I was reading through Leary and Kristiansen's A Friendly Introduction to Mathematical Logic, and had some trouble understanding the idea of variable/term assignment functions. The authors give the following definitions:
Given a language $\mathcal{L}$, and an $\mathcal{L}-$structure $\mathfrak{A}$ (whose universe is $A$), a variable assignment function is a map $s$ from the set of variables to A. Furthermore, a term assignment function is a function $\overline{s}$ mapping the set of $\mathcal{L}-$terms into $A$, such that:
(1) if $t$ is a variable, then $\overline{s}(t) = s(t)$
(2) if $t$ is a constant symbol $c$, then $\overline{s}(t) = c^{\mathfrak{A}}$
(3) if $t := ft_{1}t_{2}...t_{n}$, then $\overline{s}(t) = f^{\mathfrak{A}}(\overline{s}(t_{1}), \overline{s}(t_{2}), ..., \overline{s}(t_{n}))$.
My question is what these variable and term assignment functions are supposed to represent. My initial impression is that they are supposed to represent ways of interpreting variables. For example, if $\mathcal{L}_{NT} = \{0, S, +, \cdot, E, <\}$ is the language of number theory, and we're talking about its standard structure $\mathfrak{N} = \{\mathbb{N}, 0, S, +, \cdot, E, <\}$, then the formula $x=y$ may or may not be true, depending on what we "decide" $x$ and $y$ stand for. If we're talking about a variable assignment function where $s(x) = 1$ and $s(y) = 2$, then the formula will be false. But if $s(x) = 1$ and $s(y) = 1$, then the formula will be true. So it seems to me that these assignment functions are meant to realize a particular choice of what each variable represents, so that we can talk about the truth of formulas involving these variables.
Is this assessment correct?