# \( \mathcal{L} \)-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 nonempty set \( A \), which is called the
*universe*. - A map, from constant symbols → elements of \( A \).
- A map, from relation symbols → relations over \( A^n \), where \( n \) is the arity of the relation.
- 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.

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 variable symbols of \( \mathcal{L} \) to elements of \( A
\).

### Recap some definitions.

#### Languages, first-order languages and formula

A *language* in model theory is a collection of symbols.

A *first-order language* has specific symbols
\( \symbolq{(} \),
\( \symbolq{)} \),
\( \symbolq{\lnot} \),
\( \symbolq{\lor} \),
and
\( \symbolq{\forall} \).
There are other symbols split into classes
such as variable symbols and function symbols. In can be interpreted as an
encoding scheme, which in turn can be seen as just a couple of
machines/algorithms that decode certain symbol sequences.

A recursive set of requirements narrows down the valid strings of a
first-order language to a smaller set of *formulas*.
Requirements are rules like \( aRb \)
is a formula, where
\( R \) is a relation symbol and
\( a \)
and
\( b \)
are *terms*. Terms also are defined
recursively in a similar way.

### Symbols, but not sequences

If a language is given an \( \mathcal{L} \)-structure and a variable
assignment function, then *almost* every symbol of the
language is mapped to a set theoretic object. The symbols that do not have
such an interpretation are:
\( \symbolq{(} \),
\( \symbolq{)} \),
\( \symbolq{\lnot} \),
\( \symbolq{\lor} \),
and
\( \symbolq{\forall} \).
Furthermore, no symbol sequence has been given an interpretation. Specifically,
terms and \( \mathcal{L} \)-formula have not yet been given an
interpretation.