This is a natural follow-up to my previous question on Peano structures and positive integer structures, here: Are positive integer structures axiomatizable?. In that question, I gave the Peano axioms for structures of the form $(N;0,S,+,*)$:
- $\neg \exists x Sx = 0$
- $\forall x \forall y (Sx = Sy \rightarrow x = y)$
- The axiom schema of induction (which are really an infinite set of axioms)
- $\forall x (x+0)=x$
- $\forall x \forall y (x+Sy)=S(x+y)$
- $\forall x (x*0)=0$
- $\forall x \forall y (x*Sy)=((x*y)+x)$
Define the constant $n$ by $S...S0$, where there are $n$ $S$'s, and define the relation $\leq$ by $x \leq y \iff \exists z (x+z)=y$. It can be easily proven that the set $N_n := \{x \in N | n \leq x\}$ is closed under the constant $n$, the unary operation $S$, and also the binary operations $+$ and $*$. Therefore, we can form the structure $(N_n;n,S,+,*)$. I want to know, for each positive integer $n$, is the class of structures isomorphic to those structures a first-order axiomatizable class of structures? In my previous question, I asked this same question for the specific case of the positive integer $1$. I believe that the answer to my current question is also affirmative. If it is, I would also like an explicit axiomatization of that class of structures.