Given $x,y\subset \omega$, define the equivalence relation $$x=^*y\iff x\Delta y\text{ is finite.}$$ Let $M$ be a ctm of ZFC and $G$ an $M$-generic filter over $P$=Fn$(\omega\times\omega,2)$. Put $g=\bigcup G:\omega\times\omega\to2$ and let $$a_i=\{n<\omega: g(i,n)=1\}$$ be the $i^\text{th}$ Cohen real.
I know that each $a_i$ has a canonical name, namely (no pun intended) $$ \dot a_i:=\{(\check n,p):p(i,n)=1\}$$ Can we find any similarly easy names for the equivalence class $[a_i]=\{x\subset\omega: x=^*a_i\}$?
The context for this question is: I'm considering automorphisms of $P$, and I need names for which I can simply describe the action of the automorphism on then.
Yes, of course. Let's focus on adding just a single Cohen real $\dot a$.
If $s$ is a finite binary sequence, $\dot a^s$ would be $\{(\check n,p)\mid p\subseteq s\lor s\subseteq p\land p(n)=1\}$.
Now $[\dot a]=\{\dot a^s\mid s\in 2^{<\omega}\}^\bullet$, where $\{\dot x_i\mid i\in I\}^\bullet$ is the name given by $\{(\dot x_i,1)\mid i\in I\}$.