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I was reading chapter in Todorčević's book Topics in Topology (LNM 1652, DOI: 10.1007/BFb0096295) which deals with the semigroup $\beta\mathbb N$.

Several results about left-topological compact semigroups are shown in this chapter. For example, Auslander-Ellis–Numakura lemma, which states that every such semigroup has an idempotent element or some results on left and right ideals in such semigroups. These facts are then applied to $(\beta\mathbb N,+)$ and used to obtain some combinatorial results.

The definition of left-topological semigroup is that that it is a semigroup $(S,\cdot)$ in which left translations $x\mapsto a\cdot x$ are continuous.

Since $\beta\mathbb N$ is rather complicated structure, I thought that it might be useful for me to check what these results say for some other compact left-topological semigroups. I was able to come up with these examples:

  • Any finite semigroup with the discrete topology.
  • The one point compactification of $\mathbb N$ with the discrete topology, i.e., $(\mathbb N\cup\{\infty\},+)$ with addition defined in the natural way.
  • Any compact space with the operation $a\cdot b=a$.
  • Any compact topological group.

Are there some other interesting examples of left-topological compact semigroups? Are there some other examples which are particularly simple?

Are there also some non-compact left-topological semigroups which might be interesting for me in relation to the notions studied in this chapter (i.e., notions like idempotents, left and right ideals, minimal ideals, ...)?

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  • $\begingroup$ Are you also interested in non-compact examples? $\endgroup$ Oct 2, 2015 at 17:10
  • $\begingroup$ I am mainly interested in the compact examples. (Basically the reason is just have some illustration for the results from that chapter for semigroups other than $\beta\mathbb N$.) But if you have some examples of non-compact left-topological semigroups which you consider for some reason interesting, I will be glad to read about them. (I have tried to edit the question in a way which would include non-compact examples - see the last paragraph.) $\endgroup$ Oct 2, 2015 at 17:25
  • $\begingroup$ Some related books $\endgroup$ Oct 2, 2015 at 17:31

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Here is one family of examples. They are continuous in the other argument, but it does not really matter that much.

If $S$ is any semigroup of continuous maps $X\to X$ for a topological space $X$, then the left translations $s\mapsto ss_0$ are continuous with respect to the pointwise convergence topology, i.e. when we regard $S$ as a subset of $X^X$ with the Tychonoff topology. Moreover, the same is true about the closure of $S$ (with respect to the pointwise convergence topology).

If $X$ is Hausdorff, the same is true about $S$, and if $S$ is closed and $X$ is compact, then $S$ is compact as well.

The typical context in which this is used is when this happens is when you have either a group acting on a compact Hausdorff space by homeomorphisms, or a single continuous map $T\colon X\to X$ (in which case you also have the semigroup generated by $T$).

This is called a dynamical system, and the closure of the group/semigroup is called the Ellis or enveloping semigroup. By the above remarks, such Ellis semigroups are always compact semigroups in which $s\mapsto ss_0$ is continuous (so right topological according to your terminology).

In this case (at least when you start with a group of homeomorphisms, I'm not sure about the general case), the groups comprising minimal left ideals (which, in a compact $T_2$ right topological semigroup, always exist and are always disjoint unions of groups) have a natural compact $T_1$ topology which makes them semitopological groups (note that the induced topology on them is of course Hausdorff, but is not usually compact, since they need not be closed, and multiplication is only continuous on one side), the (semitopological group) isomorphism type of which is independent of the choice of a particular group.

Note that in particular, if $G$ is a compact group acting continuously on $X$, then this all degenerates and the Ellis semigroup is simply the image of $G$ in the group of homeomorphisms of $X$ (and this is of course jointly continuous).

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