Saturated model

A model $N$ is $\kappa$-saturated when every complete type over a subset of size less than $\kappa$ is realized in $N$.

A model is omega-saturated when it realizes every complete finite-tuple type over any finite parameter set. Realizing all one-variable types over finite parameter sets suffices by successively adjoining coordinates. This is weaker than saturation over every countable parameter set.

Let $M$ be a countable omega-saturated model and $D\subseteq M$ finite. Then $D$ is a parameter-free definable set exactly when every structure automorphism preserves it setwise. One direction is automorphism invariance of parameter-free definability. For the converse, saturation implies homogeneity and the back-and-forth method show that equal empty-set types are in the same automorphism orbit. For $a\in D$, if no formula in $\mathrm{tp}(a)$ has its realization set contained in $D$, the partial type $\mathrm{tp}(a)\cup \{x\ne d:d\in D\}$ is finitely satisfiable. Its parameters form a finite set, so saturation realizes it outside $D$, contradicting invariance. Choose a defining neighbourhood for each $a\in D$ and take their finite disjunction. The empty set is defined by $x\ne x$. Setwise invariance permits automorphisms to permute points within $D$.

For every infinite regular cardinal $\kappa$, every first-order structure has a $\kappa$-saturated elementary extension. At one stage use the compactness theorem with the elementary diagram to realize all complete types over subsets of the stage having size less than $\kappa$. There is a set of such types; finite subsets of the enlarged theory are satisfiable because their finitely many witnesses are independent. Iterate for $\kappa$ stages, taking elementary unions at limits. Regularity puts any parameter set of size less than $\kappa$ inside one stage, and its type is realized at the next one. Passing to a larger regular cardinal covers singular requested degrees of saturation as well.

In a countable language, an ultraproduct over a nonprincipal ultrafilter on $\omega$ realizes every countable finitely satisfiable type, including types over countably many parameters. Enumerate the type ${\phi }_1,{\phi }_2,\dots$, and let $A_n$ be the ultrafilter-large set of coordinates where the first $n$ first-order formulas have a simultaneous witness. Put $B_n=\{i:i\ge n\}\cap {\bigcap }_{j\le n}{A}_j$. At coordinate $i$, choose a witness for the largest $n\le i$ with $i\in B_n$, or an arbitrary element when there is none. For each fixed $n$, the resulting function satisfies the first $n$ first-order formulas on $B_n$. The Łoś theorem then realizes the entire type. Countability of the language and parameter set makes the enumeration possible.

A $\kappa$-saturated model has the one-point extension property for partial elementary maps with domain of size less than $\kappa$. Given $f:A\to B$ and $a$, transport the complete type of $a$ over $A$ along $f$. Every finite subset is satisfiable by elementarity of $f$, so saturation realizes the transported type by some $b$. Then $f\cup \{(a,b)\}$ is a partial elementary map; inequalities in the type ensure that a new $a$ gets a new image.

When the model itself has infinite cardinality $\kappa$, the back-and-forth method upgrades this local property to automorphisms extending all partial elementary maps of size less than $\kappa$. Enumerate the model in $\kappa$ stages and alternately extend the domain and range. Every intermediate map still has size less than $\kappa$. For a model of larger cardinality, $\kappa$-saturation alone gives the local extension property; full automorphism extension requires additional size or homogeneity hypotheses.

If an aleph-zero-homogeneous model realizes every finite type over the empty set, it is aleph-zero-saturated. Realize the joint empty-set type of the parameters and a desired element, then use homogeneity to move the realized parameter tuple to the given one.

Fix an infinite cardinal $\kappa$ and a complete theory $T$. A model $N\models T$ of cardinality $\kappa$ is $\kappa$-universal when every model of $T$ of cardinality at most $\kappa$ has an elementary embedding into $N$. The bound includes models of size exactly $\kappa$.

Some conventions use a separate weaker notion requiring embeddings only for models of size less than $\kappa$. That convention must be distinguished here: at $\kappa ={\aleph }_0$, it can be vacuous for a theory with no finite models. A countable dense linear order without endpoints is universal for the countable models of its complete theory, since each such model is isomorphic to the rational order.

A model is strongly $\kappa$-homogeneous when every partial elementary map between its subsets of size less than $\kappa$ extends to an automorphism. Here the unqualified expression homogeneous model uses this automorphism extension convention. For finite parameter tuples it says, in particular, that equal complete types can be carried to one another by an automorphism.

A weaker local convention asks only for one-point extensions of small partial elementary maps. A $\kappa$-saturated model always has that local property. If its cardinality is $\kappa$, the back-and-forth method yields the strong version too. For example, order-preserving maps between finite subsets of the rational dense linear order without endpoints extend to automorphisms by back-and-forth.

A first-order structure has finite-tuple homogeneity when any two finite tuples of the same parameter-free complete type are related by an automorphism. For countably infinite first-order structures this agrees with extension of finite partial elementary maps to automorphisms. Countable atomic models have this property by a back-and-forth method using isolating logical formulas. Uncountable atomic models can fail it.

Let $T$ be a complete theory, let $\kappa \ge |T|+{\aleph }_0$, and let $N\models T$ have cardinality $\kappa$. If $N$ is a universal model for all models of $T$ of size at most $\kappa$, and every partial elementary map of size less than $\kappa$ extends to an automorphism, then $N$ is $\kappa$-saturated.

Let $p(x)$ be a complete type over $A\subseteq N$, with $|A|<\kappa$. By compactness theorem it is realized by an element $b$ of an elementary extension of $N$. By the Downward Lowenheim-Skolem theorem, there is an elementary substructure $M$ containing $A\cup \{b\}$ with $|M|\le \kappa$. Universality gives an elementary embedding $f:M\to N$. The inverse map from $f(A)$ to $A$ is partial elementary, so an automorphism $\sigma$ of $N$ extends it. Then $\sigma (f(b))$ realizes $p$ over the original set $A$.

Using only universality for models strictly smaller than $\kappa$ would leave a gap when the language itself has size $\kappa$, and can be vacuous for countable models.

Let $N$ be uncountable and saturated, and let a formula define an equivalence relation. If every small elementary submodel of $N$ meets every class, then there are only finitely many classes. Otherwise a small elementary submodel $M$ yields the finitely satisfiable type $\{\neg E(x,m):m\in M\}$, contradicting saturation.