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 $A$ if and only if 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.

The term Definition:Logical Formula as used here has been identified as being ambiguous. If you are familiar with this area of mathematics, you may be able to help improve $\mathsf{Pr} \infty \mathsf{fWiki}$ by determining the precise term which is to be used. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Disambiguate}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Saturated Model

Let $\MM$ be a saturated model.

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

Also see

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

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.