Language Structure

Last updated Nov 1, 2022

An $\mathcal{L}$-structure $\mathcal{M}$ is a Tuple $(M, \mathcal{F}^{\mathcal{M}}, \mathcal{R}^{\mathcal{M}}, \mathcal{C}^{\mathcal{M}})$ where

1. $M$ is a Nonempty Set called the universe, domain, or underlying set of $\mathcal{M}$.
2. $\mathcal{F}^{\mathcal{M}}: \mathcal{F} \to M$ that maps each Function Symbol $f$ of $\mathcal{L}$ to a Function $f^{\mathcal{M}} : M^{n_{f}} \to M$.
3. $\mathcal{R}^{\mathcal{M}} : \mathcal{R} \to M$ that maps each Relation Symbol $R$ of $\mathcal{L}$ to a Relation $R^{\mathcal{M}} \subset M^{n_{R}}$.
4. $\mathcal{C}^{\mathcal{M}}: \mathcal{C} \to M$ that maps each Constant Symbol $c$ of $\mathcal{L}$ to an element $c^{\mathcal{M}}\in M$.

1. Each $f^{\mathcal{M}}, R^{\mathcal{M}}, c^\mathcal{M}$ are referred to as interpretations of $f, R,$ and $c$ respectively.

1. Language