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(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!

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  • $\begingroup$ It seems that the "magma" tag is mostly used to refer to questions about MAGMA, not the algebraic structure - is there a different tag I should use? $\endgroup$
    – Noah Schweber
    Commented Feb 18, 2013 at 2:18
  • $\begingroup$ I prefer groupoid myself, but that raises problems elsewhere. Binar can be used, but as for tags, I suggest general algebra or universal algebra or algebraic structures or algebraic systems. Gerhard "That's Four Different Tags, Actually" Paseman, 2013.02.17 $\endgroup$
    – Gerhard Paseman
    Commented Feb 18, 2013 at 2:48
  • $\begingroup$ Re: tags, I've gone with "universal-algebra" - if anyone feels strongly differently, feel free to retag. $\endgroup$
    – Noah Schweber
    Commented Feb 18, 2013 at 2:58
  • $\begingroup$ @quid: when I posted on stackexchange, it suggested examples-counterexamples and general-topology; I assumed those were extant tags. Fixed. $\endgroup$
    – Noah Schweber
    Commented Feb 18, 2013 at 18:06
  • $\begingroup$ Thank you, Noah S! (I delete the longish earlier comment to avoid cluttering the comments.) $\endgroup$
    – user9072
    Commented Feb 18, 2013 at 18:18

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A similar question was studied by Justin Moore, see his unsuccessful attempt to prove amenability of R. Thompson group $F$ https://arxiv.org/abs/1209.2063. The question about amenability can be reformulated in a "Ramsey form" which, in turn, can be formulated as existence of idempotents in certain compact magmas. In any way, reading his paper may help you find what you want.

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  • $\begingroup$ This sounds interesting, but it appears that the paper was taken down. I'm looking through "Amenability and Ramsey Theory," which seems to be its descendant. $\endgroup$
    – Noah Schweber
    Commented Feb 18, 2013 at 4:44
  • $\begingroup$ It seems that this is an earlier paper. I am not sure this is what you need. I would ask Justin Moore for the file of the wrong paper where idempotents in magmas are discussed. In fact if his magma contains an idempotent, then $F$ is amenable and if not - it will be the example you want. $\endgroup$
    – user6976
    Commented Feb 18, 2013 at 5:24
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    $\begingroup$ You can get older versions of papers posted at the arXiv from the arXiv itself. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 18, 2013 at 6:28
  • $\begingroup$ @Mariano: How can one do that? All the versions of the paper seem to be removed from arXiv. I have a copy but I think it would be more appropriate if the OP asks Justin Moore. $\endgroup$
    – user6976
    Commented Feb 18, 2013 at 11:54
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    $\begingroup$ Mark, there are v numbers (v1, v2, etc.) on the abstract page which lead to other abstract pages that have the older versions. I use it to look at manuscripts (sometimes after they have been withdrawn). Gerhard "Ask Me About System Design" Paseman, 2013.02.18 $\endgroup$
    – Gerhard Paseman
    Commented Feb 18, 2013 at 15:34

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