On the lattice of weak topologies on the bicyclic monoid with adjoined zero
A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-...
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irk-123456789-1885512023-03-06T01:27:25Z On the lattice of weak topologies on the bicyclic monoid with adjoined zero Bardyla, S. Gutik, O. A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-invariant filters on ! with an attached element 1 endowed with the following partial order: F ≤ G if and only if G = 1 or F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2ᶜ and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t. 2020 Article On the lattice of weak topologies on the bicyclic monoid with adjoined zero / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 26–43. — Бібліогр.: 30 назв. — англ. 1726-3255 DOI:10.12958/adm1459 2010 MSC: 22A15, 06B23 http://dspace.nbuv.gov.ua/handle/123456789/188551 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-invariant filters on ! with an attached element 1 endowed with the following partial order: F ≤ G if and only if G = 1 or F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2ᶜ and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t. |
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Bardyla, S. Gutik, O. On the lattice of weak topologies on the bicyclic monoid with adjoined zero Algebra and Discrete Mathematics |
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Bardyla, S. Gutik, O. |
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Bardyla, S. |
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On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
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On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_full |
On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_fullStr |
On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_full_unstemmed |
On the lattice of weak topologies on the bicyclic monoid with adjoined zero |
| title_sort |
on the lattice of weak topologies on the bicyclic monoid with adjoined zero |
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Інститут прикладної математики і механіки НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188551 |
| citation_txt |
On the lattice of weak topologies on the bicyclic monoid with adjoined zero / S. Bardyla, O. Gutik // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 26–43. — Бібліогр.: 30 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bardylas onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero AT gutiko onthelatticeofweaktopologiesonthebicyclicmonoidwithadjoinedzero |
| first_indexed |
2025-07-16T10:39:20Z |
| last_indexed |
2025-07-16T10:39:20Z |
| _version_ |
1837799688912240640 |
| fulltext |
“adm-n3” — 2021/1/5 — 10:05 — page 26 — #32
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 30 (2020). Number 1, pp. 26–43
DOI:10.12958/adm1459
On the lattice of weak topologies
on the bicyclic monoid with adjoined zero
S. Bardyla∗ and O. Gutik
Communicated by V. Mazorchuk
Abstract. A Hausdorff topology τ on the bicyclic monoid
with adjoined zero C0 is called weak if it is contained in the coarsest
inverse semigroup topology on C0. We show that the lattice W of all
weak shift-continuous topologies on C0 is isomorphic to the lattice
SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on
ω with an attached element 1 endowed with the following partial
order: F 6 G if and only if G = 1 or F ⊂ G. Also, we investigate
cardinal characteristics of the lattice W. In particular, we prove
that W contains an antichain of cardinality 2c and a well-ordered
chain of cardinality c. Moreover, there exists a well-ordered chain
of first-countable weak topologies of order type t.
1. Introduction and preliminaries
In this paper all topological spaces are assumed to be Hausdorff. The
cardinality of a set X is denoted by |X|. Further we shall follow the
terminology of [22,27,28]. By ω we denote the first infinite ordinal. The
set of integers is denoted by Z. By c we denote the cardinality of the
family of all subsets of ω.
A semigroup S is called inverse if for every x ∈ S there exists a unique
y ∈ S such that xyx = x and yxy = y. Such an element y is denoted by
∗The work of the author is supported by the Austrian Science Fund FWF (grant
I3709 N35).
2010 MSC: 22A15, 06B23.
Key words and phrases: lattice of topologies, bicyclic monoid, shift-continuous
topology.
https://doi.org/10.12958/adm1459
“adm-n3” — 2021/1/5 — 10:05 — page 27 — #33
S. Bardyla, O. Gutik 27
x−1 and called the inverse of x. The map which associates every element
of an inverse semigroup to its inverse is called an inversion.
Given a semigroup S, we shall denote the set of all idempotents of S
by E(S). A semigroup S with an adjoined zero will be denoted by S0.
A poset is a set endowed with a partial order which is usually denoted
by 6. A poset X is called upper (lower, resp.) semilattice if each finite non-
empty subset of X has a supremum (infimum, resp.). An upper (lower,
resp.) semilattice X is called complete if each non-empty subset of X
has a supremum (infimum, resp.). Elements x, y of a poset X are called
incomparable if neither x 6 y nor y 6 x. For any poset X and x ∈ X put
↓x = {y ∈ X | y 6 x}, ↑x = {y ∈ X | x 6 y},
↓◦x = ↓x \ {x}, ↑◦x = ↑x \ {x}.
A poset X is called a lattice if each finite non-empty subset of X has
infimum and supremum. A lattice X is called complete if any non-empty
subset of X has infimum and supremum. A poset X is called a chain
if for any distinct a, b ∈ X either a 6 b or b 6 a. A chain X is called
well-ordered if any non-empty subset A of X contains the smallest element.
If Y is a subset of a topological space X, then by Y we denote the
closure of Y in X.
A family F of subsets of a set X is called a filter if it satisfies the
following conditions:
(1) ∅ /∈ F ;
(2) If A ∈ F and A ⊂ B then B ∈ F ;
(3) If A,B ∈ F then A ∩B ∈ F .
A family B is called a base of a filter F if for each element A ∈ F there
exists an element B ∈ B such that B ⊂ A. A filter F is called free if
∩F∈F = ∅.
A semitopological (topological, resp.) semigroup is a topological space
together with a separately (jointly, resp.) continuous semigroup operation.
An inverse semigroup with continuous semigroup operation and inversion
is called a topological inverse semigroup.
A topology τ on a semigroup S is called:
• shift-continuous, if (S, τ) is a semitopological semigroup;
• semigroup, if (S, τ) is a topological semigroup;
• inverse semigroup, if (S, τ) is a topological inverse semigroup.
The bicyclic monoid C is a semigroup with the identity 1 generated
by two elements p and q subject to the relation pq = 1.
“adm-n3” — 2021/1/5 — 10:05 — page 28 — #34
28 Lattice of weak topologies on the bicyclic monoid
The bicyclic monoid is isomorphic to the set ω×ω endowed with the
following semigroup operation:
(a, b) · (c, d) =
{
(a+ c− b, d), if b 6 c;
(a, d+ b− c), if b > c;
The bicyclic monoid plays an important role in the algebraic theory
of semigroups as well as in the theory of topological semigroups. For
example, the well-known Andersen’s result [1] states that a (0–)simple
semigroup with an idempotent is completely (0–)simple if and only if it
does not contain an isomorphic copy of the bicyclic semigroup. The bicyclic
semigroup admits only the discrete semigroup topology [21]. In [14] this
result was extended to the case of semitopological semigroups. Compact
topological semigroups cannot contain an isomorphic copy of the bicyclic
monoid [2]. The problem of an embedding of the bicyclic monoid into
compact-like topological semigroups was discussed in [4, 5, 12, 25,26].
However, it is natural to consider the bicyclic monoid with an adjoined
zero C0. It is well-known that the bicyclic monoid with an adjoined zero is
isomorphic to the polycyclic monoid P1 which is isomorphic to the graph
inverse semigroup G(E) over the graph E which consists of one vertex
and one loop. The monoid C0 is a building block of α-bicyclic monoids,
polycyclic monoids and some graph inverse semigroups. For example:
Theorem 1.1 ([10, Theorem 6]). Let a graph inverse semigroup G(E) be
a dense subsemigroup of a CLP-compact topological semigroup S. Then
the following statements hold:
(1) there exists a cardinal κ such that E = (⊔α∈κEα) ⊔ F where the
graph F is acyclic and for each α ∈ κ the graph Eα consists of one
vertex and one loop;
(2) if the graph F is non-empty, then the semigroup G(F ) is a compact
subset of G(E);
(3) each open neighborhood of 0 contains all but finitely many subsets
G(Eα) ⊂ G(E), α ∈ κ.
In [7] it was proved the following:
Theorem 1.2 ([7, Theorem 1]). Each graph inverse semigroup G(E) is
a subsemigroup of the polycyclic monoid P|G(E)|.
This result leads us to the following problem:
Problem 1.3 ([7, Question 1]). Is it true that each semitopological
(topological, topological inverse, resp.) graph inverse semigroup G(E) is a
“adm-n3” — 2021/1/5 — 10:05 — page 29 — #35
S. Bardyla, O. Gutik 29
subsemigroup of a semitopological (topological, topological inverse, resp.)
polycyclic monoid P|G(E)|.
So, to solve this problem we need to know more about a topologization
of polycyclic monoids and their subsemigroups. Locally compact shift-
continuous and semigroup topologies on polycyclic monoids and graph
inverse semigroups have been actively investigated in recent years. For
instance, in [23] it was proved that a Hausdorff locally compact semitopo-
logical bicyclic semigroup with an adjoined zero C0 is either compact or
discrete. In [6] and [8] this result was extended to polycyclic monoids
and graph inverse semigroups over strongly connected graphs with finitely
many vertices, respectively. Similar dichotomy also holds for other gener-
alizations of the bicyclic monoid (see [9, 24,30]).
There exists the coarsest inverse semigroup topology τmin on C0 which
is defined as follows: elements of C are isolated and the family Bmin(0) =
{Cn : n ∈ ω}, where Cn = {0} ∪ {(k,m) | k,m > n)}, forms an open
neighborhood base at 0 of the topology τmin (see [7, Theorem 3.6]).
Let X be any semigroup. By SCT (X) we denote the set of all Hausdorff
shift-continuous topologies on X endowed with the following natural partial
order: τ1 6 τ2 if and only if τ1 ⊂ τ2. For any topologies τ1, τ2 on X by
τ1∨τ2 we denote a topology on X whose subbase is τ1 ∪ τ2.
Lemma 1.4. For any semigroup X, the poset SCT (X) is a complete upper
semilattice. Moreover, if the poset SCT (X) contains the least element
then SCT (X) is a complete lattice.
Proof. Let T = {τα}α∈A be an arbitrary subset of SCT (X). Obviously, the
topology τ which is generated by the subbase Bτ = ∪T is the supremum of
T in the lattice T of all topologies on X. Since SCT (X) is a subposet of T ,
it is sufficient to show that τ ∈ SCT (X). Fix arbitrary x, y ∈ X and a basic
open neighborhood U of xy in the topology τ . Then there exist a finite
subset {α1, . . . , αn} ⊂ A and open neighborhoods Uα1
∈ τα1
, . . . , Uαn
∈
ταn
of xy such that ∩n
i=1Uαi
⊂ U . Since for every i 6 n the topology ταi
is shift-continuous there exists an open neighborhood Vαi
∈ ταi
of y such
that x · Vαi
⊂ Uαi
. Then V = ∩n
i=1Vαi
∈ τ and x · V ⊂ U . Analogously it
can be shown that there exists an open neighborhood W ∈ τ of x such
that W · y ⊂ U . Hence τ ∈ SCT (X).
Assume that the poset SCT (X) contains the least element υ and
T = {τα}α∈A is an arbitrary subset of SCT (X). The set S = ∩α∈A↓τα is
non-empty, because it contains υ. The above arguments imply that there
exists a topology τ such that τ = supS. The topology τ is generated by
“adm-n3” — 2021/1/5 — 10:05 — page 30 — #36
30 Lattice of weak topologies on the bicyclic monoid
the subbase ∪S. Fix any α ∈ A and a basic open set U = ∩n
i=1Vi ∈ τ
where Vi ∈ τi ∈ S, i 6 n. Then Vi ∈ τα for each i 6 n witnessing that
U ∈ τα. Hence τ ⊂ τα for each α ∈ A which implies that τ = inf T .
By SCT we denote the poset SCT (C0).
Let τc be a topology on C0 such that each non-zero point is isolated
in (C0, τc) and an open neighborhood base at 0 of the topology τc consists
of cofinite subsets of C0 which contain 0. We remark that τc is the unique
compact shift-continuous topology on C0 which implies that τc = inf SCT
(see [23, Theorem 1] and [8, Lemma 3] for a more general case). Hence
Lemma 1.4 implies the following:
Corollary 1.5. The poset SCT is a complete lattice.
A topology τ on C0 is called weak if it is contained in the coarsest
inverse semigroup topology τmin on C0. By W we denote the sublattice
↓τmin ⊂ SCT of all weak shift-continuous topologies on C0.
In this paper we investigate properties of the lattice W . More precisely,
we show that W is isomorphic to the lattice SIF1×SIF1 where SIF1
is the set of all shift-invariant filters on ω with an attached element 1
endowed with the following partial order: F 6 G if and only if G = 1 or
F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In
particular, we prove that W contains an antichain of cardinality 2c and a
well-ordered chain of cardinality c. Moreover, there exists a well-ordered
chain of first-countable weak topologies of order type t.
Cardinal characteristics of chains and antichains, and other properties
of a poset of group topologies were investigated in [3, 13,15–19].
First we define two frequently used weak topologies τL and τR on the
semigroup C0. All non-zero elements are isolated in both of the above
topologies and
• the family BL(0) = {An | n ∈ ω}, where An = {0} ∪ {(k,m) | k >
n,m ∈ ω}, is an open neighborhood base at 0 of the topology τL;
• the family BR(0) = {Bn | n ∈ ω}, where Bn = {0} ∪ {(k,m) | m >
n, k ∈ ω}, is an open neighborhood base at 0 of the topology τR.
Observe that τmin = τL∨τR.
A semigroup topology τ on C0 is called minimal if there exists no
semigroup topology on C0 which is strictly contained in τ .
Lemma 1.6. τL and τR are minimal semigroup topologies on C0.
Proof. We shall prove the minimality of τL. In the case of τR the proof is
similar.
“adm-n3” — 2021/1/5 — 10:05 — page 31 — #37
S. Bardyla, O. Gutik 31
It is easy to check that that (i, j) · An+j ⊆ An, An · (i, j) ⊆ An and
An ·An ⊆ An for all i, j, n ∈ ω. Hence (C0, τL) is a topological semigroup.
Suppose to the contrary that there exists a Hausdorff semigroup
topology τ on C0 which is strictly contained in τL. Then there exists
An ∈ BL(0) such that the set U \An is infinite for any open neighbourhood
U of zero in (C0, τ). The pigeonhole principle implies that there exists
a non-negative integer i0 6 n such that the set U ∩ {(i0, j) | j ∈ ω} is
infinite for each open neighborhood U ∈ τ of 0. The continuity of the
semigroup operation in (C0, τ) yields an open neighborhood V ∈ τ of 0
such that V · V ⊂ C0 \ {(i0, i0)}. Since V contains some Am and the set
V ∩ {(i0, j) | j ∈ ω} is infinite, there exists a positive integer j > m such
that (i0, j) ∈ V and (j, i0) ∈ V . Hence (i0, i0) = (i0, j) · (j, i0) ∈ V · V ⊂
C0 \ {(i0, i0)} which provides a contradiction.
Problem 1.7. Do there exist other minimal semigroup topologies on C0?
Lemma 1.8. If τ is a semigroup topology on C0 such that τ ∈ ↓◦τmin
then either τ = τL or τ = τR.
Proof. Let τ be a semigroup topology on C0 such that τ ∈ ↓◦τmin. Then
there exists Cn ∈ Bmin such that the set U \ Cn is infinite for each open
neighborhood U ∈ τ of 0. The pigeonhole principle implies that there
exists a non-negative integer i0 6 n such that at least one of the following
two cases holds:
(1) the set U ∩ {(i0, j) | j ∈ ω} is infinite for each open neighborhood
U of 0;
(2) the set U ∩ {(j, i0) | j ∈ ω} is infinite for each open neighborhood
U of 0.
Consider case (1). Fix an arbitrary n ∈ ω and an open neighborhood
U ∈ τ of 0. Since (n, i0) · 0 = 0 the continuity of the semigroup operation
in (C0, τ) yields an open neighborhood V of 0 such that (n, i0) · V ⊂ U .
Since the set V ∩ {(i0, j) | j ∈ ω} is infinite and (n, i0) · (i0, j) = (n, j) for
each n, j ∈ ω, the set U ∩ {(n, j) | j ∈ ω} is infinite as well. Hence 0 is
an accumulation point of the set {(n, j) | j ∈ ω} for each n ∈ ω. Using
one more time the continuity of the semigroup operation in (C0, τ) we
can find an open neighborhood W ∈ τ of 0 such that W ·W ⊂ U . Since
τ ⊂ τmin we obtain that there exists m ∈ ω such that Cm ⊂ W . Recall
that 0 is an accumulation point of the set {(n, j) | j ∈ ω} for each n ∈ ω.
Hence for each n ∈ ω we can find a positive integer jn > m such that
(n, jn) ∈ W . Observe that for each n ∈ ω the set {(jn, k) | k > m} ⊂ W .
“adm-n3” — 2021/1/5 — 10:05 — page 32 — #38
32 Lattice of weak topologies on the bicyclic monoid
Then for each n ∈ ω the following inclusion holds:
{(n, k) | k > m} = (n, jn) · {(jn, k) | k > m} ⊂ W ·W ⊂ U.
Hence for each open neighborhood U ∈ τ of 0 there exists m ∈ ω such that
Bm ⊂ U which implies that U ∈ τR and τ ⊂ τR. By lemma 1.6, τ = τR.
Similar arguments imply that τ = τL provided that case (2) holds.
Now we are going to describe the sublattices ↓τL and ↓τR of W . Let τ be
an arbitrary shift-continuous topology on C0 such that τ ∈ ↓◦τL. For each
open neighborhood U ∈ τ of 0 and i ∈ ω put FU
i = {n ∈ ω | (i, n) ∈ U}.
Lemma 1.9. Let τ ∈ ↓◦τL. Then for each i ∈ ω the set Fi = {FU
i | 0 ∈
U ∈ τ} is a filter on ω. Moreover, Fi = Fj for each i, j ∈ ω.
Proof. Since τ ∈ ↓◦τL there exists n ∈ ω such that the set U \ An is
infinite for each open neighborhood U ∈ τ of 0. Similar arguments as in
the proof of Lemma 1.8 imply that 0 is an accumulation point of the set
{(k, n) | n ∈ ω} for each k ∈ ω. Fix any i ∈ ω. Since the intersection of two
neighborhoods of 0 is a neighborhood of 0, the family Fi is closed under
finite intersections. Fix an arbitrary subset A ⊂ ω such that FU
i ⊂ A
for some U ∈ τ . Since each non-zero point is isolated in (C0, τ) the set
V = U ∪ {(i, n) | n ∈ A} is an open neighborhood of 0 such that F V
i = A.
Hence the family Fi is a filter for each i ∈ ω.
Fix arbitrary i, j ∈ ω. Without loss of generality we can assume
that i < j. To prove that Fi = Fj it is sufficient to prove the following
statements:
(1) for each FU
i ∈ Fi there exists F V
j ∈ Fj such that F V
j ⊂ FU
i ;
(2) for each FU
j ∈ Fj there exists F V
i ∈ Fi such that F V
i ⊂ FU
j .
Consider statement (1). Fix an arbitrary element FU
i ∈ Fi. The
separate continuity of the semigroup operation in (C0, τ) yields an open
neighborhood V of 0 such that (0, j − i) · V ⊂ U . Observe that (0, j − i) ·
(j, n) = (i, n) for each n ∈ ω which implies that F V
j ⊂ FU
i .
Consider statement (2). Fix an arbitrary element FU
j ∈ Fj . The
separate continuity of the semigroup operation in (C0, τ) yields an open
neighborhood V of 0 such that (j − i, 0) · V ⊂ U . Observe that (j − i, 0) ·
(i, n) = (j, n) for each n ∈ ω which implies that F V
i ⊂ FU
j .
Hence Fi = Fj , for each i, j ∈ ω.
By Lemma 1.9, each shift-continuous topology τ ∈ ↓◦τL generates a
unique filter on ω which we denote by Fτ .
“adm-n3” — 2021/1/5 — 10:05 — page 33 — #39
S. Bardyla, O. Gutik 33
For each n ∈ Z and A ⊂ ω the set {n+ x | x ∈ A} ∩ ω is denoted by
n+ A. A filter F on the set ω is called shift-invariant if it satisfies the
following conditions:
• each cofinite subset of ω belongs to F ;
• for each F ∈ F and n ∈ Z there exists H ∈ F such that n+H ⊂ F .
For each filter F on ω and n ∈ Z the filter generated by the family
{n+ F | F ∈ F} is denoted by n+ F .
Lemma 1.10. A free filter F on the set ω is shift-invariant if and only if
F = n+ F for each n ∈ Z.
Proof. Let F be a shift-invariant filter on ω and n ∈ Z. Fix any F ∈ F .
There exists an element H ∈ F such that n+H ⊂ F . Hence F ⊂ n+ F .
Fix any set n+ F ∈ n+ F . Since F is shift-invariant there exists H ∈ F
such that −n + H ⊂ F and k > n for each k ∈ H. Then H ⊂ n + F
witnessing that n+ F ⊂ F . Hence F = n+ F for each n ∈ Z.
Let F be a free filter on ω such that F = n+F for each n ∈ Z. Since
F is free every cofinite subset of ω belongs to F . Fix any n ∈ Z and
F ∈ F . Since F = n+ F there exists H ∈ F such that n+H ⊂ F .
By SIF we denote the set of all shift-invariant filters on ω endowed
with the following partial order: F1 6 F2 if and only if F1 ⊆ F2, for each
F1,F2 ∈ SIF .
Lemma 1.11. Each τ ∈ ↓◦τL generates a shift-invariant filter Fτ on ω.
Moreover, Fτ1 6= Fτ2 for any distinct shift-continuous topologies τ1 and
τ2 on C0 which belong to ↓◦τL.
Proof. Observe that each open neighborhood U ∈ τ of 0 is of the form
U = ∪n
i=0{(i, n) | n ∈ FU
i } ∪ An where FU
i ∈ Fτ for each i 6 n and
n ∈ ω. By Lemma 1.9, the set F = ∩n
i=0F
U
i belongs to Fτ . Then the set
UF,n = {(i, k) | i 6 n and k ∈ F}∪An is an open neighborhood of 0 which
is contained in U . Hence the family B(0) = {UF,n | F ∈ Fτ and n ∈ ω}
forms an open neighborhood base at 0 of the topology τ .
The Hausdorffness of (C0, τ) implies that each cofinite subset of ω
belongs to F .
Fix an arbitrary n ∈ Z and a basic open neighborhood UF,0 of 0.
If n > 0, then the separate continuity of the semigroup operation in
(C0, τ) yields a basic open neighborhood UH,m of 0 such that (0, n)·UH,m ⊂
UF,0. Observe that {(0, k) | k ∈ H} ⊂ UH,m and
(0, n) · {(0, k) | k ∈ H} = {(0, k + n) | k ∈ H}
= {(0, k) | k ∈ H + n} ⊂ UF,0.
“adm-n3” — 2021/1/5 — 10:05 — page 34 — #40
34 Lattice of weak topologies on the bicyclic monoid
Hence H + n ⊂ F .
Assume that n < 0. Since (C0, τ) is a Hausdorff semitopological
semigroup there exists a basic open neighborhood UH,m of 0 such that
H ∩ {0, . . . , n} = ∅ and UH,m · (|n|, 0) ⊂ UF,0. Observe that {(0, k) | k ∈
H} ⊂ UH,m and
{(0, k) | k ∈ H} · (|n|, 0) = {(0, k − |n|) | k ∈ H}
= {(0, k) | k ∈ H + n} ⊂ UF,0.
Hence H + n ⊂ F .
Observe that the second part of the lemma follows from the description
of topologies τ1, τ2 ∈ ↓◦τL and from the definition of filters Fτ1 ,Fτ2 .
Let F be a shift-invariant filter on ω. By τLF we denote the topology on
C0 which is defined as follows: each non-zero element is isolated in (C0, τLF )
and the family B = {UF,n | F ∈ F and n ∈ ω} where UF,n = {(i, k) | i 6
n and k ∈ F} ∪An forms an open neighborhood base of 0 in (C0, τLF ).
Lemma 1.12. For each shift-invariant filter F on ω the topology τLF is
shift-continuous and belongs to ↓◦τL. Moreover, if F1 and F2 are distinct
shift-invariant filters on ω then τLF1
6= τLF2
.
Proof. Observe that the second part of the statement of the lemma follows
from the definition of topologies τLF1
and τLF2
.
The definition of the topology τLF implies that τLF ∈ ↓◦τL.
Let F be a shift-invariant filter on ω. Since the filter F contains all
cofinite subsets of ω the definition of the topology τLF implies that the
space (C0, τLF ) is Hausdorff. Recall that the bicyclic monoid C is generated
by two elements (0, 1) and (1, 0) and C is the discrete subset of (C0, τLF ).
Hence to prove the separate continuity of the semigroup operation in
(C0, τLF ) it is sufficient to check it in the following four cases:
(1) (0, 1) · 0 = 0;
(2) (1, 0) · 0 = 0;
(3) 0 · (0, 1) = 0;
(4) 0 · (1, 0) = 0.
Fix an arbitrary basic open neighborhood UF,n of 0.
(1) Since the filter F is shift-invariant there exists H ∈ F such that
H ⊂ F and H + 1 ⊂ F . It is easy to check that (0, 1) · UH,n+1 ⊂ UF,n.
(2) It is easy to check that (1, 0) · UF,n ⊂ UF,n.
(3) Since the filter F is shift-invariant there exists H ∈ F such that
H ⊂ F and H + 1 ⊂ F . It is easy to check that UH,n · (0, 1) ⊂ UF,n.
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S. Bardyla, O. Gutik 35
(4) Since the filter F is shift-invariant there exists H ∈ F such that
H ⊂ F and H − 1 ⊂ F . It is easy to check that UH,n · (1, 0) ⊂ UF,n.
Hence the topology τLF is shift-continuous.
Put SIF1 = SIF ⊔ {1} and extend the partial order 6 to SIF1 as
follows: F 6 1 for each F ∈ SIF .
Theorem 1.13. The posets ↓τL and SIF1 are order isomorphic.
Proof. By Lemma 1.11, for each topology τ ∈ ↓◦τL there exists a shift-
invariant filter F ∈ SIF such that τ = τLF . We define the map f : ↓τL →
SIF1 as follows:
• f(τLF ) = F for each topology τLF ∈ ↓◦τL;
• f(τL) = 1.
Let F1 ∈ SIF and F2 ∈ SIF . Observe that F1 6 F2 if and only if
τLF1
6 τLF2
. Lemmas 1.11 and 1.12 imply that the map f is an order
isomorphism between posets ↓τL and SIF1.
Corollary 1.14. The poset SIF1 is a complete lattice.
Let τ be an arbitrary shift-continuous topology on C0 such that τ ∈
↓◦τR. For each open neighborhood U ∈ τ of 0 and i ∈ ω put GU
i = {n ∈
ω | (n, i) ∈ U}.
The proof of the following lemma is similar to that of Lemma 1.9.
Lemma 1.15. Let τ ∈ ↓◦τR. Then for each i ∈ ω the set Gi = {GU
i | 0 ∈
U ∈ τ} is a filter on ω. Moreover, Gi = Gj for each i, j ∈ ω.
Let G be a shift-invariant filter on ω. By τRG we denote the topology on
C0 which is defined as follows: each non-zero element is isolated in (C0, τRG )
and the family B = {UG,n | G ∈ G and n ∈ ω} where UG,n = {(k, i) | i 6
n and k ∈ G} ∪Bn forms an open neighborhood base of 0 in (C0, τRG ).
Analogues of Lemmas 1.11 and 1.12 hold for the topology τ ∈ ↓◦τR.
Hence we obtain the following:
Theorem 1.16. The posets ↓τR and SIF1 are order isomorphic.
Proposition 1.17. For each topology τ ∈ W there exists a unique pair
of topologies (τ1, τ2) ∈ ↓τL×↓τR such that τ = τ1∨τ2.
Proof. Let τ ∈ ↓◦τmin. Then there exists n ∈ ω such that for each open
neighborhood U ∈ τ of 0 the set U \ Cn is infinite.
It is easy to see that τ satisfies one of the following three conditions.
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36 Lattice of weak topologies on the bicyclic monoid
(1) For every i 6 n the set {(i,m) | m ∈ ω}∩U is infinite for each open
neighborhood U ∈ τ of 0 and there exists an open neighborhood
U0 ∈ τ of 0 such that {(m, j) | m ∈ ω} ∩ U0 = ∅ for any j 6 n.
(2) For every j 6 n the set {(m, j) | m ∈ ω}∩U is infinite for each open
neighborhood U ∈ τ of 0 and there exists an open neighborhood
U0 ∈ τ of 0 such that {(i,m) | m ∈ ω} ∩ V = ∅ for any i 6 n.
(3) For every i, j 6 n the sets {(i,m) | m ∈ ω} ∩ U and {(m, j) | m ∈
ω} ∩ U are infinite for each open neighborhood U ∈ τ of 0.
Assume that τ satisfies condition (1). Fix a non-negative integer k and
an arbitrary open neighborhood U ∈ τ of 0. Similar arguments as in the
proofs of Lemmas 1.9, 1.11 imply that there exists a unique shift-invariant
filter F on ω such that the family B(0) = {CF,n | F ∈ F , n ∈ ω}, where
CF,n = Cn ∪ {(i, k) | i 6 n and k ∈ F}, is an open neighborhood base
at 0 of the topology τ .
It is easy to check that τ = τLF∨τR. We denote such a topology τ
by τF ,1.
Assume that τ satisfies condition (2). Similar arguments imply that
there exists a unique shift stable filter G on ω such that the family B(0) =
{CG,n | G ∈ G, n ∈ ω}, where CG,n = Cn ∪ {(k, i) | i 6 n and k ∈ G}, is
an open neighborhood base at 0 of the topology τ .
Then τ = τL∨τ
R
G . We denote such a topology τ by τ1,G .
If the topology τ satisfies condition (3), then there exist unique shift-
invariant filters F ,G on ω such that τ = τLF∨τ
R
G . We denote such a
topology τ by τF ,G .
Recall that τmin = τL∨τR. For convenience we denote τmin by τ1,1.
Theorem 1.18. The poset W is order isomorphic to the poset
SIF1×SIF1.
Proof. By Proposition 1.17, each topology τ ∈ W is of the form τx,y,
where x, y ∈ SIF1. The routine verifications show that the map f : W →
SIF1×SIF1, f(τx,y) = (x, y) is an order isomorphism.
An inverse semigroup S is called quasitopological if it is semitopological
and the inversion is continuous in S. A topology τ on an inverse semigroup
S is called quasisemigroup if (S, τ) is a quasitopological semigroup.
By Wq we denote the set of all weak quasisemigroup topologies on
C0. Obviously, Wq is a sublattice of W. Fix the weak topology τ = τx,y
where x, y ∈ SIF1 (see the proof of Proposition 1.17). Observe that
(n,m)−1 = (m,n) for each element (n,m) ∈ C. At this point it is easy to
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S. Bardyla, O. Gutik 37
see that the topology τx,y is quasisemigroup if and only if x = y. Hence
we obtain the following.
Proposition 1.19. The lattice Wq is isomorphic to the lattice SIF1.
At the end of this section we prove a nice complete-like property of
weak topologies on C0.
A semitopological semigroup X is called absolutely H-closed if for any
continuous homomorphism h from X into a Hausdorff semitopological
semigroup Y the image h(X) is closed in Y .
The next proposition complements results about complete polycyclic
monoids Pk, k > 1 obtained in [11].
Proposition 1.20. Let (C0, τ) be a semitopological semigroup such that
0 is an accumulation point of the set E(C). Then (C0, τ) is absolutely
H-closed.
Proof. By σ we denote the least group congruence on C. According to
Theorem 3.4.5 from [28], (a, b)σ(c, d) if and only if a − b = c − d for
any a, b, c, d ∈ ω, every congruence on the bicyclic monoid is a group
congruence and C/σ is isomorphic to the additive group of integers. For
each k ∈ Z put [k] = {(a, b) ∈ C | a− b = k}.
Fix any k ∈ Z and an open neighborhood U of 0. If k > 0, then the
separate continuity of the semigroup operation in (C0, τ) yields an open
neighborhood V of 0 such that (k, 0) · V ⊂ U . Since 0 is an accumulation
point of the set E(C) there exists an infinite subset A ⊂ ω such that
{(n, n) | n ∈ A} ⊂ V . Then (k, 0) · {(n, n) | n ∈ A} = {(k + n, n) | n ∈
A} ⊂ U . Hence 0 is an accumulation point of the set [k] for each k > 0.
If k < 0 then the separate continuity of the semigroup operation in
(C0, τ) yields an open neighborhood V of 0 such that (0, k) · V ⊂ U .
Similarly it can be shown that 0 is an accumulation point of the set [k]
for each k < 0.
Assume that h is a continuous homomorphism from C0 into a Haus-
dorff semitopological semigroup X. If there exists (n,m) ∈ C such that
h(n,m) = h(0) then
h((0, 0)) = h((0, n) · (n,m) · (m, 0)) = h(0, n) · h(0) · h(m, 0)
= h((0, n) · 0 · (m, 0)) = h(0).
In this case the map h is annihilating.
Otherwise, by Theorem 3.4.5 from [28], there are three cases to consider:
(1) the image h(C0) is finite;
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38 Lattice of weak topologies on the bicyclic monoid
(2) the image h(C0) is isomorphic to the additive group of integers with
an adjoint zero;
(3) h is injective, i.e., h(C0) is isomorphic to C0.
(1) The Hausdorffness of X implies that h(C0) is a closed subset of X.
(2) Observe that 0 is an accumulation point of each equivalence class
[k] of the least group congruence σ. Hence each open neighborhood U of
h(0) in X contains the set h(C) which contradicts the Hausdorffness of X.
Hence this case is not possible.
(3) To obtain a contradiction, assume that h(C0) is not closed in X.
Hence there exists an element x ∈ h(C0) \ h(C0). Each open neighbor-
hood of x contains infinitely many elements from C. Hence for each open
neighborhood V of x at least one of the following two subcases holds:
(3.1) the set LV = {n | there exists m such that (n,m) ∈ V } is infinite;
(3.2) the set RV = {m | there exists n such that (n,m) ∈ V } is infinite.
(3.1) We claim that (k, k) ·x = x for each k ∈ ω. Indeed, fix any k ∈ ω
and observe that (k, k) · (n,m) = (n,m) for each n > k. Hence the set
((k, k) · V ) ∩ V is infinite for each open neighborhood V of x witnessing
that (k, k) · x = x.
Fix any open neighborhood U of 0 which does not contain x. The
separate continuity of the semigroup operation in X implies that 0 · x =
x · 0 = 0. Hence there exists an open neighborhood W ⊂ U of 0 such that
W · x ⊂ U . Fix any idempotent (k, k) ∈ W (it is possible since 0 is an
accumulation point of the set E(C) = {(k, k) | k ∈ ω}). The above claim
implies that x = (k, k) · x ∈ W · x ⊂ U which contradicts the choice of
the set U .
(3.2) Analogously it can be showed that x · (k, k) = x for each k ∈ ω.
At this point the contradiction can be obtained similarly as in (3.1).
Proposition 1.20 provides the following.
Corollary 1.21. For each weak topology τ the semitopological semigroup
(C0, τ) is absolutely H-closed.
2. Cardinal characteristics of the lattice W
By [ω]ω we denote the family of all infinite subsets of ω. We write
A ⊂∗ B if |A \ B| < ω. Let F be a filter on ω. The cardinal χ(F) =
min{|B| : B is a base of the filter F} is called the character of the filter F .
A filter F is called first-countable if χ(F) = ω.
For each a, b ∈ ω by [a, b] we denote the set {n ∈ ω | a 6 n 6 b}.
Observe that [a, b] = ∅ if a > b. Let A be an infinite subset of ω. For
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S. Bardyla, O. Gutik 39
each k ∈ ω put FA,k = ∪n∈A[n!− n+ k, n! + n− k]. Since FA,k ∩ FA,n =
FA,max{k,n} the family {FA,k | k ∈ ω} is closed under finite intersections.
By FA we denote the filter on ω whose base consists of the sets FA,k,
k ∈ ω.
Lemma 2.1. Let A,B ∈ [ω]ω. Then the following statements hold:
(1) the filter FA is shift-invariant;
(2) if A and B are almost disjoint, then the filters FA and FB are
incomparable in the lattice SIF .
(3) if A ⊂∗ B then FB 6 FA;
(4) if |A \B| = ω then FA 6= FB
(5) (C0, τLFA
) and (C0, τRFA
) are first-countable topological spaces.
Proof. (1). Fix any FA,n ∈ FA, m ∈ Z and k ∈ ω. It is easy to check
that m+ FA,n+|m| ⊂ FA,n and FA,k+1 ⊂ ω \ [0, k]. Hence the filter FA is
shift-invariant.
(2). There exists n ∈ ω such that A ∩ B ⊂ [0, n]. It is easy to check
that FA,n+1 ∩ FB,n+1 = ∅ which implies that the filters FA and FB are
incomparable in the poset SIF .
(3). There exists n ∈ ω such that A \ B ⊂ [0, n]. It is easy to check
that FA,k ⊂ FB,k for each k > n+ 1 which implies that FB 6 FA.
Statement (4) follows from the definition of the filters FA and FB.
Observe that the filter FA is first-countable. At this point statement
(5) follows from the definition of the topologies τLFA
and τRFA
.
By SCT ω (Wω, resp.) we denote the set of all (weak, resp.) Hausdorff
shift-continuous first-countable topologies on C0. It is easy to check that
SCT ω is a sublattice of SCT . A subset A of a poset X is called an antichain
if each two distinct elements of A are incomparable in X.
A set A is called a pseudo-intersection of a family F ⊂ [ω]ω if A ⊂∗ F
for each F ∈ F . A tower is a set T ⊂ [ω]ω which is well-ordered with
respect to the relation defined by x 6 y if and only if y ⊂∗ x. It is called
maximal if it cannot be further extended, i.e. it has no pseudointersection.
Denote t = min{|T | : T is a maximal tower}. By [20, Theorem 3.1],
ω1 6 t 6 c. Put t̂ = sup{|T | : T is a maximal tower}. Obviously, t 6 t̂ 6 c.
Theorem 2.2. The poset Wω has the following properties:
(1) Wω contains an antichain of cardinality c;
(2) For each ordinal κ ∈ t̂ the poset Wω contains a well-ordered chain
of order type κ;
(3) |Wω| = |SCT ω| = c.
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40 Lattice of weak topologies on the bicyclic monoid
Proof. (1) Fix any almost disjoint family A ⊂ [w]ω such that |A| = c.
By [27, Theorem 1.3], such a family exists. Statement (2) of Lemma 2.1
implies that the set {FA | A ∈ A} forms an antichain in the poset SIF . By
Theorem 1.13 (resp., 1.16), the set {τLFA
| A ∈ A} (resp., {τRFA
| A ∈ A})
is an antichain in the lattice W . By statement (5) of Lemma 2.1, the sets
{τLFA
| A ∈ A} and {τRFA
| A ∈ A} are contained in Wω
(2) Fix any ordinal κ ∈ t̂. By the definition of t̂ there exists a tower
T = {Tα}α∈λ of length λ > κ. Observe that |Tα \ Tβ | = ω for each
α < β < λ. Statements (3), (4) and (5) of Lemma 2.1 imply that the sets
{τLFTα
| α ∈ κ} and {τRFTα
| α ∈ κ} are well-ordered chains in Wω of order
type κ.
(3) Observe that the cardinality of the set of all first countable filters
on ω is equal to c. Statement (1) implies that |SCT ω| = |Wω| = c.
Now we are going to show that each free filter on ω generates a
shift-invariant filter on ω. Fix an arbitrary free filter G on ω. For each
G ∈ G and k ∈ ω put FG,k = ∪n∈G[n! − n + k, n! + n − k]. Observe
that FG,k ∩ FH,n = FH∩G,max{k,n} for each G,H ∈ G and k, n ∈ ω. Since
H ∩ G ∈ G we obtain that the family {FG,k | G ∈ G, k ∈ ω} is closed
under finite intersections. By FG we denote the filter on ω whose base
consists of the sets FG,k, G ∈ G and k ∈ ω. A filter F on ω is called an
ultrafilter if for each subset A ⊂ ω either A ∈ F or there exists F ∈ F
such that A ∩ F = ∅.
Lemma 2.3. Let G,H be free filters on ω. Then the following statements
hold:
(1) the filter FG is shift-invariant;
(2) if G and H are ultrafilters, then the filters FG and FH are incompa-
rable in the lattice SIF .
(3) if G ⊂ H then FG ⊂ FH;
(4) if G 6= H then FG 6= FH;
(5) The character of the spaces (C0, τLFG
) and (C0, τRFG
) is equal to the
character of the filter FG .
Proof. (1) Fix any element FG,n ∈ FG , m ∈ Z and k ∈ ω. It is easy to
check that m+FG,n+|m| ⊂ FG,n and FG,(k+1)! ⊂ ω \ [0, k]. Hence the filter
FG is shift-invariant.
(2) Assume that G and H are ultrafilters on ω. Then there exist A ∈ G
and B ∈ H such that A∩B = ∅. It is easy to see that the sets FA,2 ∈ FG
and FB,2 ∈ FH are disjoint which implies that the filters FG and FH are
incomparable in the poset SIF .
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S. Bardyla, O. Gutik 41
Statement (3) follows from the definition of the filters FG and FH.
The proof of statement (4) is straightforward.
Statement (5) follows from the definition of the topologies τLFG
and τRFG
.
By FF we denote the set of all free filters on ω endowed with the
natural partial order: F 6 G if and only if F ⊂ G.
Lemma 2.4. The poset FF contains a well-ordered chain of cardinality c.
Proof. By Theorem 4.4.2 from [29], there exists a free ultrafilter F on ω
such that χ(F) = c. Fix any base B = {Fα}α∈c of F .
Now we are going to construct a well-ordered chain L = {Gα}α∈c in
FF . Let G0 be a filter which consists of cofinite subsets of ω. Since the
filter F is free, G0 ⊂ F . Assume that for all ordinals β < α < c the filters
Gβ are already constructed. There are two cases to consider:
(1) α is a limit ordinal;
(2) α is a successor ordinal.
In case (1) put Gα = ∪β∈αGβ .
In case (2) α = δ+1 for some ordinal δ. Let γδ be the smallest ordinal
such that Fγ /∈ Gδ. Let Gα be a filter generated by the smallest family X
of subsets of ω such that Gδ ⊂ X, Fγδ ∈ X and X is closed under finite
intersections.
Observe that |Gδ| 6 max{ω, |δ|} < c. Hence for each δ ∈ c the ordinal
γδ exists. Otherwise the family Gδ is a base of the filter F of cardinality
less then c which contradicts our assumption.
It is easy to see that the family L = {Gα}α∈c is a well-ordered chain
in FF .
Theorem 2.5. The poset W has the following properties:
(1) W contains an antichain of cardinality 2c;
(2) W contains a well-ordered chain of cardinality c;
(3) |W| = |SCT | = 2c.
Proof. It is well-known that there are 2c ultrafilters on ω. Hence statement
(1) follows from statements (1) and (2) of Lemma 2.3 and Theorem 1.13.
Consider statement (2). By Lemma 2.4, there exists a well-ordered
chain L ⊂ FF of cardinality c. Statements (3) and (4) of Lemma 2.3 imply
that the family Z = {FG | G ∈ L} is a well-ordered chain in SIF . By
Theorem 1.13, the poset W contains a well-ordered chain of cardinality c.
Consider statement (3). Obviously, |W| 6 |SCT | 6 2c. Then statement
(1) yields the desired equality.
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42 Lattice of weak topologies on the bicyclic monoid
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Contact information
Serhii Bardyla Institute of Mathematics, Kurt Gödel Research
Center, Vienna, Austria
E-Mail(s): sbardyla@yahoo.com
Oleg Gutik Department of Mechanics and Mathematics,
National University of Lviv, Universytetska 1,
Lviv, 79000, Ukraine
E-Mail(s): oleg.gutik@lnu.edu.ua,
ovgutik@yahoo.com
Received by the editors: 17.09.1019
and in final form 26.11.2019.
mailto:sbardyla@yahoo.com
mailto:oleg.gutik@lnu.edu.ua
mailto:ovgutik@yahoo.com
S. Bardyla, O. Gutik
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