nLab ultraroot

Idea

An elementary substructure $N \preceq M$ of a model of a first-order theory is an ultraroot of $M$ if $M$ is isomorphic to an ultrapower of $N$.

Definition

Formally, recall that any structure $N$ admits a diagonal elementary embedding $N \overset{\Delta}{\to} N^{\mathcal{U}}$ to any ultrapower $N^{\mathcal{U}}$ of $N$.

If $i \colon N \to M$ is an elementary embedding, we say that $N$ is an ultraroot of $M$ if there is some ultrafilter $\mathcal{U}$ on some index set such that there is an isomorphism

$$\left(N \overset{i}{\to} M\right) \simeq \left(N \overset{\Delta}{\to} N^{\mathcal{U}}\right)$$

in the co-slice category under $N$.

Examples

• $N$ is an ultraroot of $N^{\mathcal{U}}$.

• Assuming the continuum hypothesis, any countable model of a complete-first order theory will be an ultraroot of that theory’s $\aleph_1$-saturated continuum-sized model.

• Similarly, taking a countably-indexed ultrapower of a slightly-expanded copy (like a nonstandard model generated by throwing in a single infinitesimal) of the reals still yields the hyperreal numbers.

Remarks

Closure under taking ultraroots, along with closure under ultraproducts and elementary embeddings, gives a criterion for detecting elementary classes. See also at Birkhoff's HSP theorem.

Related concepts

• ultraproduct

• ultrapower

References

• See e.g. page 7 of these notes by Ward Henson on continuous model theory,