(I am looking at the first part of Antonio Montalban's manuscript on Computable Structure Theory, for those that want specifics or to read along.)
In computable structure theory it is common to take the union of finite structures to get an infinite one. A nested sequence of strings with characters in $\Bbb N$, $\sigma_0 \subseteq \sigma_1 \subseteq ...$, can be unioned over, $\bigcup\limits_{i \in \Bbb N} \sigma_i$, to get a possibly infinite string of these characters, and this could be seen as a function. For a Turing functional or c.e. operator with oracle $f$, $\Phi^f_e$, encoding them (their domains, $W^f_e$) is done similarly, writing $W^\sigma_e = \{ n \in \Bbb N \vert \langle \sigma , n \rangle \in W_e \}$, we have $W^f_e = \bigcup\limits_{\sigma \subset f}W^\sigma_e$.
An $\omega-$presentation of a structure $\mathcal{A}$ (or of a copy of $\mathcal{A}$) is just a structure whose domain is $\Bbb N$, and we say it is computable when the set (or string)
$\tau^{\mathcal{M}} := \bigoplus\limits_{i \in I_R}R_i^{\mathcal{M}}\oplus \bigoplus\limits_{i \in I_F}F_i^{\mathcal{M}}\oplus\bigoplus\limits_{i \in I_C}\{ c_i^\mathcal{M}\}$
is computable. The atomic diagram of an $\omega-$presentation $\mathcal M$ is the infinite binary string $D(\mathcal{M}) \in 2^{\Bbb N}$ defined as:
$D(\mathcal{M})(i) :=$ $\begin{cases} 1 \text{ if } \mathcal{M} \models \phi_i^{at}[x_j \mapsto j : j \in \Bbb N] \\ 0 \text{ otherwise,} \end{cases}$
where $\phi_i^{at}$ is the $i^{th}$ atomic $\tau$-formula in an effective enumeration of all the atomic $\tau$-formulas of the set $\{ x_0, x_1, ... \}$.
More generally, for subsets of $\Bbb N$, we can talk about $( \subseteq \omega) -$presentations, which can be used to present finite structures as well as infinite sequences of finite structures.
This being established, the same sort of "finite approximation" approach with atomic diagrams is applied, where the sequence $\{ \mathcal{M}_s : s \in \Bbb N \}$ is identified with the sequence of codes $\{ D(\mathcal{M}_s) : s \in \Bbb N \}$.For these finite approximations, we have
$D(\mathcal{M_0}) \subseteq D(\mathcal{M_1}) \subseteq D(\mathcal{M_2}) \subseteq \cdots \text{ so } D(\mathcal{M}) = \bigcup\limits_{s \in \Bbb N} D(\mathcal{M}_s)$.
I imagine this union sort of as circles based at $0$ including more and more characters of the atomic diagram's string, these characters arranged on a line going to the right. Bigger circles capture more of the structure. I was wondering if this analogy could be "rephrased" in the following sense. So just focusing on the last bit of the string, i.e., ignoring the segment going all the way to $D(\mathcal{M}_{i-1}$), let's call this $D'(\mathcal{M}_i)$. It seems to me you could get the same string by just taking the disjoint union over all these $D'(\mathcal{M}_i), \coprod\limits_{i \in \Bbb N} D'(\mathcal{M}_i)$, and it would just look like the string, but with each of these primed segments "marked" in a way.
Q. Is this notion equivalent, or does something go wrong?
EDIT: So, in light of some good questions from Noah Schweber, I think I should clear up definitions. The atomic diagram for the full $\omega-$presentation of the structure $\mathcal{M}$ given by finite approximations $\{ \mathcal{M}_s : s \in \Bbb N \}$ which are each "nested" in the full diagram of $D(\mathcal{M})$ in the sense that you write $D(\mathcal{M}_0)$, and then you write what "remains" to write $D(\mathcal{M}_1)$, and so on, concatenating each extra "remaining" part of the next finite approximation as you go. I want to write them all out "next to each other" rather than "nested", so the disjoint union makes sense. So I would write out $D(\mathcal{M}_0)$, and next to it $D(\mathcal{M}_1)$ in its entirety, and so on.
What I wonder is if this disjoint union gives the same atomic diagram when taken in this way as the union taken in the "nested" way (that I believe is how it is done, given the exchange in the comments). For a visual aid I drew what I am thinking about here:
As a side question I wonder if this changes any of the computability theoretic aspects of it. I would like to believe that it doesn't (since you can just do the same thing as before but "copy" what you see behind you as an extra step).
EDIT2: One thing I just considered is that this must be done uniformly so that you can know which positions the characters on which you are writing the structure are because those are linked to which atomic sentences get considered for the structure. The mapping has to be uniform so you can ever hope to write it down, I think.
(This worried me because of an exercise (1.5) that Montalban writes. He says there could exist a computable list of presentations of finite structures with sizes that cannot be computed uniformly, despite having uniformly computable domains and relations. Trying to think of an example, could we consider the "structure" with no relations and domain $K$, the Halting set? So the $n$th finite substructure would be the set containing just the $n$th element of the Halting set? Obviously each finite natural is computable, but there is no computable function $f$ such that $f(n)$ is the size of the $n$th substructure. <- My rationale)