In this paper we investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterize graph inverse semigroups which admit a compact ...semigroup topology and describe graph inverse semigroups which can be embedded densely into CLP-compact topological semigroups.
We recall various notions of size for a topological semigroup
T
, not necessarily discrete, such as
GH
-syndetic set, syndetic set, and positive Følner density set. We give new information about ...these sets and give some examples to study the relation between them. Let
φ
:
T
×
X
→
X
, or simply (
T
,
X
), be any dynamical system on a space
X
with a topological semigroup
T
. We say that
x
∈
X
is a uniformly recurrent point, almost periodic point of von Neumann, or weakly uniformly recurrent point, if the return time set
N
(
x
,
U
) is syndetic,
GH
-syndetic, or
d
F
ϕ
(
N
(
x
,
U
)
>
0
, respectively, where
U
is a neighborhood of the point
x
and
N
(
x
,
U
)
=
{
t
:
t
x
∈
U
}
. It is known that there is no relation between the set of uniformly recurrent points and the set of almost periodic points of von Neumann for a semiflow (
T
,
X
). We give further examples for it. We introduce a notion of
f
-uniformly recurrent point, where
f
:
X
→
R
+
is upper semicontinuous and show that
x
∈
X
is uniformly recurrent if and only if it is an
f
-uniformly recurrent point for every upper semicontinuous
f
:
X
→
R
+
on the regular space
X
. In the case of metric space
X
,
f
:
X
→
R
+
is a continuous function. Also,
x
∈
X
is a uniformly recurrent point if and only if
x
∈
Ax
¯
for every thick set
A
of
T
. Assume that
S
is a closed normal non-trivial subsemigroup of
T
. We prove that every uniformly recurrent point of (
S
,
X
) is a uniformly recurrent point of (
T
,
X
). The converse holds if
T
is a discrete semigroup. Let (
T
,
X
) be a semiflow on topological space
X
. Then we show that every two nonempty open sets in
X
share an orbit of a weakly uniformly recurrent point of (
T
,
X
) if and only if (
T
,
X
) is a topologically transitive with a dense set of weakly uniformly recurrent points. Finally, we give topological version of sensitive dependence on the initial condition for semiflow (
T
,
X
) on topological space
X
and we show that if the semiflow (
T
,
X
) is nonminimal and every two non-empty open sets share an orbit of a weakly uniformly recurrent point, then (
T
,
X
) is syndetic-sensitive.
Let
S
be a foundation topological semigroup and
M
a
(
S
)
the space of all measures
μ
∈
M
(
S
)
for which the maps
x
⟼
|
μ
|
∗
δ
x
and
x
⟼
δ
x
∗
|
μ
|
from
S
into
M
(
S
) are weakly continuous. In ...the present paper, we introduce and study the concept of
ϕ
-amenability for
S
and investigate the relations between
ϕ
-amenability of
S
and essential
ϕ
^
-amenability of
M
a
(
S
)
, where
ϕ
is a character on
S
and
ϕ
^
is the extension of
ϕ
to
M
a
(
S
)
.
We show, contrary to some published statements, that spectral synthesis does not generally hold for commutative semigroups that are not groups. On the positive side we prove that it holds if the ...semigroup is a monoid with no prime ideal. For semigroups with a prime ideal, the picture is not so clear. On the negative side we provide a variety of examples illustrating the failure of spectral synthesis for many semigroups with prime ideals, but we also give examples of semigroups with prime ideals on which spectral synthesis holds.
In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let (X,d⋆) be a complete ⋆-metric space ...containing a semigroup (group) G that is a dense subset of X. If the restriction of d⋆ on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and d⋆ is invariant on X. (2) Let (G,d⋆) be a ⋆-metric group such that d⋆ is invariant on G. Then, (G,d⋆) is complete if and only if (G,τd⋆) is Raǐkov complete.
The sine and cosine addition laws on a (not necessarily commutative) semigroup are
f
(
x
y
)
=
f
(
x
)
g
(
y
)
+
g
(
x
)
f
(
y
)
, respectively
g
(
x
y
)
=
g
(
x
)
g
(
y
)
-
f
(
x
)
f
(
y
)
. Both of ...these have been solved on groups, and the first one has been solved on semigroups generated by their squares. Quite a few variants and extensions with more unknown functions and/or additional terms have also been studied. Here we extend these results and solve the Levi–Civita functional equation
f
(
x
y
)
=
g
1
(
x
)
h
1
(
y
)
+
g
2
(
x
)
h
2
(
y
)
by elementary methods on groups and monoids generated by their squares, assuming that
f
is central. We also find the continuous solutions in the case of topological groups and monoids.
In this paper we generalize the retracting property in homotopy theory for topological semigroups by introducing the notions of deformation S-retraction with its weaker forms and ES-homotopy ...extension property. Furthermore, the covering homotopy theorems for S-maps into Sχ-fibrations and Sχ-cofibrations are introduced and pullbacks for Sχ-fibrations behave properly.
In this paper we define pointwise eventually non-expansive action of semi-topological semigroups and prove fixed point theorems for it. As a result we give an affirmative answer (without any extra ...condition on the Banach space) to an open problem raised by Kirk and Xu (2008) 7. Our results also extend and improve fixed point theorems of Lim and Lau–Mah.
Let 𝒞
be a monoid which is generated by the partial shift
:
+1 of the set of positive integers ℕ and its inverse partial shift
:
+ 1 ↦
. In this paper we prove that if
is a submonoid of the monoid
...of all partial cofinite isometries of positive integers which contains Cscr;
as a submonoid then every Hausdorff locally compact shift-continuous topology on
with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup
with an adjoined compact ideal.
We show that an Abelian topological group
G
is absolutely closed in the class of topological semigroups if and only if
G
is complete and there is
n
∈
N
such that the subgroup
n
G
=
{
n
x
:
x
∈
G
}
is ...totally bounded. If for every
n
∈
N
, the subgroup
nG
is not totally bounded, then the topology of
G
can be extended to a semigroup topology
T
on
G
×
Z
+
in which
G
is open and dense, and if
G
is locally compact, so can be chosen
T
. In particular, the topology of
R
can be extended to a locally compact semigroup topology on
R
×
Z
+
in which
R
is dense. We also show that the topology of
G
can be extended to a regular (equivalently, Tychonoff) semigroup topology on
G
×
Z
+
in which
G
is open and dense if and only if there is a neighborhood
U
of
0
∈
G
such that for every
n
∈
N
, the subgroup
nG
is
U
-unbounded.
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