nLab definable and algebraic closure (in model theory)

Idea

In model theory, the definable closure of a subset of a model is a way of “generating a substructure” around that subset. The algebraic closure of a subset always contains the definable closure and generalizes the usual algebraic closure of fields.

Definition

Let $A$ be a small parameter set (i.e. a subset) of a monster model $\mathbb{M} \models T.$ The definable closure $\operatorname{dcl}(A)$ of $A$ is the set of all tuples $b \in \mathbb{M}$ such that there exists a formula $\varphi(x,y)$ and a tuple $a$ from $A$ such that $b$ is the unique solution to $\varphi(a,y)$, i.e. $\varphi(\mathbb{M}_x,y) = \{b\}$.

To obtain the algebraic closure $\operatorname{acl}(A)$ of $A$, we relax “unique solution” to “one of finitely many.”

Inside a monster

Inside a sufficiently saturated and homogeneous model $\mathbb{M}$, tuples $b$ are definable over $A$ if and only if they are fixed by the stabilizer $\operatorname{Aut}(\mathbb{M}/A)$ of $A$, and are algebraic over $A$ if and only if they have finite orbit under $\operatorname{Aut}(\mathbb{M}/A).$

Proof. To see nonobvious direction of the first claim, try the contrapositive: if every formula in the type of $b$ fails to uniquely pick out $b$, then the type of some $b'$ which is not $b$ but has the same type as $b'$ is finitely consistent, therefore realized, and by homogeneity there is an automorphism interchanging $b$ and $b'$.

To see the nonobvious direction of the second claim, try the contrapositive again: if $b$ isn’t algebraic over $A$, then every formula in its type over $A$ is infinite, in particular all conjunctions of finite fragments of its type. So we can iteratively obtain infinitely many disjoint realizations of its type, and again by homogeneity these must all lie in the same orbit. $\square$

Saturation ensures that infinite definable sets become big—as big as the universe $\mathbb{M}$, while finite sets stay the same size, because the theory knows exactly how big they are.

Examples

• In $\mathsf{ACF}_p$, the definable closure inside a monster $\mathbb{M}$ of a subset $A$ is the perfect hull of $\mathbb{F}(A)$, where $\mathbb{F}$ is $\mathbb{Q}$ for $p = 0$ or the prime field of the characteristic $p > 0$, and algebraic closure coincides with the separable closure.

• In the theory of an infinite set without structure, definable and algebraic closures coincide, and are trivial.

• In the relational structure consisting of the disjoint union of the cycle graphs of length $n$ for each $n \geq 1$, $\operatorname{dcl}(\emptyset)$ is just the cycle graph of length $1$, while $\operatorname{acl}(\emptyset)$ is the entire structure.

Related concepts

• type (in model theory)
• model-theoretic Galois theory

References

• Lou van den Dries, An Introduction to Model-Theoretic Stability.
• Dave Marker, Model Theory: An Introduction.