$\textstyle\smile$${M^{*}}$${N_{0}}$${N}$${M_{b}}$${M_{0}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  2. (2)

If $N_{i}$ is already defined, by construction

$\textstyle\smile$${M^{*}}$${N_{i}}$${M_{b}}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

As $\mathscr{A}$ is regular, it is also the case that

$\textstyle\smile$${N_{i}}$${N}$${M_{i}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Since $M_{i}\leq M_{i+1}\leq M$, again by regularity, there is $N_{i+1}\leq N$
such that

$\textstyle\smile$${N_{i}}$${N_{i+1}}$${N}$${M_{i}}$${M_{i+1}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  3. (3)

At limit stage $\alpha$, we have

$\textstyle\smile$${M^{*}}$${N_{0}}$${N_{1}}$${\cdots}$${N_{i}}$${N_{i+1}}$${\cdots}$${M_{b}}$${M_{0}}$${M_{1}}$${\cdots}$${M_{i}}$${M_{i+1}}$${\cdots}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

As $\mathscr{A}$ is continuous and $(M_{i})_{i<\alpha}$ is an increasing
continuous chain, letting $N_{\alpha}=\bigcup_{i<\alpha}N_{i}$, we get that

$\textstyle\smile$${M^{*}}$${N_{\alpha}}$${M_{b}}$${M_{\alpha}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

∎

##  3\. Sequential amalgamation

From a model-theoretic perspective, that the class of vector spaces over a
fixed (countable) field is uncountably categorical stems from the exchange
property of vectors and the fact that all vector spaces are direct sums of
1-dimensional spaces. In order to mimic this structure (or equivalently, the
structure of models with a pregeometry), we must first define the amalgam of
not only two models but of a possibly infinite sequence of models. We thus
devote this section to showing that under the assumptions of $\mathscr{A}$
being absolutely minimal, regular, and continuous, then sequential
amalgamation under $\mathscr{A}$ behaves as one would expect from the example
of direct sums.

######  Notation 3.1.

For an ordinal $\alpha$, we define the ordinal $s(\alpha)$ by:

  * •

$s(\alpha)=\alpha$ for limit $\alpha$

  * •

$s(\alpha)=\alpha+1$ otherwise

######  Definition 3.2.

Let $M_{b}\in K$, and let $(M_{i})_{i<\alpha}$ be a sequence of models such
for each $i<\alpha$, $M_{b}\leq M_{i}$. We say that $N$ is an
$\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ if there exists a
sequence of models $(N_{i})_{i<s(\alpha)}$ and $K$-embeddings
$(f_{i}:M_{i}\longrightarrow N_{i+1})_{i<\alpha}$ such that:

  1. (1)

$N_{0}=M_{b}$ and $N_{1}=f_{0}[M_{0}]$

  2. (2)

For each $i<\alpha$, $f_{i}[M_{b}]=M_{b}$ and $f_{i}[M_{i}]\leq N_{i+1}$

  3. (3)

$(N_{i})_{i<s(\alpha)}$ is a continuous resolution of $N$ i.e. it is an
increasing continuous chain with $N=\bigcup_{i<s(\alpha)}N_{i}$.

  4. (4)

For every $i\geq 1$, the following diagram is an $\mathscr{A}$-amalgam:

$\textstyle\smile$${N_{i}}$${N_{i+1}}$${M_{b}}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$

Paralleling the two-model case, we say that $N$ is an $\mathscr{A}$-amalgam by
inclusion of $(M_{i})_{i<\alpha}$ over $M_{b}$ if $N$ is an
$\mathscr{A}$-amalgam as above with each $f_{i}$ being an inclusion map
$\iota_{i}:M_{i}\hookrightarrow N_{i+1}$. When each $M_{i}\leq N$, we say that
$(M_{i})_{i<\alpha}$ is $\mathscr{A}$-subamalgamated over $M_{b}$ inside $N$
if there is some $N^{\prime}\leq N$ such that $N^{\prime}$ is an
$\mathscr{A}$-amalgam by inclusion.

In order to understand what properties of sequential amalgams is desirable for
our analysis, recall that any divisible group can be uniquely decomposed as a
direct sum of (copies of) the rationals and Prufer $p$-groups. Using this as a
guiding example, ideally the amalgamation of a sequence of models should be
independent from the order of amalgamation, and moreover it should be possible
to take subsets of a “basis” to construct smaller models. In order to prove
this claim (Theorem 3.14), we proceed by a number of lemmas:

######  Lemma 3.3.

If $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$, then
for any $\beta\leq\alpha$, there exists some $L\leq N$ such that:

  1. (1)

$L$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\beta}$ over $M_{b}$; and

  2. (2)

$N$ is an $\mathscr{A}$-amalgam of the sequence
$(L)^{\frown}(M_{i})_{\beta\leq i<\alpha}$ over $M_{b}$

###### Proof.

Let $(N_{i})_{i<s(\alpha)}$ be a continuous resolution of $N$ witnessing that
$N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ via the
maps $(f_{i}:M_{i}\longrightarrow N_{i+1})_{i<\alpha}$, and let
$L=\bigcup_{i<\beta}N_{i}$. Then the resolution $(N_{i})_{i<s(\beta)}$
witnesses that $L$ is the desired $\mathscr{A}$-amalgam, and moreover the
sequence $(L)^{\frown}(N_{i})_{\beta\leq i<s(\alpha)}$ witnesses that $N$ is
also an $\mathscr{A}$-amalgam of $(L)^{\frown}(M_{i})_{\beta\leq i<\alpha}$
over $M_{b}$ (via the maps $(\iota:L\longrightarrow
N)^{\frown}(f_{i})_{\beta\leq i<\alpha}$). ∎

######  Lemma 3.4.

Suppose that $\mathscr{A}$ is a notion of amalgamation which is regular and
continuous. If $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over
$M_{b}$ via the maps $(f_{i}:M_{i}\longrightarrow N)_{i<\alpha}$, then there
is some $L\leq N$ such that:

  * •

$L$ is an $\mathscr{A}$-amalgam of $(M_{i})_{1\leq i<\alpha}$ over $M_{b}$ via
the same maps; and

  * •

$N$ is an $\mathscr{A}$-amalgam of $L$ and $M_{0}$ over $M_{b}$ in the
following diagram:

$\textstyle\smile$${M_{0}}$${N}$${M_{b}}$${L}$$\textstyle\smile$$\scriptstyle{f_{0}}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

###### Proof.

Fix $(N_{i})_{i<s(\alpha)}$ a continuous resolution of $N$ witnessing that $N$
is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ via
$(f_{i})_{i<\alpha}$. Let us first construct the model $L$ as the union of an
increasing continuous chain $(L_{i})_{1\leq i<s(\alpha)}$, with the following
conditions:

  1. (1)

$L_{1}=f_{1}[M_{1}]$, and each $L_{i}\leq N_{i}$

  2. (2)

For limit $\delta$, $L_{\delta}=\bigcup_{1\leq i<\delta}L_{i}$

  3. (3)

For $i\geq 1$, the following diagram is an $\mathscr{A}$-amalgam:

$\textstyle\smile$${M_{i+1}}$${L_{i+1}}$${M_{b}}$${L_{i}}$$\textstyle\smile$$\scriptstyle{f_{i+1}}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  4. (4)

For $i\geq 1$, the following diagram is an $\mathscr{A}$-amalgam:

$\textstyle\smile$${L_{i}}$${N_{i}}$${M_{b}}$${M_{0}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{0}}$

For the successor step, recall that as $(N_{i})_{i<s(\alpha)}$ witnesses that
$N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$, in
particular for each $i<\alpha$, the following diagram is an
$\mathscr{A}$-amalgam:

$\textstyle\smile$${M_{i}}$${N_{i+1}}$${M_{b}}$${N_{i}}$$\textstyle\smile$$\scriptstyle{f_{i}}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Hence, as $L_{i}\leq N_{i}$ by assumption, by regularity there exists some
$L_{i+1}\leq N_{i+1}$ such that:

$\textstyle\smile$${M_{i}}$${L_{i+1}}$${N_{i+1}}$${M_{b}}$${L_{i}}$${N_{i}}$$\textstyle\smile$$\scriptstyle{f_{i}}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

It remains to show that (4) is satisfied. We note that combining the above
diagram and assuming (4) holds for $L_{i}$, we get the following diagram:

$\textstyle\smile$${M_{i+1}}$${L_{i+1}}$${N_{i+1}}$${M_{b}}$${L_{i}}$${N_{i}}$${M_{b}}$${M_{0}}$$\textstyle\smile$$\scriptstyle{f_{i}}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$id$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{0}}$

Applying regularity to the two commutative squares on the right, we see that
(4) is satisfied at the $i+1$ step:

$\textstyle\smile$${L_{i+1}}$${N_{i+1}}$${M_{b}}$${M_{0}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{0}}$

For the limit step, it suffices to check again that $L_{\delta}$ satisfies
(4). Since $L_{i}$ satisfies (4) by assumption for $i<\delta$, we have the
diagram:

$\textstyle\smile$${f_{0}[M_{0}]}$${N_{1}}$${N_{2}}$${\cdots}$${N_{i}}$${N_{i+1}}$${\cdots}$${M_{b}}$${L_{1}}$${L_{2}}$${\cdots}$${L_{i}}$${L_{i+1}}$${\cdots}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Hence by continuity (and invariance), we get that

$\textstyle\smile$${M_{0}}$${\bigcup_{i<\delta}N_{i}}$${N_{\delta}}$${M_{b}}$${\bigcup_{i<\delta}L_{i}}$${L_{\delta}}$$\textstyle\smile$$\scriptstyle{f_{0}}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$id$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$id$\textstyle\smile$$\scriptstyle{\iota}$

This completes the definition of $(L_{i})_{1\leq i<s(\alpha)}$. Note then that
this resolution of $L=\bigcup_{i<s(\alpha)}L_{i}$ is a witness to the fact
that $L$ is an $\mathscr{A}$-amalgam of $(M_{i})_{1\leq i<\alpha}$ over
$M_{b}$, and moreover the proof for (4) in the limit case also shows that
$N=\bigcup_{i<s(\alpha)}N_{i}$ is an $\mathscr{A}$-amalgam of $M_{0},L$ over
$M_{b}$, as desired. ∎

######  Corollary 3.5.

Suppose $\mathscr{A}$ is regular and continuous. If $N$ is an
$\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by inclusion, then
for any $0<j<\alpha$, there are $L_{1},L_{2}\leq N$ such that:

  * •

$L_{1}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<j}$ over $M_{b}$ by
inclusion

  * •

$L_{2}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{j<i<\alpha}$ over $M_{b}$ by
inclusion; and

  * •

$N$ is an $\mathscr{A}$-amalgam of $(L_{1},M_{j},L_{2})$ over $M_{b}$ by
inclusion

###### Proof.

That $L_{1}$ exists by Lemma 3.3 and $L_{2}$ exists by Lemma 3.4. ∎

######  Lemma 3.6.

Suppose $\mathscr{A}$ is a notion of amalgamation that is regular and
continuous. Let $N$ be an $\mathscr{A}$-amalgam of $M^{*},M^{\prime}$ over
$M_{b}$ by the following diagram:

$\textstyle\smile$${M^{*}}$${N}$${M_{b}}$${M^{\prime}}$$\textstyle\smile$$\scriptstyle{g}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

If $M^{\prime}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over
$M_{b}$ (via the $K$-embeddings $(f_{i}:M_{i}\longrightarrow M)_{i<\alpha}$),
then the sequence $(M^{*})^{\frown}(M_{i})_{i<\alpha}$ is
$\mathscr{A}$-subamalgamated over $M_{b}$ inside $N$.

###### Proof.

Since $M^{\prime}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over
$M_{b}$ via the maps $(f_{i})_{i<\alpha}$, there is a continuous resolution
$(M_{i}^{\prime})_{i<s(\alpha)}$ of $M^{\prime}$ such that
$M_{0}^{\prime}=M_{b},M_{1}^{\prime}=f_{0}[M_{0}]$ and for each $1\leq
i<\alpha$,

$\textstyle\smile$${M_{i}^{\prime}}$${M_{i+1}^{\prime}}$${M_{b}}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$

So let us define an increasing continuous chain $(N_{i})_{i<\beta}$ such that

  1. (1)

$\beta=\alpha+2$ iff $\alpha<\omega$; otherwise $\beta=s(\alpha)$

  2. (2)

$N_{0}=M_{b}$ and $N_{1}=g[M^{*}]$

  3. (3)

For limit $\delta$, $N_{\delta}=\bigcup_{i<\delta}N_{i}$

  4. (4)

For each $i<\omega$, $M_{i}^{\prime}\leq N_{i+1}\leq N$ and the commutative
squares in the following diagram are $\mathscr{A}$-amalgams:

$\textstyle\smile$${N_{1}}$${N_{i+2}}$${N}$${M_{b}}$${M_{i+1}^{\prime}}$${M}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$

  5. (5)

For each $i$ such that $\omega\leq i<\alpha$, $M_{i}^{\prime}\leq N_{i}\leq N$
and the commutative squares in the following diagram are
$\mathscr{A}$-amalgams:

$\textstyle\smile$${N_{1}}$${N_{i+1}}$${N}$${M_{b}}$${M_{i+1}^{\prime}}$${M}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$

We will define $N_{i}$ inductively to satisfy the above conditions:

  * •

For $i=0$ and $i=1$, the construction of $N_{i}$ is specfied as above.

  * •

For $i=2$, note that since $N_{1}=g[M^{*}]$ by definition, we have (by Side
Invariance) that

$\textstyle\smile$${g[M^{*}]}$${N_{1}}$${N}$${M_{0}^{\prime}}$${M_{b}}$${M}$$\textstyle\smile$id$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$id$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

As $M_{b}\leq M_{1}^{\prime}\leq M$, by regularity there exists some
$N_{2}\leq N$ such that

$\textstyle\smile$${N_{1}}$${N_{2}}$${N}$${M_{b}}$${M_{1}^{\prime}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  * •

If $1\leq i<\omega$, then by the inductive hypothesis, we have:

$\textstyle\smile$${N_{1}}$${N_{i+1}}$${N}$${M_{b}}$${M_{i}^{\prime}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

As $M_{i}^{\prime}\leq M_{i+1}^{\prime}\leq M$, again by regularity there is
some $N_{i+1}\leq N$ such that

$\textstyle\smile$${N_{1}}$${N_{i+1}}$${N_{i+2}}$${N}$${M_{b}}$${M_{i}^{\prime}}$${M_{i+1}^{\prime}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Furthermore, apply regularity to the two commutative squares on the left, we
also get:

$\textstyle\smile$${N_{1}}$${N_{i+2}}$${N}$${M_{b}}$${M_{i+1}^{\prime}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  * •

If $i=\omega$, then by the inductive hypothesis we have

$\textstyle\smile$${N_{1}}$${N_{2}}$${N_{3}}$${\cdots}$${N_{i+1}}$${N_{i+2}}$${\cdots}$${M_{b}}$${M_{1}^{\prime}}$${M_{2}^{\prime}}$${\cdots}$${M_{i}^{\prime}}$${M_{i+1}^{\prime}}$${\cdots}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Defining $N_{\omega}=\bigcup_{i<\omega}N_{i}$, by continuity we have that

$\textstyle\smile$${N_{1}}$${N_{\omega}}$${M_{b}}$${M_{\omega}^{\prime}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Now, since $N_{\omega}\leq N$, by regularity (specifically, the “moreover”
part of condition (3), see Definition 2.1), it is also true that

$\textstyle\smile$${N_{\omega}}$${N}$${M_{\omega}^{\prime}}$${M}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  * •

For successor and limit $i$’s beyond $\omega$, the construction is the same as
above except for the shifted indices.

Letting $N^{\prime}=\bigcup_{i<\beta}N_{i}$, it remains to show that
$N^{\prime}$ is an $\mathscr{A}$-amalgam of
$(M^{*})^{\frown}(M_{i})_{i<\alpha}$ over $M_{b}$ (via the maps
$(g)^{\frown}(f_{i})_{i<\alpha}$) i.e. that for each $i<\omega$ and $j$ such
that $\omega\leq j<\alpha$,

$\textstyle\smile$${N_{i+1}}$${N_{i+2}}$${N_{j}}$${N_{j+1}}$${M_{b}}$${M_{i}}$${M_{b}}$${M_{j}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{j}}$

For the $i<\omega$ case, recall that $(M_{i}^{\prime})_{i<\alpha}$ witnesses
that $M$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$, and
hence for each $i$,

$\textstyle\smile$${M_{i}^{\prime}}$${M_{i+1}^{\prime}}$${M_{b}}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$

Combining this with condition (4) above and the construction of $N_{i}$, we
get the diagram

$\textstyle\smile$${N_{1}}$${N_{i+1}}$${N_{i+2}}$${N}$${M_{b}}$${M_{i}^{\prime}}$${M_{i+1}^{\prime}}$${M}$${M_{b}}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$id$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{f_{i}}$

Note that apply regularity to the two commutative squares in the middle column
gives us the desired result. As the same argument applies to the case of
$\omega\leq j<\alpha$ with shifted indices, this completes the proof. ∎

######  Lemma 3.7.

Suppose $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i_{<}\alpha}$ over
$M_{b}$ via the maps $(f_{i}:M_{i}\longrightarrow N)_{i<\alpha}$. If
additionally $M_{0}$ is an $\mathscr{A}$-amalgam of $(L_{j})_{j<\beta}$ over
$M_{b}$ via the maps $(g_{j}:L_{j}\longrightarrow M_{0})_{j<\beta}$, then $N$
is an $\mathscr{A}$-amalgam of the concatenated sequence
$(L_{j}:j<\beta)^{\frown}(M_{i}:i<\alpha)$ over $M_{b}$.

###### Proof.

Fix $(M_{j}^{\prime})_{j<s(\beta)}$ a continuous resolution of $M_{0}$
witnessing that it is an $\mathscr{A}$-amalgam of $(L_{j})_{j<\alpha}$ over
$M_{b}$, and also fix $(N_{i})_{i<s(\alpha)}$ a continuous resolution of $N$
witnessing that it is an $\mathscr{A}$-amalgam of $(M_{i})_{i<s(\alpha)}$ over
$M_{b}$. Consider then the sequence
$S=(f_{0}[M_{j}^{\prime}])_{j<s(\beta)}^{\frown}(N_{i}^{\prime})_{1\leq
i<s(\alpha)}$: it is a continuous resolution of $N$ since
$N_{0}=f_{0}[M_{0}]=\bigcup_{j<s(\beta)}$. Since $f_{0}\upharpoonright
M_{j}^{\prime}$ is a $K$-isomorphism between $M_{j}^{\prime}$, and
$f_{0}[M_{j}^{\prime}]$, Invariance of $\mathscr{A}$ implies the desired
result. ∎

######  Lemma 3.8.

Suppose $\mathscr{A}$ is absolutely minimal. Let $N$ be an
$\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by inclusion, and
suppose that $N^{\prime}\geq N$, $N^{*}\leq N^{\prime}$, and each
$M_{i}\subseteq N^{*}$. Then $N\leq N^{*}$.

###### Proof.

By induction on $\alpha$:

  * •

If $\alpha=2$, then this is true by definition of $\mathscr{A}$ being
absolutely minimal.

  * •

Assuming the statement is true for $\alpha$. Given $N$ an
$\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha+1}$ over $M_{b}$ (by inclusion)
and $N^{\prime},N^{*}$ as above, let $N_{\alpha}\leq N$ be such that
$N_{\alpha}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$,
and hence by induction $N_{\alpha}\leq N^{*}$. But $N$ is an
$\mathscr{A}$-amalgam of $N_{\alpha},M_{\alpha}$ over $M_{b}$, and as
$M_{\alpha}\leq N^{*}$ by assumption, hence $N\leq N^{*}$ as $\mathscr{A}$ is
absolutely minimal.

  * •

For limit $\delta$, if $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\delta}$
over $M_{b}$, then fix $(N_{\alpha})_{\alpha<\delta}$ a continuous resolution
of $N$ witnessing that $N$ is an $\mathscr{A}$-amalgam. In particular, each
$N_{\alpha}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$.
Now, given $N^{\prime},N^{*}$ as above, by induction each $N_{\alpha}\leq
N^{*}$. Since $N=\bigcup_{\alpha<\delta}N_{\alpha}$, hence $N\subseteq N^{*}$.
Furthermore, as $N,N^{*}\leq N^{\prime}$, by Coherence we have that $N\leq
N^{*}$.

∎

######  Corollary 3.9.

Suppose $\mathscr{A}$ is absolutely minimal. If each $M_{i}\leq N$ and the
sequence $(M_{i})_{i<\alpha}$ is $\mathscr{A}$-subamalgamated over $M_{b}$
inside $N$, then there is a unique $N^{\prime}\leq N$ which is an
$\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$.

######  Notation 3.10.

If $(M_{i})_{i<\alpha}$ are such that each $M_{b}\leq M_{i}\leq N$ and the
sequence $(M_{i})_{i<\alpha}$ is $\mathscr{A}$-subamalgamated inside $N$ via
inclusion, then we denote the unique $\mathscr{A}$-amalgam inside $N$ by
$\bigoplus_{M_{b},i<\alpha}^{N}M_{i}$.

######  Lemma 3.11.

Suppose $\mathscr{A}$ is regular, continuous, and absolutely minimal. Let
$\alpha$ be a limit ordinal, and
$(M^{1}_{i})_{i\leq\alpha},(M^{2}_{i})_{i\leq\alpha},(N_{i})_{i\leq\alpha}$ be
increasing continuous chains such that for each $i<\alpha$, $N_{i}$ is an
$\mathscr{A}$-amalgam of $M^{1}_{i},M^{2}_{i}$ over $M_{b}$ by inclusion. Then
for any $i,j<\alpha$, there is $M_{ij}\leq N_{\text{max}(i,j)}$ which is an
$\mathscr{A}$-amalgam of $M^{1}_{i},M^{2}_{j}$ over $M_{b}$ by inclusion.
Moreover, we can choose the system of $M_{ij}$’s such that if $i$ is a limit
ordinal, then $M_{ij}=\bigcup_{k<i}M_{kj}$, and similarly if $j$ is a limit
ordinal.

###### Proof.

Let $M^{1}_{i},M^{2}_{i},N_{i}$ be as above, so that we have the diagram

$\textstyle\smile$${\vdots}$${\iddots}$${M^{2}_{1}}$${N_{1}}$${M^{2}_{0}}$${N_{0}}$${M_{b}}$${M^{1}_{0}}$${M^{1}_{1}}$${\cdots}$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$

where all the arrows are inclusions and all the commutative squares with a
vertex at $M_{b}$ are $\mathscr{A}$-amalgams. Letting $M_{ii}$ be defined as
$N_{i}$, we will define $M_{ij}$ by induction on $\text{max}(i,j)<\alpha$ such
that in addition to the requirements above, we have additionally that the
condition $(A(ij))$ holds when $i,j$ are not limits:

(A(i,j))
$\textstyle\smile$${M_{i-1,j}}$${M_{i,j}}$${M_{i-1,j-1}}$${M_{i,j-1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

(where $M_{-1,-1}=M_{b},M_{-1,j}=M^{2}_{j},M_{i,-1}=M^{1}_{i}$)

  * •

For $M_{01}$, note that $N_{1}$ is an $\mathscr{A}$-amalgam of
$M^{1}_{1},M^{2}_{1}$ over $M_{b}$ (by inclusion). As $M_{b}\leq M^{1}_{0}\leq
M^{1}_{1}$, by regularity there is $M_{01}\leq N_{1}$ such that

$\textstyle\smile$${M^{2}_{1}}$${M_{01}}$${N_{1}}$${M_{b}}$${M^{1}_{0}}$${M^{1}_{1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

$M_{10}$ is defined symmetrically.

  * •

If $M_{ij}$ is defined for all $i,j\leq\alpha$, then for any $i\leq\alpha$, by
regularity there is $M_{i,\alpha+1}$ such that

$\textstyle\smile$${M^{2}_{\alpha+1}}$${M_{i,\alpha+1}}$${N_{\alpha+1}}$${M_{b}}$${M^{1}_{i}}$${M^{1}_{\alpha+1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

It is straightforward to see that condition $(A(i,\alpha+1))$ by induction on
$i$ (and using regularity for the base case). We define $M_{\alpha+1,j}$ which
satisfies $(A(\alpha+1,j))$ by the symmetric argument. Finally, to see that
condition $(A(\alpha+1,\alpha+1))$ holds, note that by definition of
$M_{\alpha,\alpha+1}$, we have

$\textstyle\smile$${M^{2}_{\alpha+1}}$${M_{\alpha,\alpha+1}}$${N_{\alpha+1}}$${M_{b}}$${M^{1}_{\alpha}}$${M^{1}_{\alpha+1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Apply regularity to the commutative square on the right (and symmetry), we get
that

$\textstyle\smile$${M^{1}_{\alpha+1}}$${M_{\alpha+1,\alpha}}$${N_{\alpha+1}}$${M_{b}}$${N_{\alpha}}$${M_{\alpha,\alpha+1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

The $\mathscr{A}$-amalgam on the right shows that $(A(\alpha+1,\alpha+1))$ is
indeed satisfied.

  * •

If $\delta$ is a limit and $M_{ij}$ are defined for $i,j<\delta$, then by
regularity let $M_{i,\delta}\leq N_{\delta}$ be an $\mathscr{A}$-amalgam of
$M^{1}_{i},M^{2}_{\delta}$ over $M_{b}$. We need to show that:

######  Claim.

$M_{i,\delta}=\bigcup_{j<\delta}M_{ij}$

To prove the claim, note that since $(M^{2}_{j})_{j<\delta}$ is a continuous
resolution of $M^{2}_{\delta}$, by Lemma 2.13 there is a continuous resolution
$(M^{\prime}_{ij})_{j<\delta}$ of $M_{i,\delta}$ such that each
$M^{\prime}_{ij}$ is an $\mathscr{A}$-amalgam of $M^{1}_{i},M^{2}_{j}$ over
$M_{b}$. But then each $M^{\prime}_{ij},M_{ij}\leq N_{\delta}$, and as
$\mathscr{A}$ is absolutely minimal, by Lemma 2.7 we have that
$M_{ij}=M^{\prime}_{ij}$. This proves the claim. Additionally, this
construction implies that when $\gamma$ is also a limit, then
$M_{\gamma,\delta}=\bigcup_{i<\gamma}M_{i,\delta}$.

Symmetrically, we define $M_{\delta,j}$. To finish the construction, we need
to show that:

$\bigcup_{i<\delta}M_{i,\delta}=N_{\delta}=\bigcup_{j<\delta}M_{\delta,j}$

But this is true since each $N_{\alpha}\leq
M_{\alpha,\delta},M_{\delta,\alpha}\leq N_{\delta}$, and
$\bigcup_{\alpha<\delta}N_{\alpha}=N_{\delta}$.

∎

######  Corollary 3.12.

Suppose $\mathscr{A}$ is absolutely minimal, regular, and continuous. If
$\alpha$ is a limit ordinal, and
$(M^{1}_{i})_{i\leq\alpha},(M^{2}_{i})_{i\leq\alpha},(N_{i})_{i\leq\alpha}$
are increasing continuous chains such that for each $i<\alpha$, $N_{i}$ is an
$\mathscr{A}$-amalgam of $M^{1}_{i},M^{2}_{i}$ over $M_{b}$ by inclusion, then
$N_{\alpha}$ is an $\mathscr{A}$-amalgam of $M^{1}_{\alpha},M^{2}_{\alpha}$
over $M_{b}$.

###### Proof.

For $i,j<\alpha$, let $M_{ij}\leq N_{\alpha}$ be constructed as in the above
Lemma, and for each $i<\alpha$, let $M_{i,\alpha}=\bigcup_{j<\alpha}M_{ij}$.

######  Claim.

$\textstyle\smile$${M^{2}_{\alpha}}$${M_{0,\alpha}}$${M_{b}}$${M^{1}_{0}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

###### Proof.

Note that by condition $(A(0,j))$ for each $j<\alpha$, we have that

$\textstyle\smile$${M^{1}_{0}}$${M_{0,0}}$${M_{0,1}}$${\cdots}$${M_{0,i}}$${M_{0,i+1}}$${\cdots}$${M_{b}}$${M^{2}_{0}}$${M^{2}_{1}}$${\cdots}$${M^{2}_{i}}$${M^{2}_{i+1}}$${\cdots}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Hence the claim holds as $\mathscr{A}$ is continuous. ∎

######  Claim.

For each $i<\alpha$,

$\textstyle\smile$${M_{i,\alpha}}$${M_{i+1,\alpha}}$${M^{1}_{i}}$${M^{1}_{i+1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

This holds by the same argument.

######  Claim.

For limit $\delta<\alpha$, $M_{\delta,\alpha}=\bigcup_{i<\delta}M_{i,\alpha}$,
and moreover

$\textstyle\smile$${M^{2}_{\alpha}}$${M_{\delta,\alpha}}$${M_{b}}$${M^{1}_{\delta}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

###### Proof.

Note that

$\bigcup_{i<\delta}M_{i,\alpha}=\bigcup_{i<\delta}\bigcup_{j<\alpha}M_{ij}=\bigcup_{j<\alpha}M_{\delta,j}=M_{\delta,\alpha}$

For the moreover part, combining the above claims and induction along
$\delta$, we get that

$\textstyle\smile$${M^{2}_{\alpha}}$${M_{0,\alpha}}$${M_{1,\alpha}}$${\cdots}$${M_{i,\alpha}}$${M_{i+1,\alpha}}$${\cdots}$${M_{b}}$${M^{1}_{0}}$${M^{1}_{1}}$${\cdots}$${M^{1}_{i}}$${M^{1}_{i+1}}$${\cdots}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

As $\mathscr{A}$ is continuous, hence
$\bigcup_{i<\delta}M_{i,\alpha}=M_{\delta,\alpha}$ is an $\mathscr{A}$-amalgam
of $M^{2}_{\alpha},M^{1}_{\delta}$ over $M_{b}$. ∎

Combining the above claims, we get the diagram (for all $i<\alpha$)

$\textstyle\smile$${M^{2}_{\alpha}}$${M_{0,\alpha}}$${M_{1,\alpha}}$${\cdots}$${M_{i,\alpha}}$${M_{i+1,\alpha}}$${\cdots}$${M_{b}}$${M^{1}_{0}}$${M^{1}_{1}}$${\cdots}$${M^{1}_{i}}$${M^{1}_{i+1}}$${\cdots}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

As $\mathscr{A}$ is continuous, hence $\bigcup_{i<\alpha}M_{i,\alpha}$ is an
$\mathscr{A}$-amalgam of $M^{1}_{\alpha},M^{2}_{\alpha}$ over $M_{b}$. But
since for any $i<j<\alpha$, $N_{i}\leq M_{ij},M_{ji}\leq N_{j}$, we have that
$N_{\alpha}=\bigcup_{i<\alpha}M_{i,\alpha}$. This completes the proof. ∎

######  Theorem 3.13.

Suppose $\mathscr{A}$ is absolutely minimal, regular, and continuous. Let $N$
be an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by inclusion.
Then for any subsequence $S\subseteq\alpha$, there is some $M_{S}\leq N$ which
is an $\mathscr{A}$-amalgam of $(M_{S(j)})_{j<|S|}$ over $M_{b}$ by inclusion.
Moreover, if $\bar{S}$ is the complement of $S$ in $\alpha$ (and considered as
an increasing sequence), then there is $M_{\bar{S}}$ such that additionally,
$N$ is an $\mathscr{A}$-amalgam of $M_{S},M_{\bar{S}}$ over $M_{b}$ by
inclusion.

###### Proof.

We will proceed by induction on the length of $\alpha$:

  * •

When $\alpha=2$, this is trivial.

  * •

Assume the claim holds for $\alpha$. Given $N$ an $\mathscr{A}$-amalgam of
$(M_{i})_{i<\alpha+1}$ over $M_{b}$, suppose that $S$ is a subsequence of
$\alpha+1$. This breaks down into three cases:

    1. (1)

If $S=\\{\alpha\\}$, then the case is trivial.

    2. (2)

If $S\subseteq\alpha$, then consider $N_{\alpha}\leq N$ which is an
$\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ (as guaranteed by
Lemma 3.3): by the inductive hypothesis, $M_{S}\leq N_{\alpha}$ exists, and so
does $M_{\bar{S}_{\alpha}}$, where $\bar{S}_{\alpha}$ is the complement of $S$
w.r.t. $\alpha$. Now, since $N$ is an $\mathscr{A}$-amalgam of $N_{\alpha}$
and $M_{\alpha}$ over $M_{b}$, we get the $\mathscr{A}$-amalgams

$\textstyle\smile$${M_{S}}$${N_{\alpha}}$${N_{\alpha}}$${N}$${M_{b}}$${M_{\bar{S}_{\alpha}}}$${M_{b}}$${M_{\alpha}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

By Lemma 2.6, hence there is some $M_{\bar{S}}$ such that:

       * –

$M_{\bar{S}}$ is an $\mathscr{A}$-amalgam of $M_{\bar{S}_{\alpha}},M_{\alpha}$
over $M_{b}$; and

       * –

$N$ is an $\mathscr{A}$-amalgam of $M_{S},M_{\bar{S}}$ over $M_{b}$

Furthermore, by Lemma 3.7, $M_{\bar{S}}$ is also an $\mathscr{A}$-amalgam of
$(M_{j}:j<|\bar{S}_{\alpha}|)^{\frown}(M_{\alpha})$ over $M_{b}$.

    3. (3)

If $S\ni\alpha$ and $S\cap\alpha\neq\varnothing$, then $\bar{S}$ satisfies the
above case (2), so the same construction gives the required submodels.

  * •

Let $\delta$ be a limit, and suppose the claim holds for all $\alpha<\delta$.
Given $N$ an $\mathscr{A}$-amalgam of $(M_{i})_{i<\delta}$ over $M_{b}$ by
inclusion, let $(N_{i})_{i<\delta}$ be a continuous resolution of $N$ such
that each $N_{i}$ is an $\mathscr{A}$-amalgam of $(M_{j})_{j<i}$ over $M_{b}$.
Now, if $S$ is a subsequence of $\delta$, denote
$S_{\alpha}:=S\upharpoonright\alpha$ and
$\bar{S}_{\alpha}:=\bar{S}\upharpoonright\alpha$. Note then that for each
$\alpha$, $\bar{S}_{\alpha}$ is the complement of $S_{\alpha}$ relative to
$\alpha$, and hence the inductive hypothesis implies that there are models
$M^{\alpha}_{S_{\alpha}},M^{\alpha}_{\bar{S}_{\alpha}}\leq N_{\alpha}$ such
that:

    * –

$M^{\alpha}_{S_{\alpha}}$ is an $\mathscr{A}$-amalgam of
$(M_{S(j)})_{j<|S_{\alpha}|}$ over $M_{b}$

    * –

$M^{\alpha}_{\bar{S}_{\alpha}}$ is an $\mathscr{A}$-amalgam of
$(M_{\bar{S}(j)})_{j<|\bar{S}_{\alpha}|}$ over $M_{b}$

    * –

$N_{\alpha}$ is an $\mathscr{A}$-amalgam of
$M^{\alpha}_{S_{\alpha}},M^{\alpha}_{\bar{S}_{\alpha}}$ over $M_{b}$

Moreover, by Lemma 3.8, $M^{\alpha}_{S_{\alpha}}$ is the unique
$\mathscr{A}$-amalgam of $(M_{S(j)})_{j<|S|}$ over $M_{b}$ inside
$N_{\delta}$, and similarly for $M^{\alpha}_{\bar{S}_{\alpha}}$. Hence we will
drop the superscript, and define
$M_{S}:=\bigcup_{\alpha<\delta}M_{S_{\alpha}},M_{\bar{S}}:=\bigcup_{\alpha<\delta}M_{\bar{S}_{\alpha}}$.
Note then that the chains
$(M_{S_{\alpha}})_{\alpha\leq\delta},(M_{\bar{S}_{\alpha}})_{\alpha\leq\delta},(N_{\alpha})_{\alpha\leq\delta}$
satisfies the hypothesis of Corollary 3.12 above, and hence $N_{\delta}$ is an
$\mathscr{A}$-amalgam of $M_{S},M_{\bar{S}}$ over $M_{b}$. Moreover, the
continuous resolution $(M_{S_{\alpha}})_{\alpha<\delta}$ witnesses that
$M_{S}$ is an $\mathscr{A}$-amalgam of $(M_{S(j)})_{j<|S|}$ over $M_{b}$ as
desired.

∎

######  Remark.

It should be noted that if $K$ is assumed to be an AEC rather than a weak AEC
(i.e. $K$ has Smoothness), then the proof of the above theorem can be
simplified considerably: if $N$ is an $\mathscr{A}$-amalgam of
$(M_{i})_{i<\alpha}$ over $M_{b}$ by inclusion and $S$ is a subsequence of
$\alpha$, then the $\mathscr{A}$-amalgam of $(M_{S(i)})_{i<\text{otp}(S)}$
over $M_{b}$ can be easily defined by induction. This works even at limit
stages when $K$ is assumed to have Smoothness; otherwise, the above argument
seems to be necessary.

######  Theorem 3.14.

Suppose $\mathscr{A}$ is absolutely minimal, regular, and continuous. Let
$\alpha\geq 2$ be an ordinal, and $\sigma:|\alpha|\longrightarrow\alpha$ be
any enumeration of $\alpha$. Then $N$ is an $\mathscr{A}$-amalgam of
$(M_{i})_{i<\alpha}$ over $M_{b}$ by inclusion iff $N$ is also an
$\mathscr{A}$-amalgam of $(M_{\sigma(j)})_{j<|\alpha|}$.

###### Proof.

Let $N$ be an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by
inclusion. We proceed by induction on $\alpha$:

  * •

When $\alpha=2$, this is just Lemma 2.9.

  * •

Suppose the claim holds for $n$, which is finite. If $\sigma$ is an
enumeration of $n+1$, then $\sigma$ is a permutation of $n+1$. There are two
cases to consider:

    * –

If $\sigma(n)=n$, then $\sigma\upharpoonright n$ is a permutation of $n$, and
the claim follows from the inductive hypothesis.

    * –

Otherwise, let $m=\sigma(n)<n$. By Corollary 3.5, there are models
$N_{1},N_{2}\leq N$ such that:

      * *

$N_{1}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<m}$ over $M_{b}$

      * *

$N_{2}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{m<i\leq n}$ over $M_{b}$

      * *

$N$ is an $\mathscr{A}$-amalgam of $(N_{1},M_{m},N_{2})$ over $M_{b}$

But then by Lemma 2.9, $N$ is also an $\mathscr{A}$-amalgam of
$(N_{1},N_{2},M_{m})$ over $M_{b}$. Now, if $N^{\prime}\leq N$ is an
$\mathscr{A}$-amalgam of $N_{1},N_{2}$ over $M_{b}$, then by Lemma 3.7 and
3.6, $N^{\prime}$ is an $\mathscr{A}$-amalgam of $(M_{j})_{j\neq m}$ over
$M_{b}$. Since $\sigma(n)=m$ and $\sigma$ is a permutation, (by re-indexing)
the inductive hypothesis implies that $N^{\prime}$ is also an
$\mathscr{A}$-amalgam of $(M_{\sigma(i)})_{i<n}$ over $M_{b}$, and hence $N$
is an $\mathscr{A}$-amalgam of $(M_{\sigma(i)})_{i<n+1}$ over $M_{b}$.

  * •

Suppose the claim holds for an infinite $\alpha$, and so
$|\alpha|=|\alpha+1|$. Given $\sigma:|\alpha|\longrightarrow\alpha+1$ an
enumeration, there is some $\beta<|\alpha|$ such that $\sigma(\beta)=\alpha$.
Let $S$ be the subsequence of $\alpha$ such that $\text{ran }S=\text{ran
}\sigma\upharpoonright\beta$, and let $\bar{S}$ be its complement in $\alpha$,
so in particular $\bar{S}=S^{\prime\frown}\alpha$ for some subsequence
$S^{\prime}$ of $\alpha$. Now, since $N$ is an $\mathscr{A}$-amalgam of
$(M_{i})_{i<\alpha+1}$ over $M_{b}$, there is an $N^{*}\leq N$ such that
$N^{*}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$, and
$N$ is an $\mathscr{A}$-amalgam of $N^{*},M_{\alpha}$ over $M_{b}$. But since
$S$ is a subsequence of $\alpha$ and $S^{\prime}$ is its complement w.r.t.
$\alpha$, by Theorem 3.13 there are models $N_{S},N_{S^{\prime}}\leq N^{*}$
such that

    * –

$N_{S}$ is an $\mathscr{A}$-amalgam of $(M_{S(i)})_{i<|S|}$ over $M_{b}$

    * –

$N_{S^{\prime}}$ is an $\mathscr{A}$-amalgam of
$(M_{S^{\prime}(i)})_{i<|S^{\prime}|}$ over $M_{b}$

    * –

$N^{*}$ is an $\mathscr{A}$-amalgam of $N_{S},N_{S^{\prime}}$ over $M_{b}$

Furthermore, since $S,S^{\prime}$ are subsequences of $\alpha$,
$\text{otp}(S),\text{otp}(S^{\prime})\leq\alpha$, and so by the inductive
hypothesis $N_{S}$ is also an $\mathscr{A}$-amalgam of
$(M_{\sigma(i)})_{i<\beta}$ over $M_{b}$. Similarly, $N_{S^{\prime}}$ is an
$\mathscr{A}$-amalgam of $(M_{\sigma(i)})_{\beta<i<|\alpha|}$. Moreover, $N$
is also an $\mathscr{A}$-amalgam of $(N_{S},M_{\alpha},N_{S}^{\prime})$ over
$M_{b}$ by Lemma 2.9, and so by Lemma 3.7 and 3.6, $N$ is indeed an
$\mathscr{A}$-amalgam of $(M_{\sigma(i)})_{i<|\alpha|}$ over $M_{b}$. We also
need to show that if $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<|\alpha|}$
over $M_{b}$, then $N$ is also an $\mathscr{A}$-amalgam of
$(M_{\sigma^{-1}(j)})_{j<\alpha+1}$. Again letting $\beta$ be such that
$\sigma(\alpha)=\beta$, by Lemma 3.5 there are models $N^{1},N^{2}\leq N$ such
that

    * –

$N^{1}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\beta}$ over $M_{b}$

    * –

$N^{2}$ is an $\mathscr{A}$-amalgam of $(M_{i})_{\beta<i<|\alpha|}$ over
$M_{b}$

    * –

$N$ is an $\mathscr{A}$-amalgam of $(N^{1},M_{\beta},N^{2})$ over $M_{b}$

By Lemma 2.9 again, we see that $N$ is also an $\mathscr{A}$-amalgam of
$(N^{1},N^{2},M_{\beta})$ over $M_{b}$. If $N^{\prime}\leq N$ is such that
$N^{\prime}$ is an $\mathscr{A}$-amalgam of $N^{1},N^{2}$ over $M_{b}$, then
by the inductive hypothesis $N^{\prime}$ is also an $\mathscr{A}$-amalgam of
$(M_{\sigma^{-1}(j)})_{j<\alpha}$ over $M_{b}$. By Lemma 3.7, hence $N$ is an
$\mathscr{A}$-amalgam of $(M_{\sigma^{-1}(j)})_{j<\alpha+1}$ over $M_{b}$.

  * •

Suppose $\alpha$ is a limit ordinal, and that the claim holds for all
$\beta<\alpha$. As $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$
over $M_{b}$, let $(N_{\beta})_{\beta<\alpha}$ be a continuous resolution of
$N$ such that each $N_{\beta}$ is an $\mathscr{A}$-amalgam of
$(M_{i})_{i<\beta}$ over $M_{b}$. Now, given
$\sigma:|\alpha|\longrightarrow\alpha$ an enumeration, for $j<|\alpha|$ let
$\sigma_{j}:=\sigma\upharpoonright j$, and let $S_{j}$ be a subsequence of
$\alpha$ such that $\text{ran }S_{j}=\text{ran }\sigma_{j}$ i.e. $S_{j}$ is
the set enumerated by $\sigma_{j}$ but re-indexed by the ordinal ordering.
Note that since each $S_{j}$ is a subsequence of $\alpha$, by Theorem 3.13
there is $N_{S_{j}}\leq N$ which is an $\mathscr{A}$-amalgam of
$(M_{S_{j}(i)})_{i<\text{otp}(S_{j})}$ over $M_{b}$. Furthermore, since each
$|S_{j}|<|\alpha|$, $\text{otp}(S_{j})<\alpha$, and hence by the inductive
hypothesis $N_{S_{j}}$ is also an $\mathscr{A}$-amalgam of
$(M_{\sigma_{j}(i)})_{i<j}$ over $M_{b}$. Letting
$N^{\prime}=\bigcup_{j<|\alpha|}N_{S_{j}}$, this implies that $N^{\prime}$ is
an $\mathscr{A}$-amalgam of $(M_{\sigma(i)})_{i<|\alpha|}$ over $M_{b}$.

######  Claim.

$N^{\prime}=N$

###### Proof.

Since $(N_{i})_{i<\alpha}$ is a continuous resolution of $N$, it suffices to
show that each $N_{i}\subseteq N^{\prime}$. Now, for each $i<\alpha$, let
$\zeta_{i}$ be a subsequence of $\sigma$ such that $\text{ran }\zeta_{i}=i$,
and so by the inductive hypothesis $N_{i}$ is an $\mathscr{A}$-amalgam of
$(M_{\zeta_{i}(j)})_{j<\text{otp}(\zeta_{i})}$ over $M_{b}$. But by Theorem
3.13 there is $N^{\prime}_{i}\leq N^{\prime}$ which is an
$\mathscr{A}$-amalgam of $(M_{\zeta_{i}(j)})_{j<\text{otp}(\zeta_{i})}$ over
$M_{b}$, and as $\mathscr{A}$ is absolutely minimal hence
$N^{\prime}_{i}=N_{i}$. This proves the claim. ∎

It remains to show, that when $\alpha$ is not an initial ordinal, that if $N$
is an $\mathscr{A}$-amalgam of $(M_{i})_{i<|\alpha|}$ over $M_{b}$, then it is
also an $\mathscr{A}$-amalgam of $(M_{\sigma^{-1}(j)})_{j<\alpha}$. However,
we note that the argument analogous to the one given above also works here,
and hence the claim is proven for $\alpha$.

∎

Given Theorems 3.13 and 3.14, we see that when $\mathscr{A}$ is absolutely
minimal, regular, and continuous, then $\mathscr{A}$-amalgamation of models
indexed by a sequence is independent of the ordering, and hence can be
considered as being indexed by a set. Moreover, if $N$ is an
$\mathscr{A}$-amalgam of $(M_{i})_{i\in Y}$ over $M_{b}$ by inclusion, then
for any $X\subseteq Y$, there is $N_{X}\leq N$ which is an
$\mathscr{A}$-amalgam of $(M_{i})_{i\in X}$ over $M_{b}$.

Before moving on from sequential amalgamation, let us note that when
$\mathscr{A}$ is additionally assumed to admit decomposition, this actually
allows a model to be decomposed as the $\mathscr{A}$-amalgam of a sequence of
small models:

######  Lemma 3.15.

Suppose $\mathscr{A}$ is a notion of amalgamation which is regular and admits
decomposition. Then for any $M_{b}\lneq N$, there exists an ordinal
$\alpha<|N|^{+}$ and a sequence of models $(M_{i})_{i<\alpha}$ such that:

  * •

For every $i<\alpha$, $M_{b}\lneq M_{i}\leq N$ and
$|M_{i}|=\text{LS}(K)+|M_{b}|$

  * •

$N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by
inclusion.

###### Proof.

Let $\lambda=|N|^{+}$ and $\mu=|M_{b}|+\text{LS}(K)$. We will try to define
two sequences of models, $(M_{i})_{i<\lambda}$ and $(N_{i})_{i<\lambda}$, such
that:

  1. (1)

For each $i$, $M_{b}\leq M_{i}\leq N$, $N_{i}\leq N$, and $|M_{i}|=\mu$

  2. (2)

$(N_{i})_{i<\lambda}$ is an increasing continuous chain with $N_{0}=M_{b}$ and
$N_{1}=M_{0}$

  3. (3)

For every $i\geq 1$, the following diagram is an $\mathscr{A}$-diagram:

$\textstyle\smile$${N_{i}}$${N_{i+1}}$${M_{b}}$${M_{i}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Proceeding inductively:

  * •

For $i=0$, let $M_{0}$ be any model such that $M_{b}\lneq M_{0}\leq N$ and
$|M_{0}|=\mu$, and let $N_{0}=M_{b},N_{1}=M_{0}$.

  * •

Suppose inductively that $M_{i},N_{i+1}$ has been defined to satisfy (3).
Since $N_{i+1}\leq N$, either $N_{i+1}=N$ or $N_{i+1}\lneq N$. In the former
case, we terminate the inductive construction; otherwise, since $\mathscr{A}$
admits decomposition, there is some $M_{i+1}^{\prime}$ such that

$\textstyle\smile$${N_{i+1}}$${N}$${M_{b}}$${M_{i+1}^{\prime}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Note that as $\mathscr{A}$ is minimal, $M_{i+1}^{\prime}-N_{i+1}$ must be
nonempty as otherwise $N_{i+1}=N$. So let $M_{i+1}$ be any model of
cardinality $\mu$ such that $M_{b}\lneq M_{i+1}\leq M_{i+1}^{\prime}$ and
$M_{i+1}-N_{i+1}$ is nonempty. Then, as $M_{b}\leq M_{i+1}\leq
M_{i+1}^{\prime}$, by regularity there exists some $N_{i+2}$ such that

$\textstyle\smile$${N_{i+1}}$${N_{i+2}}$${N}$${M_{b}}$${M_{i+1}}$${M_{i+1}^{\prime}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

  * •

For limit $\delta$, let $N_{\delta}=\bigcup_{i<\delta}N_{i}$. If
$N=N_{\delta}$, then the construction terminates; otherwise, $M_{\delta}$ and
$N_{\delta+1}$ can be defined by the same procedure as in the successor case.

Note that by construction, each $N_{i}\lneq N_{i+1}\leq N$, and as
$\lambda=|N|^{+}$, hence the above procedure must terminate at some ordinal
$\alpha<\lambda$. In that case, $(N_{i})_{i<s(\alpha)}$ witnesses the fact
that $N$ is an $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by
inclusion. ∎

One last but important property of the direct sum in vector spaces and
divisible groups is that under any “basis” decomposition, any element is
contained within the “span” (or amalgam) of finitely many basis elements.
Whilst this is clearly true in the two examples because such algebraic objects
are finitary, in the present context we are also interested in classes which
are infinitary but not unboundedly; analogously, there are interesting classes
which are $\mathscr{L}_{\kappa,\omega}$ classes rather than just a
$\mathscr{L}_{\infty,\omega}$ class. To this end, we will define a cardinal
$\mu(K)$ by:

######  Definition 3.16.

Suppose that $\mathscr{A}$ is a notion of amalgamation which is regular,
continuous, absolutely minimal, and admits decomposition.

  1. (1)

For $M\in K$ and $a\in M$, we define $\mu(a,M)$ to be the least cardinal $\mu$
such that: for any $M_{b}\leq M$ and any sequence $(M_{i})_{i<\alpha}$ such
that $M$ is the $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over $M_{b}$ by
inclusion, there is a subsequence $S\subseteq\alpha$ with $|S|<\mu$ such that
$a\in\bigoplus^{M}_{M_{b},j\in S}M_{j}$.

  2. (2)

We define $\mu(M):=\text{sup}\\{\mu(a,M):a\in M\\}$

  3. (3)

We define $\mu(K):=\text{sup}\\{\mu(M):M\in K\\}$ if it exists, or
$\mu(K)=\infty$ otherwise.

  4. (4)

If $\mu(K)<\infty$, then we define $\mu_{r}(K)$ to be the least regular
cardinal $\geq\mu(K)$.

######  Remark.

Strictly speaking, $\mu(K)$ should be considered as $\mu^{\mathscr{A}}(K)$
since the definition depends on $\mathscr{A}$ and different notions of
amalgamation might give rise to different values of $\mu(K)$. However, since
in this paper we will always be considering a class $K$ with a fixed notion of
amalgamation $\mathscr{A}$, we have chosen to suppress the extra notation.

##  4\. An Independence Relation defined from $\mathscr{A}$

In the elementary class of algebraically closed fields with characteristic 0,
the forking relationship can be easily understood in terms of transcendental
degrees: $\text{gtp}(\bar{a}/F_{1})$ does not fork over $F_{0}$ iff
$\text{td}(\bar{a}/F_{1})=\text{td}(\bar{a}/F_{0})$. Since this is essentially
a characterization of forking using the concept of bases, we would expect that
a suitably well-behaved notion of amalgamation would also give rise to a
forking-like independence relation. To that end, we define:

######  Definition 4.1.

Suppose $K$ is an AEC, $\mathscr{A}$ is a notion of amalgamation in $K$. We
define a notion of $\mathscr{A}$-independence, denoted by
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$, as follows: if $M\leq N$ and
$A,B\subseteq N$, then $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ if there exists models
$M_{1},M_{2}$ with $M\leq M_{1},M_{2}\leq N$ such that $A\subseteq
M_{1},B\subseteq M_{2}$, and $M_{1},M_{2}$ are $\mathscr{A}$-subamalgamated
inside $N$ over $M$ i.e. there is some $N^{\prime}\leq N$ such that

$\textstyle\smile$${M_{2}}$${N^{\prime}}$${N}$${M}$${M_{1}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

In such a case, we say that the pair $(M_{1},M_{2})$ is a witness to
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$.

Our goal here is to establish the conditions necessary for
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ to behave as forking for stationary
types in a simple first order theories: To that end, we need to establish that
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ has the defining properties of
forking:

  * •

Invariance

  * •

Top monotonicity (i.e. forking does not depend on the ambient model)

  * •

Right monotonicity

  * •

Base monotonicity

  * •

Symmetry

  * •

Transitivity

  * •

Existence of nonforking extensions

  * •

Continuity

  * •

$\kappa$-ary character for some cardinal $\kappa$

  * •

Uniqueness of nonforking extensions

We will show that these properties hold through a series of propositions.

######  Proposition 4.2 (Top Monotonicity).

Let $\mathscr{A}$ be a notion of amalgamation.

  1. (1)

If $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ and $N^{\prime}\geq N$, then
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N^{\prime}}B$

  2. (2)

Suppose that $K$ admits finite intersection and $\mathscr{A}$ is regular,
minimal. If $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ and $N^{\prime}\leq N$ is
such that $A,B,M\subseteq N^{\prime}$, then $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N^{\prime}}B$

###### Proof.

That (1) is true is straightforward from the definition of
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$. For (2), let $(M_{1},M_{2})$ witness
that $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$; as $K$ has FI and $M\leq
M_{1},M_{2},N^{\prime}\leq N$, both $M_{1}\cap N^{\prime}$ and $M_{2}\cap
N^{\prime}$ are models of $K$, and by regularity $M_{1}\cap
N^{\prime},M_{2}\cap N^{\prime}$ are $\mathscr{A}$-subamalgamated over $M$
inside $N$. Since $K$ admits finite intersection and $\mathscr{A}$ is minimal,
hence $\mathscr{A}$ is absolutely minimal by Lemma 2.11, and so in particular
$(M_{1}\cap N^{\prime})\oplus_{M}^{N}(M_{2}\cap N^{\prime})\leq N^{\prime}$.
Hence $(M_{1}\cap N^{\prime},M_{2}\cap N^{\prime})$ is a witness to
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N^{\prime}}B$. ∎

Some straightforward observations which follow from the definition of
$\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ are:

######  Proposition 4.3.

Let $\mathscr{A}$ be a notion of amalgamation

  1. (1)

(Existence) For any $M\leq N$ and $A\subseteq N$, $A\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}M$.

  2. (2)

(Symmetry) $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ implies
$B\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}A$.

  3. (3)

(Right Monotonicity) If $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ and $B^{\prime}\subseteq B$,
then $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B^{\prime}$.

  4. (4)

(Right Normality) $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ iff $A\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}(B\cup M)$

######  Proposition 4.4 (Base Monotonicity).

Suppose $\mathscr{A}$ is regular. If $M^{\prime}$ is such that $M\leq
M^{\prime}\subseteq B$ and $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$, then $A\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M^{\prime}}^{N}B$.

###### Proof.

Let $(M_{1},M_{2})$ witness that $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$. In particular, this implies
that there is some $N^{\prime}\leq N$ such that

$\textstyle\smile$${M_{1}}$${N^{\prime}}$${M}$${M_{2}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Since $M\leq M^{\prime}\leq M_{2}$ (as $M^{\prime}\subseteq B\subseteq
M_{2}$), by regularity there exists some $N^{\prime\prime}\leq N^{\prime}$
with

$\textstyle\smile$${M_{1}}$${N^{\prime\prime}}$${N^{\prime}}$${M}$${M^{\prime}}$${M_{2}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Since $A\subseteq M_{1}\leq N^{\prime\prime}$, hence
$(N^{\prime\prime},M_{2})$ is a witness to $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M^{\prime}}^{N}B$. ∎

######  Lemma 4.5.

If $\mathscr{A}$ is regular and $M_{0}\leq M_{1}$, then $A\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$ iff there is some
$M_{2}\leq N$ such that $M_{0}\leq M_{2}$, $A\subseteq M_{2}$ and
$M_{1},M_{2}$ are $\mathscr{A}$-subamalgamated over $M_{0}$ inside $N$.

###### Proof.

For the reverse direction, note that $(M_{2},M_{1})$ is a witness to
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$. For the forward
direction, let $(M_{2},M^{\prime})$ witness that $A\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$, and so in
particular $M_{0}\leq M_{1}\leq M^{\prime}$. Hence by regularity,
$M_{1},M_{2}$ are also $\mathscr{A}$-subamalgamated over $M_{0}$ inside $N$. ∎

######  Proposition 4.6 (Transitivity).

Suppose $K$ admits finite intersection and $\mathscr{A}$ is regular,
absolutely minimal. If $M_{0}\leq M_{1}\leq M_{2}\leq N$ and $A\subseteq N$ is
such that $A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$ and
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{1}}^{N}M_{2}$, then
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{2}$.

###### Proof.

By the above lemma, there exists $M^{\prime},M^{\prime\prime}\leq N$ such that
$(M^{\prime},M_{1})$ witnesses $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$ and
$(M^{\prime\prime},M_{2})$ witnesses $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{1}}^{N}M_{2}$. Hence, there are
also models
$M^{\prime}\oplus_{M_{0}}^{N}M_{1},M^{\prime\prime}\oplus_{M_{1}}^{N}M_{2}\leq
N$ such that:

$\textstyle\smile$${M^{\prime}}$${M^{\prime}\oplus_{M_{0}}^{N}M_{1}}$${M^{\prime\prime}}$${M^{\prime\prime}\oplus_{M_{1}}^{N}M_{2}}$${M_{0}}$${M_{1}}$${M_{1}}$${M_{2}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Since $K$ has FI and $M_{0}\leq M^{\prime},M^{\prime\prime}\leq N$, there is a
model $M^{*}:=M^{\prime}\cap M^{\prime\prime}$, and in particular $M_{0}\leq
M^{*},A\subseteq M^{*}$. So by regularity, $M^{*},M_{1}$ are also
$\mathscr{A}$-subamalgamated over $M_{0}$ inside $N$ i.e.

$\textstyle\smile$${M_{1}}$${M_{1}\oplus_{M_{0}}^{N}M^{*}}$${M_{1}\oplus_{M_{0}}^{N}M^{\prime}}$${M_{0}}$${M^{*}}$${M^{\prime}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Note that since $M_{1},M^{*}\leq M^{\prime\prime}$,
$M_{1}\oplus_{M_{0}}^{N}M^{*}\leq M^{\prime\prime}$ as $\mathscr{A}$ is
absolutely minimal. Therefore, again by regularity,
$(M_{1}\oplus_{M_{0}}^{N}M^{*}),M_{2}$ are $\mathscr{A}$-subamalgamated over
$M_{1}$ inside $N$, so there is some $M^{**}$ such that:

$\textstyle\smile$${M_{2}}$${M^{**}}$${M_{2}\oplus_{M_{0}}^{N}M^{\prime\prime}}$${M_{1}}$${M_{1}\oplus_{M_{0}}^{N}M^{*}}$${M^{\prime\prime}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Combining the commutative squares on the left of the two diagrams, we get that

$\textstyle\smile$${M^{*}}$${M_{1}\oplus_{M_{0}}^{N}M^{*}}$${M^{**}}$${M_{0}}$${M_{1}}$${M_{2}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Applying regularity once more, hence $(M^{*},M_{2})$ witness that
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{2}$. ∎

######  Proposition 4.7 (Invariance).

If $\mathscr{A}$ is a notion of amalgamation, then $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is invariant under $K$-embeddings: if
$A\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N}B$ and $f:N\longrightarrow
N^{\prime}$ is a $K$-embedding, then $f(A)\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{f[M]}^{N^{\prime}}f(B)$.

###### Proof.

First, for the case where $f:N\cong N^{\prime}$ is a $K$-isomorphism, the
statement above holds due to the Invariance properties of $\mathscr{A}$. Then
Proposition 4.2 shows that this is true for the general case where $f$ is a
$K$-embedding. ∎

######  Corollary 4.8.

If $\bar{a}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$ and
$\text{gtp}(\bar{a}/M_{1},N)=\text{gtp}(\bar{b}/M_{1},N^{\prime})$, then
$\bar{b}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N^{\prime}}M_{1}$.

The above corollary shows that when $\mathscr{A}$ is a notion of amalgamation
and $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ is derived from $\mathscr{A}$, then
in fact $\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ can be extended to a form of
nonforking notion for Galois types.

######  Notation 4.9.

Let $p\in S(M_{1})$. We say that $p$ does not fork over $M_{0}$ if $M_{0}\leq
M_{1}$ and there is some $\bar{a}$ and a model $N\geq M_{1}$ such that
$(\bar{a},M_{1},N)$ realize $p$, and $\bar{a}\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$.

We say that $q\geq p$ is a nonforking extension if $q$ does not fork over
$\text{dom }p$

######  Corollary 4.10.

Suppose $K$ admits finite intersection, and $\mathscr{A}$ is regular,
absolutely minimal. If $p$ does not fork over $M$ and $q$ is a nonforking
extension of $p$, then $q$ does not fork over $M$.

######  Proposition 4.11 (Extension).

Let $p\in S(M_{0})$. If $M_{1}\geq M_{0}$, then there is $q\in S(M_{1})$ such
that $q\geq p$ and $q$ does not fork over $M_{0}$.

###### Proof.

Let $(\bar{a},M_{0},M_{2})$ realize $p$, and let $N$ be an
$\mathscr{A}$-amalgam of $M_{2},M_{1}$ over $M_{0}$ via

$\textstyle\smile$${M_{2}}$${N}$${M_{0}}$${M_{1}}$$\textstyle\smile$$\scriptstyle{f}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Then $f(\bar{a})\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N}M_{1}$ (as witnessed by
$(f[M_{2}],M_{1})$), and $\text{gtp}(f(\bar{a})/M_{0},N)=p$. Hence
$\text{gtp}(f(\bar{a})/M_{1},N)$ is the desired nonforking extension. ∎

######  Proposition 4.12 (Locality, version 1).

Suppose $\mathscr{A}$ is regular, continuous, absolutely minimal and admits
decomposition. Assume further that $\mu(K)<\infty$. If $(M_{i})_{i<\alpha}$ is
a strictly increasing continuous chain of models and
$\text{cf}(\alpha)\geq\mu_{r}(K)$, then for every $p\in
S(\bigcup_{i<\alpha}M_{i})$ a Galois type of length $<\mu(K)$, there is some
$i<\alpha$ such that $p$ does not fork over $M_{i}$.

###### Proof.

Let $M_{b}:=M_{0},M:=\bigcup_{i<\alpha}M_{i}$, and let us define a sequence of
models $(M_{i}^{\prime})_{i<\alpha}$ such that:

  1. (1)

For each $i<\alpha$, $M_{b}\leq M_{i}^{\prime}\leq M_{i+1}$.

  2. (2)

$M_{0}^{\prime}=M_{1}$

  3. (3)

For each $i<\alpha$, $M_{i}^{\prime}$ is such that
$M_{i}\oplus_{M_{b}}^{M}M_{i}^{\prime}=M_{i+1}$.

Note that $\mathscr{A}$ admitting decomposition implies that such a sequence
exists. Furthermore, by construction we have that
$\bigoplus_{M_{b},i<\alpha}^{M}M_{i}^{\prime}=M$ (as witnessed by the
resolution $(M_{i})_{i<\alpha}$).

Given $p\in S(M)$ a Galois type of length $<\mu(K)$, let $(\bar{a},M,N)$
realize $p$, and let $N^{*}\leq N$ be such that $N=M\oplus_{M_{b}}^{N}N^{*}$
(again, $N^{*}$ exists as $\mathscr{A}$ admits decomposition). Hence we also
have that $N$ is the $\mathscr{A}$-amalgam of
$\\{N^{*}\\}\cup\\{M_{i}^{\prime}\\}_{i<\alpha}$ over $M_{b}$ (by inclusion).
Since $|\bar{a}|<\mu_{r}(K)\leq\text{cf}(\alpha)$, there is some
$i_{0}<\alpha$ such that

$\bar{a}\in
N^{*}\oplus_{M_{b}}^{N}\bigg{(}\bigoplus_{M_{b},i<i_{0}}^{N}M_{i}^{\prime}\bigg{)}=N^{*}\oplus_{M_{b}}^{N}M_{i_{0}+1}$

Let $N^{\prime}:=N^{*}\oplus_{M_{b}}^{N}M_{i_{0}+1}$. Since $N$ is the
$\mathscr{A}$-amalgam of $N^{*},M$ over $M_{b}$ by inclusion, by regularity we
also have that $N$ is the $\mathscr{A}$-amalgam of $N^{\prime},M$ over
$M_{i_{0}+1}$. Diagrammatically,

$\leavevmode\hbox to65.21pt{\vbox
to49.65pt{\pgfpicture\makeatletter\hbox{\hskip
32.60439pt\lower-24.87619pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.60439pt}{-24.77635pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
10.26802pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.96248pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${N^{*}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
10.26802pt\hfil&\hfil\hskip 32.868pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.56248pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${N}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.86803pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
10.90303pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.59749pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${M_{b}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
10.90303pt\hfil&\hfil\hskip 33.70134pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.39583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${M}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
9.70137pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.23334pt}{9.87643pt}\pgfsys@lineto{13.43503pt}{9.87643pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.63501pt}{9.87643pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.06206pt}{12.2292pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.68996pt}{-9.2285pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathscr{A}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-21.70135pt}{-14.08333pt}\pgfsys@lineto{-21.70135pt}{3.11676pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-21.70135pt}{3.31674pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.53166pt}{-6.79025pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-10.59833pt}{-22.27635pt}\pgfsys@lineto{12.60168pt}{-22.27635pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.80167pt}{-22.27635pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.03711pt}{-19.92358pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{22.90302pt}{-14.08333pt}\pgfsys@lineto{22.90302pt}{3.11676pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{22.90302pt}{3.31674pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{18.07271pt}{-6.79025pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\Longrightarrow\leavevmode\hbox
to119.3pt{\vbox to50.84pt{\pgfpicture\makeatletter\hbox{\hskip
59.64816pt\lower-25.46925pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-59.64816pt}{-25.36942pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
10.26802pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.96248pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${N^{*}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
10.26802pt\hfil&\hfil\hskip 33.63799pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.33247pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${N^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
9.63802pt\hfil&\hfil\hskip 32.868pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.56248pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${N}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.86803pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
10.90303pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.59749pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${M_{b}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
10.90303pt\hfil&\hfil\hskip 39.04376pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.73825pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${M_{i_{0}+1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.0438pt\hfil&\hfil\hskip 33.70134pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.39583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${M}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
9.70137pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-38.27711pt}{9.28337pt}\pgfsys@lineto{-9.03632pt}{9.28337pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.83633pt}{9.28337pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-24.69551pt}{11.63614pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.94507pt}{-9.29301pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathscr{A}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{11.03967pt}{9.28337pt}\pgfsys@lineto{40.4788pt}{9.28337pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{40.67879pt}{9.28337pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.72044pt}{11.63614pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.69276pt}{-10.12517pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\mathscr{A}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-48.74513pt}{-14.67639pt}\pgfsys@lineto{-48.74513pt}{2.5237pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-48.74513pt}{2.72368pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-53.57544pt}{-7.38332pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-37.6421pt}{-22.86942pt}\pgfsys@lineto{-14.4421pt}{-22.86942pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.24211pt}{-22.86942pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-27.08089pt}{-20.51665pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{1.20166pt}{-14.67639pt}\pgfsys@lineto{1.20166pt}{2.5237pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{1.20166pt}{2.72368pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62865pt}{-7.38332pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{16.44545pt}{-22.86942pt}\pgfsys@lineto{39.64546pt}{-22.86942pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{39.84544pt}{-22.86942pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.00667pt}{-20.51665pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{49.9468pt}{-14.67639pt}\pgfsys@lineto{49.9468pt}{2.5237pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{49.9468pt}{2.72368pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{$\textstyle\smile$}\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{45.11649pt}{-7.38332pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Note then that $(N^{\prime},M)$ is a witness to $\bar{a}\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{i_{0}+1}}^{N}M$, and therefore $p$
does not fork over $M_{i_{0}+1}$. ∎

In fact, a related formulation of the locality property can be shown to be
true using the same proof:

######  Proposition 4.13 (Locality, version 2).

Suppose $\mathscr{A}$ is regular, continuous, absolutely minimal, admits
decomposition and is such that $\mu(K)<\infty$. If
$|M|>\mu_{r}(K)+\text{LS}(K)$ and $p$ is a Galois type over $M$ of length
$<\mu(K)$, then there is some $M^{*}\leq M$ such that
$|M^{*}|<\mu_{r}(K)+\text{LS}(K)^{+}$ and $p$ does not fork over $M^{*}$.

###### Proof.

Let $\lambda=|M|$, and take some $M_{b}\leq M$ with $|M_{b}|=\text{LS}(K)$. As
$\mathscr{A}$ admits decomposition and is absolutely minimal, by Lemma 3.15
there is a sequence $(M_{i})_{i<\alpha}$ such that:

  1. (1)

$\alpha<\lambda^{+}$

  2. (2)

For each $i<\alpha$, $M_{b}\leq M_{i}\leq M$ and $|M_{i}|=\text{LS}(K)$

  3. (3)

$M=\bigoplus_{M_{b},i<\alpha}^{M}M_{i}$

Further, as $\mathscr{A}$ is regular and continuous, by Theorem 3.14 we may
assume that $\alpha=\lambda$. Letting $(\bar{a},M,N)$ be a realization of $p$,
as in the proof for the above proposition there exists some $N^{*}$ such that
$N=N^{*}\oplus_{M_{b}}^{N}M$. Now, as $|A|<\mu(K)<\infty$, there is some
subset $S\subseteq\lambda$ such that $|S|<\mu_{r}(K)$ and $A\subseteq
N^{*}\oplus_{M_{b}}^{N}\big{(}\bigoplus_{M_{b},i\in S}^{N}M_{i}\big{)}$.
Letting $N^{\prime}=N^{*}\oplus_{M_{b}}^{N}\big{(}\bigoplus_{M_{b},i\in
S}^{N}M_{i}\big{)}$, hence (as in the above proof) $N$ is the
$\mathscr{A}$-amalgam of $N^{\prime},M$ over $\bigoplus_{M_{b},i\in
S}^{N}M_{i}$ by regularity. Therefore, letting $M^{*}=\bigoplus_{M_{b},i\in
S}^{N}M_{i}$, we have $A\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M^{*}}^{N}M$. Furthermore, as
$|S|<\mu_{r}(K)$ and each $|M_{i}|=\text{LS}(K)$, by Lemma 2.12,
$|M^{*}|<\mu_{r}(K)+\text{LS}(K)^{+}$ as desired. ∎

######  Corollary 4.14.

Suppose $K$ admits finite intersection, and $\mathscr{A}$ is regular,
continuous, absolutely minimal, admits decomposition, and is such that
$\mu(K)<\infty$. If $M\in K$, $(M_{i})_{i<\alpha}$ is continuous resolution of
$M$ with $\text{cf}(\alpha)\geq\mu_{r}(K)$, and $p\in S(M)$ is a type of
length $<\mu(K)$ such that each $p\upharpoonright M_{i}$ does not fork over
$M_{0}$, then $p$ does not fork over $M_{0}$.

###### Proof.

By the previous proposition, there is some $i<\alpha$ such that $p$ does not
fork over $M_{i}$. But $p\upharpoonright M_{i}$ does not fork over $M_{0}$ by
assumption, and so by Proposition 4.6 $p$ does not fork over $M_{0}$. ∎

######  Proposition 4.15 (Uniqueness).

Suppose $K$ admits finite intersection, $\mathscr{A}$ has uniqueness and is
regular. Then for any Galois type $p\in S(M)$ and any $N\geq M$, there is a
unique $q\in S(N)$ such that $q$ is a nonforking extension of $p$.

###### Proof.

Let $q_{1},q_{2}\in S(N)$ be nonforking extensions of $p$, and let
$(\bar{a},N,N_{1}),(\bar{b},N,N_{2})$ be realizations of the types
respectively. Since $q_{1}\upharpoonright M=q_{2}\upharpoonright M=p$ and $K$
has AP (since $\mathscr{A}$ is a notion of amalgamation), we may assume that
there is a $K$-isomorphism $f:N_{1}\longrightarrow_{M}N_{2}$ such that
$f(\bar{a})=\bar{b}$. Now, as each $q_{i}$ is a nonforking extension of $p$,
there exists $M_{1}\leq N_{1}$ such that $(M_{1},N)$ is a witness to
$\bar{a}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N_{1}}N$, and similarly
$M_{2}\leq N_{2}$. Letting $M^{\prime\prime}=f[M_{1}]\cap M_{2}$, note then
that $M\leq M^{\prime\prime},\bar{b}\in M^{\prime\prime}$. Hence, by
regularity, $(M^{\prime\prime},N)$ is also a witness for
$\bar{b}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N_{2}}N$. Further, let
$M^{\prime}\leq N_{1}$ be such that $f[M^{\prime}]=M^{\prime\prime}$, and
similarly $(M^{\prime},N)$ is a witness for $\bar{a}\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M}^{N_{1}}N$. But as $f$ is also an
isomorphism between $M^{\prime}$ and $M^{\prime\prime}$ over $M$, by
uniqueness of $\mathscr{A}$ there is an isomorphism $g$ satisfying the
following commutative diagram (where all the unlabelled maps are inclusions):

$\textstyle\smile$${N\oplus_{M}^{N_{1}}M^{\prime}}$${N}$${N\oplus_{M}^{N_{2}}M^{\prime\prime}}$${M^{\prime}}$${M}$${M^{\prime\prime}}$$\textstyle\smile$$\scriptstyle{g}$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\scriptstyle{f}$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$

In particular, $g(\bar{a})=f(\bar{a})=f(\bar{b})=g(\bar{b})$ and $g[N]=N$, and
hence $\text{gtp}(\bar{a},N,N_{1})=\text{gtp}(\bar{b},N,N_{2})$. This
completes the proof. ∎

######  Corollary 4.16.

Suppose $K$ admits finite intersection, $\mathscr{A}$ has uniqueness and is
regular. If $(M_{i})_{i<\alpha}$ is an increasing continuous chain, and
$(p_{i})_{i<\alpha}$ is an increasing chain of types (with each $p_{i}\in
S(M_{i})$) such that each $p_{i+1}$ is a nonforking extension of $p_{i}$, then
there is $p\in S\big{(}\bigcup_{i<\alpha}M_{i}\big{)}$ such that
$p\upharpoonright M_{i}=M_{i}$ and $p$ does not fork over $M_{0}$.

###### Proof.

Denote $M_{\alpha}:=\bigcup_{i<\alpha}M_{i}$. By Proposition 4.11, let $p\in
S(M_{\alpha})$ be a nonforking extension of $p_{0}$. Note then that for each
$i<\alpha$, $p\upharpoonright M_{i}$ also does not fork over $M_{0}$, and
hence is a nonforking extension of $p_{0}$. By the above proposition, hence
$p\upharpoonright M_{i}=p_{i}$. ∎

######  Definition 4.17.

We say that $\mathscr{A}$ is a notion of geometric amalgamation if
$\mathscr{A}$ is regular, continuous, absolutely minimal, and admits
decomposition with $\mu(K)<\infty$. We say that $\mathscr{A}$ is a notion of
free amalgamation if additionally $\mathscr{A}$ has uniqueness.

######  Conclusion 4.18.

Suppose $K$ is an AEC admitting finite intersections, $\mathscr{A}$ is a
notion of free amalgamation on $K$. Then the relation $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ satisfies:

  * •

Invariance (Proposition 4.7)

  * •

Existence (Proposition 4.3.1)

  * •

Symmetry (Proposition 4.3.2)

  * •

Top monotonicity (Proposition 4.2)

  * •

Right (and left) monotonicity (Proposition 4.3.3 and symmetry)

  * •

Base monotonicity (Proposition 4.4)

  * •

Normality (Proposition 4.3.4)

  * •

Transitivity (Proposition 4.6)

  * •

Extension (Proposition 4.11)

  * •

$\mu(K)$-local character for types of length $<\mu(K)$ (Proposition 4.12 and
4.13)

  * •

Continuity (Proposition 4.14)

  * •

Uniqueness of nonforking extensions (Proposition 4.15)

This completes the propositions needed to prove that $\mathop{\hbox{\lower
2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ has the desired properties (under
certain assumptions on $K$ and $\mathscr{A}$). A nontrivial example of such an
independence relation comes from the class of free groups:

######  Example 4.19.

Let $K$ be the class of free groups, with an ordering $\leq_{f}$ such that
$G\leq_{f}H$ iff $G$ is a free factor in $H$ i.e. there is some set $Y$ such
that we can consider $H=F(Y)$ (the free group with $Y$ as the set of
generators), and moreover there is some $X\subseteq Y$ such that $G=F(X)$.

Note then that $K$ is a weak AEC which admits finite intersections (see the
Appendix for details), and taking $\mathscr{A}$ to be the notion of free
amalgamation gives us that $\mathscr{A}$ is minimal (hence absolutely
minimal), regular, continuous, admits decomposition, and has uniqueness. It is
also clear that $\mu(K)=\aleph_{0}$. In this case,
$\bar{a}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{F}^{G}\bar{b}$ iff there is a free
basis $X$ of $G$ (so $F(X)=G$) along with subsets $X_{0},X_{1},X_{2}\subseteq
X$ such that:

  * •

$F_{0}=F(X_{0})$

  * •

$X_{1}\cap X_{2}=X_{0}$

  * •

$\bar{a}\in F(X_{1})$ and $\bar{b}\in F(X_{2})$

Moreover, the above lemmas show that $\mathop{\hbox{\lower 2.0pt\hbox
to12.77773pt{\hfil\vrule width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits$ for the class of free groups behaves
as if for a superstable first order theory; this is not surprising since by
defining superstability in terms of uniqueness of limit models, the
uncountable categoricity of the class implies that it is indeed superstable as
an AEC.

On the other hand, this example is notable for two reasons:

  1. (1)

The free factors of a free group are not closed under infinite intersections
(for an example, see [BCS77]), and in particular the class of free groups do
not admit arbitrary intersection. This is in contrast to classes such as
vector spaces and algebraically closed fields, where the pregeometry is used
to define independence but implies that the class admits intersections.

  2. (2)

The first order theory of free groups is known to be not superstable (see, for
example, [Poi83]), whereas $(K,\leq_{f})$ is indeed superstable as an AEC.
Furthermore, since $G\leq_{f}H$ imlies that $G$ is an elementary substructure
of $H$ (see the Appendix), this implies that the free groups lies within the
superstable part of the theory of free groups. This fact is, of course,
trivial given that the free groups are uncountably categorical, but does show
how different the class of free groups is from the class of elementarily free
groups.

Before ending this section, let us demonstrate the known fact that the
existence of a superstable-like independence notion implies that the class is
tame:

######  Definition 4.20.

Let $I$ be a linear order. We say that $K$ is $(<\lambda)$-tame for $I$-types
if for any model $M$ and $p,q\in S^{I}(M)$, $p\neq q$ iff there exists some
$N\leq M$ such that $|N|<\lambda$ and $p\upharpoonright N\neq q\upharpoonright
N$. We say that $K$ is $\lambda$-tame if it is $(<\lambda^{+})$-tame.

######  Corollary 4.21.

If $K$ admits finite intersection and $\mathscr{A}$ is a notion of free
amalgamation, then $K$ is $(\mu_{r}(K)+\text{LS}(K))^{+}$-tame for types of
length $<\mu(K)$.

###### Proof.

Let $M\in K$ with $|M|>\mu_{r}(K)+\text{LS}(K)$, $p,q\in S(M)$ be types of
length $l<\mu(K)$, and let $(\bar{a},M,N_{1}),(\bar{b},M,N_{2})$ realize $p,q$
respectively. By Proposition 4.13, there is $M_{a}\leq M$ such that
$\bar{a}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{a}}^{N_{1}}M$ and
$|M_{a}|=\mu_{r}(K)+\text{LS}(K)$. Define $M_{b}\leq M$ similarly, and (by the
Löwenheim-Skolem property) let $M_{0}\leq M$ be such that $M_{a},M_{b}\leq
M_{0}$ and $|M_{0}|=\mu_{r}(K)+\text{LS}(K)$. Then by Proposition 4.4,
$\bar{a}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N_{1}}M$ and
$\bar{b}\mathop{\hbox{\lower 2.0pt\hbox to12.77773pt{\hfil\vrule
width=0.3pt,depth=-2.0pt,height=10.0pt\hfil}\lower
2.0pt\hbox{$\textstyle\smile$}}}\limits_{M_{0}}^{N_{2}}M$. Now, if $p,q$ are
such that $p\upharpoonright M^{\prime}=q\upharpoonright M^{\prime}$ for every
$M^{\prime}\leq M$ with $|M^{\prime}|\leq\mu_{r}(K)+\text{LS}(K)$, then in
particular $p\upharpoonright M_{0}=q\upharpoonright M_{0}$. But $p$ is a
nonforking extension of $p\upharpoonright M_{0}$ and similarly $q$, so by
Proposition 4.15, $p=q$. ∎

######  Remark.

The statement of the the above Corollary begs comparison to a result of Boney
that appears as Theorem 3.7 of [Vas17], stating that a pseudouniversal AEC is
$(\aleph_{0})$-tame.111The actual result is slightly stronger, but difficult
to state here precisely due to small conflicts of notation. Since
pseudouniversality is a strengthening222To quote [Vas17], the extra
requirement is that “the isomorphism characterizing equality of Galois types
is unique”. of admitting intersections with $\mu(K)=\aleph_{0}$ (when a
suitable notion of amalgamation is defined), at first glance our result
appears to be comparable. Besides the slightly different cardinal arithmetic,
the main difference is that our result here relaxes the intersection
requirement to only finite intersections, but at the expense of requiring
$\mathscr{A}$ to have uniqueness (which, as Section 6 explores, has strong
implications regarding the structure of $K$ and is not a trivial assumption).

##  5\. Categoricity Transfer Using Unique Amalgams

Up until this point, we have three primary examples of classes with a notion
of free amalgamation which have guided our exploration:

  * •

The class of vector spaces over a fixed field with the subspace (equivalently,
elementary submodel) ordering

  * •

The class of (torsion) divisible groups with the subgroup ordering

  * •

The class of free groups with the “free factor” ordering (see Example 4.19)

The key characteristic shared, and indeed the driving intuition for this
study, is that such classes have some notion of “basis” which generates each
model. Now, in the case of the class of vector spaces, the eventual
categoricity of the class can be derived from the fact that any bijection
between two bases extends to an isomorphism between the spanned spaces. An
analogous principle clearly holds also for the free groups, and the same
argument can be applied more generally to the cases of strongly minimal first
order theories and quasiminimal excellent classes with the countable closure
property. On the other hand, this does not apply to the class of divisible
groups, and the torsion divisible groups are not categorical in any cardinal
whereas the class of free groups are uncountably categorical. In this sense,
we will formalize the intuitive argument above to establish sufficient
conditions for a categoricity transfer theorem.

One aspect of the argument above for vector spaces is that two superspaces
$V,W$ of $U$ are isomorphic over $U$ if $V/U,W/U$ have the same dimension.
Although there is no notion of dimensionality within the current context, we
note that the dimension of a vector space only differs from its cardinality
for spaces of small dimension. This allows us to formalize the notion of two
extensions being “isomorphic” when they are of sufficiently large cardinality:

######  Definition 5.1.

Suppose $\mathscr{A}$ is a notion of free amalgamation in $K$.

  1. (1)

We define $\theta(K)=\mu(K)+\text{LS}(K)$

  2. (2)

Given $M_{b}\leq M_{t}$ and $\alpha$ an ordinal, for a model $N\geq M_{b}$ we
write “$N\cong M_{t}^{\lambda}/M_{b}$” to indicate that
$N=\bigoplus_{M_{b},i<\alpha}^{N}M_{i}$, where each $M_{i}\cong_{M_{b}}M_{t}$.

  3. (3)

We define an equivalence relation $\backsim$ on pairs of models of $K$ by:
given $M_{1}\leq N_{1}$ and $M_{2}\leq N_{2}$,
$(N_{1},M_{1})\backsim(N_{2},M_{2})$ iff there is a $K$-isomorphism
$f:N_{1}^{\theta(K)}/M_{1}\cong N_{2}^{\theta(K)}/M_{2}$ with
$f[M_{1}]=M_{2}$.

######  Remark.

Note that the above definition does not construct $M_{t}^{\alpha}/M_{b}$ as a
particular model, but if $N_{1},N_{2}$ are such that both $N_{1}\cong
M_{t}^{\alpha}/M_{b}$ and $N_{2}\cong M_{t}^{\alpha}/M_{b}$, then in fact
$N_{1}\cong_{M_{b}}N_{2}$ by uniqueness of $\mathscr{A}$, and hence we may
consider $M_{t}^{\alpha}/M_{b}$ as a particular choice of representative
inside $K$. In this sense, for any ordinal $0<\beta<\alpha$ we may consider
$M_{b}\leq M_{t}\leq M_{t}^{\beta}/M_{b}\leq M_{t}^{\alpha}/M_{b}$. In this
sense, we extend the notation by defining $M_{t}^{0}/M_{b}=M_{b}$

######  Lemma 5.2.

Suppose $\mathscr{A}$ is a notion of free amalgamation. If
$(N_{1},M)\backsim(N_{2},M)$, then for any $\lambda\geq\theta(K)$, there is a
$K$-isomorphism $f:N_{1}^{\lambda}/M\cong_{M}N_{2}^{\lambda}/M$.

###### Proof.

Decompose $\lambda=\bigsqcup_{j<\lambda}S_{j}$ such that each
$|S_{j}|=\theta(K)$. Defining models $N_{1}^{*},N_{1}^{i},N_{2}^{*},N_{2}^{i}$
such that $N_{l}^{*}=\bigoplus_{M,i<\lambda}^{N_{l}^{*}}N_{l}^{i}$ for
$l=1,2$, note then that for each $j,j^{\prime}<\lambda$,

$\bigoplus_{M,i\in S_{j}}^{N_{1}^{*}}N_{1}^{i}\cong N_{1}^{\theta(K)}/M\cong
N_{2}^{\theta(K)}/M\cong\bigoplus_{M,i\in
S_{j^{\prime}}}^{N_{2}^{*}}N_{2}^{i}$

So let us define $N_{l}^{S_{j}}=\bigoplus_{M,i\in
S_{j}}^{N_{l}^{*}}N_{1}^{i}$. Then, by applying Theorem 3.13, we get that
$N_{l}^{*}=\bigoplus_{M,j<\lambda}^{N_{l}^{*}}N_{l}^{S_{j}}$. Hence, as
$\mathscr{A}$ has uniqueness, we get that $N_{1}^{*},N_{2}^{*}$ are isomorphic
over $M$. ∎

######  Definition 5.3.

Given $K$ an AEC, we say that $M\in K$ is a prime and minimal model of $K$ if:

  1. (1)

For every $N\in K$, there is a $K$-embedding $\iota_{N}:M\longrightarrow N$;
and

  2. (2)

For every $K$-embedding $f:N_{1}\longrightarrow N_{2}$,
$f\circ\iota_{N_{1}}=\iota_{N_{2}}$

If $K$ has a prime and minimal model, we fix such a model and denote it by
$0_{K}$.

######  Theorem 5.4.

Suppose $\mathscr{A}$ is a notion of free amalgamation in $K$, and $0_{K}$ is
a prime and minimal model. If $K$ is $\lambda$-categorical in some
$\lambda\geq\theta(K)$, then for any $M_{1},M_{2}$ in $K$ with
$|M_{1}|=|M_{2}|=\text{LS}(K)$, $(M_{1},0_{K})\backsim(M_{2},0_{K})$.

###### Proof.

Given $M_{1},M_{2}$ and $\lambda$ as above, note that by Lemma 3.15,
$|M_{1}^{\lambda}/0_{K}|=|M_{2}^{\lambda}/0_{K}|=\lambda$. Hence, by
$\lambda$-categoricity, there is some $K$-isomorphism
$f:M_{1}^{\lambda}/0_{K}\cong M_{2}^{\lambda}/0_{K}$, and moreover
$f[0_{K}]=0_{K}$ as $0_{K}$ is prime and minimal. Thus, WLOG we may assume
that $N=M_{1}^{\lambda}/0_{K}=M_{2}^{\lambda}/0_{K}$, and in fact that there
exists sequence $(M_{1}^{i})_{i<\lambda},(M_{2}^{i})_{i<\lambda}$ such that:

  1. (1)

For each $i<\lambda$, $M_{1}^{i}$ is isomorphic to $M_{1}$ and $M_{2}^{i}$ is
isomorphic to $M_{2}$ (over $0_{K}$).

  2. (2)

Each $0_{K}\leq M_{1}^{i},M_{2}^{i}\leq N$; and

  3. (3)

$N=\bigoplus_{0_{K},i<\lambda}^{N}M_{1}^{i}=\bigoplus_{0_{K},i<\lambda}^{N}M_{2}^{i}$

We will construct two sequences of sets
$(S_{j})_{j<\omega},(T_{j})_{j<\omega}$, satisfying:

  1. (1)

Each $S_{j}\subseteq\lambda$ with $|S_{j}|=\theta(K)$, and similarly for
$T_{j}$

  2. (2)

$S_{0}=T_{0}=\theta(K)$

  3. (3)

$S_{j}\subseteq S_{j+1}$ and $T_{j}\subseteq T_{j+1}$

  4. (4)

For each $j<\omega$, $\bigoplus_{0_{K},i\in
T_{j}}^{N}M_{2}^{i}\leq\bigoplus_{0_{K},i\in S_{j+1}}^{N}M_{1}^{i}$; and

  5. (5)

For each $j<\omega$, $\bigoplus_{0_{K},i\in
S_{j}}^{N}M_{1}^{i}\leq\bigoplus_{0_{K},i\in T_{j+1}}^{N}M_{2}^{i}$

Let us first show that such a construction is sufficient: defining
$S:=\bigcup_{j<\omega}S_{j}$ and $T:=\bigcup_{j<\omega}T_{j}$, note that as
$\mathscr{A}$ is absolutely minimal,

$\bigoplus_{0_{K},i\in
S}^{N}M_{1}^{i}=\bigcup_{j<\omega}\Bigg{(}\bigoplus_{0_{K},i\in
S_{j}}^{N}M_{1}^{i}\Bigg{)}$

The same statement holds for $T$ and $M_{2}^{i}$. Hence, by (3) and (4) of the
construction above, we have that $\bigoplus_{0_{K},i\in
S}^{N}M_{1}^{i}=\bigoplus_{0_{K},i\in T}^{N}M_{2}^{i}$. But since
$|S|=|T|=\theta(K)$, hence we can take $\bigoplus_{0_{K},i\in S}M_{1}^{i}\cong
M_{1}^{\theta(K)}/0_{K}$, and therefore $(M_{1},0_{K})\backsim(M_{2},0_{K})$.

Let us complete the proof by constructing $S_{j},T_{j}$. Given $S_{j},T_{j}$
already defined, consider $M^{\prime}=\bigoplus_{0_{K},i\in
T_{j}}^{N}M_{2}^{i}$: by Lemma 2.12,
$|M^{\prime}|=\text{LS}(K)+|T_{j}|=\theta(K)$, and hence there is
$S_{j+1}\subseteq\lambda$ such that $|S_{j+1}|=\theta(K)+\mu_{r}(K)=\theta(K)$
and $M^{\prime}\subseteq\bigoplus_{0_{K},i\in S_{j+1}}^{N}M_{1}^{i}$.
Similarly we can define $T_{j+1}$, and this completes the proof. ∎

Note that the conclusion of the above theorem holds for the classes of vector
spaces and free groups, but not for divisible groups: for example, letting
$0_{G}$ denote the trivial group, it is clear that if $p\neq q$ are primes,
then $(\mathbb{Z}(p^{\infty}),0_{G}),(\mathbb{Z}(q^{\infty}),0_{G})$ are not
$\backsim$ equivalent.

######  Lemma 5.5.

Suppose $\mathscr{A}$ is a notion of free amalgamation in $K$. Given models
$M_{0}\leq M_{1},M_{2}$, if $(M_{1},M_{0})\backsim(M_{2},M_{0})$, then for any
ordinal $\beta$,

$(M_{1}^{\beta}/M_{0})\oplus_{M_{0}}(M_{2}^{\theta(K)}/M_{0})\cong_{M_{0}}M_{1}^{|\beta|+\theta(K)}/M_{0}\cong_{M_{0}}M_{2}^{|\beta|+\theta(K)}/M_{0}$

###### Proof.

As $(M_{1},M_{0})\backsim(M_{2},M_{0})$,
$M_{1}^{\theta(K)}/M_{0}\cong_{M_{0}}M_{2}^{\theta(K)}/M_{0}$, and hence

$(M_{1}^{\beta}/M_{0})\oplus_{M_{0}}(M_{2}^{\theta(K)}/M_{0})\cong_{M_{0}}M_{1}^{\beta+\theta(K)}/M_{0}$

Furthermore, by Theorem 3.14 and Lemma 5.2, we have that

$M_{1}^{\beta+\theta(K)}/M_{0}\cong_{M_{0}}M_{1}^{|\beta|+\theta(K)}/M_{0}\cong_{M_{0}}M_{2}^{|\beta|+\theta(K)}/M_{0}$

∎

######  Theorem 5.6.

Suppose $\mathscr{A}$ is a notion of free amalgamation in $K$, and $K$ has a
prime and minimal model. If $K$ is $\lambda$-categorical in some
$\lambda\geq\theta(K)$, then $K$ is $\kappa$-categorical in every cardinal
$\kappa\geq\theta(K)+(2^{\text{LS}(K)})^{+}$.

###### Proof.

By the previous theorem, for any $M_{1},M_{2}\in K_{\text{LS}(K)}$,
$(M_{1},0_{K})\backsim(M_{2},0_{K})$. Hence by Lemma 5.2, it suffices to show
that if $M\in K$ and $|M|=\kappa\geq\theta(K)+(2^{\text{LS}(K)})^{+}$, then
$M\cong M^{\prime\kappa}/0_{K}$ for some $M^{\prime}\in K_{\text{LS}(K)}$.

So given $M\in K$ and $|M|=\kappa$, by Lemma 3.15 we can decompose
$M=\bigoplus_{0_{K},i<\kappa}^{M}M_{i}$ such that each $|M_{i}|=\text{LS}(K)$.
Letting $\Gamma:=\\{M_{i}/\cong:i<\kappa\\}$ (where $\cong$ is the equivalence
relation of $K$-isomorphism), note that since $|\Gamma|\leq 2^{\text{LS}(K)}$,
there is some $P\in\Gamma$ which is realized
$\geq\theta(K)+(2^{\text{LS}(K)})^{+}$ times in the sequence
$(M_{i})_{i<\kappa}$. For each $Q\in\Gamma$, let us also fix some
$M_{Q}\in\\{M_{i}:i<\kappa\\}$ such that $M_{Q}\vDash Q$. Note that by the
previous theorem, for any $Q_{1},Q_{2}\in\Gamma$,
$(M_{Q_{1}},0_{K})\backsim(M_{Q_{2}},0_{K})$.

Defining $S:=\\{i\in\kappa:M_{i}\vDash P\\}$, we can decompose $S$ as a
disjoint union $S=\bigsqcup_{Q\in\Gamma}S_{Q}$ which is indexed by $\Gamma$
and such that each $|S_{Q}|\geq\theta(K)+(2^{\text{LS}(K)})^{+}$ and is a
regular cardinal (possibly except for $S_{P}$). Now, for each $Q\in\Gamma$, we
have that $\bigoplus_{0_{K},i\in S_{Q}}^{M}M_{i}\cong M_{P}^{|S_{Q}|}/0_{K}$
as each $i\in S_{Q}\subseteq S$. Defining $N_{S_{Q}}=\bigoplus_{0_{K},i\in
S_{Q}}^{M}M_{i}$, note that as $|S_{Q}|\geq\theta(K)$ and
$(M_{P},0_{K})\backsim(M_{Q},0_{K})$, by Theorem 3.13 there is a sequence
$(N_{Q}^{i})_{i<|S_{Q}|}$ such that
$N_{S_{Q}}=\bigoplus_{0_{K},i<|S_{Q}|}^{M}N_{Q}^{i}$ and such that each
$N_{Q}^{i}\vDash Q$.

Now, for each $Q\in\Gamma$ such that $Q\neq P$, let
$T_{Q}:=\\{i\in\kappa:M_{i}\vDash Q\\}$, and define
$N^{*}_{Q}:=\bigoplus_{0_{K},i\in T_{Q}}^{M}M_{i}$. By Theorems 3.13 and 3.14,
each $N_{S_{Q}},N^{*}_{Q}$ are $\mathscr{A}$-subamalgamated inside $M$ over
$0_{K}$, and so we have that

$N_{S_{Q}}\oplus_{0_{K}}^{M}N^{*}_{Q}=\Bigg{(}\bigoplus_{0_{K},i<|S_{Q}|}^{M}N_{Q}^{i}\Bigg{)}\oplus_{0_{K}}^{M}\Bigg{(}\bigoplus_{0_{K},i\in
T_{Q}}^{M}M_{i}\Bigg{)}$

In other words, $N_{S_{Q}}\oplus_{0_{K}}^{M}N^{*}_{Q}\cong
M_{Q}^{|S_{Q}|+|T_{Q}|}/0_{K}$ by Lemma 5.2. In particular, as
$(M_{Q},0_{K})\backsim(M_{P},0_{K})$, we also have that
$N_{S_{Q}}\oplus_{0_{K}}^{M}N^{*}_{Q}\cong M_{P}^{|S_{Q}|+|T_{Q}|}/0_{K}$.

This implies that

$\displaystyle M=\bigoplus_{0_{K},i<\kappa}^{M}M_{i}$
$\displaystyle=\Bigg{(}\bigoplus_{0_{K},i\in
S}^{M}M_{i}\Bigg{)}\oplus_{0_{K}}^{M}\Bigg{(}\bigoplus_{0_{K},Q\neq
P}^{M}\Bigg{(}\bigoplus_{0_{K},i\in T_{Q}}^{M}M_{i}\Bigg{)}\Bigg{)}$
$\displaystyle=N_{S_{P}}\oplus_{0_{K}}^{M}\Bigg{(}\bigoplus_{0_{K},Q\neq
P}^{M}N_{S_{Q}}\oplus_{0_{K}}^{M}N^{*}_{Q}\Bigg{)}$

Since $N_{S_{P}}\cong M_{P}^{|S_{P}|}/0_{K}$ and
$N_{S_{Q}}\oplus_{0_{K}}^{M}N^{*}_{Q}\cong M_{P}^{|S_{Q}|+|T_{Q}|}/0_{K}$,
thus we get that $M\cong M_{P}^{\kappa}/0_{K}$. This completes the proof. ∎

Note that in the above argument, the fact that $\lambda>2^{\text{LS}(K)}$ was
used to ensure that $|\Gamma|<\lambda$, and hence some $P\in\Gamma$ is
realized by many $M_{i}$’s. In particular, since each $|M_{i}|=\text{LS}(K)$,
in fact we can bound $|\Gamma|\leq I(K,\text{LS}(K))$, where $I(K,\theta)$ is
the number of non-isomorphic models in $K_{\theta}$. This gives the following
strengthening:

######  Theorem 5.7.

Suppose $\mathscr{A}$ is a notion of free amalgamation in $K$, and $K$ has a
prime and minimal model. If $K$ is $\lambda$-categorical in some
$\lambda\geq\theta(K)$, then $K$ is $\kappa$-categorical in every cardinal
$\kappa\geq\theta(K)+I(K,\text{LS}(K))^{+}$.

This concludes our study of categoricity transfer in the case where there is a
prime and minimal model, which for most algebraic examples is the trivial
object inside the class. On the other hand, this is a very strong assumption
from a model-theoretic point of view; for example, intuitively the class of
saturated algebraically closed fields (equivalently, the algebreically closed
fields of infinite transcendental degree) should also allow the same argument
for categoricity transfer, but the class lacks a prime and minimal model. In
order to modify the above argument to work in this case, we need to strengthen
the notion of amalgmation with an additional property:

######  Definition 5.8.

Let $\mathscr{A}$ be a notion of amalgamation that is regular and absolutely
minimal. We say that $\mathscr{A}$ is 3-monotonic if the following condition
is satisfied: Given models $M_{0}\leq M_{1},M_{2},M_{3}\leq N$ such that

  1. (1)

$M_{1},M_{2}$ are $\mathscr{A}$-subamalgamated inside $N$ over $M_{0}$; and

  2. (2)

$N$ is the $\mathscr{A}$-amalgam of $M_{3},M_{1}\oplus_{M_{0}}^{N}M_{2}$ over
$M_{0}$ via inclusion

Then $N$ is the $\mathscr{A}$-amalgam of
$M_{1}\oplus_{M_{0}}^{N}M_{3},M_{2}\oplus_{M_{0}}^{N}M_{3}$ over $M_{3}$.

Diagrammatically, if the following commutative squares are
$\mathscr{A}$-amalgams:

$\textstyle\smile$${M_{1}}$${M_{1}\oplus_{M_{0}}^{N}M_{2}}$${M_{3}}$${N}$${M_{0}}$${M_{2}}$${M_{0}}$${M_{1}\oplus_{M_{0}}^{N}M_{2}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

Then we also have the $\mathscr{A}$-amalgam

$\textstyle\smile$${M_{1}\oplus_{M_{0}}^{N}M_{3}}$${N}$${M_{3}}$${M_{2}\oplus_{M_{0}}^{N}M_{3}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$${\mathscr{A}}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$$\textstyle\smile$$\scriptstyle{\iota}$

In particular, these models also form the following commutative diagram
(simplifying $M_{ij}:=M_{i}\oplus_{M_{0}}^{N}M_{j}$ and where all the arrows
are inclusion maps), where each face of the cube is an $\mathscr{A}$-amalgam:

$\textstyle\smile$${M_{13}}$${N}$${M_{3}}$${M_{23}}$${M_{1}}$${M_{12}}$${M_{0}}$${M_{2}}$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$$\textstyle\smile$

######  Lemma 5.9.

Suppose $\mathscr{A}$ is regular, continuous, absolutely minimal and
3-monotonic. If $M=\bigoplus_{M_{b},i<\alpha}^{M}M_{i}$ and
$N=N^{*}\oplus_{M_{b}}^{N}M$, then
$N=\bigoplus_{N^{*},i<\alpha}^{N}(N^{*}\oplus_{M_{b}}^{N}M_{i})$.

###### Proof.

Let $(M^{\prime}_{i})_{i<s(\alpha)}$ be a continuous resolution of $M$
witnessing that $M$ is the $\mathscr{A}$-amalgam of $(M_{i})_{i<\alpha}$ over
$M_{b}$ by inclusion. As $\mathscr{A}$ is absolutely minimal, we have that
each $M^{\prime}_{\alpha}=\bigoplus_{M_{b},i<\alpha}^{N}M_{i}$. We will prove
the statement by induction on $\alpha$:

  1. (1)

When $\alpha=1$, the statement is trivially true.

  2. (2)

If the $\delta$ is a limit ordinal and the statement is true for all
$\alpha<\delta$, then for each $\alpha<\delta$, we have

$N^{\prime}_{\alpha}:=N^{*}\oplus_{M_{b}}^{N}M^{\prime}_{\alpha}=N^{*}\oplus_{M_{b}}^{N^{\prime}_{\alpha}}\Bigg{(}\bigoplus_{M_{b},i<\alpha}^{N^{\prime}_{\alpha}}M_{i}\Bigg{)}=\bigoplus_{N^{*},i<\alpha}^{N^{\prime}_{\alpha}}(N^{*}\oplus_{M_{b}}^{N^{\prime}_{\alpha}}M_{i})$

Note that as $\mathscr{A}$ is absolutely minimal, we can replace all the
superscript $N^{\prime}_{\alpha}$ by $N$. As a result, we thus have:

    1. (a)

$N^{\prime}_{0}=N^{*}\oplus_{M_{b}}^{N}M^{\prime}_{0}=N^{*}\oplus_{M_{b}}^{N}M_{b}=N^{*}$

    2. (b)

$N^{\prime}_{1}=N^{*}\oplus_{M_{b}}^{N}M_{1}$

    3. (c)

For $1<\alpha<\delta$,
$N^{\prime}_{\alpha}=\bigoplus_{N^{*},i<\alpha}^{N}(N^{*}\oplus_{M_{b}}^{N}M_{i})$

Hence, letting $N^{\prime}_{\delta}:=\bigcup_{i<\delta}N^{\prime}_{\alpha}$,
the sequence $(N^{\prime}_{\alpha})_{\alpha<\delta}$ is a witness to

$N^{\prime}_{\delta}=\bigoplus_{N^{*},\alpha<\delta}^{N}(N^{*}\oplus_{M_{b}}^{N}M_{i})$

But as
$N=N^{*}\oplus_{M_{b}}^{N}\big{(}\bigcup_{\alpha<\delta}M^{\prime}_{\alpha})$,
that $\mathscr{A}$ is continuous and absolutely minimal implies that

$N=\bigcup_{\alpha<\delta}N^{*}\oplus_{M_{b}}^{N}M^{\prime}_{\alpha}=\bigcup_{\alpha<\delta}N^{\prime}_{\alpha}=N^{\prime}_{\delta}$

This completes the proof for the limit step.

  3. (3)

If the inductive hypothesis is true for $\alpha$, then we have

$N^{*}\oplus_{M_{b}}^{N}\Bigg{(}\bigoplus_{M_{b},i<\alpha}^{N}M_{i}\Bigg{)}=\bigoplus_{N^{*},i<\alpha}^{N}(N^{*}\oplus_{M_{b}}^{N}M_{i})$