# Are there functions (or category of functions) that satisfy following conditions?

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 $U(x) \neq x$.

• 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 $T$? – Eric Wofsey Feb 9 '18 at 5:55
• @EricWofsey Ideally for all T, but is there a solution for just one T? – Meekaa Saangoo Feb 9 '18 at 10:59

No. Assume that $S(x) \ne x$ for some specific value $x$, and take $T$ to be the constant function $T \equiv x$