First-Order Structure

Definition

Given a formal language $\mathcal{L}$, a first-order structure $\mathcal{M}$ consists of a non-empty underlying set $M$ together with interpretations of the constant symbols, function symbols, and relation symbols of $\mathcal{L}$.

If $c$ is a constant symbol, it is interpreted as an element of $M$. If $f$ is an $n$-ary function symbol, it is interpreted as a function $M^n\to M$. If $R$ is an $n$-ary relation symbol, it is interpreted as a subset of $M^n$.

Remarks

The notation $\mathcal{M}$ refers to the whole structure, while $M$ refers only to its underlying set.

If a first-order structure $\mathcal{M}$ satisfies every sentence of a theory $T$, then $\mathcal{M}$ is a model of $T$.

This use of “structure” belongs to Model Theory and is distinct from a Quillen Model Structure.

Related Concepts

• Formal Language
• Model Theory
• theory
• Tarski Semantics