Let $T_{X}$ be the full transformation semigroup on $X$. For $\alpha$, $\beta \in T_{X}$ \alpha \mathcal{R}\beta \text { if and only if there exist }\gamma,\gamma' \in T_{X}:\alpha\gamma=\beta\gamma' .
This question that looks trivial, takes us into about an hour with my course mates. We argue that by definition $\alpha R\beta$ implies $\alpha T_{X}^1=\beta T_{X}^1$.
So, there exist \gamma,\gamma' \in T_{X} such that \alpha\gamma=\beta\gamma'. Hence the result.
But our professor rejected our proof since \gamma,\gamma' \in T_{X} not in $T_{X}^1$ as given in the statement of the problem. The lecture notes by Tero Harju are here, chapter 5 page 52.
Note that: In any semigroups S the relation $\mathcal{L}$, $ \mathcal{R}$ and $\mathcal{J}$ are define by $x \mathcal{L}y \Leftrightarrow S^1x=S^1y$ $x \mathcal{R}y \Leftrightarrow xS^1=yS^1$ $x \mathcal{J}y \Leftrightarrow S^1xS^1=S^1yS^1$.
The set $T_{X}$ is the set of all mappings from $X$ to $X$ known as the full transformation semigroup on X with the operation of composition of mappings.