\( \mathcal{L} \)-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?].