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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 ultrafilters), with the usual topology and the semigroup structure inherited (extended) from the naturals.

Also, let $\mathcal{C}$ be the (topological) Cantor set, and $\mathcal{C}^\mathcal{C}$ be the space of all functions from $\mathcal{C}$ to $\mathcal{C}$, endowed with the Tychonoff product topology (or the topology of pointwise convergence) and the semigroup structure of composition of functions.

Do you know of any characterisation of the possible topological embeddings of $\beta\mathbb{N}$ into $\mathcal{C}^\mathcal{C}$? Are there any concrete (and non-trivial) examples? Are any of these embeddings also homomorphisms?

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  • $\begingroup$ @JosephVanName, I think the OP is interested in homomorphisms (preserving semigroup structure) rather than homeomorphisms. $\endgroup$ – Ramiro de la Vega Sep 16 '15 at 16:59
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    $\begingroup$ @RamirodelaVega The question ends with "Are any of these embeddings also homomorphisms?" So I think the earlier parts of the question were intended to be about arbitrary, not necessarily homomorphic, embeddings. $\endgroup$ – Andreas Blass Sep 16 '15 at 17:04
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    $\begingroup$ Continuous homomorphisms from $h:\beta\mathcal N\to S$ to any compact semigroups $S$ are uniquely determined by the image $s=h(1)$ of the unit $1\in\mathcal N$. For any ultrafilter $p$ the image $h(p)$ is just the $p$-limit of the sequence $(s^n)_{n=1}^\infty$. So, the question is to find $s\in S$ such that the continuous extension $\bar h:\beta\mathbb N\to S$ of the map $h:n\mapsto s^n$, is an (injective) semigroup homomorphism. It seems that $\mathcal C^{\mathcal C}$ is a compact left-topological semigroup, so there is a non-zero probability that this approach can work. $\endgroup$ – Taras Banakh Sep 18 '15 at 16:32
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    $\begingroup$ What happens if we take for the image $h(1)$ a minimal homeomorphism on the Cantor set $\mathcal C$. Is the continuous extension $\bar h:\beta\mathbb N\to\mathcal C^{\mathcal C}$ of the map $n\mapsto s^n$, a semigroup homomorphism? $\endgroup$ – Taras Banakh Sep 18 '15 at 16:42
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    $\begingroup$ By Theorem 19.11 in Hindman-Strauss book, for any continuous self-map $f:X\to X$ of a compact Hausdorff space the unique continuous map $h:\beta\mathbb N\to X^X$ with $h(n)=f^n$ for all $n$ is a semigroup homomorphism. So, the question is to find $f$ making $h$ is injective. It is certainly is not injective if the set $\{f^n:n\in\mathbb N\}$ is dense in $X^X$ and $X$ is Dugundji (e.g. metrizable) compact. It may happen that no continuous $f$ making $h$ injective exists at all. Then one should try non-continuous generator $f$. But in this case it is not clear why $h$ should be a homomorphism. $\endgroup$ – Taras Banakh Sep 21 '15 at 19:58

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