Here is a somewhat positive answer in terms of

### germs and equivalence modulo finite differences.

There has been a good deal of impressive lattice-theoretical work on the lattice of recursively enumerable sets. (This topic is a subtopic of recursively enumerable sets and degrees (AMS classification 03D25); these are also known as r.e. sets, computably enumerable sets, and c.e. sets.)

This work in particular has dealt with the lattice of c.e. subsets of a given c.e. set $E$, $\mathcal L(E)$, which has different properties depending on properties of $E$ such as hypersimplicity and creativity.

About half of the work concerns the quotient structures modulo finite differences. So in a sense they are studying **germs** and there is some analogy with what you are asking for. That is, one studies $\mathcal E$, the lattice of r.e. sets, and $\mathcal E^*$, the lattice of equivalence classes of elements of $\mathcal E$, where $A$ and $B$ are equivalent if
$$
\{n : A(n)\ne B(n)\}
$$
is a finite subset of $\omega=\mathbb N$.

The focus is on the automorphism group, its orbits, and first order definability.
See for instance

Cholak, Peter A.; Downey, Rodney; Harrington, Leo A. On the orbits of computably enumerable sets. J. Amer. Math. Soc. 21 (2008), no. 4, 1105–1135.