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If by "$=$" you mean $\beta$-equality, then the answer is yes, $MX=X$ for all $X$ is a stronger property than $MM=M$. For example, let $$A := \lambda a.aa(aa)$$ (to save parentheses, I am using the standard left-associative notation for application; in your notation, the above term would be $\lambda a.a(a)(a(a))$) and take $$M := AA.$$ We clearly ...


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