All Questions
11 questions
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 ...
10
votes
1
answer
579
views
Group completion of topological monoids
Let $M$ be an abelian monoid. For sake of simplicity we shall assume that in $M$ the cancellation law holds true. With this last assumption we define the group completion $G$ of $M$ as $$G:=M\times M/\...
- 1,252
9
votes
1
answer
743
views
Why is choice needed in Ellis' Lemma?
Ellis Lemma on idempotent elements asserts that:
Lemma (Ellis). Every compact semigroup has an idempotent.
The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.
...
- 1,442
9
votes
0
answers
164
views
Parallelizability of Lie monoids
A Lie monoid is a monoid, together with a structure of a smooth manifold (possibly with a boundary), such that the monoid multiplication is smooth.
If all left (or right) translations in a Lie monoid $...
- 61
3
votes
0
answers
161
views
Making the powerset into a topological monoid
Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via
$$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$
Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...
- 11.8k
3
votes
0
answers
157
views
Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
- 85
2
votes
1
answer
194
views
Continuity of Moore-Penrose generalized inversion
Any matrix $A\in\mathbb{C}^{m\times n}$ has a unique generalized inverse $A^{\dagger}\in\mathbb{C}^{n\times m}$ with the properties $$AA^{\dagger}A=A,\qquad A^{\dagger}AA^{\dagger}=A^{\dagger},\qquad (...
- 1,093
2
votes
0
answers
73
views
Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
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 ...
- 1,093
1
vote
0
answers
43
views
Interleaving in Viennot's Heaps models?
I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
- 11
1
vote
0
answers
121
views
Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as
$$
\mathcal{U} * \mathcal{V} = \left\{ A \...
- 59