
monoid Pλ. By [4, Theorem 1.3] for any topological semigroup S with
the pseudocompact square S × S the semigroup operation µ : S × S → S
extends to a continuous semigroup operation βµ : βS × βS → βS, so S
is a subsemigroup of the compact topological semigroup βS. Therefore
S. Bardyla, O. Gutik 179
the λ-polycyclic monoid Pλ is a subsemigroup of compact topological
semigroup βS which contradicts Corollary 4.1. The first statement of
the proposition implies from the statement that Pλ is a congruence-free
semigroup.
Recall [12] that a Bohr compactification of a topological semigroup S
is a pair (β, B(S)) such that B(S) is a compact topological semigroup,
β : S → B(S) is a continuous homomorphism, and if g : S → T is a
continuous homomorphism of S into a compact semigroup T , then there
exists a unique continuous homomorphism f : B(S) → T such that the
diagram
S
β
//
g
��
B(S)
f
}}
T
commutes.
By Theorem 2.5 for every infinite cardinal λ the polycyclic monoid Pλ
is a congruence-free inverse semigroup and hence Corollary 4.1 implies
the following corollary.
Corollary 4.3. For every cardinal λ > 2 the Bohr compactification of a
topological λ-polycyclic monoid Pλ is a trivial semigroup.
The following theorem generalized Theorem 5 from [23].
Theorem 4.4. For every infinite cardinal λ the semigroup of λ×λ-matrix
units Bλ does not densely embed into a Hausdorff feebly compact topological
semigroup.
Proof. Suppose to the contrary that there exists a Hausdorff feebly com-
pact topological semigroup S which contains the semigroup of λ×λ-matrix
units Bλ as a dense subsemigroup.
First we shall show that the subsemigroup of idempotents E(Bλ) of
the semigroup λ×λ-matrix units Bλ with the induced topology from
S is compact. Suppose to the contrary that E(Bλ) is not a compact
subspace of S. Then there exists an open neighbourhood U(0) of the zero
0 of S such that E(Bλ) \ U(0) is an infinite subset of E(Bλ). Since the
closure of semilattice in a topological semigroup is subsemilattice (see
[21, Corollary 19]) and every maximal chain of E(Bλ) is finite, Theorem 9
of [38] implies that the band E(Bλ) is a closed subsemigroup of S. Now,
by Lemma 2 from [22] every non-zero element of the semigroup Bλ is an
180 On a semitopological polycyclic monoid
isolated point in the space S, and hence by Theorem 3.3.9 of [14], Bλ \{0}
is an open discrete subspace of the topological space S. Therefore we get
that E(Bλ) \ U(0) is an infinite open-and-closed discrete subspace of S.
This contradicts the condition that S is a feebly compact space.
If the subsemigroup of idempotents E(Bλ) is compact then by The-
orem 1 from [23] the semigroup of λ×λ-matrix units Bλ is closed sub-
semigroup of S and since Bλ is dense in S, the semigroup Bλ coincides
with the topological semigroup S. This contradicts Theorem 2 of [22]
which states that there exists no a feebly compact Hausdorff topology τ
on the semigroup of λ×λ-matrix units Bλ such that (Bλ, τ) is a topologi-
cal semigroup. The obtained contradiction implies the statement of the
theorem.
Lemma 4.5. Every Hausdorff feebly compact topological space with a
dense discrete subspace is countably pracompact.
Proof. Suppose to the contrary that there exists a feebly compact topo-
logical space X with a dense discrete subspace D such that X is not
countably pracompact. Then every dense subset A in the topological
space X contains an infinite subset BA such that BA hasn’t an accumu-
lation point in X. Hence the dense discrete subspace D of X contains
an infinite subset BD such that BD hasn’t an accumulation point in the
topological space X. Then BD is a closed subset of X. By Theorem 3.3.9
of [14], D is an open subspace of X, and hence we have that BD is a
closed-and-open discrete subspace of the space X, which contradicts the
feeble compactness of the space S. The obtained contradiction implies
the statement of the lemma.
Theorem 4.6. For arbitrary cardinal λ > 2 there exists no Hausdorff fee-
bly compact topological semigroup which contains the λ-polycyclic monoid
Pλ as a dense subsemigroup.
Proof. By Proposition 3.1 and Lemma 4.5 it is suffices to show that there
does not exist a Hausdorff countably pracompact topological semigroup
which contains the λ-polycyclic monoid Pλ as a dense subsemigroup.
Suppose to the contrary that there exists a Hausdorff countably
pracompact topological semigroup S which contains the λ-polycyclic
monoid Pλ as a dense subsemigroup. Then there exists a dense subset A
in S such that every infinite subset B ⊆ A has an accumulation point
in the topological space S. By Proposition 3.1, Pλ \ {0} is a discrete
dense subspace of S and hence Theorem 3.3.9 of [14] implies that Pλ \ {0}
S. Bardyla, O. Gutik 181
is an open subspace of S. Therefore we have that Pλ \ {0} ⊆ A. Now,
by Proposition 2.6 the λ-polycyclic monoid Pλ contains an isomorphic
copy of the semigroup of ω×ω-matrix units Bω. Then the countable
pracompactness of the space S implies that every infinite subset C of the
set Bω{0} has an accumulating point in X, and hence the closure clS(Bω)
is a countably pracompact subsemigroup of the topological semigroup
S. This contradicts Theorem 4.4. The obtained contradiction implies the
statement of the theorem.
Acknowledgements
We acknowledge Alex Ravsky for his comments and suggestions.
References
[1] O. Andersen, Ein Bericht über die Struktur abstrakter Halbgruppen, PhD Thesis,
Hamburg, 1952.
[2] L.W. Anderson, R.P. Hunter, R.J. Koch, Some results on stability in semigroups,
Trans. Amer. Math. Soc. 117 (1965), 521–529.
[3] A. V. Arkhangel’skii, Topological Function Spaces, Kluwer Publ., Dordrecht, 1992.
[4] T. O. Banakh and S. Dimitrova, Openly factorizable spaces and compact extensions
of topological semigroups, Commentat. Math. Univ. Carol. 51:1 (2010), 113–131.
[5] T. Banakh, S. Dimitrova, and O. Gutik, The Rees-Suschkiewitsch Theorem for
simple topological semigroups, Mat. Stud. 31:2 (2009), 211–218.
[6] T. Banakh, S. Dimitrova, and O. Gutik, Embedding the bicyclic semigroup into
countably compact topological semigroups, Topology Appl. 157:18 (2010), 2803–
2814.
[7] M. O. Bertman and T. T. West, Conditionally compact bicyclic semitopological
semigroups, Proc. Roy. Irish Acad. A76:21–23 (1976), 219–226.
[8] J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of Topological
Semigroups, Vol. I, Marcel Dekker, Inc., New York and Basel, 1983; Vol. II, Marcel
Dekker, Inc., New York and Basel, 1986.
[9] I. Chuchman and O. Gutik, Topological monoids of almost monotone injective
co-finite partial selfmaps of the set of positive integers, Carpathian Math. Publ.
2:1 (2010), 119–132.
[10] I. Chuchman and O. Gutik, On monoids of injective partial selfmaps almost
everywhere the identity, Demonstr. Math. 44:4 (2011), 699–722.
[11] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vols. I
and II, Amer. Math. Soc. Surveys 7, Providence, R.I., 1961 and 1967.
[12] K. DeLeeuw and I. Glicksberg, Almost-periodic functions on semigroups, Acta
Math. 105 (1961), 99–140.
[13] C. Eberhart and J. Selden, On the closure of the bicyclic semigroup, Trans. Amer.
Math. Soc. 144 (1969), 115–126.
182 On a semitopological polycyclic monoid
[14] R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.
[15] I. R. Fihel and O. V. Gutik, On the closure of the extended bicyclic semigroup,
Carpathian Math. Publ. 3:2 (2011), 131–157.
[16] J. A. Green, On the structure of semigroups, Ann. Math. (2) 54 (1951), 163-–172.
[17] I. Guran, O. Gutik, O. Ravs’kyj, and I. Chuchman, Symmetric topological groups
and semigroups, Visn. L’viv. Univ., Ser. Mekh.-Mat. 74 (2011), 61–73.
[18] O. Gutik, On closures in semitopological inverse semigroups with continuous
inversion, Algebra Discr. Math. 18:1 (2014), 59-–85.
[19] O. Gutik, On the dichotomy of a locally compact semitopological bicyclic monoid
with adjoined zero, Visn. L’viv. Univ., Ser. Mekh.-Mat. 80 (2015), 33–41.
[20] O. Gutik, J. Lawson, and D. Repovs, Semigroup closures of finite rank symmetric
inverse semigroups, Semigroup Forum 78:2 (2009), 326–336.
[21] O. Gutik and K. Pavlyk, Topological Brandt λ-extensions of absolutely H-closed
topological inverse semigroups, Visn. L’viv. Univ., Ser. Mekh.-Mat. 61 (2003),
98–105.
[22] O. V. Gutik and K. P. Pavlyk, Topological semigroups of matrix units, Algebra
Discrete Math. no. 3 (2005), 1–17.
[23] O. Gutik, K. Pavlyk, and A. Reiter, Topological semigroups of matrix units and
countably compact Brandt λ
0-extensions, Mat. Stud. 32:2 (2009), 115–131.
[24] O. Gutik and I. Pozdnyakova, On monoids of monotone injective partial selfmaps
of Ln ×lex Z with co-finite domains and images, Algebra Discrete Math. 17:2
(2014), 256–279.
[25] O. V. Gutik and A. R. Reiter, Symmetric inverse topological semigroups of finite
rank 6 n, Math. Methods and Phys.-Mech. Fields 52:3 (2009), 7–14; reprinted
version: J. Math. Sc. 171:4 (2010), 425–432.
[26] O. Gutik and A. Reiter, On semitopological symmetric inverse semigroups of a
bounded finite rank, Visn. L’viv. Univ., Ser. Mekh.-Mat. 72 (2010), 94–106 (in
Ukrainian).
[27] O. Gutik and D. Repovš, On countably compact 0-simple topological inverse
semigroups, Semigroup Forum 75:2 (2007), 464–469.
[28] O. Gutik and D. Repovš, Topological monoids of monotone injective partial selfmaps
of N with cofinite domain and image, Stud. Sci. Math. Hung. 48:3 (2011), 342–353.
[29] O. Gutik and D. Repovš, On monoids of monotone injective partial selfmaps of
integers with cofinite domains and images, Georgian Math. J. 19:3 (2012), 511–532.
[30] J. A. Hildebrant and R. J. Koch, Swelling actions of Γ-compact semigroups,
Semigroup Forum 33:1 (1986), 65–85.
[31] R. J. Koch, On monothetic semigroups, Proc. Amer. Math. Soc. 8 (1957), 397–401.
[32] M. Lawson, Inverse Semigroups. The Theory of Partial Symmetries, Singapore:
World Scientific, 1998.
[33] Z. Mesyan, J. D. Mitchell, M. Morayne, and Y. H. Péresse, Topological graph
inverse semigroups, Topology Appl. 208 (2016), 106–126.
[34] W. D. Munn, Uniform semilattices and bisimple inverse semigroups, Quart. J.
Math. 17:1 (1966), 151–159.
S. Bardyla, O. Gutik 183
[35] M. Nivat and J.-F. Perrot, Une généralisation du monoide bicyclique, C. R. Acad.
Sci., Paris, Sér. A 271 (1970), 824–827.
[36] M. Petrich, Inverse Semigroups, John Wiley & Sons, New York, 1984.
[37] W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lect.
Notes Math., 1079, Springer, Berlin, 1984.
[38] J. W. Stepp, Algebraic maximal semilattices. Pacific J. Math. 58:1 (1975), 243–248.
Contact information
S. Bardyla,
O. Gutik
Faculty of Mathematics, National University of
Lviv, Universytetska 1, Lviv, 79000, Ukraine
E-Mail(s): sbardyla@yahoo.com,
o gutik@franko.lviv.ua,
ovgutik@yahoo.com
Received by the editors: 29.01.2016
and in final form 16.02.2016.
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