Let $f$ and $g$ are binary relations (on some set $\mho$). The function $f\times^{C} g$ is defined by the formula: $(f\times^{C} g) a = g\circ a \circ f^{-1}$ (for every binary relation $a$ on $\mho$.


Suppose $f$ and $g$ are non-empty. Can we restore $f$ and $g$ knowing only the value of $f\times^{C} g$?



Restore multipliers from the product
Let $a\_{u,v} = \{(u,v)\}$. Then $g\circ a\_{u,v}\circ f^{-1} = f^{-1}(u)\times g(v)$. Assuming $g$ is nonempty, pick $v\in \text{dom}(g)$. Then $\text{dom} \left[(f\times^C g)a\_{u,v}\right] = f^{-1}(u)$, so varying $u$ allows you to recover $f$. Similarly picking $u\in\text{range}(f)$ and varying $v$ you can recover $g$.


