Definition:Algebraic (Model Theory)

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Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$.

Let $A$ be a subset of $M$.

and let $\bar b$ be an ordered $n$-tuple of elements from $M$.

Let $\mathcal{L}_A$ be the language formed by adding constant symbols to $\mathcal{L}$ for each element of $A$.

$\bar b$ is algebraic over $A$ if there is an $\mathcal{L}_A$-formula $\phi(\bar x)$ with $n$ free variables such that $\mathcal{M}\models \phi(\bar b)$ and the set $\{\bar m \in M^n : \mathcal{M}\models \phi(\bar m)\}$ has only finitely many elements.

Alternative Definition

The following definition is sometimes used. It is not equivalent in general. However, these definitions are equivalent when working in a saturated model. This is proved in Algebraic iff Finite Orbit.

$\bar b$ is algebraic over $A$ if it has only finitely many images under $A$-automorphisms.