Short recursively saturated models and boundedly saturated models

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 boundedly saturated.

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

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.