# \( \mathcal{L} \)-structure

In model theory, the definition of a language involves syntax only. An \( \mathcal{L} \)-structure moves beyond syntax and gives a first-order language a set theoretic interpretation.

The back side has a recap for concepts referenced in the below definition.

### \( \mathcal{L}\)-structure

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

- A [something], which is called the
*universe*. - A map, from constant symbols → [to what?].
- A map, from relation symbols → relations over [what sets?], where \( n \) is the arity of the relation.
- A map, from function symbols → functions with signature [\( \; ? \, \to \, ? \; \)], 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.

An \( \mathcal{L} \)-structure doesn't give an interpretation to the
variable symbols of a language. Variable symbols are mapped to elements of
the universe by a *variable assignment function*.

### 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 [what?] of \( \mathcal{L} \) to [what?].