Definition:Age (Model Theory)

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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 it contains exactly one representative from each isomorphism type of the finitely-generated structures that embed into $\MM$.