Short recursively saturated models and boundedly saturated models

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Short Recursively Saturated Models

Short Models

A model is short if there is an element in the model whose Skolem closure is cofinal in the model. That is, $M$ is short iff there is $a\in M$ such that $M=Sup(Scl(a))$. If a model is not short, it is called tall.

The gap of an element $a$ is the set of all $b\in M$ such that $t(b)>a$ and $t(a)>b$ for some Skolem term $t$. Short models have a last gap while tall models do not have a last gap.

Bounded Recursively Saturated Models

A type $p(v,b)$ is bounded if it contains the formula $v<b$.

A model $M$ is bounded recursively saturated if for every $b$ in $M$, every bounded recursive type $p(v,b)$ is realized in $M$.

Every elementary cut of a recursively saturated model is bounded recursively saturated.

An elementary cut of an arithmetically saturated model is bounded arithmetically saturated.

A model is short recursively saturated if it is short and bounded recursively saturated.

A model is short arithmetically saturated if it is short and bounded arithmetically saturated.

Countable Short Recursively Saturated Models

If an elementary cut of a countable recursively saturated model is tall, it is recursively saturated (and hence isomorphic to the model). If the elementary cut is short it is bounded recursively saturated, and hence short recursively saturated.

Two countable short recursively saturated models $M$ and $N$ are isomorphic if and only if $SSy(M)=SSy(N)$, $Th(M)=Th(N$), and there is an element $a$ in the last gap of $M$ and $b$ in the last gap on $N$ such that $tp(a)=tp(b)$.

Automorphisms

Every automorphism of a countable recursively saturated model fixes elements in the last gap of the model.

Problem: Do countable short recursively saturated models have the small index property?

We know that countable arithmetically saturated models have the small index property so we can ask the following:

Problem: Do countable short arithmetically saturated models have the small index property?

Boundedly Saturated Models

A model is boundedly saturated if it realizes every bounded type.

Every elementary cut of a saturated models is boundedly saturated.

Short Saturated Models

A model is short saturated if it is short and boundedly saturated.

Every short elementary cut of a saturated model is short saturated.