nLab definable and algebraic closure (in model theory)



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.


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

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

Inside a monster

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

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

To see the nonobvious direction of the second claim, try the contrapositive again: if bb isn’t algebraic over AA, then every formula in its type over AA 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 𝕄\mathbb{M}, while finite sets stay the same size, because the theory knows exactly how big they are.


  • In ACF p\mathsf{ACF}_p, the definable closure inside a monster 𝕄\mathbb{M} of a subset AA is the perfect hull of 𝔽(A)\mathbb{F}(A), where 𝔽\mathbb{F} is \mathbb{Q} for p=0p = 0 or the prime field of the characteristic p>0p > 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 nn for each n1n \geq 1, dcl()\operatorname{dcl}(\emptyset) is just the cycle graph of length 11, while acl()\operatorname{acl}(\emptyset) is the entire structure.


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

Last revised on March 8, 2017 at 05:27:09. See the history of this page for a list of all contributions to it.