Help with logic notation, what are $S$ and $\underline0$?

Question

This is a more or less literal translation from German, hopefully it's understandable, it's all the information available:

Let $L_N=\{\underline0,S,+,\cdot~,<\}$ be the language of the natural numbers and let $\mathcal A$ be the $L_N$-structure with underlying set $\mathbb N$.
Furthermore, $\beta$ is a variable assignment with $\beta(v_n)=2n ~\forall n \in \mathbb N_0$.

Calculate

• $t_1^{\mathcal A}[\beta]$, where $t_1=\underline0$
• $t_2^{\mathcal A}[\beta]$, where $t_2=(S(S(\underline0))\cdot v_{2016} +(S(v_{2015})+t_1))$
• $t_3^{\mathcal A}[\beta]$, where $t_3=S(t_1)\cdot t_1$

Now, are $S$ and $\underline0$ so common I should know what is meant or are they "irrelevant" to solving the task? I can remember seeing something similar at the Peano axioms, so $S(S(\underline0))$ would be $2$, but then why the underline?

Answer 1

This $\beta$ appears to be what is here called a variable assignment. $S$ is a unary function symbol that is intended to be interpreted as the successor function in $\Bbb N$, and $\underline{0}$ is a constant symbol that is intended to be interpreted as $0\in\Bbb N$.

Added: And on cross-checking with German Wikipedia, I see that in this context a Belegung is indeed a variable assignment.