Structure

Definition

Let $L$ be a signature: a set of function symbols, relation symbols and constant symbols. Each function symbol and each relation symbol has an arity $n \ge 1$. An $L$-structure $\mathcal{M}$ consists of the following data.

1. A nonempty set $M$, the domain or universe of $\mathcal{M}$.
2. For each $n$-ary function symbol $f$ of $L$, a function $f^{\mathcal{M}} \colon M^n \to M$, where $M^n$ is the Cartesian product of $n$ copies of $M$.
3. For each $n$-ary relation symbol $R$ of $L$, a subset $R^{\mathcal{M}} \subseteq M^n$.
4. For each constant symbol $c$ of $L$, an element $c^{\mathcal{M}} \in M$.

The objects $f^{\mathcal{M}}$, $R^{\mathcal{M}}$ and $c^{\mathcal{M}}$ are the interpretations of the symbols in $\mathcal{M}$.