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Are there functions(or category of functions) S and U such that

$S(T(U(k))) = T(k)$ for any function T where

$S(T(U(k))) \neq U(T(S(k)))$ and, S and U are not identity functions i.e $S(x) \neq x$ and $U(x) \neq x$.

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  • $\begingroup$ Is $S\circ T\circ U=T$ supposed to hold for all $T$ such that $S\circ T\circ U\neq U\circ T\circ S$, or just one such $T$? $\endgroup$ – Eric Wofsey Feb 9 '18 at 5:55
  • $\begingroup$ @EricWofsey Ideally for all T, but is there a solution for just one T? $\endgroup$ – Meekaa Saangoo Feb 9 '18 at 10:59
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No. Assume that $S(x) \ne x$ for some specific value $x$, and take $T$ to be the constant function $T \equiv x$

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