I recently met the problem as indicated in the title: find an $F \in \Lambda$ such that $Fx = xF$ for arbitrary variable $x \in \Lambda$. I am not only seeking a solution, but also a systematic way to think about such problems. Could someone help?
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What have you tried? What is the context? – gist076923 Nov 25 '22 at 04:35
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2Hint: $F = \lambda x. x F$, which is a fixpoint equation. – Trebor Nov 25 '22 at 04:36
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1@gist076923 Actually, I got what Trebor told me. I will proceed to think about it. – Ziqi Fan Nov 25 '22 at 04:49
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What does $\Lambda$ mean? – Gerry Myerson Dec 03 '22 at 09:31
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@GerryMyerson Set of lambda expressions. – Ziqi Fan Dec 03 '22 at 20:30
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Finally I got the solution after reviewing materials from Harvard. From \begin{equation} F = \lambda x. xF, \end{equation} we may define \begin{equation} F^{\prime} = \lambda t. \lambda x. x\left(tt\right). \end{equation} Then \begin{equation} F = F^{\prime}F^{\prime}. \end{equation}
Indeed, without reading the materials, I couldn't have solved this problem.
Ziqi Fan
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