We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup ...have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections between subsets of N. We also study whether our examples satisfy automatic continuity.
We find (completeness type) conditions on topological semilattices X,Y guaranteeing that each continuous homomorphism h:X→Y has closed image h(X) in Y.
It is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero
C
ℤ
0
is discrete. At the same time, on
C
ℤ
0
, there exist 𝔠 different ...Hausdorff locally compact shift-continuous topologies. In addition, on
C
ℤ
0
, we construct a unique minimal shift-continuous topology and a unique minimal inverse semigroup topology.
We obtain conditions on fuzzy quasi-pseudometrics on either semigroups or groups which imply that they are either fuzzy topological semigroups or topological groups. Our main results are: (1) Let
(
S
...,
M
,
∗
)
be a fuzzy quasi-pseudometric right topological semigroup (resp., group) such that
(
M
,
∗
)
is left weakly invariant; then
(
S
,
M
,
∗
)
is a fuzzy quasi-pseudometric topological semigroup (resp., group); (2) Suppose that
(
M
,
∗
)
is a left weakly invariant fuzzy quasi-pseudometric on a monoid
G
such that each left translation of
G
is open and every right translation is continuous at the identity
e
of
(
G
,
M
,
∗
)
; then
(
G
,
M
,
∗
)
is a fuzzy quasi-pseudometric topological semigroup. Many results in Sánchez and Sanchis (Fuzzy Sets Syst 330:79–86, 2018) are improved. We also study complete weakly invariant fuzzy metrics (in the sense of Kramosil and Michálek) on semigroups.
Following the tremendous reception of our first volume on topological groups called "Topological Groups: Yesterday, Today, and Tomorrow", we now present our second volume. Like the first volume, this ...collection contains articles by some of the best scholars in the world on topological groups. A feature of the first volume was surveys, and we continue that tradition in this volume with three new surveys. These surveys are of interest not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. Well-known researchers who contributed to this volume include Taras Banakh, Michael Megrelishvili, Sidney A. Morris, Saharon Shelah, George A. Willis, O'lga V. Sipacheva, and Stephen Wagner.
For any affine semigroup S, the set
has a natural semigroup structure; additionally, if S is endowed with the discrete topology, then the semigroup
can be studied as the one-point compactification of ...S. In this article, we study the derivations on semigroup algebra
in relation to the derivations on semigroup algebra
considering the metrizable topology on
induced by the one-point compactification topology of
On the Cohomology of Topological Semigroups MAYSAMİ SADR, Maysam; BOUZARJOMEHRİ AMNİEH, Danial
Communications in Advanced Mathematical Sciences,
09/2019, Letnik:
2, Številka:
3
Journal Article
Recenzirano
Odprti dostop
In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology ...group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
We investigate the bounded derivations and bounded crossed homomorphisms from S into
(the first dual of X). We show that the innerness of these bounded derivations implies that S is inner amenable. ...We prove that every left (right) crossed homomorphism on a semigroup is principal if and only if it is left (right) amenable. Finally, we show that every bounded left (right) crossed homomorphism from S into M(X), the Banach space of all Borel measures on X, is principal. In the locally compact group case, this is an answer for the derivation problem on locally compact groups.
We give a pointwise version of sensitivity in terms of open covers for a semiflow (
T
,
X
) of a topological semigroup
T
on a Hausdorff space
X
and call it a Hausdorff sensitive point. If
(
X
,
U
)
...is a uniform space with topology
τ
, then the definition of Hausdorff sensitivity for
(
T
,
(
X
,
τ
)
)
gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow (
T
,
X
) on a compact Hausdorff space
X
, these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are
T
-invariant if
T
is a
C
-semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow (
T
,
X
) on a topological space
X
and show that if (
T
,
X
) is a topologically equicontinuous pair in (
x
,
y
), for all
y
∈
X
, then
Tx
¯
=
D
T
(
x
)
where
D
T
(
x
)
=
⋂
{
TU
¯
:
for all open neighborhoods
U
of
x
}
.
We prove for a topologically transitive semiflow (
T
,
X
) of a
C
-semigroup
T
on a regular space
X
with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of
C
-semigroup
T
on a regular space
X
with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if
X
is a regular space and (
T
,
X
) is not a topologically equicontinuous pair in (
x
,
y
), then
x
is a Hausdorff sensitive point for (
T
,
X
). Hence, a minimal semiflow of a
C
-semigroup
T
on a regular space
X
is either topologically equicontinuous or topologically sensitive.
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