Informal idea: I'll visualize Peano arithmetic "PA" as arithmetic rising from a linear structure, we can call it in graph terms a linear directed path with a beginning and no end. The numbers are the nodes of the path, and the successor relation is the directed edges between the nodes, the various arithmetical operators are just functions defined on nodes of that path, and of course we can have relations defined on those nodes as well.
Now I here want to ask about whether there are other forms of arithmetic rising from non linear structures, and whether those are interpretable in PA? Just a simple example is an arithmetic which have the following structure
$1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 8 \rightarrow 9 \rightarrow ..\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nearrow \ \ \searrow \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7 \leftarrow 6\leftarrow \ 5 $
Lets say a loop of size four nodes happens every $4$ nodes on the upper linear path, so the first one at $4$, the other one at $11$, the next at $18$, and so on...
Now the arrow can be thought of representing a binary successor function, so we have $\mathcal S (a,b)=c$ to mean that $c$ is the successor of $b$ coming from $a$. So here we have $\mathcal S (3,4)= 5$, while $\mathcal S(7,4)= 8 $, and so on...(of course this successor function is a partial function, and its the theory in question that would specify which numbers have successors).
We can have an induction principle stating that if a property hold of the first node $0$, and when it holds of a node $a$, then it hold of all nodes $b,c$ such that $\mathcal S(a,b,c)$; then it holds of all nodes.
I'm aware that modular arithmetic has loops, but what I'm asking here is if there is a first order logical axiomatization handling all (or most) such non-linear variants of arithmetic?
My question here had such forms of non-linear arithmetic had been defined as first order axiomatic systems in a manner generally similar to how PA is defined? And if there is a general theory about all or most kinds of non-linear structure based arithmetic? or whether it is actually possible to define such a mother theory of all such variants?
$$S_1(n)=\begin{cases}S(n)\quad n\not\equiv 0,4\bmod 7\\P^3(n)\quad n\equiv 0\bmod 7\\S(P_1(n))\quad n\equiv 4\bmod 7 \end{cases}$$
Not quite basic, but only had to use a new predecessor function once. it needs only 4 cases to be defined. Once that's done pass it back to Peano.