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I need to find such F, so that for any M $FM = MF$. I can't figure out, how to bring this equation to the form like this: $F = TF$, so that I could just apply Y combinator

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We want a definition like:

$F = \lambda x.xF$

Instead of using $Y$, we can use $R=\lambda x.xx$ for recursion.

$F = R(\lambda fM.M(Rf))$

So:

$$ FM = R(\lambda fM.M(Rf)) \\ = (\lambda x.xx)(\lambda fM.M(Rf))M \\ = (\lambda fM.M(Rf))(\lambda fM.M(Rf))M \\ = M(R(\lambda fM.M(Rf))) \\ = MF $$

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