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A colleague and I are interested in idempotent relations from $I=[0,1]$ to $I$ - relations such that $R\circ R(x)=R(x)$ for all $x\in I$. Specifically, the graphs of the relations we care about must be closed subsets of the square.
Is there any work in the literature addressing these objects from a topological point of view? Thanks!
We worked out the details to get what we needed in that paper. Specifically, if $f$ is an idempotent upper-semicontinuous continuum-valued function from $I$ to $I$ (equivalently, idempotent with a graph which is closed and connected satisfying $f(x)=[l(x),u(x)]$), then the graph of $f$ satisfies what we called condition $\Gamma$: there exists $x,y\in I$ such that $\langle x,x\rangle,\langle y,y\rangle,\langle x,y\rangle$ are all in the graph of $f$.
It would be interesting if continuum-valued was not necessary.
Edit: In https://arxiv.org/pdf/1805.06827.pdf we weakened the assumptions to any weakly countably compact space, and any idempotent relation that's a closed subset of the square and yields a non-empty image and pre-image for each point.
