In this article, "type" will by default mean "complete type," rather than "partial type." Also, types will be finitary (*n*-types) rather than finitary, by default.

Roughly speaking, a structure is *saturated* if all types over it are realized. This is practically impossible, so instead we only consider types over small sets. If is an infinite cardinal, a structure *M* is said to be **-saturated** if every type over a subset *A* ⊆ *M* with is realized in *M*. It turns out that an equivalent condition is that every *1*-type over a subset of size less than is realized.

The structure *M* is said to be **saturated** if it is |*M*|-saturated: every type over a subset of smaller cardinality than *M* is realized.

A closely related concept is *strong homogeneity*. A structure *M* is said to be **-strongly homogeneous** if whenever *a* and *b* are possibly infinite tuples from *M* having the same type over the empty set Ø, and *a* and *b* have length less than , then some automorphism of *M* sends *a* to *b*. Equivalently, every partial elementary map on *M* of size less than can be extended to an automorphism of *M*. **Warning:** Some authors use the term "-homogeneous" rather than "-strongly homogeneous."

In stability theory, one frequently works with a "monster model", by which one means a -saturated and -strongly homogeneous model of the theory for some cardinal much bigger than any cardinals we expect to run into.

Saturation can be viewed as an analogue of compactness:

**Theorem:** Let *M* be -saturated. Let *D* be some definable subset of *M* (or of a power of *M*). Let be some infinite family of definable sets, with

If the family has size less than , i.e., if , then *D* is covered by some finite collection of the *E*_{i}'s.

*Proof:* Let be the partial type over *M* asserting that *x* is in *D* but in none of the 's. Then consists of fewer than formulas, so it is a partial type over a subset with . If is inconsistent, then it is finitely inconsistent. Therefore there exists such that

or equivalently

. Then we are done.

Otherwise, is consistent. Let *p*(*x*) be a complete type over *A* extending . By saturation, *p*(*x*) is realized in *M*, hence so is . But a realization of is a point which is in *D* but in no *E*_{i}, contradicting the hypotheses.

QED.

A basic fact about -saturated and -strongly homogeneous structures is that enough of them exist:

- Every consistent theory
*T*has a model which is both -saturated and -strongly homogeneous. - In fact, if
*M*is any structure and is any infinite cardinal, then*M*has an elementary extension which is -saturated and -strongly homogeneous.

Unfortunately it is hard to guarantee the existence of saturated models of set-size. Here are some results in these directions:

- Countable ultrapowers of countable structures in countable languages are always -saturated, and typically have cardinality . If the continuum hypothesis holds, it follows that countable theories have saturated models.
- More generally, if the generalized continuum hypothesis holds, there is some way to construct saturated models using ultrapowers, if I recall correctly.
- If
*M*has size less than some inaccessible cardinal , then*M*has a saturated elementary extension of size . In particular, if there are a proper class of inaccessible cardinals, then every structure has a saturated elementary extension. - Stable theories have saturated models. In particular, if a structure
*M*is stable, then*M*has a saturated elementary extension.

An easy work around this problem is to take a conservative extension of ZFC. Take for example the BGC (Bernays-Goedel-Global Choice) were we have class-size elements. In that we can construct a monster model of class-size. For details see the Tent and Ziegler book. Now if one wants to realise global-types of this class size monster model in a bigger, one can construct a monster of size "class of all classes". But this has to be done in a conservative of BGC again.

ZFC cannot show that every consistent first-order theory has a saturated model (Hodges 1993 p.506)

Saturated models are strongly homogeneous:

**Theorem:** If *M* is saturated (i.e., |*M*|-saturated), then *M* is |*M*|-strongly homogeneous.

Saturated structures are determined up to isomorphism by their complete theory and cardinality:

**Theorem:** If *T* is a complete theory, any two saturated models of *T* of the same cardinality are isomorphic. Equivalently, if *M* and *N* are two elementarily equivalent saturated structures, of the same cardinality, then *M* and *N* are isomorphic.

The main use of strong homogeneity lies in the following basic facts:

**Theorem:** Let *M* be a -strongly homogeneous structure. Let *A* ⊆ *M* have size less than . Let *a* and *b* be two tuples of length less than . Then (i.e., tp(*a*/*A*) = tp(*b*/*A*)) if and only if there is an automorphism sending *a* to *b*. Here is the group of automorphisms of *M* fixing *A* pointwise.

*Proof:* If some sends *a* to *b*, then for any formula over *A*, we certainly have

,

by symmetry. This does not require strong homogeneity.

Conversely, suppose . Then , where we are using the standard notation to denote the concatenation of *A* and *a*. At any rate, there is then a partial elementary map *f* from to fixing *A* pointwise and sending *a* to *b*. By strong homogeneity, this can be extended to an automorphism in .

QED.

**Theorem:** Let *M* be a -saturated and -strongly homogeneous structure. Let *A* ⊆ *M* be a subset of size less than , and *a* be a finite tuple from *M*. Then *a* is in the definable closure of *A* if and only if *a* is fixed by Aut(*M*/*A*). Similarly, *a* is in the algebraic closure of *A* if and only if *a* has finite orbit under Aut(*M*/*A*).

See the article on definable and algebraic closure for the proof. One uses -saturation to verify that

*a*∈ dcl(*A*) if and only if*a*is the unique realization of tp(*a*/*A*) in*M*.*a*∈ acl(*A*) if and only if tp(*a*/*A*) has finitely many realizations in*M*.

Then one uses the previous Theorem to conclude that the set of realizations of tp(*a*/*A*) is exactly the orbit of *a* under Aut(*M*/*A*).

More generally, the definable closure part of this Theorem remains true without the assumption that *a* is finite, but merely that *a* has size less than .

**Lemma 1:** Let *M* be a structure. Then there exists an elementary extension of *M* in which every type over *M* is realized.

*Proof:* Let *T* be the union of the elementary diagram together with the statements

where we have added a new constant symbol *c*_{p} for each type *p* over *M*. If *T*_{0} is a finite subset of *T*, then *M* can be made into a model of *T*_{0} by choosing the *c*_{p} appropriately. Indeed, for each type *p*, the set of statements in *T*_{0} concerning *c _{p}* is a finite subset of

So by compactness, *T* has a model *N*. Since *T* contains the elementary diagram of *M*, . Also, for each type *p* over *M*, *N* contains an element *c _{p}* satisfying

**Theorem:** Let *M* be a structure and be an infinite cardinal. Then *M* has a -saturated elementary extension.

*Proof:* A -saturated elementary extension of *M* will certainly be -saturated, so replacing with , we may assume that is regular.

Build an ascending chain of structures inductive as follows:

*M*_{0}= M- is an elementary extension of in which all types over are realized. We can find such an elementary extension by the Lemma.
- If is a limit ordinal, .

On easily proves by induction on that for all , using the Tarski-Vaught Theorem at the limit ordinals. So this is an elementary chain of models. Let *N* be the union of this elementary chain. Again, by Tarski-Vaught, *N* is an elementary extension of every . In particular, it is an elementary extension of *M*_{0} = *M*.

It suffices to show that *N* is -saturated. Let *A* be a subset of *N* of size less than . Because is regular, for some . Now every type over *A* can be extended to a (complete) type over . Since every type over is realized in , so is every type over *A*. Consequently, every type over *A* is realized in *N*. Thus 'N *is -saturated. QED*

Recall that if *M*, *N* are two structures, a partial elementary map from *M* to *N* is a map *f* : *A* -> *N* for some subset *A* ⊆ *M* with the property that for every tuple *a* from *A* and every formula ,

Partial elementary maps are always injective (consider the case where is the formula *x* = *y*). The inverse of a partial elementary map is always a partial elementary map.

The notion of a partial elementary map is closely related to the notion of type. Specifically, if is an (infinite) tuple enumerating *A*, then *f* is a partial elementary map if and only if and have the same type over the empty set, i.e.,

.

In particular, if and are tuples from *M* and *N* having the same type, then there is a partial elementary map mapping to for each *i* in the index set.

If *f* : *A* -> *B* is a partial elementary map from *A* ⊆ *M* onto *B* ⊆ *N*, then we can *push forward* types from *A* to *B*. Specifically, let *p* be a type on *A*. The **push-foward** *f***p*(x) is the type on *B* given as follows:

The fact that the type *f***p* is consistent follows from the fact that *f* is a partial elementary map. This definition even makes sense when *p* is an infinitary type.

**Observation:** Let *a* and *b* be possibly infinite tuples in *M* and *N* respectively, having the same type over the empty set. So there is an elementary map *f* sending *a* coordinatewise to *b*. If *c* is a (possibly infinite) tuple from *M*, and *p* = tp(*c*/*a*), then a tuple *d* from *N* realizes *f***p* if and only if

,

i.e., we can extend the partial elementary map *f* to *a* ∪ *c* by mapping *c* to *d*. (Here, we are using *ac* to denote the concatenation of *a* and *c*, and similarly for *bd*.)

*Proof:* This really follows by unwinding the definitions. The type *p* is the set

So *f***p* is

And in particular, *d* realizes *f***p* if

for all formulas . Replacing with its negation, we get the implication in the other direction. So *d* realizes *f***p* if and only if

for all formulas . This is equivalent to

QED.

From this, we conclude:

**Lemma 2:** Let *M* and *N* be structures, and *f* be a partial elementary map from a subset *A* ⊆ *M* onto a subset *B* ⊆ *N*. Suppose *c* is a singleton from *A*, and every 1-type over *B* is realized in *N*. Then we can extend *f* to a partial elementary map from *A* ∪ {*c*} to (some subset of) *N*.

*Proof:* Let *p* = tp(*c*/*A*). This is a 1-type because *c* is a singleton. The pushforward *f***p* is also a 1-type; by assumption it is realized by some *d* in *N*. Then by the Observation, we can extend *f* to *A* ∪ {*c*} by declaring *f*(*c*) = *d*.

QED.

**Theorem:** Let *M* be a structure, and be an infinite cardinal. The following are equivalent:

- For every subset
*A*⊆*M*of size less than , every 1-type over*A*is realized in*M*. - For every subset
*A*⊆*M*of size less than , every*n*-type over*A*is realized in*M*, for all*n*. - For every subset
*A*⊆*M*of size less than , and every (complete) infinitary type*p*(*x*) over*A*, with*x*a tuple of length*at most*,*p*is realized in*M*.

*Proof:* Each successive condition is clearly stronger than the previous ones, so it suffices to prove the third condition from the first.

Assume that *M* has the first property, so every 1-type over a subset of *M* of size less than is realized in *M*. Let *A* ⊆ *M* have size less than . Let *p*(*x*) be a finitary or infinitary type over *A*. We may assume that the tuple of variables *x* is indexed by some cardinal number . Let *N* be an elementary extension of *M* in which *p*(*x*) is realized by some tuple *c*. We want to find a tuple *d* in *M* such that

or equivalently

.

The identity map on *A* is a partial elementary map from a subset of *N* to *M*, and we basically want to extend the domain of this partial elementary map from *A* to *Ac*.

Let denote the initial segment of the tuple consisting of for .

Inductively build an increasing sequence of partial elementary maps

as follows

- is the identity map from
*A*to*A* - Given , apply Lemma 2 above to extend the domain of this map to the element , yielding

Here we are using the fact that is a set of size less than , so that every 1-type over is realized in *M*, by assumption.

- If is a limit ordinal, let be the union of for . An easy exercise shows that this is a partial elementary map. (A function
*f*is a partial elementary map if and only if its restriction to every finite set is a partial elementary map.)

The union of this increasing chain of partial elementary maps will itself be a partial elementary map . Then

so that

In particular, tp(*f*(*c*)/*A*) = tp(*c*/*A*) = *p*. Since *f* mapped into *M*, we have realized the type *p* inside *M*.

QED.

**Lemma 3:** Let

be an elementary chain of models, of length , and let *N* be the union of this chain. Suppose that for each *i*, *M*_{i+1} is |*M _{i}*|

*Proof:* We use a back-and-forth argument. First of all, by Tarski-Vaught, each *M _{i}* is an elementary substructure of

*Claim:* Let *g* be a partial elementary map from some subset of *M*_{i} to some subset of *M*_{i}. Then we can extend *g* to a partial elementary map whose domain is all of *M*_{i} and whose range is contained in *M*_{i+1}. Alternatively, we can extend *g* to a partial elementary map whose domain is contained in *M*_{i+1} and whose range is exactly *M*_{i}.

*Proof:* Let *A* be the domain of *g*. Let *p(x)* be the type of *M*_{i} over *A*. If , then is \kappa^+-saturated. Since and *x* is a tuple of length at most , by the Theorem above we have that the pushforward type *g***p* (an infinitary type over ) is realized in . If *D* realizes this, then

.

In particular, the map sending coordinatewise to *D* is a partial elementary map, extending *g*. This map has domain exactly and has range contained in .

If we instead wanted the range to be exactly and the domain to be contained in , apply the same argument to . QED_{claim}.

Now build an increasing sequence of partial elementary maps

with each being a partial elementary map from some subset of to some subset of . Do so as follows:

- is the original map
*f* - For
*n*> 0 even, let be a partial elementary map from to a subset of extending . This is doable by the Claim. - For
*n*> 0 odd, let be a partial elementary map from a subset of onto extending . This is doable by the claim.

Let be the union of this increasing chain of partial elementary maps on subsets of *N*. Then is itself a partial elementary map. However, the even steps insure that the domain of is

while the odd steps insure that the range of is

.

In particular, is defined on all of *N*, and is surjective onto *N*. So is in fact an automorphism of *N*.

QED.

**Lemma 4:** Let *M* be a structure. We can find an elementary extension such that every type over *M* is realized in *N*, and every partial elementary map from a subset of *M* to *M* can be extended to an automorphism of *N*.

*Proof:* Build an increasing elementary chain of models

by inductively choosing *M*_{i+1} to be an |*M _{i}*|

QED.

**Theorem:** Let *M* be a structure, and be an infinite cardinal. Then *M* has an elementary extension which is both -saturated and -strongly homogeneous.

*Proof:* As in the proof that -saturated models exist, we may replace with and assume that is a regular cardinal.

Build an elementary chain as follows.

- is
*M* - is an elementary extension of such that every type over is realized in and every partial elementary map from a subset of to extends to an automorphism of . Such an elementary extension of exists by Lemma 4.
- If is a limit cardinal, take to be the union of the for .

Let *N* be the union of this chain. By Tarski-Vaught, the form an elementary chain and for each .

If *A* is a subset of *N* of size less than , then by regularity of , for some . Then every type over *A* is realized in , hence in *N*. So *N* is -saturated.

For -strong homogeneity, let *f* be a partial elementary map from *A* to *B* (subsets of *N*), with . By regularity of , there is some such that . By choice of , there is an automorphism of extending *f*.

Recursively define for (where ) as follows:

- is an automorphism of extending
*f*. This is doable by choice of . - For , is an automorphism of extending the automorphism of .
- For a limit ordinal, is the union of the increasing chain of for .

Then the form an increasing chain of automorphisms, each extending *f*, and is an automorphism of *N* extending *f*.

QED.

Recall that *M* is *saturated* if *M* is |*M*|-saturated.

**Theorem:** Suppose *M* and *N* are two saturated models of the same size, which are elementarily equivalent. Then *M* and *N* are isomorphic.

*Proof:* The type over *M* over the empty set is consistent with *N*, because *M* and *N* are elementarily equivalent. (The assertion that *M* and *N* are elementarily equivalent is equivalent to assertion that the empty map between Ø ⊆ *M* and Ø ⊆ *N* is a partial elementary map. Push forward tp(*M*/Ø) along this map.) Since *M* is a tuple of length at most |*N*|, the type of *M* is realized in *N*. This means that there is an elementary embedding of *M* into *N*. Similarly, we can embed *N* into *M*. With a bit more work (a back-and-forth argument) we can obtain an isomorphism between *M* and *N*. This is done as follows:

Let . Choose enumerations and of *M* and *N*, respectively. Build up by induction on a sequence of partial elementary maps from subsets of *M* to subsets of *N*. The 's will form an increasing chain, and will have domain and range of cardinality at most . Proceed as follows:

- will be the empty map. This is a partial elementary map because .
- If is a limit ordinal, define to be the union of the for . This will be a partial elementary map because its restriction to any finite subset is a partial elementary map. Its domain and range will have size at most , because the domain of is a union of -many sets of size at most .
- Given , define as follows. Since , the domain and range of have size at most , and therefore must not be all of or , respectively. In particular, every 1-type over the range of is realized in
*N*. Hence, by Lemma 2, we can extend to a partial elementary map on . Calling the resulting map , we now run the same argument on to extend to a partial elementary map whose range includes . Let be this new partial elementary map. The domain and range of are bigger than the domain and range of by at most two, so the condition on the cardinalities of the domain and range of are satisfied.

Let *f* be the union of all the . This is a partial elementary map from some subset of *M* to some subset of *N*. By the construction, is in the domain of and hence in the domain of *f*, for every . In particular, the domain of *f* contains all of *M*. Similarly, the range of *f* contains all of *N*. As a partial elementary map, *f* is an injection, so *f* is a bijection from *M* to *N*. Then *f* is an isomorphism from *M* to *N*.

QED.

**Corollary:** Let *T* be a complete theory. Then *T* has at most one saturated model, up to isomorphism, of each cardinality.

**Corollary:** Let *M* be a saturated structure. Then *M* is |*M*|-strongly homogeneous.

*Proof:* Let f : *A* -> *B* be a partial elementary map between two subsets of *M* of cardinality less than |*M*|. Let *L* be the language/signature of *M*, and let *L*(*A*) be the language obtained by adding a constant *c _{a}* for every element

We claim that *M*_{1} is saturated. If *S* ⊆ *M* is a subset of size less than |*M*|, then *S* ∪ *A* also has size less than |*M*|. An *L*-formula over *S* ∪ *A* is the same thing as an *L*(*A*)-formula over *S*, so a 1-type over *S* ∪ *A* in *M* is equivalent to a 1-type over *S* in *M*_{1}. In particular, the fact that all 1-types over *S* ∪ *A* is realized in *M* implies that all 1-types over *S* are realized in *M*_{1}. So *M*_{1} is saturated.

We can also make *M* into an *L*(*A*)-structure in another way, by interpreting *c*_{a} as *f*(*a*). Call this expansion *M*_{2}. By the same argument, *M*_{2} is saturated.

Moreover, *M*_{1} and *M*_{2} are elementarily equivalent, essentially because *f* is a partial elementary map. Indeed, for any *L*-formula ,

where the middle equivalence follows from the fact that *f* is an elementary map.

So *M*_{1} and *M*_{2} are elementarily equivalent saturated structures of the same size (since they have the same underlying set). By the Theorem, they must be isomorphic. Let be an isomorphism witnessing this. Then induces an isomorphism of the reducts to *L*, which are both *M*. So induces an automorphism of *M*. Moreover, to be a morphism of *L*(*A*)-structures, must send the interpretation of *c*_{a} in *M*_{1} to the interpretation of *c*_{a} in *M*_{2}. In other words, must send *a* to *f*(*a*). So extends *f*.

QED.

**Theorem:** Let be a countable list of structures in some countable language *L*. Let be a non-principal ultrafilter on . Then the ultraproduct is -saturated.

*Proof:* Since the language is countable, any type over a countable subset of *M* must consist of countably many formulas. So it suffices to show that any countable consistent partial type over *M* is realized in *M*. A countable partial type can be enumerated as . Each can be represented by some element

,

where each is a tuple in of the same length and sort as in .

The fact that this partial type is consistent means that for each *n*,

In particular, by Łoś's theorem, for "most" values of *j*,

.

Let

Then

and each (by Łoś's theorem). Let

,

and let . Then , because both and are in (the latter because is non-principal). Also, the form a descending sequence:

For each *j*, let *n(j)* be the largest *n* such that . A largest such *n* exists because if , then . Choose a singleton as follows:

- If
*n(j) = -1*, choose randomly. - Otherwise, , so

Let be an witnessing this. So

Let *c* be the class of

in the ultraproduct *M*.

Note that if , then , by definition of . Then by choice of , we have

,

and in particular

.

Consequently,

.

Since is "big" with respect to the ultraproduct , so is the even-bigger set

.

By Łoś's Theorem, it follows that

.

This holds for each *m*, so *c* satisfies our original partial type . In particular, the type is realized in *M*.

QED.

**Corollary:** Assume the continuum hypothesis. Then every consistent theory *T* in a countable language has a saturated model. *Proof:* Without loss of generality, *T* is complete. If the models of *T* are finite, then they are automatically -saturated for all , and there is nothing to show. So assume the models of *T* are infinite. Let *M* be some countable model. Let be a non-principal ultrafilter on , and let be the corresponding ultrapower. Then by the Theorem, is -saturated. But is a quotient of which has size . By the continuum hypothesis, this is at most , so

Therefore, *N* is |*N*|-saturated. Also, because *N* is an ultrapower of a model of *T*.

QED.

Some variant of these facts works in other cardinalities, though one has to be more careful about the choice of the ultrafilter. In particular, if one assumes the Generalized Continuum Hypothesis, one can show that every structure has a saturated elementary extension. (Somebody who knows about good ultrafilters should verify this.)

**Lemma 5:** Let *M* be a structure in a language *L*. Then there is an elementary extension such that every type over *M* is realized in *N*, and .

*Proof:* Let *N*_{0} be an elementary extension of *M* in which all types are realized (Lemma 1 above). A type *p*(*x*) over *M* is determined by which *L*(*M*) formulas are true of the variable *x*. There are only -many *L*(*M*)-formulas, so there are at most types over *M*. For each type over *M*, choose a realization in *N*_{0}. Let *S* be the set of all these realizations. Thus . By Downward Löwenheim–Skolem, we can find an elementary substructure containing , of size . Then *N* is an elementary extension of *M*,

and every type over *M* is realized in *S*, hence in *N*.

QED.

**Theorem:** Assume there exists a proper class of strongly inaccessible cardinals. Then every structure *M* has an elementary extension which is saturated.

*Proof:* Let be an inaccessible cardinal greater than the size of *M* and greater than the size of the language. Build an increasing elementary chain as follows:

- If is a limit ordinal, then

- Given , let be an elementary extension of in which all types over are realized, but . Such a model can be found by Lemma 5.

By induction on , we see that . The base case is by choice of . The limit ordinal case follows from the regularity of . The successor ordinal step follows from the fact that is a strong limit cardinal:

,

because

by induction.

Let *N* be the limit structure. Then *N* is -saturated, by the same arguments used in the theorems above (using the fact that is regular). Since *N* is the union of -many structures of size less than , *N* has size (or less). Therefore, *N* is |*N*|-saturated.

QED.

**Theorem:** Let *T* be a complete theory, and suppose *T* is stable. Then every model of *T* has a saturated elementary extension.

*Proof:* TODO. It's in Poizat's Model Theory book.

- Monster model ::: ::: :::