Definition:Algebraic (Model Theory)

Definition
Let $\MM$ be an $\LL$-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 $\LL_A$ be the language formed by adding constant symbols to $\LL$ for each element of $A$.

$\bar b$ is algebraic over $A$ there is an $\LL_A$-formula $\phi(\bar x)$ with $n$ free variables such that $\MM \models \map \phi {\bar b}$ and the set $\set {\bar m \in M^n : \MM \models \map \phi {\bar m} }$ 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 $A$ it has only finitely many images under $A$-automorphisms.