I know that a relation on a set $S$ is a subset $R \subseteq S \times S$ such that for all $(s,s') \in S \times S$, $(s,s') \in R$ iff $sRs'$, therefore the set $T$ of all relations on $S$ is the set $\mathcal{P}(S \times S)$ of all subsets of $S \times S$, in other words $T=\mathcal{P}(S \times S)$.
I'm wondering if there is a particular name for "the set of all relations", something condensed, less wordy, like 'the omnirelational set'.
Side question: Say $T^0=T$. Is there anything interesting that comes from considering $T^1=\mathcal{P}(T^0 \times T^0)$ or furthermore $T^{n+1}=\mathcal{P}(T^n \times T^n)$?