I am trying to write the output of a universal Turing machine using the map notation. Let $L$ be the set of all sentences of the binary language. Then the map:
$$ \operatorname{UTM}: L \to L $$
and I implement the map as:
$$ \operatorname{UTM}[s]=\cases{r & If UTM[s] halts \\ \nexists & otherwise} $$
where $s$ and $r$ are elements of $L$.
However, since $\nexists$ is not an element of $L$, the implementation is not allowed by the definition of the map. Note that the empty sentence $\varnothing \in L$ is not the same as $\nexists$.
Is there a standard way to define maps that can include $\nexists$ in the image? Otherwise, I would like to consider the pros and cons of these two possible fixes:
1. Define the map as:
$$ \operatorname{UTM}:L \to L \cup \{\nexists\} $$
In the case, the implementation can live within the definition. But it feels like its a loophole to the requirement that a map must map each element of its domain? Is it allowed?
2. Define the domain of $\operatorname{UTM}$ as a suitable subset of $L$
I define a subset of $L$ which only includes the sentences that halt for $\operatorname{UTM}$. We can call this subset the domain of the universal Turing machine: $\operatorname{Dom}(\operatorname{UTM}) \subset L$. Then the map is defined as:
$$ \operatorname{UTM}:\operatorname{Dom}[\operatorname{UTM}] \to L $$
The implementation would simply be
$$ \operatorname{UTM}[s]=r $$
But this feels odd because the complexity of the implementation is pushed into the definition of the map, leaving the implementation very simple and the definition relatively more complex.