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1 vote
0 answers
48 views

Neighborhoods of idempotents in topological inverse semigroups

In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
3 votes
0 answers
161 views

Making the powerset into a topological monoid

Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via $$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$ Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...
3 votes
0 answers
157 views

Closure of the inverse image under the projection map

Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
4 votes
0 answers
74 views

Is each TS-topologizable group TG-topologizable?

Definition 1. A topology $\tau$ on a group $X$ is called $\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$; $\bullet$ a ...
12 votes
1 answer
624 views

Stone–Čech compactification as a semigroup

Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
3 votes
0 answers
31 views

Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup

A semigroup $X$ endowed with a topology is called $\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous; $\bullet$ a semitopological semigroup if for every $a,b\...
7 votes
0 answers
138 views

The smallest cardinality of a cover of a group by algebraic sets

$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
29 votes
1 answer
1k views

Is the Golomb countable connected space topologically rigid?

The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
-8 votes
1 answer
351 views

Are there overwhelmingly more finite monoids than finite spaces? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
4 votes
0 answers
72 views

When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open. Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace. Under what ...
6 votes
3 answers
472 views

Spaces with unique endomorphism monoids

If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$. We ...
25 votes
3 answers
1k views

What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?

This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
10 votes
0 answers
314 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
6 votes
0 answers
132 views

Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
6 votes
0 answers
117 views

Closedness of the partial order in complete Hausdorff semitopological semilattices

First some definitions. A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the ...
3 votes
1 answer
244 views

Category of continuous self maps

Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)? How can we tell whether a category is the category of continuous ...
3 votes
0 answers
80 views

On the compactification of partial semigroups

We begin by introducing some relevant definitions. Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...
9 votes
0 answers
373 views

Embedding $\beta\mathbb{N}$ into a product of Cantor sets

Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
9 votes
4 answers
1k views

When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)

The title has it all. I'm looking for a reference to the following: Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
8 votes
1 answer
229 views

Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
6 votes
1 answer
1k views

Who coined "mob" and "clan" and why these words?

A mob is a word used for a topological semigroup which is a Hausdorff space. A clan is a compact connected mob with a two-sided identity element. Who used these words with these meanings first and ...
2 votes
0 answers
122 views

First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions. We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
2 votes
2 answers
241 views

If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?

Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post). Basically following some ideas of W. Lawvere (but not his ...
7 votes
2 answers
417 views

Does every commutative monoid admit a translation-invariant measure?

Let $T$ be a commutative monoid, written additively. The set $T$ is equipped with a canonical pre-order, defined by $s \le t$ when there exists $s' \in T$ so that $s + s' = t$. Consequently, $T$ may ...