Is cyclic PA interpretable in PA? If we remove the axiom that zero doesn't have a predecessor, and stipulate that every natural number has a predecessor, and that no number is the successor of itself. And keep all other axioms of $\small \sf PA$, and introduce a new primitive relation symbol $``<"$ to signify strictly smaller than relation, and axiomatize that each $n^{th}$ successor of zero  [for a meta-theoretic $n$] is strictly smaller than its successor, and that $<$ is asymmetric and transitive.
Formally this is the first order theory with axioms of:
axioms of first order $\small \sf PA$ minus $\not \exists x (S(x)=0)$ plus:


*

*$\forall x \exists y: S(y)=x$

*$S(x) \neq x$
Define recursively: 
$S_0(0) = 0 \\ S_{i+1}(0)=S(S_i(0))$


*For $n=1,2,3,....\\ S_n(0) < S_{n+1}(0)$ 

*$x < y \to \neg \ y < x$

*$ x < y < z \to x < z$
This is what I term as $\small \sf cyclic \ PA$ or $\small \sf cPA$

Question: is there an interpretation of $\small \sf cPA$ in $\small \sf PA$?

 A: The theory cPA is interpretable already in Robinson’s theory $R$ by a result of Albert Visser [1], because it is recursively axiomatized and locally finitely satisfiable, meaning that every finite subtheory of cPA has a finite model. Indeed, a finite subtheory of cPA only mentions finitely many axioms 3, hence it is satisfied for sufficiently large $n\in\mathbb N$ in the model $(\{0,\dots,n-1\},0,S,{+},{\cdot},{<})$, where the arithmetic operations are computed modulo $n$.
An explicit interpretation of cPA in PA (or even in $I\Delta_0$, which is itself well-known to be interpretable in $Q$) can be constructed as follows. First, PA interprets its own extension $\mathrm{PA}+\phi$ which proves the existence of a nonstandard definable element $a$: for example, we may take $\phi=\neg\mathrm{Con_{PA}}$, and let $a$ be the least code of a proof of contradiction in PA. Then we interpret cPA by making the domain of interpretation be the interval $\{x:x<a\}$ with the induced ordering, and operations computed modulo $a$.
Reference:
[1] Albert Visser: Why the theory R is special. In: Foundational
  Adventures: Essays in Honor of Harvey M. Friedman (N. Tennant, ed.),
  Tributes vol. 22, College Publications, London, 2014. LGPS preprint no. 279.
