$\mathcal{L}$-structure

$\mathcal{L}$-structure

Let $\mathcal{L}$ be a language. An $\mathcal{L}$-structure consists of the following:

1. A nonempty set $A$, which is called the universe.
2. A map, from constant symbols → elements of $A$.
3. A map, from relation symbols → relations over $A^n$, where $n$ is the arity of the relation.
4. A map, from function symbols → functions with signature $A^n\to A$, where $n$ is the arity of the function.

An $\mathcal{L}$-structure is often denoted by the Fraktur symbol $\mathfrak{U}$.

$c$ is a symbol often used to represent some "constant" symbol of a language, and $c^{\mathfrak{U}}$ is often used to represent an element of the universe mapped to by the symbol represented by $c$. For the similar purposes, the symbols $f$, $f^{\mathfrak{U}}$, $R$ and $R^{\mathfrak{U}}$ are used.

Variable assignment function

Let $\mathcal{L}$ be a language, and let $\mathfrak{U}$ be an $\mathcal{L}$-structure for the language. Let $A$ be the universe of $\mathfrak{U}$.

A variable assignment function is a mapping from the variable symbols of $\mathcal{L}$ to elements of $A$.