Let $X$ be a compact Hausdorff topological space, and $*: X^2\rightarrow X$ an associative map (so that $(X, *)$ is a semigroup) which is left continuous (for all $s\in X$, the map $t\mapsto ts$ is continuous). Then $(X, *)$ has idempotent elements: there is some $x\in X$ such that $x*x=x$. This is, for example, the argument used to show that there are idempotent (under addition) ultrafilters, which result can then be used to prove Hindman's theorem.

This argument crucially uses the fact that $*$ is associative, and it is relatively easy to come up with non-associative counterexamples. However, I don't know of any nice ones. Right now, I'm in the position of having a left continuous binary operation on a compact Hausdorff space, and I want to know if it has any idempotent elements; I'm hoping that some nice examples of left continuous magmas without idempotents will help me get a feel for whether my example has any. So my question is: what is a "nice" example of a left continuous magma with no idempotents? By "nice" I mean "coming from some reasonably natural operation on the underlying set," not necessarily "simple," although obviously simplicity is good.


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