We define a tree $T$ to be a poset $(T,\leq)$ such that $\forall x\in T$, the set $\{y\in T|y\leq x \}$ is well-ordered.
Consider partial functions $f:T\rightarrow T$ such that the domain of $f$ is totally ordered.If $F$ is the graph of such a function, consider the characteristic function $\chi_F$ and let $\chi_T=\{\chi_F\in\{0,1\}^{T\times T}| F \text { is the graph of said function}\}$
Is it true that each element in $\chi_T$ has a finite support?
Is it also true that, if we replace T with $\omega_1$ and $f$ with strictly decreasing partial functions $f:\omega_1 \rightarrow\omega_1$, $f$ has a finite domain?