Definition:Age (Model Theory)

This page is about age in the context of model theory. For other uses, see age.

Definition

Let $\MM$ be an $\LL$-structure.

An age of $\MM$ is a class $K$ of $\LL$-structures such that:

• if $\AA$ is a finitely generated $\LL$-structure such that there is an $\LL$-embedding $\AA \to \MM$, then $\AA$ is isomorphic to some structure in $K$,
• no two structures in $K$ are isomorphic, and
• $K$ does not contain any structures which are not finitely generated or do not embed into $\MM$.

That is, $K$ is an age of $\MM$ if and only if it contains exactly one representative from each isomorphism type of the finitely-generated structures that embed into $\MM$.

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