Existentially closed partial orders

Question

Existentially closed linear orders are dense linear orders without endpoints, which are finitely axiomatizable, and occur as order-types of natural mathematical structures such as the rationals or reals. What about existentially closed partial orders? Is the theory finitely axiomatizable, and are there natural mathematical structures with an existentially closed partial order structure?

Answer 1

A model $M$ of a theory $T$ is existentially closed with respect to that theory, if for any quantifier-free formula $\varphi$ and any objects $\vec a$ in $M$, if there is model $N$ of the theory $T$ extending $M$ in which there is an object $z$ for which $N\models \varphi(z,\vec a)$, then there is already such an object $z$ inside $M$.

For the theory of linear orders, this property implies density, since if $a<b$, then there is a larger linear order with some $z$ in between them, so there must already be something between. Similarly, it implies that there is no largest element and no smallest element, since you can always add a new element above or below.

In a partial order, what you get is that any finite partial order can be extended by adding points of any type that can occur in any particular partial order.

Theorem. There is a unique countable existentially closed partial order, and it has a computable presentation.

Proof. (Existence) Start with a single point; then add a point above, below, and to the side. Continuing in stages, at every stage you have finitely many points. Add points in all possible ways that can be realized in any partial order extending what you have so far. The result will be existentially closed, since for any finitely many elements, if a point realizes some pattern in a partial order extending it, you've already added a point just like that. This process gives a computable presentation of the order.

(Uniqueness) This follows from a back-and-forth argument. Every finite partial isomorphism can be extended one more step, since whatever type is realized by the next point in one of the models, the other model will also have a point realizing the corresponding type. QED

This partial order is the unique homogeneous countable partial order that is universal for all countable partial orders. It is also the Fraïssé limit of all finite partial orders.

The slides for my talk on the Hypnagogic digraph at the JMM 2016 special session on the surreal numbers (slides) contain an elucidation of the countable universal homogeneous partial order---look at the section on universality to see an partial animation of the countable existentially closed partial order being constructed.