(Disclaimer: below, when I say "compact" I mean "compact Hausdorff.")

I asked a version of this question on math stackexchange (https://math.stackexchange.com/questions/305186/left-continuous-magmas-with-no-fixed-points) a couple days ago, but received no answer; I apologize if this question is too low-level for this site.

Background: Mostly for fun, I'm looking at a particular binary operation on the set of ultrafilters on the natural numbers, $\beta\mathbb{N}$ - this was inspired by reading the proof via idempotent ultrafilters of Hindman's theorem. The key component of this proof is the easy result that any left continuous compact semigroup (that is, any compact topological space $X$ together with a binary operation $*: X^2\rightarrow X$ such that for all $s\in X$ the map $t\mapsto t*s$ is continuous) has an idempotent element. A while ago, I naively concluded that this meant that my operation had idempotents; however, this turned out to be very unjustified, since the operation I'm looking at turned out to be non-associative (which surprised me), and associativity is crucial to the proof of the existence of idempotents: there are plenty of left continuous compact magmas without idempotents (consider the set $\lbrace a, b\rbrace$ in the discrete topology with the operation $x*y=z\iff z\not=y$).

I still have no idea whether the magma I'm studying has any idempotents. But more to the point, I realized that I have no idea in general how to go about trying to tell whether a given magma has idempotents.

My question is twofold: first, what are some necessary/sufficient conditions for a (let's say left continuous and compact) magma to have idempotents? Second, what are some instructive examples of such magmas without idempotents? I don't think the two-element example is particularly instructive, since - despite being very simple! - it doesn't really resemble anything I can think of encountering in practice. As far as examples go, I'm particularly interested in ones of large cardinality - a naturally occuring left continuous compact magma of size $2^{2^{\aleph_0}}$ without idempotents would be fantastic!