Questions tagged [topological-semigroups]

A semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.

Filter by
Sorted by
Tagged with
0 votes
0 answers
11 views

A question about Semiflow on $[0, 1]$

Let $(T, [0, 1])$ be a semiflow, it means that $T$ is topological semigroup and $e\in T$, $t:[0, 1]\to [0, 1]$ is a continuous map for all $t\in T$, $t_0(t_1 x)= (t_0t_1)x$ for all $t_0, t_1\in T$ ...
user avatar
0 votes
0 answers
10 views

A dense orbit of semiflow $(T, X)$ on $X=\{0\}\cup \{\frac{1}{n}: n=1, 2, \ldots \}$

Let $X=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{0\}$ be subspace of $\mathbb{R}$ with usual metric and let $(T, X)$ be a semiflow on $X$, this means that $\varphi:T\times X\to X$ with $\varphi(t, x)= tx$...
user avatar
1 vote
0 answers
17 views

What conditions on topological semigroup $T$ implies that if $t_n\to \infty$, then $st_n\to\infty$ for all $s\in T$?

Let $T$ be a topological semigroup. For a net $\{t_i\}$ in $T$, $t_i\to \infty$ is to mean that the net $\{t_i\}$ is ultimately outside each element of compact set $K$ in $T$. In my research, I need ...
user avatar
0 votes
0 answers
36 views

$T=[0, 1]$ with operation $xy=\max \{x, y \}$ has a Folner net?

It is known that every abelian semigroup $T$ does have a Folner net, which means that there is a net $\{F_n\}_{n\in D}$ in $\mathcal{P}_f(T)$, where $\mathcal{P}_f(T)=\{A: A \text{ is finite set of } ...
user avatar
0 votes
0 answers
13 views

Upper density of a syndetic set in semigroup

Let $T$ be a topological semigroup. and $\mathcal{F}=\langle F_n\rangle_{n\in D}$ be a net in $\mathcal{P}_f(T)$, where $\mathcal{P}_f(T)= \{A : A \text{ is finite set of } T\}$. Then $\mathcal{F}$ ...
user avatar
0 votes
1 answer
22 views

$T=[0, \frac{1}{2}]\cup\{1\}$ with usual topology and multiplication operation is amenable semigroup?

Let $T$ be a topological semigroup. We say that $T$ satisfies $SFC$-condition, if for every finite set $H\subseteq T$ and every $\epsilon>0$, there is a finite set $K\subseteq T$ such that $|K\...
user avatar
0 votes
0 answers
20 views

A question about syndetic-preserving by subgroups

Let $T$ be a topological discrete semigroup and $S\leq T$ be a subgroup of $T$ which is closed set in $T$. For $A\subseteq S$ there is a finite set $K$ in $T$ such that $T=KA$. Is there a subset $K_0$ ...
user avatar
0 votes
0 answers
20 views

Is it true that if $\{F_n\}$ is a Folner net, then $\{F_nx_n\}$ is Folner net?

Let $T$ be a right cancelative semigroup and let $\mathcal{F}=\{F_n\}_{n\in D}$ be a net in $\mathcal{P}_f(T)= \{A: A \text{ is finite subsets of } T\}$. $\mathcal{F}$ is called a Folner net if for ...
user avatar
1 vote
1 answer
31 views

$|F\cap K^{-1}A|\leq \sum_{k\in K}|F\cap k^{-1}A|$?, where $K$ is compact set, $F$ is finite set in topological semigroup $T$

Let $T$ be a topological semigroup and $F\subseteq T$ be a non-empty finite set, $K\subseteq T$ be a compact set and $A$ be a non-empty subset in $T$. What can say about relation between $|F\cap K^{-1}...
user avatar
0 votes
2 answers
34 views

Is there a compact set $S$ in topological semigroup $T$ with $t^{-1}K\subseteq S$, where $K$ is compact set in $T$? [closed]

Let $T$ be a topological semigroup with $e\in T$ and for $t\in T$, $A\subseteq T$, define $t^{-1}A=\{r: tr\in A\}$. If $K\subseteq T$ is a compact set then $tK$ is a compact set in $T$. It seems that $...
user avatar
1 vote
0 answers
17 views

For entourage $U$ and continuous map $t:X\to X$, is there an entourage $V$ with $t^{-1}V[x]\subseteq U[x]$?

Let $(X, \mathcal{U})$ be a uniform space, $T$ be a topological semigroup and $(T, X)$ be a semiflow. This means that $t:X\to X$ is continuous and $(t_0t_1)(x)=t_0(t_1x)$ and if $e\in T$, then $ex=x$. ...
user avatar
4 votes
1 answer
45 views

$\langle a\rangle \simeq \langle b \rangle$ if and only if $a$ and $b$ have the same index and period

If $a$ and $b$ are elements of finite order in the same or in different semigroups, the $\langle a\rangle \simeq \langle b \rangle$ if and only if $a$ and $b$ have the same index and period. $\...
user avatar
  • 494
1 vote
2 answers
43 views

Invariance of syndetic set in topological semigroup

Let $T$ be a topological semigroup. $A\subseteq T$ is syndetic, if there is compact set $K\subseteq T$ with $T=KA$. Let syndetic set $A\subseteq T$ and $g\in T$ be given. I think that $Ag^{-1}=\{y: ...
user avatar
2 votes
2 answers
84 views

Natural Choice of Topology for Free Monoid on a Space

Suppose $X$ is a metric space, and let $X^*$ denote the free monoid on $X$, that is the monoid consisting of all finite strings of elements of $X$, with string concatenation as the monoid operation (...
user avatar
0 votes
1 answer
34 views

Particular property of an open subsemigroup in a given topological group

In chapter $\rm{III}$ of the "Topologie générale" of Bourbaki -- which is the chapter they dedicate to a brief introduction to generalities concerning topological groups -- I have ...
user avatar
  • 3,747
8 votes
0 answers
154 views

Topological nature of IEEE floating-point numbers

If IEEE floating-point numbers had countably infinite precisions, its domain would be: $$ \{-\infty\}\cup\mathbb{R}^-\cup\{-0,+0\}\cup\mathbb{R}^+\cup\{+\infty\}\cup\{\text{NaN}\} $$ Let's denote ...
user avatar
  • 1,345
2 votes
1 answer
128 views

What is an involutorial homeomorphism?

Having searched the term "involutorial" here, on Google, and in the references of the source paper where I first have met this term (in fact the only time), I am wondering whether could anyone give a ...
user avatar
1 vote
0 answers
36 views

Is there a typo in the theorem?

I have downloaded an article of A.D. Wallace named The Structure of Topological Semigroups, https://projecteuclid.org/euclid.bams/1183519418. Theorem 2.1 in the text reads: "Each subgroup of $S$ ...
user avatar
0 votes
0 answers
34 views

About the definition of topological semigroups

I have two questions about the definition of topological semigroup. Thank you very much to all the respondents. Q1. Why topological space has to be Hausdorff when defining topological semigroups? Q2....
user avatar
1 vote
1 answer
65 views

Regarding Compact Semigroup

I have a problem in understanding the proof of following Theorem 2 (b), which is given in the book "The Theory of Topological Semigroups" by J. H. Carruth, J. A. Hildebrant and R. J. Koch as Theorem 1....
user avatar
  • 1,184