More Constructions - Model Theory

8 More Constructions

Let $L$ be a language, $M$ an $L$ -structure. Fix a collection ${(M_{i})}_{i \in I}$ of substructures of $M$ . Let $N = ⋂_{i \in I} M_{i}$ , and assume $N$ is non-empty.

Then we have a canonical $L$ -structure, with universe $N$ and interpretiation:

For $f$ a function, $f^{N} = f^{μ} |_{N}$ (which equals $f^{M_{i}} |_{N}$ for each $i \in I$ )
For $R$ an $n$ -ary relation, $R^{N} = R^{M} \cap N^{n}$ (which equals $R^{M_{i}}$ for each $i \in I$ )
For $c$ a constant, $c^{N} = c^{M}$ (which equals $c^{M_{i}}$ for each $i \in I$ )

Note $N$ is also a substructure.

Definition 8.1 (Generated by). Given an $L$ -structure $M$ , a non-empty $A \subseteq M$ , the substructure generated by $A$ is the intersection of all substructures containing $A$ .

Definition 8.2 (Chain, Elementary chain). Let $α$ be a limit ordinal.

A collection ${(M_{i})}_{i < α}$ of $L$ -structures is a chain if $M_{i} \subseteq M_{j}$ (substructure) for all $i < j$ , and is an elementary chain if $M_{i} ≼ M_{j}$ for all $i < j$ .

If ${(M_{i})}_{i < α}$ is a chain then $⋃_{i < α} M_{j}$ is a well-defined $L$ -structure.