Model theory: Find an example for an infinite structure with only finite substructures

Question

So I tried solving this for a long time: Find an example for an infinite structure with only finite substructures.

So I tried looking at group signatures and infinite groups, but couldn't find an example that I was taught at my Abstract Algebra course that I could use - so I can't actually justify those examples.

Any assistance would be great! Thank you.

Answer 1

Let $S=\{+,f,0\}$ and $\mathfrak{A}$ be the S-structure $(\mathbb{N},0, +_\mathbb{N}, f^\mathfrak{A})$, where $f^\mathfrak{A}(0)=0$ and $f^\mathfrak{A}(n)=1$ for $n>0$

Now the only proper substructure of $\mathfrak{A}$ is $(\{0\}, 0, +_\mathbb{N}\restriction_{\{0\}\times \{0\}}, f^\mathfrak{A}\restriction_{\{0\}})$ which is finite.