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

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

If ${(M_i)}_{i<\alpha }$ is a chain then ${\bigcup }_{i<\alpha }{M}_j$ is a well-defined $L$-structure.