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 $\map \phi {\bar x}$ 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.

Also see

 * Algebraic iff Finite Orbit: proving that the definitions are equivalent when working in a saturated model.