00footnotetext: Version 2023-04-10. See https://shelah.logic.at/papers/E108/
for possible updates.

# Stable frames and weights
E108

Saharon Shelah  Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 91904, Israel
and
Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA<EMAIL_ADDRESS>http://shelah.logic.at

(Date: December 23, 2015)

###### Abstract.

Was paper 839 in the author-s list till winter 2023 when it was divided to
three.

Part I: We would like to generalize imaginary elements, weight of ${\rm
ortp}(a,M,N),{\mathbf{P}}$-weight, ${\mathbf{P}}$-simple types, etc. from
[She90, Ch.III,V,§4] to the context of good frames. This requires allowing the
vocabulary to have predicates and function symbols of infinite arity, but it
seemed that we do not suffer any real loss.

Part II: Good frames were suggested in [She09d] as the (bare bones) right
parallel among a.e.c. to superstable (among elementary classes). Here we
consider $(\mu,\lambda,\kappa)$-frames as candidates for being the right
parallel to the class of $|T|^{+}$-saturated models of a stable theory (among
elementary classes). A loss as compared to the superstable case is that going
up by induction on cardinals is problematic (for cardinals of small
cofinality). But this arises only when we try to lift. For this context we
investigate the dimension.

Part III: In the context of Part II, we consider the main gap problem for the
parallel of somewhat saturated model; showing we are not worse than in the
first order case.

###### Key words and phrases:

model theory, classification theory, stability, a.e.c., stability,
orthogonality, weight, main gap

###### 2010 Mathematics Subject Classification:

Primary 03C45, 03C48; Secondary: 03C55

The author thanks Alice Leonhardt for the beautiful typing. The author would
like to thank the Israel Science Foundation for partial support of this
research (Grant No. 242/03). First Typed - 2002/Sept/10. Was paper [839] till
winter 203 when by an editor request it was divided to [839], [1248], [1249]

Anotated Content

Part I, pg.Stable frames and weights  E108

§0 Introduction, pg. ‣ 0\. Introduction

§1 Weight and $\mathbf{P}$-weight, pg.5 (labels w(dot),wp(dot) and without
dot), pg.1

  1. [For ${\mathfrak{s}}$ a good $\lambda$-frame with some additional properties we define placed and $\mathbf{P}$-weight.]

§2 Imaginary elements, an ${\rm ess}-(\mu,\lambda)$-a.e.c. and frames, (labels
$m(dot,e(dot),b)$, pg.2

  1. [Define an ${\rm ess}-(\mu,\lambda)$-a.e.c. allowing infinitary functions. Then get ${\mathfrak{s}}$ with type bases.]

§3 $\mathbf{P}$-simple types, pg.18 (label 3.1, on (dot)), pg. 3

Part II Generalizing stable classes, pg.3

§4 Introduction, pg.4

§5 Axiomatize a.e.c. without full continuity, (label f), pg.5

  1. [Smooth out: generalize [She09c, §1].]

§(5A) a.e.c., pg.5(A)

§(5B) Basic Notions, (label 5.15 on), pg.5(B)

§(5C) Liftings, (labels 5.20 on), pg.5(C)

§6 PR frames, (labels pr(dot)), pg.6

  1. [Seems better with NF, here, so earlier;

    1. $(a)$

dominated appear

    2. $(b)$

missing reference

    3. $(c)$

“$P$ based on $\mathbf{a}$”, see I, but by

    4. $(d)$

use places $K_{(M,A)}$ or monsters ${\mathfrak{C}}$.]

Part III, pg.6

§7 Introduction, (7.1-7.4 give labels!), pg.7

§8 Analysis of dimension for $\mathbf{P}$, (label g(dot)), pg.8

  1. [Question: use NF? Place $\mathbf{a}$ or ${\mathfrak{a}}$?

  2. $(a)\quad K_{A}$ or $K_{(M,A)}$ or $K_{M,\infty}$

  3. $(b)\quad$ use monster ${\mathfrak{C}}$ or play…

  4. $(c)\quad$ define $M<_{{\mathfrak{k}}_{A}}N$

  5. $(d)\quad$ m.d. candidate (multi-dimensional)

  6. $(e)\quad(<\kappa)$-based.]

Part IV, pg.8

§9 Strong stability: weak form of superstability [continue [She14], (label
uq(dot)), pg.9

§10 Decomposition (more on main gap + decomposition, continue §7,§8), (label
h(dot)), pg.10

§11 Decompositions, (label dc(dot)), pg.11

Part I: Beautiful frames: weight and simplicity

##  0\. Introduction

We consider here the directions listed in the abstract111As we have started
this in 2002 and have not worked on it for long, we intend to make public what
is in reasonable state.

In part I we assume ${{\mathfrak{s}}}$ is a good $\lambda$-frame with some
extra properties from [She09e], e.g. as in the assumption of [She09e, §12], so
we shall assume knowledge of [She09e], and the basic facts on good
$\lambda$-frames from [She09c].

We can look at results from [She90] which were not regained in beautiful
$\lambda$-frames. Well, of course, we are far from the main gap for the
original ${{\mathfrak{s}}}$ ([She90, Ch.XIII]) and there are results which are
obviously more strongly connected to elementary classes, particularly
ultraproducts. This leaves us with parts of type theory: semi-regular types,
weight, ${\mathbf{P}}$-simple 222The motivation is for suitable $\mathbf{P}$
(e.g. a single regular type) that on the one hand
stp$(a,A)\pm\mathbf{P}\Rightarrow{\rm stp}(a/E,A)$ is $\mathbf{P}$-simple for
some equivalence relation definable over $A$ and on the other hand if ${\rm
stp}(a_{i},A)$ is $\mathbf{P}$-simple for $i<\alpha$ then
$\Sigma\\{w(a_{i},A)\cup\\{a_{j}:j<i\\}):i<\alpha\\}$ does not depend on the
order in which we list the $a_{i}$’s. Note that $\mathbf{P}$ here is
${{\mathscr{P}}}$ there. types, “hereditarily orthogonal to $\mathbf{P}$” (the
last two were defined and investigated in [She78, Ch.V,§0 + Def4.4-Ex4.15],
[She90, Ch.V,§0,pg.226,Def4.4-Ex4.15,pg.277-284]).

Note that “a type $q$ is $p$-simple (or ${\mathbf{P}}$-simple)” and “$q$ is
hereditarily orthogonal to $p$ (or ${\mathbf{P}}$)” are essentially the
333Note, “foreign to $\mathbf{P}$” and “hereditarily orthogonal to
$\mathbf{P}$” are equivalent. Now ($\mathbf{P}=\\{p\\}$ for ease)  $(a)$
$q(x)$ is $p(x)$-simple when for some set $A$, in ${{\mathfrak{C}}}$ we have
$q({{\mathfrak{C}}})\subseteq{\rm acl}(A\cup\bigcup p_{i}({{\mathfrak{C}}}))$
$(b)$ $q(x)$ is $p(x)$-internal when for some set $A$, in ${{\mathfrak{C}}}$
we have $q({{\mathfrak{C}}})\subseteq{\rm dcl}(A\cup p({{\mathfrak{C}}}))$.
Note  $(\alpha)$ internal implies simple $(\beta)$ if we aim at computing
weights it is better to stress acl as it covers more $(\gamma)$ but the
difference is minor and $(\delta)$ in existence it is better to stress dcl,
also it is useful that $\\{F\restriction(p({{\mathfrak{C}}})\cup
q({{\mathfrak{C}}}):F$ an automorphism of ${{\mathfrak{C}}}$ over
$p({{\mathfrak{C}}})\cup{\rm Dom}(p)\\}$ is trivial when $q(x)$ is
$p$-internal but not so for $p$-simple (though form a pro-finite group).
“internal” and “foreign” in Hrushovshi’s profound works.

##  1\. I, Weight and ${\mathbf{P}}$-weight

Recalling [She09c], [She09e]

######  Context 1.1.

1) ${{\mathfrak{s}}}$ is a full good${}^{+}\,\lambda$-frame, with primes,
$K^{3,{\rm vq}}_{{\mathfrak{s}}}=K^{3,{\rm
qr}}_{{\mathfrak{s}}},\bot={\underset{{\rm wk}}{\bot}}$ and $p\bot
M\Leftrightarrow p{\underset{{\rm a}}{\perp}}M$, note that as
${{\mathfrak{s}}}$ is full, ${{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)={{\mathscr{S}}}^{{\rm na}}_{{\mathfrak{s}}}(M)$; also
${\mathfrak{k}}_{{\mathfrak{s}}}={\mathfrak{k}}[{\mathfrak{s}}]=(K^{{\mathfrak{s}}},\leq_{{\mathfrak{k}}_{{\mathfrak{s}}}})$
is the a.e.c.

2) ${{\mathfrak{C}}}$ is an ${{\mathfrak{s}}}$-monster so it is
$K^{{\mathfrak{s}}}_{\lambda^{+}}$-saturated over $\lambda$ and
$M<_{{\mathfrak{s}}}{{\mathfrak{C}}}$ means
$M\leq_{{{\mathfrak{k}}}[{{\mathfrak{s}}}]}{{\mathfrak{C}}}$ and $M\in
K_{{\mathfrak{s}}}$. As ${\mathfrak{s}}$ is full, it has regulars.

######  Observation 1.2.

${{\mathfrak{s}}}^{{\rm reg}}$ satisfies all the above except being full.

###### Proof.

See [She09e, 10.18=L10.p19tex] and Definition [She09e, 10.17=L10.p18tex]. ∎

######  Claim 1.3.

1) If $p\in{{\mathscr{S}}}^{{\rm bs}}_{\mathfrak{s}}(M)$ then we can find
$b,N$ and a finite $\mathbf{J}$ such that:

  1. $\circledast$

$(a)\quad M\leq_{{\mathfrak{s}}}N$

  2. $(b)\quad\mathbf{J}\subseteq N$ is a finite independent set in $(M,N)$

  3. $(c)\quad c\in\mathbf{J}\Rightarrow{\rm ortp}(c,M,N)$ is regular, recalling ${\rm ortp}$ stands for orbital type

  4. $(d)\quad(M,N,\mathbf{J})\in K^{3,{\rm qr}}_{{\mathfrak{s}}}$

  5. $(e)\quad b\in N$ realizes $p$.

2) We can add, if $M$ is brimmed, that

  1. $(f)$

$(M,N,b)\in K^{3,{\rm pr}}_{{\mathfrak{s}}}$.

3) In (2), $|\mathbf{J}|$ depends only on $(p,M)$.

4) If $M$ is brimmed, then we can work in ${\mathfrak{s}}({\rm brim})$ and get
the same $\|\mathbf{J}\|$ and $N$ (so $N\in K_{{\mathfrak{s}}}$) brimmed.

###### Proof..

1) By induction on $\ell<\omega$, we try to choose
$N_{\ell},a_{\ell},q_{\ell}$ such that:

  1. $(*)$

$(a)\quad N_{0}=M$

  2. $(b)\quad N_{\ell}\leq_{{\mathfrak{s}}}N_{\ell+1}$

  3. $(c)\quad q_{\ell}\in{{\mathscr{S}}}_{{\mathfrak{s}}}(N_{\ell})$, so possibly $q_{\ell}\notin{\mathscr{S}}^{{\rm na}}_{{\mathfrak{s}}}(N_{\ell})$

  4. $(d)\quad q_{0}=p$

  5. $(e)\quad q_{\ell+1}\restriction N_{\ell}=q_{\ell}$

  6. $(f)\quad q_{\ell+1}$ forks over $N_{\ell}$ so now necessarily $q_{\ell}\notin{{\mathscr{S}}}^{{\rm na}}_{{\mathfrak{s}}}(N_{\ell})$

  7. $(g)\quad(N_{\ell},N_{\ell+1},a_{\ell})\in K^{3,{\rm pr}}_{{\mathfrak{s}}}$

  8. $(h)\quad r_{\ell}={\rm ortp}(a_{\ell},N_{\ell},N_{\ell+1})$ is regular

  9. $(i)\quad r_{\ell}$ either is $\perp M$ or does not fork over $M$.

If we succeed to carry the induction for all $\ell<\omega$ let
$N=\cup\\{N_{\ell}:\ell<\omega\\}$; as this is a countable chain, there is
$q\in{{\mathscr{S}}}(N)$ such that $\ell<\omega\Rightarrow q\restriction
N_{\ell}=q$ and as $q$ is not algebraic (because each $q_{n}$ is not), and
${{\mathfrak{s}}}$ is full, clearly $q\in{{\mathscr{S}}}_{{\mathfrak{s}}}(N)$;
but $q$ contradicts the finite character of non-forking. So for some $n\geq 0$
we are stuck, but this cannot occur if $q_{n}\in{{\mathscr{S}}}^{{\rm
na}}_{{\mathfrak{s}}}(N_{n})$. [Why? By 1.2, equivalently
${{\mathfrak{s}}}^{{\rm reg}}$ has enough regulars and then we can apply
[She09e, 8.3=L6.1tex].] So for some $b\in N_{n}$ we have $q_{n}={\rm
ortp}(b,N_{n},N_{n})$, i.e., $b$ realizes $q_{n}$ hence it realizes $p$.

Let $\mathbf{J}=\\{a_{\ell}:{\rm ortp}(a_{\ell},N_{\ell},N_{\ell+1})$ does not
fork over $N_{0}\\}$. By [She09e, 6.2] we have
$(M,N_{n},\mathbf{J})=(N_{0},N_{n},\mathbf{J})\in K^{3,{\rm
vq}}_{{\mathfrak{s}}}$ hence $\in K^{3,{\rm qr}}_{{\mathfrak{s}}}$ by [She09e]
so we are done.

2) Let $N,b,\mathbf{J}$ be as in part (1) with $|\mathbf{J}|$ minimal. We can
find $N^{\prime}\leq_{{\mathfrak{s}}}N$ such that $(M,N^{\prime},b)\in
K^{3,{\rm pr}}_{{\mathfrak{s}}}$ and we can find $\mathbf{J}^{\prime}$ such
that $\mathbf{J}^{\prime}\subseteq N^{\prime}$ is independent regular in
$(M,N^{\prime})$ and maximal under those demands. Then we can find
$N^{\prime\prime}\leq_{{\mathfrak{s}}}N^{\prime}$ such that
$(M,N^{\prime\prime},\mathbf{J}^{\prime})\in K^{3,{\rm qr}}_{{\mathfrak{s}}}$.
If ${\rm
ortp}_{{\mathfrak{s}}}(b,N^{\prime\prime},N^{\prime})\in{{\mathscr{S}}}^{{\rm
na}}_{{\mathfrak{s}}}(N^{\prime\prime})$ is not orthogonal to $M$ we can
contradict the maximality of $\mathbf{J}^{\prime}$ in $N^{\prime}$ as in the
proof of part (1), so ${\rm
ortp}_{{\mathfrak{s}}}(b,N^{\prime\prime},N^{\prime})\perp M$ (or
$\notin{{\mathscr{S}}}^{{\rm na}}_{{\mathfrak{s}}}(N)$). Also without loss of
generality $(N^{\prime\prime},N^{\prime},b)\in K^{3,{\rm
pr}}_{{\mathfrak{s}}}$, so by [She09e] we have
$(M,N^{\prime},\mathbf{J}^{\prime})\in K^{3,{\rm qr}}_{{\mathfrak{s}}}$. Hence
there is an isomorphism $f$ from $N^{\prime}$ onto $N^{\prime\prime}$ which is
the identity of $M\cup\mathbf{J}^{\prime}$ (by the uniqueness for $K^{3,{\rm
qr}}_{{\mathfrak{s}}}$). So using $(N^{\prime},f(b),\mathbf{J}^{\prime})$ for
$(N,b,\mathbf{J})$ we are done.

3) If not, we can find $N_{1},N_{2},\mathbf{J}_{1},\mathbf{J}_{2},b$ such that
$M\leq_{{\mathfrak{s}}}N_{\ell}\leq_{{\mathfrak{s}}}N$ and the quadruple
$(M,N_{\ell},\mathbf{J}_{\ell},b)$ is as in (a)-(e)+(f) of part (1)+(2) for
$\ell=1,2$. Assume toward contradiction that
$|\mathbf{J}_{1}|\neq|\mathbf{J}_{2}|$ so without loss of generality
$|\mathbf{J}_{1}|<|\mathbf{J}_{2}|$.

By “$(M,N_{\ell},b)\in K^{3,{\rm pr}}_{{\mathfrak{s}}}$” without loss of
generality $N_{2}\leq_{{\mathfrak{s}}}N_{1}$.

By [She09e, 10.15=L10b.11tex(3)] for some
$c\in\mathbf{J}_{2}\backslash\mathbf{J}_{1},\mathbf{J}_{1}\cup\\{c\\}$ is
independent in $(M,N_{1})$, contradiction to $(M,N,\mathbf{J}_{1})\in
K^{3,{\rm vq}}_{{\mathfrak{s}}}$ by [She09e, 10.15=L10b.11tex(4)].

4) Similarly. ∎

######  Definition 1.4.

1) For $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$, let the weight of
$p,w(p)$ be the unique natural number such that: if
$M\leq_{{\mathfrak{s}}}M^{\prime},M^{\prime}$ is brimmed,
$p^{\prime}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M^{\prime})$ is a
non-forking extension of $p$ then it is the unique $|\mathbf{J}|$ from Claim
1.3(3), it is a natural number.

2) Let $w_{{\mathfrak{s}}}(a,M,N)=w({\rm ortp}_{{\mathfrak{s}}}(a,M,N))$.

######  Claim 1.5.

1) If $p\in{{\mathscr{S}}}^{bs}(M)$ regular, then $w(p)=1$.

2) If $\mathbf{J}$ is independent in $(M,N)$ and $c\in N$, then for some
$\mathbf{J}^{\prime}\subseteq\mathbf{J}$ with $\leq w_{{\mathfrak{s}}}(c,M,N)$
elements, $\\{c\\}\cup(\mathbf{J}\backslash\mathbf{J}^{\prime})$ is
independent in $(M,N)$.

###### Proof..

Easy by now. ∎

Note that the use of ${\mathfrak{C}}$ in Definition 1.6 is for transparency
only and can be avoided, see 1.10 below.

######  Definition 1.6.

1) We say that ${\mathbf{P}}$ is an $M^{*}$-based family (inside
${{\mathfrak{C}}}$) when :

  1. $(a)$

$M^{*}<_{{\mathfrak{k}}[{\mathfrak{s}}]}{{\mathfrak{C}}}$ and $M^{*}\in
K_{{\mathfrak{s}}}$

  2. $(b)$

${\mathbf{P}}\subseteq\cup\\{{\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}(M):M\leq_{{\mathfrak{k}}[{\mathfrak{s}}]}{{\mathfrak{C}}}$
and $M\in K_{{\mathfrak{s}}}\\}$

  3. $(c)$

${\mathbf{P}}$ is preserved by automorphisms of ${{\mathfrak{C}}}$ over
$M^{*}$.

2) Let $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ where
$M\leq_{{\mathfrak{k}}[{\mathfrak{s}}]}{{\mathfrak{C}}}$

  1. $(a)$

we say that $p$ is hereditarily orthogonal to ${\mathbf{P}}$ (or
${\mathbf{P}}$-foreign) when :

if
$M\leq_{{\mathfrak{s}}}N\leq_{{{\mathfrak{k}}}[{{\mathfrak{s}}}]}{{\mathfrak{C}}},q\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(N),q\restriction M=p$, then $q$ is orthogonal to
${\mathbf{P}}$

  2. $(b)$

we say that $p$ is ${\mathbf{P}}$-regular when $p$ is regular, not orthogonal
to ${\mathbf{P}}$ and if $q\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M^{\prime}),M\leq_{{\mathfrak{s}}}M^{\prime}<_{{{\mathfrak{k}}}[{{\mathfrak{s}}}]}{{\mathfrak{C}}}$
and $q$ is a forking extension of $p$ then $q$ is hereditarily orthogonal to
${\mathbf{P}}$

  3. $(c)$

$p$ is weakly ${\mathbf{P}}$-regular if it is regular and is not orthogonal to
some ${\mathbf{P}}$-regular $p^{\prime}$.

3) ${\mathbf{P}}$ is normal when ${\mathbf{P}}$ is a set of regular types and
each of them is ${\mathbf{P}}$-regular.

4) For $q\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M),M<_{{{\mathfrak{k}}}[{{\mathfrak{s}}}]}{{\mathfrak{C}}}$
let $w_{\mathbf{P}}(q)$ be defined as the natural number satisfying the
following

  1. $\circledast$

if
$M\leq_{{\mathfrak{s}}}M_{1}\leq_{{\mathfrak{s}}}M_{2}\leq_{{\mathfrak{s}}}{{\mathfrak{C}}},M_{\ell}$
is $(\lambda,*)$-brimmed, $b\in M_{2},{\rm
ortp}_{{\mathfrak{s}}}(b,M_{1},M_{2})$ is a non-forking extension of
$q,(M_{1},M_{2},b)\in K^{3,{\rm
pr}}_{{\mathfrak{s}}},(M_{1},M_{2},\mathbf{J})\in K^{3,{\rm
qr}}_{{\mathfrak{s}}}$ and $\mathbf{J}$ is regular in $(M_{1},M_{2})$, i.e.
independent and $c\in\mathbf{J}\Rightarrow{\rm
ortp}_{{\mathfrak{s}}}(c,M_{1},M_{2})$ is a regular type then
$w_{\mathbf{P}}(q)=|\\{c\in\mathbf{J}:{\rm ortp}_{{\mathfrak{s}}}(c,M_{1},M)$
is weakly ${\mathbf{P}}$-regular$\\}|$.

5) We replace ${\mathbf{P}}$ by $p$ if ${\mathbf{P}}=\\{p\\}$, where
$p\in{{\mathscr{S}}}^{{\rm bs}}(M^{*})$ is regular (see 1.7(1)).

######  Claim 1.7.

1) If $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ is regular then
$\\{p\\}$ is an $M$-based family and is normal.

2) Assume $\mathbf{P}$ is an $M^{*}$-based family. If
$q\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ and
$M^{*}\leq_{{\mathfrak{s}}}M\leq_{{{\mathfrak{k}}}[{{\mathfrak{s}}}]}{{\mathfrak{C}}}$
then $w_{\mathbf{P}}(q)$ is well defined (and is a natural number).

3) In Definition 1.6(4) we can find $\mathbf{J}$ such that for every
$c\in\mathbf{J}_{1}$ we have: ${\rm ortp}(c,M_{1},M)$ is
weakly-$\mathbf{P}$-regular $\Rightarrow{\rm ortp}(c,M_{1},M)$ is
$\mathbf{P}$-regular.

###### Proof..

Should be clear. ∎

######  Discussion 1.8.

It is tempting to try to generalize the notion of ${\mathbf{P}}$-simple
(${\mathbf{P}}$-internal in Hrushovski’s terminology) and semi-regular. An
important property of those notions in the first order case is that: e.g. if
$(\bar{a}/A)\pm p,p$ regular then for some equivalence relation $E$ definable
over $A,{\rm ortp}(\bar{a}/E,A)$ is $\pm p$ and is $\\{p\\}$-simple. So assume
that $p,q\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ are not
orthogonal, and we can define an equivalence relation
${{\mathscr{E}}}^{p,q}_{M}$ on $\\{c\in{{\mathfrak{C}}}:c$ realizes $p\\}$,
defined by

$\begin{array}[]{clcr}c_{1}{{\mathscr{E}}}^{p,q}_{M}c_{2}&\text{
{\text@underline{iff}}\,\,for every }d\in{{\mathfrak{C}}}\text{ realizing
}q\text{ we have}\\\ &{\rm
ortp}_{{\mathfrak{s}}}(c_{1}d,M,{{\mathfrak{C}}})={\rm
ortp}_{{\mathfrak{s}}}(c_{2}d,M,{{\mathfrak{C}}}).\end{array}$

This may fail (the desired property) even in the first order case: suppose
$p,q$ are definable over $a^{*}\in M$ (on getting this, see later) and we have
$\langle c_{\ell}:\ell\leq n\rangle,\langle M_{\ell}:\ell<n\rangle$ such that
${\rm ortp}(c_{\ell},M_{\ell},{{\mathfrak{C}}})=p_{\ell}$ each $p_{\ell}$ is
parallel to $p,c_{\ell}{{\mathscr{E}}}^{p,q}_{M_{\ell}}c_{\ell+1}$ but
$c_{0},c_{n}$ realizes $p,q$ respectively and $\\{c_{0},c_{n}\\}$ is
independent over $M_{0}$. Such a situation defeats the attempt to define a
${\mathbf{P}}-\\{q\\}$-simple type $p/{{\mathscr{E}}}$ as in [She90, Ch.V].

In first order logic we can find a saturated $N$ and $a^{*}\in N$ such that
${\rm ortp}(M,\bigcup\limits_{\ell}M_{\ell}\cup\\{c_{0},\dotsc,c_{n}\\})$ does
not fork over $a^{*}$ and use “average on the type with an ultrafilter $c$
over $q({{\mathfrak{C}}})+a^{*}_{t}$” (for suitable $a^{*}_{t}$’s). But see
below.

######  Discussion 1.9.

: 1) Assume (${\mathfrak{s}}$ is full and) every $p\in{{\mathscr{S}}}^{{\rm
na}}_{{\mathfrak{s}}}(M)$ is representable by some $a_{p}\in M$ (in [She90],
e.g. the canonical base ${\rm Cb}(p)$). We can define for
$\bar{a},\bar{b}\in{}^{\omega>}{{\mathfrak{C}}}$ when ${\rm
ortp}(\bar{a},\bar{b},{{\mathfrak{C}}})$ is stationary (and/or non-forking).
We should check the basic properties. See §3.

2) Assume $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ is regular,
definable over $\bar{a}^{*}$ (in the natural sense). We may wonder if the
niceness of the dependence relation hold for $p\restriction\bar{a}^{*}$?

If you feel that the use of a monster model is not natural in our context, how
do we “translate” a set of types in ${{\mathfrak{C}}}^{{\rm eq}}$ preserved by
every automorphism of ${{\mathfrak{C}}}$ which is the identity on $A$? by
using a “place” defined by:

######  Definition 1.10.

1) A local place is a pair $\mathbf{a}=(M,A)$ such that $A\subseteq M\in
K_{{\mathfrak{s}}}$ (compare with 8.2.

2) The places $(M_{1},A_{1}),(M_{2},A_{2})$ are equivalent if $A_{1}=A_{2}$
and there are $n$ and $N_{\ell}\in K_{{\mathfrak{s}}}$ for $\ell\leq n$
satisfying $A\subseteq N_{\ell}$ for $\ell=0,\ldots,n$ such that
$M_{1}=N_{0},M_{2}=N_{n}$ and for each
$\ell<n,N_{\ell}\leq_{{\mathfrak{s}}}N_{\ell+1}$ or
$N_{\ell+1}\leq_{{\mathfrak{s}}}N_{\ell}$. We write
$(M_{1},A_{1})\sim(M_{2},A_{2})$ or $M_{1}\sim_{A_{1}}M_{2}$.

3) For a local place $\mathbf{a}=(M,A)$ let
$K_{\mathbf{a}}=K_{(M,A)}=\\{N:(N,A)\sim(M,A)\\}$; so in $(M,A)/\sim$ we fix
both $A$ as a set and the type it realizes in $M$ over $\emptyset$.

4) We call such class $K_{\mathbf{a}}$ a place.

5) We say that ${\mathbf{P}}$ is an invariant set444Really a class of types in
a place $K_{(M,A)}$ when :

  1. $(a)$

${\mathbf{P}}\subseteq\\{{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(N):N\sim_{A}M\\}$

  2. $(b)$

membership in ${\mathbf{P}}$ is preserved by isomorphism over $A$

  3. $(c)$

if $N_{1}\leq_{{\mathfrak{s}}}N_{2}$ are both in
$K_{(M,A)},p_{2}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(N_{2})$ does
not fork over $N_{1}$ then $p_{2}\in{\mathbf{P}}\Leftrightarrow
p_{2}\restriction N_{1}\in{\mathbf{P}}$.

6) We say $M\in K_{{\mathfrak{s}}}$ is brimmed over $A$ when for some $N$ we
have $A\subseteq N\leq_{{\mathfrak{s}}}M$ and $M$ is brimmed over $N$.

######  Claim/Definition 1.11.

1) If $A\subseteq M\in K_{{\mathfrak{s}}}$ and
$\mathbf{P}_{0}\subseteq{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ then
there is at most one invariant set $\mathbf{P}^{\prime}$ of types in the place
$K_{(M,A)}$ such that $\mathbf{P}^{+}\cap{\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)=\mathbf{P}_{0}$ and $M\leq_{{\mathfrak{s}}}N\wedge
p\in\mathbf{P}^{+}\cap{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(N)\Rightarrow$
($p$ does not fork over $M$).

2) If in addition $M$ is brimmed555$M$ is brimmed over $A$ means that for some
$M_{1},A\subseteq M_{1}\leq_{{\mathfrak{s}}}M$ and $M$ is brimmed over
$M_{1}$. over $A$ then we can omit the last demand in part (1).

3) If $\mathbf{a}=(M_{1},A),(M_{2},A)\in K_{\mathbf{a}}$ then
$K_{(M_{2},A)}=K_{\mathbf{a}}$.

###### Proof..

Easy. ∎

######  Definition 1.12.

1) If in 1.11 there are such $\mathbf{P}$, we denote it by ${\rm
inv}(\mathbf{P}_{0})={\rm inv}(\mathbf{P}_{0},M)$.

2) If $\mathbf{P}_{0}=\\{p\\}$, then let ${\rm inv}(p)={\rm inv}(p,M)={\rm
inv}(\\{p\\})$.

3) We say $p\in{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ does not split
(or is definable) over $A$ when ${\rm inv}(p)$ is well defined.

##  2\. I Imaginary elements, an essential-$(\mu,\lambda)$-a.e.c. and frames

###  2(A). Essentially a.e.c.

We consider revising the definition of a.e.c. ${{\mathfrak{k}}}$, by allowing
function symbols in $\tau_{{\mathfrak{k}}}$ with infinite number of places
while retaining local characters, e.g., if $M_{n}\leq
M_{n+1},M=\cup\\{M_{n}:n<\omega\\}$ is uniquely determined. Before this we
introduce the relevant equivalence relations. In this context, we can give
name to equivalence classes for equivalence relations on infinite sequences.

######  Definition 2.1.

We say that ${{\mathfrak{k}}}$ is an essentially $[\lambda,\mu)$-a.e.c. or
ess-$[\lambda,\mu)$-a.e.c. and we may write $(\mu,\lambda)$ instead of
$[\lambda,\mu)$ if ($\lambda<\mu$ and) it is an object consisting of:

  1. $I(a)$

a vocabulary $\tau=\tau_{{\mathfrak{k}}}$ which has predicates and function
symbols of possibly infinite arity but $\leq\lambda$

  2. $(b)$

a class of $K=K_{{\mathfrak{k}}}$ of $\tau$-models

  3. $(c)$

a two-place relation $\leq_{{\mathfrak{k}}}$ on $K$

such that

  1. $II(a)$

if $M_{1}\cong M_{2}$ then $M_{1}\in K\Leftrightarrow M_{2}\in K$

  2. $(b)$

if $(N_{1},M_{1})\cong(N_{2},M_{2})$ then
$M_{1}\leq_{{\mathfrak{k}}}N_{1}\Leftrightarrow
M_{2}\leq_{{\mathfrak{k}}}N_{2}$

  3. $(c)$

every $M\in K$ has cardinality $\in[\lambda,\mu)$

  4. $(d)$

$\leq_{{\mathfrak{k}}}$ is a partial order on $K$

  5. $III_{1}$

if $\langle M_{i}:i<\delta\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing and
$|\bigcup\limits_{i<\delta}M_{i}|<\mu$ then there is a unique $M\in K$ such
that $|M|=\cup\\{|M_{i}|:i<\delta\\}$ and $i<\delta\Rightarrow
M_{i}\leq_{{\mathfrak{k}}}M$

  6. $III_{2}$

if in addition $i<\delta\Rightarrow M_{i}\leq_{{\mathfrak{k}}}N$ then
$M\leq_{{\mathfrak{k}}}N$

  7. $IV$

if $M_{1}\subseteq M_{2}$ and $M_{\ell}\leq_{{\mathfrak{k}}}N$ for $\ell=1,2$
then $M_{1}\leq_{{\mathfrak{k}}}M_{2}$

  8. $V$

if $A\subseteq N\in K$, then there is $M$ satisfying $A\subseteq
M\leq_{{\mathfrak{k}}}N$ and $\|M\|\leq\lambda+|A|$ (here it is enough to
restrict ourselves to the case $|A|\leq\lambda$).

######  Definition 2.2.

1) We say ${{\mathfrak{k}}}$ is an ess-$\lambda$-a.e.c. if it is an
ess-$[\lambda,\lambda^{+})$-a.e.c.

2) We say ${{\mathfrak{k}}}$ is an ess-a.e.c. if there is $\lambda$ such that
it is an ess-$[\lambda,\infty)$-a.e.c., so $\lambda={\rm
LST}({{\mathfrak{k}}})$.

3) If ${\mathfrak{k}}$ is an ess-$[\lambda,\mu)$-a.e.c. and
$\lambda\leq\lambda_{1}<\mu_{1}\leq\mu$ then let
$K^{{\mathfrak{k}}}_{\lambda_{1}}=(K_{{\mathfrak{k}}})_{\lambda_{1}}=\\{M\in
K_{{\mathfrak{k}}}:\|M\|=\lambda_{1}\\},K^{{\mathfrak{k}}}_{\lambda_{1},\mu_{1}}=\\{M\in
K_{{\mathfrak{k}}}:\lambda_{1}\leq\|M\|<\mu_{1}\\}$.

4) We define $\Upsilon^{{\rm or}}_{{\mathfrak{k}}}$ as in [She09b,
0.8=L11.1.3A(2)].

5) We may omit the “essentially” when ${\rm
arity}(\tau_{{\mathfrak{k}}})=\aleph_{0}$ where ${\rm
arity}({\mathfrak{k}})={\rm arity}(\tau_{{\mathfrak{k}}})$ and for vocabulary
$\tau,{\rm arity}(\tau)=\min\\{\kappa$: every predicate and function symbol
have ${\rm arity}<\kappa\\}$.

We now consider the claims on ess-a.e.c.

######  Claim 2.3.

Let ${{\mathfrak{k}}}$ be an ess-$[\lambda,\mu)$-a.e.c.

1) The parallel of ${\rm Ax}(III)_{1},(III)_{2}$ holds with a directed family
$\langle M_{t}:t\in I\rangle$.

2) If $M\in K$ we can find $\langle
M_{\bar{a}}:\bar{a}\in{}^{\omega>}M\rangle$ such that:

  1. $(a)$

$\bar{a}\subseteq M_{\bar{a}}\leq_{{\mathfrak{k}}}M$

  2. $(b)$

$\|M_{\bar{a}}\|=\lambda$

  3. $(c)$

if $\bar{b}$ is a permutation of $\bar{a}$ then $M_{\bar{a}}=M_{\bar{b}}$

  4. $(d)$

if $\bar{a}$ is a subsequence of $\bar{b}$ then
$M_{\bar{a}}\leq_{{\mathfrak{k}}}M_{\bar{b}}$.

3) If $N\leq_{{\mathfrak{k}}}M$ we can add in (2) that
$\bar{a}\in{}^{\omega>}N\Rightarrow M_{\bar{a}}\subseteq N$.

4) If for simplicity
$\lambda_{*}=\lambda+\sup\\{\Sigma\\{|R^{M}|:R\in\tau_{{\mathfrak{k}}}\\}+\Sigma\\{|F^{M}|:F\in\tau_{{\mathfrak{k}}}\\}:M\in
K_{{\mathfrak{k}}}$ has cardinality $\lambda\\}$ then $K_{{\mathfrak{k}}}$ and
$\\{(M,N):N\leq_{{\mathfrak{k}}}M\\}$ essentially are ${\rm
PC}_{\chi,\lambda_{*}}$-classes where $\chi=|\\{M/\cong:M\in
K^{{\mathfrak{k}}}_{\lambda}\\}|$, noting that $\chi\leq 2^{2^{\theta}}$. That
is, $\langle M_{\bar{a}}:\bar{a}\in{}^{\omega>}A\rangle$ satisfying clauses
(b),(c),(d) of part (2) such that
$A=\cup\\{|M_{\bar{a}}|:\bar{a}\in{}^{\omega>}A\\}$ represent a unique $M\in
K_{{\mathfrak{k}}}$ with universe $A$ and similarly for
$\leq_{{\mathfrak{k}}}$, (on the Definition of ${\rm PC}_{\chi,\lambda_{*}}$,
see [She09a, 1.4(3)]). Note that if in $\tau_{{\mathfrak{k}}}$ there are no
two distinct symbols with the same interpretation in every $M\in
K_{{\mathfrak{k}}}$ then $|\tau){k_{*}}|\leq 2^{2^{\lambda}}$.

5) The results on omitting types in [She99] or [She09b, 0.9=L0n.8,0.2=0n.11]
hold, i.e., if $\alpha<(2^{\lambda_{*}})^{+}\Rightarrow
K^{{\mathfrak{k}}}_{\beth_{\alpha}}\neq\emptyset$ then
$\theta\in[\lambda,\mu)\Rightarrow K_{\theta}\neq\emptyset$ and there is an
${\rm EM}$-model, i.e., $\Phi\in\Upsilon^{{\rm or}}_{{\mathfrak{k}}}$ with
$|\tau_{\Phi}|=|\tau_{{\mathfrak{k}}}|+\lambda$ and ${\rm EM}(I,\Phi)$ having
cardinality $\lambda+|I|$ for any linear order $I$.

6) The lemma on the equivalence of being universal model homogeneous and of
being saturated (see [She09f, 3.18=3.10] or [She09c, 1.14=L0.19]) holds.

7) We can generalize the results of [She09c, §1] on deriving an
ess-$(\infty,\lambda)$-a.e.c. from an ess $\lambda$-a.e.c.

###### Proof..

The same proofs, on the generalization in 2.3(7), see in §5 below. The point
is that, in the term of §5, our ${\mathfrak{k}}$ is a
$(\lambda,\mu,\kappa)$-a.e.c. (automatically with primes). ∎

######  Remark 2.4.

1) In 2.3(4) we can decrease the bound on $\chi$ if we have more nice
definitions of $K_{\lambda}$, e.g., if ${\rm arity}(\tau)\leq\kappa$ then
$\chi=2^{(\lambda^{<\kappa}+|\tau|)}$ where ${\rm arity}(\tau)=\min\\{\kappa$:
every predicate and function symbol of $\tau$ has ${\rm arity}<\kappa\\}$.

2) We may use above $|\tau_{{\mathfrak{s}}}|\leq\lambda,{\rm
arity}(\tau_{{\mathfrak{k}}})=\aleph_{0}$ to get that $\\{(M,\bar{a})/\cong
M\in K^{{\mathfrak{k}}}_{\lambda},\bar{a}\in{}^{\lambda}M$ list $M\\}$ has
cardinality $\leq 2^{\lambda}$. See also 2.18.

3) In 2.10 below, if we omit “${\mathbb{E}}$ is small” and
$\lambda_{1}=\sup\\{|{\rm seq}(M)/{\mathbb{E}}_{M}|:M\in
K^{{\mathfrak{k}}}_{\lambda}\\}$ is $<\mu$ then
${\mathfrak{k}}_{[\lambda_{1},\mu)}$ is an ess-$[\lambda_{1},\mu)$-a.e.c.

4) In Definition 2.1, we may omit axiom V and define ${\rm
LST}({\mathfrak{k}})\in[\lambda,\mu]$ naturally, and if $M\in
K^{{\mathfrak{k}}}_{\lambda}\Rightarrow\mu>|{\rm seq}(M)/{\mathbb{E}}_{M}|$
then in 2.10(1) below we can omit “${\mathbb{E}}$ is small”.

5) Can we preserve in such “transformation” the arity finiteness? A natural
candidate is trying to code $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)$ by $\\{\bar{a}:\bar{a}\in{}^{\omega>}M\\}$ and there
are $M_{0}\leq_{{\mathfrak{s}}}M_{1}$ such that $M\leq_{{\mathfrak{s}}}M_{1}$
and ${\rm ortp}(a_{\ell},M_{0},M_{1})$ is parallel to $p$ and $\bar{a}$ is
independent in $(M_{0},M_{1})\\}$. If e.g., $K_{{\mathfrak{s}}}$ is saturated
this helps but still we suspect it may fail.

6) What is the meaning of ess-$[\lambda,\mu)$-a.e.c.? Can we look just at
$\langle M_{t}:t\in I\rangle,I$ directed, $t\leq_{I}s\Rightarrow
M_{t}\leq_{{\mathfrak{s}}}M_{{\mathfrak{s}}}\in K_{\lambda}$? But for
isomorphism types we take a kind of completion and so make more pairs
isomorphic but $\bigcup\limits_{t\in I}M_{t}$ does not determine
$\bar{M}=\langle M_{t}:t\in I\rangle$ and the completion may depend on this
representation.

7) If we like to avoid this and this number is $\lambda^{\prime}$, then we
should change the definition of ${\rm seq}(N)$ (see 2.5(b)) to ${\rm
seq}^{\prime}(N)=\\{\bar{a}:\ell g(\bar{a})=\lambda$ and for some
$M\leq_{{\mathfrak{s}}}N$ from $K^{{\mathfrak{k}}}_{\lambda},\langle
a_{1+\alpha}:\alpha<\lambda\rangle$ list the members of $M$ and
$a_{0}\in\\{\gamma:\gamma<\mu_{*}\\}$.

###  2(B). Imaginary Elements and Smooth Equivalent Relations

Now we return to our aim of getting canonical base for orbital types.

######  Definition 2.5.

Let ${{\mathfrak{k}}}=(K_{{\mathfrak{k}}},\leq_{{\mathfrak{k}}})$ be a
$\lambda$-a.e.c. or just ess-$[\lambda,\mu)$-a.e.c. (if
${{\mathfrak{k}}}_{\lambda}={{\mathfrak{k}}}_{{\mathfrak{s}}}$ we may write
${{\mathfrak{s}}}$ instead of ${{\mathfrak{k}}}_{\lambda}$, see 2.11). We say
that ${\mathbb{E}}$ is a smooth ${{\mathfrak{k}}}_{\lambda}$-equivalence
relation when :

  1. $(a)$

${\mathbb{E}}$ is a function with domain $K_{{\mathfrak{k}}}$ mapping $M$ to
${\mathbb{E}}_{M}$

  2. $(b)$

for $M\in K_{{\mathfrak{k}}},{\mathbb{E}}_{M}$ is an equivalence relation on a
subset of ${\rm seq}(M)=\\{\bar{a}:\bar{a}\in{}^{\lambda}M$ and
$M\restriction{\rm Rang}(\bar{a})\leq_{{\mathfrak{k}}}M\\}$ so $\bar{a}$ is
not necessarily without repetitions; note that ${\mathfrak{k}}$ determines
$\lambda$, pedantically when non-empty

  3. $(c)$

if $M_{1}\leq_{{\mathfrak{k}}}M_{2}$ then
${\mathbb{E}}_{M_{2}}\restriction{\rm seq}(M_{1})={\mathbb{E}}_{M_{1}}$

  4. $(d)$

if $f$ is an isomorphism from $M_{1}\in K_{{\mathfrak{s}}}$ onto $M_{2}$ then
$f$ maps ${\mathbb{E}}_{M_{1}}$ onto ${\mathbb{E}}_{M_{2}}$

  5. $(e)$

if $\langle M_{\alpha}:\alpha\leq\delta\rangle$ is
$\leq_{{\mathfrak{s}}}$-increasing continuous then
$\\{\bar{a}/{\mathbb{E}}_{M_{\delta}}:\bar{a}\in{\rm
seq}(M_{\delta})\\}=\\{\bar{a}/{\mathbb{E}}_{M_{\delta}}:\bar{a}\in\bigcup\limits_{\alpha<\delta}{\rm
seq}(M_{\alpha})\\}$.

2) We say that ${\mathbb{E}}$ is small if each ${\mathbb{E}}_{M}$ has
$\leq\|M\|$ equivalence classes.

######  Remark 2.6.

1) Note that if we have $\langle{\mathbb{E}}_{i}:i<i^{*}\rangle$, each
${\mathbb{E}}_{i}$ is a smooth ${{\mathfrak{k}}}_{\lambda}$-equivalence
relation and $i^{*}<\lambda^{+}$ then we can find a smooth
${{\mathfrak{k}}}_{\lambda}$-equivalence relation ${\mathbb{E}}$ such that
essentially the ${\mathbb{E}}_{M}$-equivalence classes are the
$\mathbb{E}_{i}$-equivalence classes for $i<i^{*}$; in detail: without loss of
generality $i^{*}\leq\lambda$ and $\bar{a}{\mathbb{E}}_{M}\bar{b}$ iff $\ell
g(\bar{a})=\ell g(\bar{b})$ and

  1. $\circledast_{1}$

$i(\bar{a})=i(\bar{b})$ and if $i(\bar{a})<i^{*}$ then
$\bar{a}\restriction[1+i(\bar{a})+1,\lambda){\mathbb{E}}_{i(\bar{a})}\bar{b}\restriction[1+i(\bar{b})+1,\lambda)$
where $i(\bar{a})={\rm Min}\\{j:(j+1<i(*))\wedge a_{0}\neq a_{1+j}$ or
$j=\lambda\\}$.

2) In fact $i^{*}\leq 2^{\lambda}$ is O.K., e.g. choose a function
$\mathbf{e}$ from $\\{e:e$ an equivalence relation on $\lambda$ to $i^{*}$ and
for $\bar{a},\bar{b}\in{\rm seq}(M)$ we let
$i(\bar{a})=\mathbf{e}(\\{(i,j):a_{2i+1}=a_{2j+1}\\}$ and

  1. $\circledast_{2}$

$\bar{a}\in{\mathbb{E}}_{M}\bar{b}$ iff $i(\bar{a})=i(\bar{b})$ and $\langle
a_{2i}:i<\lambda\rangle{\mathbb{E}}_{i(\bar{a})}\langle
b_{2i}:i<\lambda\rangle$.

3) We can redefine ${\rm seq}(M)$ as ${}^{\lambda\geq}M$, then have to make
minor changes above.

######  Definition 2.7.

Let ${{\mathfrak{k}}}$ be a $\lambda$-a.e.c. or just
ess-$[\lambda,\mu)$-a.e.c. and ${\mathbb{E}}$ a small smooth
${{\mathfrak{k}}}$-equivalence relation and the reader may assume for
simplicity that the vocabulary $\tau_{{\mathfrak{k}}}$ has only predicates.
Also assume $F_{*},c_{*},P_{*}\notin\tau_{{\mathfrak{k}}}$. We define
$\tau_{*}$ and
${{\mathfrak{k}}}_{*}={{\mathfrak{k}}}\langle{\mathbb{E}}\rangle=(K_{{\mathfrak{k}}_{*}},\leq_{{\mathfrak{k}}_{*}})$
as follows:

  1. $(a)$

$\tau^{*}=\tau\cup\\{F_{*},c_{*},P_{*}\\}$ with $P_{*}$ a unary predicate,
$c_{*}$ an individual constant and $F_{*}$ a $\lambda$-place function symbol

  2. $(b)$

$K_{{\mathfrak{k}}_{*}}$ is the class of $\tau^{*}_{{\mathfrak{k}}}$-models
$M^{*}$ such that for some model $M\in K_{{\mathfrak{k}}}$ we have:

    1. $(\alpha)$

$|M|=P^{M^{*}}_{*}$

    2. $(\beta)$

if $R\in\tau$ then $R^{M^{*}}=R^{M}$

    3. $(\gamma)$

if $F\in\tau$ has arity $\alpha$ then $F^{M^{*}}\restriction M=F^{M}$ and for
any $\bar{a}\in{}^{\alpha}(M^{*}),\bar{a}\notin{}^{\alpha}M$ we have
$F^{M^{*}}(\bar{a})=c^{M^{*}}_{*}$ (or allow partial function or use
$F^{M^{*}}(\bar{a})=a_{0}$ when $\alpha>0$ and $F^{M^{*}}(\langle\rangle)$
when $\alpha=0$, i.e. $F$ is an individual constant);

    4. $(\delta)$

$F_{*}$ is a $\lambda$-place function symbol and:

    5. $(i)\quad$ if $\bar{a}\in{\rm seq}(M)$ then $F^{M^{*}}_{*}(\bar{a})\in|M^{*}|\backslash|M|\backslash\\{c^{M^{*}}_{*}\\}$

    6. $(ii)\quad$ if $\bar{a},\bar{b}\in{\rm Dom}({\mathbb{E}})\subseteq{\rm seq}(M)$ then $F^{M^{*}}_{*}(\bar{a})=F^{M^{*}}_{*}(\bar{b})\Leftrightarrow\bar{a}{\mathbb{E}}_{M}\bar{b}$

    7. $(iii)\quad$ if $\bar{a}\in{}^{\lambda}(M^{*})$ and $\bar{a}\notin{\rm Dom}({\mathbb{E}})\subseteq{\rm seq}(M)$ then $F^{M^{*}}_{*}(\bar{a})=c_{*}^{M^{*}}$

    8. $(\varepsilon)$

$c^{M^{*}}_{*}\notin|M|$ and if
$b\in|M^{*}|\backslash|M|\backslash\\{c^{M^{*}}_{*}\\}$ then for some
$\bar{a}\in{\rm Dom}({\mathbb{E}})\subseteq{\rm seq}(M)$ we have
$F^{M^{*}}_{*}(\bar{a})=b$

  3. $(c)$

$\leq_{{\mathfrak{k}}_{*}}$ is the two-place relation on
$K_{{\mathfrak{k}}_{*}}$ defined by: $M^{*}\leq_{{\mathfrak{k}}_{*}}N^{*}$ if

    1. $(\alpha)$

$M^{*}\subseteq N^{*}$ and

    2. $(\beta)$

for some $M,N\in{{\mathfrak{k}}}$ as in clause (b) we have
$M\leq_{{\mathfrak{k}}}N$.

######  Definition 2.8.

1) In 2.7(1) we call $M\in{{\mathfrak{k}}}$ a witness for $M^{*}\in
K_{{\mathfrak{k}}_{*}}$ if they are as in clause (b) above.

2) We call $M\leq_{{\mathfrak{k}}}N$ witness for
$M^{*}\leq_{{{\mathfrak{k}}}^{*}_{\lambda}}N^{*}$ if they are as clause (c)
above.

######  Discussion 2.9.

Up to now we have restricted ourselves to vocabularies with each predicate and
function symbol of finite arity, and this restriction seems very reasonable.
Moreover, it seems a priori that for a parallel to superstable, it is quite
undesirable to have infinite arity. Still our desire to have imaginary
elements (in particular canonical basis for types) forces us to accept them.
The price is that inthe class of $\tau$-models the union of increasing chains
of $\tau$-models is not a well defined $\tau$-model, more accurately we can
show its existence, but not smoothness; however inside the class
${{\mathfrak{k}}}$ it will be.

######  Claim 2.10.

1) If ${{\mathfrak{k}}}$ is a $[\lambda,\mu)$-a.e.c. or just an
ess-$[\lambda,\mu)$-a.e.c. and ${\mathbb{E}}$ a small smooth
${{\mathfrak{k}}}$-equivalence relation then
${{\mathfrak{k}}}\langle{\mathbb{E}}\rangle$ is an ess-$[\lambda,\mu)$-a.e.c.

2) If ${{\mathfrak{k}}}$ has amalgamation and ${\mathbb{E}}$ is a small
${{\mathfrak{k}}}$-equivalence class then
${{\mathfrak{k}}}\langle{\mathbb{E}}\rangle$ has amalgamation property.

###### Proof..

The same proofs. Left as an exercise to the reader. ∎

###  2(C). Good Frames

Now we return to good frames.

######  Definition 2.11.

1) We say that ${{\mathfrak{s}}}$ is a good ess-$[\lambda,\mu)$-frame if
Definition [She09c, 2.1=L1.1tex] is satisfied except that:

  1. $(a)$

in clause (A),
${\mathfrak{K}}_{{\mathfrak{s}}}=(K_{{\mathfrak{s}}},\leq_{{\mathfrak{s}}}),{\mathfrak{k}}$
is an ess-$[\lambda,\mu)$-a.e.c. and ${\mathfrak{K}}[{\mathfrak{s}}]$ is an
ess-$(\infty,\lambda)$-a.e.c.

  2. $(b)$

$K_{{\mathfrak{s}}}$ has a superlimit model in $\chi$ in every
$\chi\in[\lambda,\mu)$

  3. $(c)$

$K^{{\mathfrak{s}}}_{\lambda}/\cong$ has cardinality $\leq 2^{\lambda}$, for
convenience.

######  Discussion 2.12.

We may consider other relatives as our choice and mostly have similar results.
In particular:

  1. $(a)$

we can demand less: as in [SV, §2] we may replace ${\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}$ by a formal version of ${\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}$

  2. $(b)$

we may demand goodness only for ${\mathfrak{s}}_{\lambda}$, i.e.
${\mathfrak{s}}$ restriction the class of models to
$K^{{\mathfrak{s}}}_{\lambda}$ and have only the formal properties above so
amalgamation and JEP are required only for models of cardinality $\lambda$.

######  Claim 2.13.

All the definitions and results in [She09c], [She09e] and §1 here work for
good ess-$[\lambda,\mu)$-frames.

###### Proof..

No problem. ∎

######  Definition 2.14.

If ${{\mathfrak{s}}}$ is a $[\lambda,\mu)$-frame or just an
ess-$[\lambda,\mu)$-frame and ${\mathbb{E}}$ a small smooth
${{\mathfrak{s}}}$-equivalence relation then let
${{\mathfrak{t}}}={{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$ be defined by:

  1. $(a)$

${{\mathfrak{k}}}_{{\mathfrak{t}}}={{\mathfrak{k}}}_{{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$

  2. $(b)$

${{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{t}}}(M^{*})=\\{{\rm
ortp}_{{{\mathfrak{k}}}_{{\mathfrak{t}}}}(a,M^{*},N^{*}):M^{*}\leq_{{{\mathfrak{k}}}_{{\mathfrak{t}}}}N^{*}$
and if $M\leq_{{\mathfrak{k}}}N$ witness
$M^{*},N^{*}\in{{\mathfrak{k}}}_{{\mathfrak{t}}}$ then $a\in N\backslash M$
and ${\rm ortp}_{{\mathfrak{s}}}(a,M,N)\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)\\}$

  3. $(c)$

non-forking similarly.

######  Remark 2.15.

We may add: if ${{\mathfrak{s}}}$ is666The reader may ignore this version. an
NF-frame we define
${{\mathfrak{t}}}={{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$ as an NF-frame
similarly, see [She09e].

######  Claim 2.16.

1) If ${{\mathfrak{s}}}$ is a good ess-$[\lambda,\mu)$-frame, ${\mathbb{E}}$ a
small, smooth ${{\mathfrak{s}}}$-equivalence relation then
${{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$ is a good
ess-$[\lambda,\mu)$-frame.

2) In part (1) for every
$\kappa,\dot{I}(\kappa,K^{{{\mathfrak{s}}}<{\mathbb{E}}>})=\dot{I}(\kappa,K^{{\mathfrak{s}}})$.

3) If ${{\mathfrak{s}}}$ has primes/regulars then
${{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$ has.

######  Remark 2.17.

We may add: if ${{\mathfrak{s}}}$ is an NF-frame then so is
${{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$, hence
$({{\mathfrak{s}}}\langle{\mathbb{E}}\rangle)^{{\rm full}}$ is a full NF-
frame; see [She09e].

###### Proof..

Straightforward. ∎

Our aim is to change ${{\mathfrak{s}}}$ inessentially such that for every
$p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ there is a canonical
base, etc. The following claim shows that in the context we have presented
this can be done.

######  Claim 2.18.

The imaginary elements Claim

Assume ${{\mathfrak{s}}}$ a good $\lambda$-frame or just a good
ess-$[\lambda,\mu)$-frame.

1) If $M_{*}\in K_{{\mathfrak{s}}}$ and $p^{*}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M_{*})$, then777note that there may well be an
automorphism of $M^{*}$ which maps $p^{*}$ to some
$p^{**}\in{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(M^{*})$ such that
$p^{**}\neq p^{*}$. there is a small, smooth
${\mathfrak{k}}_{{\mathfrak{s}}}$-equivalence relation
${\mathbb{E}}={\mathbb{E}}_{{\mathfrak{s}},M_{*},p^{*}}$ and function
$\mathbf{F}$ such that:

  1. $(*)$

if $M_{*}\leq_{{\mathfrak{s}}}N$ and $\bar{a}\in{\rm seq}(N)$ so
$M=:N\restriction{\rm Rang}(\bar{a})\leq_{{\mathfrak{s}}}N$ and $M\cong
M_{*}$, then

    1. $(\alpha)$

$\mathbf{F}(N,\bar{a})$ is well defined iff $\bar{a}\in{\rm
Dom}({\mathbb{E}}_{N})$ and then $\mathbf{F}(N,\bar{a})$ belongs to
${{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(N)$

    2. $(\beta)$

$S\subseteq\\{(N,\bar{a},p):N\in K_{{\mathfrak{s}}},\bar{a}\in{\rm
Dom}({\mathbb{E}}_{N})\\}$ is the minimal class such that:

    3. $(i)\quad$ if $\bar{a}\in{\rm seq}(M_{*})$ and $p$ does not fork over $M_{*}{\restriction}{\rm Rang}(\bar{a})$ then

$(M_{*},\bar{a},p)\in S$

    4. $(ii)\quad S$ is closed under isomorphisms

    5. $(iii)\quad$ if $N_{1}\leq_{{\mathfrak{s}}}N_{2},p_{2}\in{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(N_{2})$ does not fork over $\bar{a}\in{\rm seq}(N_{1})$ then

$(N_{2},\bar{a},p_{2})\in
S\Leftrightarrow(N_{1},\bar{a},p_{2}{\restriction}N_{1})\in S$

    6. $(iv)\quad$ if $\bar{a}_{1},\bar{a}_{2}\in{\rm seq}(N),p\in{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(N)$ does not fork over $N{\restriction}{\rm Rang}(\bar{a}_{\ell})$

for $\ell=1,2$ then $(N_{2},\bar{a}_{1},p)\in
S\Leftrightarrow(N_{2},\bar{a}_{2},p)\in S$

    7. $(\gamma)$

$\mathbf{F}(N,\bar{a})=p$ iff $(N,\bar{a},p)\in S$ hence if
$\bar{a},\bar{b}\in{\rm seq}(N)$ then: $\bar{a}{\mathbb{E}}_{N}\bar{b}$ iff
$\mathbf{F}(\bar{a},N)=\mathbf{F}(\bar{b},N)$.

2) There are unique small888for small we use stability in $\lambda$ smooth
${\mathbb{E}}$-equivalence relation ${\mathbb{E}}$ called
${\mathbb{E}}_{{\mathfrak{s}}}$ and function $\mathbf{F}$ such that:

  1. $(**)(\alpha)$

$\mathbf{F}(N,\bar{a})$ is well defined iff $N\in K_{{\mathfrak{s}}}$ and
$\bar{a}\in{\rm seq}(N)$

  2. $(\beta)$

$\mathbf{F}(N,\bar{a})$, when defined, belongs to ${{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(N)$

  3. $(\gamma)$

if $N\in K_{{\mathfrak{s}}}$ and $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(N)$ then there is $\bar{a}\in{\rm seq}(N)$ such that
${\rm Rang}(\bar{a})=N$ and $\mathbf{F}(N,\bar{a})=p$

  4. $(\delta)$

if $\bar{a}\in{\rm seq}(M)$ and $M\leq_{{\mathfrak{s}}}N$ then
$\mathbf{F}(N,\bar{a})$ is (well defined and is) the non-forking extension of
$\mathbf{F}(M,\bar{a})$

  5. $(\varepsilon)$

if $\bar{a}_{\ell}\in{\rm seq}(N)$ and $\mathbf{F}(N,\bar{a}_{\ell})$ is well
defined for $\ell=1,2$ then
$\bar{a}_{1}{\mathbb{E}}_{N}\bar{a}_{2}\Leftrightarrow\mathbf{F}(N,\bar{a}_{1})=\mathbf{F}(N,\bar{a}_{2})$

  6. $(\zeta)$

$\mathbf{F}$ commute with isomorphisms.

3) For ${{\mathfrak{t}}}={{\mathfrak{s}}}\langle{\mathbb{E}}\rangle$ where
${\mathbb{E}}$ as in part (2) and $M^{*}\in K_{{\mathfrak{t}}}$ as witnessed
by $M\in K_{{\mathfrak{s}}}$ and $p^{*}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{t}}}(M^{*})$ is projected to $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)$ let ${\rm bas}(p^{*})={\rm
bas}(p)=\mathbf{F}(\bar{a},M^{*})/{\mathbb{E}}$ whenever
$\mathbf{F}(M,\bar{a})=p$. That is, assume $M_{\ell}$ witness that
$M^{*}_{\ell}\in K_{{\mathfrak{t}}}$, for $\ell=1,2$ and
$(M^{*}_{1},M^{*}_{2},a)\in K^{3,{\rm bs}}_{{\mathfrak{t}}}$ then
$(M_{1},M_{2},a)\in K^{3,{\rm bs}}_{{\mathfrak{s}}}$ and $p^{*}={\rm
ortp}_{{\mathfrak{t}}}(a,M^{*}_{1},M^{*}_{2}),p={\rm
ortp}_{{\mathfrak{s}}}(a,M_{1},M_{2})$; then in ${{\mathfrak{t}}}$:

  1. $(\alpha)$

if $M^{*}_{\ell}\leq_{{\mathfrak{s}}}M^{*},p_{\ell}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{t}}}(M^{*}_{\ell})$, then
$p^{*}_{1}\|p^{*}_{2}\Leftrightarrow{\rm bas}(p^{*}_{1})={\rm bas}(p^{*}_{2})$

  2. $(\beta)$

$p^{*}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{t}}}(M^{*})$ does not split
over ${\rm bas}(p^{*})$, see Definition 1.12(3) or [She09e, §2 end].

###### Proof..

1) Let $M^{**}\leq_{{\mathfrak{s}}}M^{*}$ be of cardinality $\lambda$ such
that $p^{*}$ does not fork over $M^{**}$. Let $\bar{a}^{*}=\langle
a_{\alpha}:\alpha<\lambda\rangle$ list the element of $M^{**}$.

We say that $p_{1}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M_{1})$ is a
weak copy of $p^{*}$ when there is a witness $(M_{0},M_{2},p_{2},f)$ which
means:

  1. $\circledast_{1}$

$(a)\quad M_{0}\leq_{{\mathfrak{s}}}M_{2}$ and
$M_{1}\leq_{{\mathfrak{s}}}M_{2}$

  2. $(b)\quad$ if $\|M_{1}\|=\lambda$ then $\|M_{2}\|=\lambda$

  3. $(c)\quad f$ is an isomorphism from $M^{**}$ onto $M_{0}$

  4. $(d)\quad p_{2}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M_{2})$ is a non-forking extension of $p_{1}$

  5. $(e)\quad p_{2}$ does not fork over $M_{0}$

  6. $(f)\quad f(p^{*}\restriction M^{**})$ is $p_{2}\restriction M_{0}$.

For $M_{1}\in K^{{\mathfrak{s}}}_{\lambda},p_{1}\in{\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}(M_{1})$ which is a weak copy of $p^{*}$, we say that
$\bar{b}$ explicate its being a weak copy when for some witness
$(M_{0},M_{2},p_{2},f)$ and $\bar{c}$

  1. $\circledast_{2}$

$(a)\quad\bar{b}=\langle b_{\alpha}:\alpha<\lambda\rangle$ list the elements
of $M_{1}$

  2. $(b)\quad\bar{c}=\langle c_{\alpha}:\alpha<\lambda\rangle$ list the element of $M_{2}$

  3. $(c)\quad\\{\alpha:b_{2\alpha}=b_{2\alpha+1}\\}$ code the folowing sets

    1. $(\alpha)\quad$ the isomorphic type of $(M_{2},\bar{c})$

    2. $(\beta)\quad\\{(\alpha,\beta):b_{\alpha}=c_{\beta}\\}$

    3. $(\gamma)\quad\\{(\alpha,\beta):f(a^{*}_{\alpha})=c_{\beta}\\}$

Now

  1. $\circledast_{3}$

if $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ is a weak copy of
$p^{*}$ then for some $\bar{a}\in{\rm seq}(M)$, there is a
$M_{1}\leq_{{\mathfrak{s}}}M$ over which $p$ does not fork such that $\bar{a}$
list $M_{1}$ and explicate $p\restriction M_{1}$ is a weak copy of $p^{*}$

  2. $\circledast_{4}$

$(a)\quad$ if $M\in K^{{\mathfrak{s}}}_{\lambda}$ and $\bar{b}$ explicate
$p^{*}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ is a weak copy of
$p^{*}$,

then from $\bar{b}$ and $M$ we can reconstruct $p_{1}$

  3. $(b)\quad$ call it $p_{M,\bar{b}}$

  4. $(c)\quad$ if $M\leq_{{\mathfrak{s}}}N$ let $p_{N,\bar{b}}$ be its non-forking extension in ${{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(N)$ we also

call it $\mathbf{F}(N,\bar{b})$.

Now we define ${\mathbb{E}}$, so for $N\in K_{{\mathfrak{s}}}$ we define a
two-place relation ${\mathbb{E}}_{N}$

  1. $\circledast_{5}$

$(\alpha)\quad{\mathbb{E}}_{N}$ is on $\\{\bar{a}$: for some
$M\leq_{{\mathfrak{s}}}N$ of cardinality $\lambda$ and
$p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$

which is a copy of $p^{*}$, the sequence $\bar{a}$ explicates $p$ being

a weak copy of $p^{*}\\}$

  2. $(\beta)$

$\bar{a}_{1}{\mathbb{E}}_{N}\bar{a}_{2}$ iff $(\bar{a}_{1},\bar{a}_{2}$ are as
above and) $p_{N,\bar{a}_{1}}=p_{N,\bar{a}_{2}}$.

Now

  1. $\odot_{1}$

for $N\in K_{{\mathfrak{s}}},{\mathbb{E}}_{N}$ is an equivalence relation on
${\rm Dom}(E_{N})\subseteq{\rm seq}(N)$

  2. $\odot_{2}$

if $N_{1}\leq_{{\mathfrak{s}}}N_{2}$ and $\bar{a}\in{\rm seq}(N_{1})$ then
$\bar{a}\in{\rm Dom}({\mathbb{E}}_{N_{1}})\Leftrightarrow\bar{a}\in{\rm
Dom}({\mathbb{E}}_{N_{2}})$

  3. $\odot_{3}$

if $N_{1}\leq_{{\mathfrak{s}}}N_{2}$ and $\bar{a}_{1},\bar{a}_{2}\in{\rm
Dom}({\mathbb{E}}_{N_{1}})$ then
$\bar{a}_{2}{\mathbb{E}}_{N_{1}}\bar{a}_{2}\Leftrightarrow\bar{a}_{1}{\mathbb{E}}_{N_{2}}\bar{a}_{2}$

  4. $\odot_{4}$

if $\langle N_{\alpha}:\alpha\leq\delta\rangle$ is
$\leq_{{\mathfrak{s}}}$-increasing continuous and $\bar{a}_{1}\in{\rm
Dom}({\mathbb{E}}_{N_{\delta}})$ then for some $\alpha<\delta$ and
$\bar{a}_{2}\in{\rm Dom}({\mathbb{E}}_{N_{\alpha}})$ we have
$\bar{a}_{1}{\mathbb{E}}_{N_{\delta}}\bar{a}_{2}$.

[Why? Let $\bar{a}_{2}$ list the elements of
$M_{1}\leq_{{\mathfrak{s}}}N_{\delta}$ and let $p=p_{N_{\delta},\bar{a}_{1}}$
so $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(N_{\delta})$, hence for
some $\alpha<\delta,p$ does not fork over $M_{\alpha}$ hence for some
$M^{\prime}_{1}\leq_{{\mathfrak{s}}}M_{\alpha}$ of cardinality $\lambda$, the
type $p$ does not fork over $M^{\prime}_{1}$. Let $\bar{a}_{2}$ list the
elements of $M^{\prime}_{1}$ such that it explicates $p\restriction
M^{\prime}_{1}$ being a weak copy of $p^{*}$. So clearly $\bar{a}_{2}\in{\rm
Dom}({\mathbb{E}}_{N_{\alpha}})\subseteq{\rm Dom}({\mathbb{E}}_{N_{\delta}})$
and $\bar{a}_{1}{\mathbb{E}}_{N_{\delta}}\bar{a}_{2}$.]

Clearly we are done.

2) Similar only we vary $(M^{*},p^{*})$ but it suffices to consider
$2^{\lambda}$ such pairs.

3) Should be clear. ∎

######  Definition/Claim 2.19.

Assume that ${{\mathfrak{s}}}$ is a good ess-$[\lambda,\mu)$-frame so without
loss of generality is full. We can repeat the operations in 2.18(3) and
2.16(2), so after $\omega$ times we get ${{\mathfrak{t}}}_{\omega}$ which is
full (that is ${{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{t}}_{\omega}}(M^{\omega})={{\mathscr{S}}}^{{\rm
na}}_{{\mathfrak{t}}_{\omega}}(M^{\omega}))$ and ${{\mathfrak{t}}}_{\omega}$
has canonical type-bases as witnessed by a function ${\rm
bas}_{{\mathfrak{t}}_{\omega}}$, see Definition 2.20.

###### Proof..

Should be clear. ∎

######  Definition 2.20.

We say that ${{\mathfrak{s}}}$ has type bases if there is a function ${\rm
bas}(-)$ such that:

  1. $(a)$

if $M\in K_{{\mathfrak{s}}}$ and $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)$ then ${\rm bas}(p)$ is (well defined and is) an
element of $M$

  2. $(b)$

$p$ does not split over bas$(p)$, that is any automorphism999there are
reasonable stronger version, but it follows that the function ${\rm bas}(-)$
satisfies them of $M$ over ${\rm bas}(p)$ maps $p$ to itself

  3. $(c)$

if $M\leq_{{\mathfrak{s}}}N$ and $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(N)$ then: ${\rm bas}(p)\in M$ iff $p$ does not fork over
$M$

  4. $(d)$

if $f$ is an isomorphism from $M_{1}\in K_{{\mathfrak{s}}}$ onto $M_{2}\in
K_{{\mathfrak{s}}}$ and $p_{1}\in{{\mathscr{S}}}^{{\rm bs}}(M_{1})$ then
$f({\rm bas}(p_{1}))={\rm bas}(f(p_{1}))$.

######  Remark 2.21.

In §3 we can add:

  1. $(e)$

strong uniqueness: if $A\subseteq
M\leq_{{\mathfrak{k}}({\mathfrak{s}})}{{\mathfrak{C}}},p\in{{\mathscr{S}}}(A,{{\mathfrak{C}}})$
well defined, then for at most one $q\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)$ do we have: $q$ extends $p$ and ${\rm bas}(p)\in A$.
(needed for non-forking extensions).

######  Definition 2.22.

We say that ${{\mathfrak{s}}}$ is equivalence-closed when :

  1. $(a)$

${{\mathfrak{s}}}$ has type bases $p\mapsto{\rm bas}(p)$

  2. $(b)$

if ${\mathbb{E}}_{M}$ is a definition of an equivalence relation on
${}^{\omega>}M$ preserved by isomorphisms and
$\leq_{{\mathfrak{s}}}$-extensions (i.e.
$M\leq_{{\mathfrak{s}}}N\Rightarrow{\mathbb{E}}_{M}={\mathbb{E}}_{N}{\restriction}{}^{\omega>}M$)
then there is a definable function $F$ from ${}^{\omega>}M$ to $M$ such that
$F^{M}(\bar{a})=F^{M}(\bar{b})$ iff $\bar{a}E_{M}\bar{b}$ (or work in
${{\mathfrak{C}}}$).

To phrase the relation between ${\mathfrak{k}}$ and ${\mathfrak{k}}^{\prime}$
we define.

######  Definition 2.23.

Assume ${\mathfrak{k}}_{1},{\mathfrak{k}}_{2}$ are ess-$[\lambda,\mu)$-a.e.c.

1) We say $\mathbf{i}$ is an interpretation in ${\mathfrak{k}}_{2}$ when
$\mathbf{i}$ consists of

  1. $(a)$

a predicate $P^{*}_{\mathbf{i}}$

  2. $(b)$

a subset $\tau_{\mathbf{i}}$ of $\tau_{{\mathfrak{k}}_{2}}$.

2) In this case for $M_{2}\in K_{{\mathfrak{k}}_{2}}$ le
$M^{[\mathbf{i}]}_{2}$ be the $\tau_{\mathbf{i}}$-model
$M_{1}=M^{[\mathbf{i}]}_{2}$ with

  1. $\bullet$

universe $P^{M_{2}}_{\mathbf{i}}$

  2. $\bullet$

$R^{M_{1}}=R^{M_{2}}{\restriction}|M_{1}|$ for $R\in\tau_{\mathbf{i}}$

  3. $\bullet$

$F^{M_{1}}$ similarly, so $F^{M_{2}}$ can be a partial function even if
$F^{M_{2}}$ is full.

3) We say that ${\mathfrak{k}}_{1}$ is $\mathbf{i}$-interpreted (or
interpreted by $\mathbf{i}$) in ${\mathfrak{k}}_{2}$ when :

  1. $(a)$

$\mathbf{i}$ is an interpretation in ${\mathfrak{k}}_{1}$

  2. $(b)$

$\tau_{{\mathfrak{k}}_{1}}=\tau_{\mathbf{i}}$

  3. $(c)$

$K_{{\mathfrak{k}}_{1}}=\\{M^{[\mathbf{i}]}_{2}:M_{2}\in
K_{{\mathfrak{k}}_{2}}\\}$

  4. $(d)$

if $M_{2}\leq_{{\mathfrak{k}}_{2}}N_{2}$ then
$M^{[\mathbf{i}]}_{2}\leq_{{\mathfrak{k}}_{1}}N^{[\mathbf{i}]}_{2}$

  5. $(e)$

if $M_{1}\leq_{{\mathfrak{k}}_{1}}N_{1}$ and $N_{1}=N^{[\mathbf{i}]}_{2}$, so
$N_{2}\in K_{{\mathfrak{k}}_{2}}$ then for some
$M_{2}\leq_{{\mathfrak{k}}}N_{2}$ we have $M_{1}=M^{[\mathbf{i}]}_{2}$

  6. $(f)$

if $M_{1}\leq_{{\mathfrak{k}}_{1}}N_{1}$ and $M_{1}=M^{[\mathbf{i}]}_{2}$, so
$M_{2}\in K_{{\mathfrak{k}}_{2}}$ then possible replacing $M_{2}$ by a model
isomorphic to it over $M_{1}$, there is $N_{2}\in K_{{\mathfrak{k}}_{2}}$ we
have $M_{2}\leq_{{\mathfrak{k}}_{2}}N_{2}$ and $N_{1}=N^{[\mathbf{i}]}_{2}$.

######  Definition 2.24.

1) Assume ${\mathfrak{k}}_{1}$ is interpreted by $\mathbf{i}$ in
${\mathfrak{k}}_{2}$. We say strictly interpreted when: if
$M^{[\mathbf{i}]}_{2}=N^{[\mathbf{i}]}_{2}$ then $M_{2},N_{2}$ are isomorphic
over $M^{[\mathbf{i}]}_{2}$.

2) We say ${\mathfrak{k}}_{1}$ is equivalent to ${\mathfrak{k}}_{2}$ if there
are $n$ and ${\mathfrak{k}}^{\prime}_{0},\dotsc,{\mathfrak{k}}^{\prime}_{n}$
such that
${\mathfrak{k}}_{1}={\mathfrak{k}}^{\prime}_{0},{\mathfrak{k}}_{2}={\mathfrak{k}}^{\prime}_{n}$
and for each $\ell<n,{\mathfrak{k}}_{\ell}$ is strictly interpreted in
${\mathfrak{k}}_{\ell+1}$ or vice versa. Actually we can demand $n=2$ and
$k_{\ell}$ is strictly interpreted in ${\mathfrak{k}}^{\prime}_{1}$ for
$\ell=1,2$.

######  Definition 2.25.

As above for (good) ess-$[\lambda,\mu)$-frame.

######  Claim 2.26.

Assume ${\mathfrak{s}}$ is a good ${\rm ess}-[\lambda,\mu)$-frame. Then there
is ${\mathfrak{C}}$ (called a $\mu$-saturated for $K_{{\mathfrak{s}}}$) such
that:

  1. $(a)$

${\mathfrak{C}}$ is a $\tau_{{\mathfrak{s}}}$-model of cardinality $\leq\mu$

  2. $(b)$

${\mathfrak{C}}$ is a union of some $\leq_{{\mathfrak{s}}}$-increasing
continuous sequence $\langle M_{\alpha}:\alpha<\mu\rangle$

  3. $(c)$

if $M\in K_{{\mathfrak{s}}}$ so $\lambda\leq\|M\|<\mu$ then $M$ is
$\leq_{{\mathfrak{s}}}$-embeddable into some $M_{\alpha}$ from clause (b)

  4. $(d)$

$M_{\alpha+1}$ is brimmed over $M_{\alpha}$ for $\alpha<\mu$.

##  3\. I ${\mathbf{P}}$-simple types

We define the basic types over sets not necessary models. Note that in
Definition 3.5 there is no real loss using $C$ of cardinality
$\in(\lambda,\mu)$, as we can replace $\lambda$ by $\lambda_{1}=\lambda+|C|$
and so replace $K_{{\mathfrak{k}}}$ to
$K^{{\mathfrak{k}}}_{[\lambda_{1},\mu)}$.

######  Hypothesis 3.1.

1) ${\mathfrak{s}}$ is a good ess-$[\lambda,\mu)$-frame, see Definition 2.11.

2) ${{\mathfrak{s}}}$ has type bases, see Definition 2.20.

3) ${\mathfrak{C}}$ denote some $\mu$-saturated model for $K_{{\mathfrak{s}}}$
of cardinality $\leq\mu$, see 2.26.

4) But $M,A,\ldots$ will be
$<_{{\mathfrak{k}}({\mathfrak{s}})}{\mathfrak{C}},\subseteq{\mathfrak{C}}$
respectively but of cardinality $<\mu$.

######  Definition 3.2.

Let $A\subseteq M\in K_{{\mathfrak{s}}}$.

1) ${\rm dcl}(A,M)=\\{a\in M$: if
$M^{\prime}\leq_{{\mathfrak{s}}}M^{\prime\prime},M\leq_{{\mathfrak{s}}}M^{\prime\prime}$
and $A\subseteq M^{\prime}$ then $a\in M^{\prime}$ and for every automorphism
$f$ of $M^{\prime},f\restriction A={\rm id}_{A}\Rightarrow f(a)=a\\}$.

2) ${\rm acl}(A,M)$ is defined similarly but only with the first demand.

######  Definition 3.3.

1) For $A\subseteq M\in K_{{\mathfrak{s}}}$ let

${{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(A,M)=\\{q\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M):{\rm bas}(q)\in{\rm dcl}(A,{{\mathfrak{C}}})\\}.$

2) We call $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(A,M)$ regular if
$p$ as a member of ${\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ is regular.

######  Definition 3.4.

1) ${\mathbb{E}}_{{\mathfrak{s}}}$ is as in Claim 2.18(2).

2) If $A\subseteq M\in K_{{\mathfrak{s}}}$ and $p\in{\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}(M)$, then $p\in{\mathscr{S}}^{{\rm
bs}}_{{\mathfrak{s}}}(A,M)$ iff $p$ is definable over $A$, see 1.12(3) iff
${\rm inv}(p)$ from Definition 1.12 is $\subseteq A$ and well defined.

######  Definition 3.5.

Let $A\subseteq{{\mathfrak{C}}}$.

1) We define a dependency relation on
good$(A,{{\mathfrak{C}}})=\\{c\in{\mathfrak{C}}:$ for some
$M<_{{\mathfrak{k}}({\mathfrak{s}})}{{\mathfrak{C}}},A\subseteq M$ and ${\rm
ortp}(c,M,{{\mathfrak{C}}})$ is definable over some finite $\bar{a}\subseteq
A\\}$ as follows:

  1. $\circledast$

$c$ depends on $\mathbf{J}$ in $(A,{\mathfrak{C}})$ iff there is no
$M<_{{\mathfrak{k}}({\mathfrak{s}})}{{\mathfrak{C}}}$ such that
$A\cup\mathbf{J}\subseteq M$ and ${\rm ortp}(c,M,{{\mathfrak{C}}})$ is the
non-forking extension of ${\rm ortp}(c,\bar{a},{{\mathfrak{C}}})$ where
$\bar{a}$ witnesses $c\in\text{ good}(A,{{\mathfrak{C}}})$.

2) We say that $C\in{}^{\mu>}[{{\mathfrak{C}}}]$ is good over $(A,B)$ when
there is a brimmed $M<_{{\mathfrak{k}}({\mathfrak{s}})}{{\mathfrak{C}}}$ such
that $B\cup A\subseteq M$ and ${\rm ortp}(C,M,{{\mathfrak{C}}})$ (see
Definition 1.12(3)) is definable over $A$. (In the first order context we
could say $\\{c,B\\}$ is independent over $A$ but here this is problematic as
${\rm ortp}(B,A,{{\mathfrak{C}}})$ is not necessary basic).

3) We say $\langle A_{\alpha}:\alpha<\alpha^{*}\rangle$ is independent over
$A$ in ${{\mathfrak{C}}}$, see [She09e, L8.8,6p.5(1)] if we can find
$M,\langle M_{\alpha}:\alpha<\alpha^{*}\rangle$ such that:

  1. $\circledast(a)$

$A\subseteq
M\leq_{{\mathfrak{k}}({\mathfrak{s}})}M_{\alpha}<_{{\mathfrak{s}}}{{\mathfrak{C}}}$
for $\alpha<\alpha^{*}$

  2. $(b)$

$M$ is brimmed

  3. $(c)$

$A_{\alpha}\subseteq M_{\alpha}$

  4. $(d)$

${\rm ortp}(A_{\alpha},M,{{\mathfrak{C}}})$ definable over $A$ (= does not
split over $A$)

  5. $(e)$

$\langle M_{\alpha}:\alpha<\alpha^{*}\rangle$ is independent over $M$.

3A) Similarly “over $(A,B)$”.

4) We define locally independent naturally, that is every finite subfamily is
independent.

######  Claim 3.6.

Assume $a\in{\mathfrak{C}},A\subseteq{{\mathfrak{C}}}$.

1) $a\in{\rm good}(A,{\mathfrak{C}})$ iff $a$ realizes
$p\in{\mathscr{S}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ for some $M$ satisfying
$A\subseteq M<_{{\mathfrak{k}}({\mathfrak{s}})}{\mathfrak{C}}$.

######  Claim 3.7.

1) If $A_{\alpha}\subseteq{{\mathfrak{C}}}$ is good over
$(A,\bigcup\limits_{i<\alpha}A_{i})$ for $\alpha<\alpha^{*},\alpha^{*}<\omega$
then $\langle A_{\alpha}:\alpha<\alpha^{*}\rangle$ is independent over $A$.

2) Independence is preserved by reordering.

3) If $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ is regular then on
$p({{\mathfrak{C}}})=\\{c:c$ realizes $p\\}$ the independence relation
satisfies:

  1. $(a)$

like (1)

  2. $(b)$

if $b^{1}_{\ell}$ depends on $\\{b^{0}_{0},\dotsc,b^{0}_{n-1}\\}$ for $\ell<k$
and $b^{2}$ depends on $\\{b^{1}_{\ell}:\ell<k\\}$ then $b^{2}$ depends on
$\\{b^{0}_{\ell}:\ell<n\\}$

  3. $(c)$

if $b$ depends on $\mathbf{J},\mathbf{J}\subseteq\mathbf{J}^{\prime}$ then $b$
depends on $\mathbf{J}^{\prime}$.

######  Remark 3.8.

1) However, we have not mentioned finite character but the local independence
satisfies it trivially.

###### Proof..

Easy. ∎

######  Definition 3.9.

1) Assume $q\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ and
$p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$.
We say that $q$ is explicitly $(p,n)$-simple when ,:

  1. $\circledast$

there are $b_{0},\dotsc,b_{n-1},c$ such that 101010clause (c) + (e) are
replacements for $c$ is algebraic over $\bar{a}+\\{b_{\ell}:\ell<n\\}$ and
each $b_{\ell}$ is necessary:

    1. $(a)$

$b_{\ell}$ realizes $p$

    2. $(b)$

$c$ realizes $q$

    3. $(c)$

$b_{\ell}$ is not good 111111not good here is a replacement to “${\rm
ortp}(b_{\ell},\bar{a}+c,{{\mathfrak{C}}})$ does not fork over $\bar{a}$” over
$(\bar{a},c)$ for $\ell<n$

    4. $(d)$

$\langle b_{\ell}:\ell<n\rangle$ is independent over $\bar{a}$

    5. $(e)$

$\langle c,b_{0},\dotsc,b_{n-1}\rangle$ is good over $\bar{a}$

    6. $(f)$

if 121212this seems a reasonable choice here but we can take others; this is
an unreasonable choice for first order $c^{\prime}$ realizes $q$ then
$c=c^{\prime}$ iff for every $b\in p({{\mathfrak{C}}})$ we have: $b$ is good
over $(\bar{a},c)$ iff $b$ is good over $(\bar{a},,c^{\prime})$.

1A) We say that $a$ is explicitly $(p,n)$-simple over $A$ if ${\rm
ortp}(a,A,{{\mathfrak{C}}})$ is; similarly in the other definitions replacing
$(p,n)$ by $p$ means “for some $n$”.

2) Assume $q\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ and ${\mathbf{P}}$ as in
Definition 1.6. We say that $q$ is ${\mathbf{P}}$-simple if we can find $n$
and explicitly ${\mathbf{P}}$-regular types
$p_{0},\dotsc,p_{n-1}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ such that: each $c\in
p({{\mathfrak{C}}})$ is definable by its type over
$\bar{a}\cup\bigcup\limits_{\ell<n}p_{\ell}({{\mathfrak{C}}})$,

3) In part (1) we say strongly explicitly $(p,n)$-simple if there are $k>a$
and $\langle\bar{a}^{*}_{\ell}:\ell<\omega\rangle$ and
$r\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$
such that [on finitely many see 3.9(3B) below]:

  1. $(a)$

$\\{\bar{a}^{*}_{\ell}:\ell<\omega\\}\in r({{\mathfrak{C}}})$ is independent
for any $c^{\prime},c^{\prime\prime}\in p({{\mathfrak{C}}})$, we have
$c^{\prime}=c^{\prime\prime}$ iff for infinitely many $m<\omega$ ($\equiv$ for
all but finitely many $\models$, see claim on average) for every $b\in
p({{\mathfrak{C}}})$ we have:

    1. $(*)$

$b$ is good over $(\bar{a},\bar{a},a^{*}_{m},c)$ iff $b$ is good over
$(\bar{a},a,a^{*}_{m},c)$, (compare with 3.13, 3.17!).

3A) In part (1) we say weakly $(p,n)$-simple if in $\circledast$, clause $(f)$
is replaced by

  1. $(f)^{\prime}$

if $b$ is good over $(\bar{a},a^{*}_{m})$ then $c^{\prime},c^{\prime}$
realizes the same type over $\bar{a}a^{*}_{m}b$.

3B) In part (1) we say $(p,n)$-simple if for some
$\bar{a}^{*}\in{}^{\omega>}{{\mathfrak{C}}}$ good over $\bar{a}$ for every
$c\in q({{\mathfrak{C}}})$ there are $b_{0},\dotsc,b_{n-1}\in
p({{\mathfrak{C}}})$ such that $c\in{\rm
dcl}(\bar{a},\bar{a}^{*},b_{0},\dotsc,b_{n-1})$ and $\bar{a}\char
94\relax\langle b_{0},\dotsc,b_{n-1}\rangle$ is good over $\bar{a}$ if simple.

4) Similarly in (2).

5) We define $gw_{p}(b,\bar{a}),p$ regular parallel to some
$p^{\prime}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(\bar{a})$ ($gw$ for
general weight). Similarly for $gw_{p}(q)$.

We first list some obvious properties.

######  Claim 3.10.

1) If $c$ is ${\mathbf{P}}$-simple over $\bar{a},\bar{a}\subseteq
A\subset{{\mathfrak{C}}}$ then $w_{p}(c,A)$ is finite.

2) The obvious implications.

######  Claim 3.11.

1) [Closures of the simple ${\rm bs}$].

2) Assume $p\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$. If $\bar{b}_{1},\bar{b}_{2}$
are $p$-simple over $A$ then

  1. $(a)$

$\bar{b}_{1}\char 94\relax\bar{b}_{2}$ is $p$-simple (of course, ${\rm
ortp}_{{\mathfrak{s}}}(\bar{b}_{2}\bar{b}_{2},\bar{a},{{\mathfrak{C}}})$ is
not necessary in ${{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ even if ${\rm
ortp}_{{\mathfrak{s}}}(\bar{b}_{\ell},\bar{a},{{\mathfrak{C}}})\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ for $\ell=1,2$)

  2. $(b)$

also ${\rm ortp}(\bar{b}_{2},\bar{a}b_{1},{{\mathfrak{C}}})$ is
${\mathbf{P}}$-simple.

2) If $\bar{b}_{\alpha}$ is $p$-simple over $\bar{a}$ for
$\alpha<\alpha^{*},\pi:\beta^{*}\rightarrow\alpha^{*}$ one to one into, then
$\sum\limits_{\alpha<\alpha^{*}}gw_{p}(b_{\alpha},\bar{a}_{*}\cup\bigcup\limits_{\ell<\alpha}b_{\alpha})=\sum\limits_{\beta<\beta^{*}}gw(b_{\pi(\beta)})\bar{a}\cup\bigcup\limits_{i<\beta}\bar{b}_{\pi(i)})$.

######  Claim 3.12.

[${{\mathfrak{s}}}$ is equivalence-closed].

Assume that $p,q\in{{\mathscr{S}}}^{{\rm bs}}(M)$ are not weakly orthogonal
(e.g. see 6.9). Then for some $\bar{a}\in{}^{\omega>}M$ we have: $p,q$ are
definable over $\bar{a}$ (works without being stationary) and for some
${{\mathfrak{k}}}_{{\mathfrak{s}}}$-definable function $\mathbf{F}$, for each
$c\in q({{\mathfrak{C}}}),{\rm
ortp}_{{\mathfrak{s}}}(\mathbf{F}(c,\bar{a}),\bar{a},{{\mathfrak{C}}})\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ and is explicitly
$(p,n)$-simple for some $n$, (if, e.g., $M$ is $(\lambda,*)$-brimmed then
$n=w_{p}(q)$.

###### Proof..

We can find $n$ and $c_{1},b_{0},\dotsc,b_{-1}\in{{\mathfrak{C}}}$ with $c$
realizing $q,b_{\ell}$ realizing $p,\\{b_{\ell},c\\}$ is not independent over
$M$ and $n$ maximal. Choose $\bar{a}\in{}^{\omega>}M$ such that ${\rm
ortp}_{{\mathfrak{s}}}(\langle
c,b_{0},\dotsc,b_{n-1}\rangle,M,{{\mathfrak{C}}}\rangle)$ is definable over
$\bar{a}$. Define $E_{\bar{a}}$, an equivalence relation on
$q({\mathfrak{C}}):c_{1}E_{\bar{a}}c_{2}$ iff for every $b\in
p{{\mathfrak{C}}})$, we have $b$ is good over $(a,c_{1})\Rightarrow b$ is good
over $(\bar{a},c_{2})$. By “${{\mathfrak{s}}}$ is eq-closed”, we are done. ∎

######  Claim 3.13.

1) Assume $p,q\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$ are weakly
orthogonal (see e.g. 6.9(1)) but not orthogonal. Then we can find
$\bar{a}\in{}^{\omega>}M$ over which $p,q$ are definable and
$r_{1}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a},{{\mathfrak{C}}})$ such that letting
$p_{1}=p\restriction\bar{a},q_{1}=q\restriction\bar{a},n=:w_{p}(q)\geq 1$ we
have:

  1. $\circledast^{n}_{\bar{a},p_{1},q_{1},r_{2}}(a)$

$p_{1},q_{1},r_{1}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(\bar{a}),\bar{a}\in{}^{\omega>}{{\mathfrak{C}}}$

  2. $(b)$

$p_{1},q_{1}$ are weakly orthogonal (see e.g. Definition 6.9(1))

  3. $(c)$

if $\\{a^{*}_{n}:n<\omega\\}\subseteq r_{1}({{\mathfrak{C}}})$ is independent
over $\bar{a}$ and $c$ realizes $q$ then for infinitely many $m<\omega$ there
is $b\in p({\mathfrak{C}})$ such that $b$ is good over $(\bar{a},a^{*}_{n})$
but not over $(\bar{a},a^{*}_{n},c)$

  4. $(d)$

in (c) really there are $n$ independent such that (but not $n+1$).

2) If $\circledast^{n}_{\bar{a},p_{1},q_{1},r_{1}}$ then (see Definition
3.9(3) for some definable function $\mathbf{F}$, if $c$ realizes
$q_{1},c^{*}=F(c,\bar{a})$ and ${\rm
ortp}_{{\mathfrak{n}}}(c^{*},\bar{a},{{\mathfrak{C}}})$ is $(p_{1},n)$-simple.

See proof below.

######  Claim 3.14.

0) Assuming $A\subseteq\mathfrak{C}$ and $a\in\mathfrak{C}$ is called finitary
when it is definable over $\\{a_{0},\dots,a_{n-1}\\}$ where each $a_{\ell}$ is
in $\mathfrak{C}$ and is good over $A$ inside $\mathfrak{C}$.

1) If $a\in{\rm dcl}(\cup\\{A_{i}:i<\alpha\\}\cup A,{{\mathfrak{C}}})$ and
${\rm ortp}(a,A,{{\mathfrak{C}}})$ is finitary over (A) and
$\\{A_{i}:i<\alpha\\}$ is independent over $A$ then for some finite
$u\subseteq\alpha$ we have $a\in{\rm dcl}(\cup\\{A_{i}:i\in u\\}\cup
A,{{\mathfrak{C}}})$.

2) If ${\rm ortp}(b,\bar{a},{{\mathfrak{C}}})$ is ${\mathbf{P}}$-simple, then
it is finitary.

3) If $\\{A_{i}:i<\alpha\\}$ is independent over $A$ and $a$ is finitary over
$A$ then for some finite $u\subseteq\alpha$ (even $|u|<{\rm
wg}(c,A)),\\{A_{i}:i\in\alpha\backslash u\\}$ is independent over
$A,A\cup\\{c\\})$ (or use
$(A^{\prime},A^{\prime\prime}),(A^{\prime},A^{\prime\prime}\cup\\{c\\})$).

######  Definition 3.15.

1) ${\rm dcl}(A)=\\{a$: for every automorphism $f$ of
${{\mathfrak{C}}},f(a)=a\\}$.

2) ${\rm dcl}_{{\rm fin}}(A)=\cup\\{{\rm dcl}(B):B\subseteq A$ finite$\\}$.

3) $a$ is finitary over $A$ if there are $n<\omega$ and
$c_{0},\dotsc,c_{n-1}\in$ good$(A)$ such that $a\in{\rm
dcl}(A\cup\\{c_{0},\dotsc,c_{n-1}\\}$ (or dcl${}_{\text{fin}}$?).

4) For such $A$ let ${\rm wg}(a,A)$ be $w({\rm tp}(a,A,{\mathfrak{C}}))$ when
well defined.

5) Strongly simple implies simple.

######  Claim 3.16.

In Definition 3.9(3), for some $m,k<\omega$ large enough, for every $c\in
q({{\mathfrak{C}}})$ there are
$b_{0},\dotsc,b_{m-1}\in\bigcup\limits_{\ell<n}p_{\ell}({{\mathfrak{C}}})$
such that $c\in\rm
dcl(\bar{a}\cup\\{a^{*}_{\ell}:\ell<k\\}\cup\\{b_{\ell}:\ell<m\\})$.

###### Proof..

Let $M_{1},M_{2}\in K_{{\mathfrak{s}}({\rm brim})}$ be such that
$M\leq_{{\mathfrak{s}}}M_{1}\leq_{{\mathfrak{s}}}M_{2},M_{1}\,(\lambda,*)$-brimmed
over $M,p_{\ell}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M_{\ell})$ a
non-forking extension of $p,q_{\ell}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M_{\ell})$ is a non-forking extension of $q,c\in M_{2}$
realizes $q_{1}$ and $(M_{1},M_{2},c)\in K^{3,{\rm pr}}_{{\mathfrak{s}}({\rm
brim})}$. Let $b_{\ell}\in p_{1}(M_{2})$ for $\ell<n^{*}=:w_{p}(q)$ be such
that $\\{b_{\ell}:\ell<n^{*}\\}$ is independent in $(M_{1},M_{2})$, let
$\bar{a}^{*}\in{}^{\omega>}(M_{1})$ be such that ${\rm
ortp}_{{\mathfrak{s}}}(\langle c,b_{0},\dotsc,b_{n-1}\rangle,M_{1},M_{2})$ is
definable over $\bar{a}^{*}$ and $r={\rm
ortp}_{{\mathfrak{s}}}(\bar{a}^{*},M_{1},M_{2}),r^{+}={\rm
ortp}(\bar{a}^{*}\char 94\relax\langle b_{0},\dotsc,b_{n-1}\rangle,M,M_{2})$.

Let $\bar{a}\in{}^{\omega>}M$ be such that ${\rm
ortp}_{{\mathfrak{s}}}(\bar{a}^{*},\langle
c,b_{0},\dotsc,b_{n-1}\rangle,M,M_{2})$ is definable over $\bar{a}$. As
$M_{1}$ is $(\lambda,*)$-saturated over $M$ there is
$\\{\bar{a}^{*}_{f}:f<\omega\\}\subseteq r({{\mathfrak{C}}})$ independent in
$(M,M_{1})$ moreover letting $a^{*}_{\omega}=\bar{a}^{*}$, we have $\langle
a^{*}_{\alpha}:\alpha\leq\omega\rangle$ is independent in $(M,M_{1})$. Clearly
${\rm ortp}_{{\mathfrak{s}}}(c\bar{a}^{*}_{n},M,M_{2})$ does not depend on $n$
hence we can find $\left<\langle
b^{\alpha}_{\ell}:\ell<n\rangle:\alpha\leq\omega\right>$ such that
$b^{\alpha}_{\ell}\in M_{2},b^{\omega}_{\ell}=b_{\ell}$ and
$\\{c\bar{a}^{*}_{\alpha},b^{\alpha}_{0}\ldots
b^{\alpha}_{n-1}:\alpha\leq\omega\\}$ (as usual as index set is independent in
$(M_{1},M_{2})$.

The rest should be clear. ∎

######  Definition 3.17.

Assume
$\bar{a}\in{}^{\omega>}{{\mathfrak{C}}},n<\omega,p,q,r\in{\mathscr{S}}^{{\rm
bs}}(M)$ are as in the definition of $p$-simple[-] but $p,q$ are weakly
orthogonal (see e.g. Definition 6.9(1)) let $p$ be a definable related
function such that for any $\bar{a}^{\nu}_{\ell}\in
r({{\mathfrak{C}}}),\ell<k^{*}$ independent mapping $c\mapsto\langle\\{b\in
q({{\mathfrak{C}}}):R{{\mathfrak{C}}}\models R(b,c,\bar{a}^{*}_{\ell})\\}$ is
a one-to-one function from $q({{\mathfrak{C}}})$ into $\\{\langle
J_{\ell}:\ell<k^{*}\rangle:J_{\ell}\subseteq p({{\mathfrak{C}}})$ is closed
under dependence and has $p$-weight $n^{*}\\}$

1) We can define $E=E_{p,q,r}$ a two-place relation over
$r({{\mathfrak{C}}}):\bar{a}^{*}_{1}E\bar{a}^{*}_{2}$ iff
$\bar{a}_{1},\bar{a}_{2}\in r({\mathfrak{C}})$ have the same projection common
to $p({\mathfrak{C}})$ and $q({\mathfrak{C}})$.

2) Define unit-less group on $r/E$ and its action on $q({{\mathfrak{C}}})$.

######  Remark 3.18.

1) A major point is: as $q$ is $p$-simple, $w_{p}(-)$ acts “nicely” on
$p({{\mathfrak{C}}})$ so if $c_{1},c_{2},c_{3}\in q({{\mathfrak{C}}})$ then
$w_{p}(\langle c_{1},c_{2},c_{3}\rangle\bar{a})\leq 3n^{*}$. This enables us
to define average using a finite sequence seem quite satisfying.
Alternatively, look more at averages of independent sets.

2) Silly Groups: Concerning interpreting groups note that in our present
context, for every definable set $P^{M}$ we can add the group of finite
subsets of $P^{M}$ with symmetric difference (as addition).

3) The axiomatization above has prototype ${\mathfrak{s}}$ where
$K_{{\mathfrak{s}}}=\\{M:M$ a $\kappa$-saturated model of
$T\\},\leq_{{\mathfrak{s}}}=\prec\restriction
K_{{\mathfrak{s}}},\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits_{\textstyle{\mathfrak{s}}}$
is non-forking, $T$ a stable first order theory with $\kappa(T)\leq{\rm
cf}(\kappa)$. But we may prefer to formalize the pair
$({{\mathfrak{t}}},{{\mathfrak{s}}}),{{\mathfrak{s}}}$ as above,
$K_{{\mathfrak{t}}}=$ models of $T,\leq_{{\mathfrak{t}}}=\prec\restriction
K_{{\mathfrak{t}}},\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits_{\textstyle{\mathfrak{t}}}$
is non-forking.

From ${{\mathfrak{s}}}$ we can reconstruct a ${{\mathfrak{t}}}$ by closing
${{\mathfrak{k}}}_{{\mathfrak{s}}}$ under direct limits, but in interesting
cases we end up with a bigger ${{\mathfrak{t}}}$.

Part II Generalizing Stable Classes

##  4\. II Introduction

In this part we try to deal with classes like “$\aleph_{1}$-saturated models
of a first order theory $T$ and even a stable one” rather than of “a model of
$T$”. The parallel problem for “model of $T$, even superstable one” is the
subject of [She09d].

$*\qquad*\qquad*$

Now some construction goes well by induction on cardinality, say by dealing
with $(\lambda,{\mathscr{P}}(n))$-system of models but not all. E.g. starting
with $\aleph_{0}$ we may consider $\lambda>\aleph_{0}$, so can find
$F:[\lambda]^{\aleph_{0}}\rightarrow\lambda$ such that there is an infinite
decreasing sequence of $F$-closed subsets of
$\lambda,u\in[\lambda]^{<\aleph_{0}}\Rightarrow F(u)=0$ maybe such that
$u\in[\lambda]^{\leq\aleph_{0}}\Rightarrow|c\ell_{F}(u)|\leq\aleph_{0}$. Let
$\langle u_{\alpha}:\alpha<\alpha_{*}\rangle$ list
$\\{c\ell_{F}(u):u\in[\lambda]^{\leq\aleph_{0}}\\}$ such that
$c\ell_{F}(u_{\alpha})\subseteq
c\ell_{F}(u_{\beta})\Rightarrow\alpha\leq\beta$ we try to choose
$M_{u_{\alpha}}$ by induction on $\alpha$.

##  5\. II Axiomatizing a.e.c. without full continuity

###  5(A). a.e.c.

Classes like “$\aleph_{1}$-saturated models of a first order $T$, which is not
superstable”, does not fall under a.e.c., still they are close and below we
suggest a framework for them. So for increasing sequences of short length we
have weaker demand.

We shall say more on primes later.

We shall lift a $(\mu,\lambda,\kappa)$-a.e.c. to
$(\infty,\lambda,\kappa)$-a.e.c. (see below), so actually $k_{\lambda}$
suffice, but for our main objects, good frames, this is more complicated as
its properties (e.g., the amalgamation property) are not necessarily preserved
by the lifting.

This section generalizes [She09c, §1], in some cases the differences are
minor, sometimes the whole point is the difference.

######  Convention 5.1.

1) In this section ${\mathfrak{k}}$ will denote a directed a.e.c., see
Definition 5.2, may write d.a.e.c. (the $d$ stands for directed).

2) We shall write (outside the definitions)
$\mu_{{\mathfrak{k}}},\lambda_{{\mathfrak{k}}},\kappa_{{\mathfrak{k}}}$.

######  Definition 5.2.

Assume $\lambda<\mu,\lambda^{<\kappa}=\lambda$ (for notational simplicity) and
$\alpha<\mu\Rightarrow|\alpha|^{<\kappa}<\mu$ and $\kappa$ is regular.

We say that ${{\mathfrak{k}}}$ is a $(\mu,\lambda,\kappa)-1$-d.a.e.c. (we may
omit or add the “$(\mu,\lambda,\kappa)$” by Ax(0)(d) below, similarly in
similar definitions; if $\kappa=\aleph_{0}$ we may omit it, instead
$\mu=\mu^{+}_{1}$ we may write $\leq\mu_{1}$) when the axioms below hold; we
write d.a.e.c. or 0-d.a.e.c. when we omit Ax(III)(b),(IV)(b) Ax(0)
${{\mathfrak{k}}}$ consists of

  1. $(a)$

$\tau_{{\mathfrak{k}}}$, a vocabulary with each predicate and function symbol
of arity $\leq\lambda$

  2. $(b)$

$K$, a class of $\tau$-models

  3. $(c)$

a two-place relation $\leq_{{\mathfrak{k}}}$ on $K$

  4. $(d)$

the cardinals
$\mu=\mu_{{\mathfrak{k}}},\mu({\mathfrak{k}}),\lambda=\lambda_{{\mathfrak{k}}}=\lambda({\mathfrak{k}})$
and $\kappa=\kappa_{{\mathfrak{k}}}=\kappa({\mathfrak{k}})$ (so
$\mu>\lambda=\lambda^{<\kappa}\geq\kappa={\rm cf}(\kappa)$ and
$\alpha<\mu\Rightarrow|\alpha|^{<\kappa}<\mu$)

such that

  1. $(e)$

if $M_{1}\cong M_{2}$ then $M_{1}\in K\Leftrightarrow M_{2}\in K$

  2. $(f)$

if $(N_{1},M_{1})\cong(N_{2},M_{2})$ then
$M_{1}\leq_{{\mathfrak{k}}}N_{1}\Rightarrow M_{2}\leq_{{\mathfrak{k}}}N_{2}$

  3. $(g)$

every $M\in K$ has cardinality $\geq\lambda$ but $<\mu$

  4. $(Ax(I)(a))$

$M\leq_{{\mathfrak{k}}}N\Rightarrow M\subseteq N$

  5. $(Ax(II)(a))$

$\leq_{{\mathfrak{k}}}$ is a partial order

  6. $Ax(III)$

assume that $\langle M_{i}:i<\delta\rangle$ is a
$\leq_{{\mathfrak{k}}}$-increasing sequence and
$\|\cup\\{M_{i}:i<\delta\\}\|<\mu$ then

    1. $(a)$

(existence of unions) if ${\rm cf}(\delta)\geq\kappa$ then there is $M\in K$
such that $i<\delta\Rightarrow M_{i}\leq_{{\mathfrak{k}}}M$ and
$|M|=\cup\\{|M_{i}|:i<\delta\\}$ but not necessarily
$M=\bigcup\limits_{i<\delta}M_{i}$

    2. $(b)$

(existence of limits) there is $M\in K$ such that $i<\delta\Rightarrow
M_{i}\leq_{{\mathfrak{k}}}M$

  7. $Ax(IV)(a)$

(weak uniqueness of limit = weak smoothness) for $\langle
M_{i}:i<\delta\rangle$ as above,

    1. $(a)$

if ${\rm cf}(\delta)\geq\kappa$ and $M$ is as in Ax(III)(a) and
$i<\delta\Rightarrow M_{i}\leq_{{\mathfrak{k}}}N$ then
$M\leq_{{\mathfrak{k}}}N$

    2. $(b)$

if $N_{\ell}\in K$ and $i<\delta\Rightarrow
M_{i}\leq_{{\mathfrak{k}}}N_{\ell}$ for $\ell=1,2$ then there are $N\in K$ and
$f_{1},f_{2}$ such that $f_{\ell}$ is a $\leq_{{\mathfrak{k}}}$-embedding of
$N_{\ell}$ into $N$ for $\ell=1,2$ and $i<\delta\Rightarrow f_{1}\restriction
M_{i}=f_{2}\restriction M_{i}$

  8. $Ax(V)$

if $N_{\ell}\leq_{{\mathfrak{k}}}M$ for $\ell=1,2$ and $N_{1}\subseteq N_{2}$
then $N_{1}\leq_{{\mathfrak{k}}}N_{2}$

  9. $Ax(VI)$

(L.S.T. property) if $A\subseteq M\in K,|A|\leq\lambda$ then there is
$M\leq_{{\mathfrak{k}}}N$ of cardinality $\lambda$ such that $A\subseteq M$.

######  Remark 5.3.

There are some more axioms listed in 5.4(5), but we shall mention them in any
claim in which they are used so no need to memorize, so 5.4(1)-(4) omit them?

######  Definition 5.4.

1) We say ${\mathfrak{k}}$ is a 4-d.a.e.c. or d.a.e.c+ when it is a
$(\lambda,\mu,\kappa)-1$-d.a.e.c. and satisfies Ax(III)(d),Ax(IV)(e) below.

2) We say ${\mathfrak{k}}$ is a 2-d.a.e.c. or d.a.e.c.± when is a
$(\lambda,\mu,\kappa)-0$-d.a.e.c. and Ax(III)(d), Ax(IV)(d) below holds.

3) We say ${\mathfrak{k}}$ is 5-d.a.e.c. when it is 1-d.a.e.c. and Ax(III)(f)
holds.

4) We say ${\mathfrak{k}}$ is 6-d.a.e.c. when it is a 1-d.a.e.c. and
Ax(III)(f) + Ax(IV)(f).

5) Concerning Definition 5.2, we consider the following axioms:

Ax(III)(c) if $I$ is $\kappa$-directed and $\bar{M}=\langle M_{s}:s\in
I\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing $s\leq_{I}$

$t\Rightarrow M_{s}\subseteq M_{s}$ and $\Sigma\\{\|M_{s}\|:s\in I\\}<\mu$
then $\bar{M}$ has a

$\leq_{{\mathfrak{s}}}$-upper bound, $M$, i.e. $s\in I\Rightarrow
M_{s}\leq_{{\mathfrak{k}}}M$.

Ax(III)(d) (union of directed system) if $I$ is $\kappa$-directed,
$|I|<\mu,\langle M_{t}:t\in I\rangle$

is $\leq_{{\mathfrak{k}}}$-increasing and $\|\cup\\{M_{t}:t\in I\\}\|<\mu$
then there is

one and only one $M$ with universe

$\cup\\{|M_{t}|:t\in I\\}$ such that $M_{s}\leq_{{\mathfrak{k}}}M$ for every
$s\in I$

we call it the $\leq_{{\mathfrak{k}}}$-union of $\langle M_{t}:t\in I\rangle$.

Ax(III)(e) like Ax(III)(c) but $I$ is just directed

Ax(III)(f) If $\bar{M}=\langle M_{i}:i<\delta\rangle$ is
$\leq_{{\mathfrak{k}}}$-increasing, ${\rm cf}(\delta)<\kappa$ and
$|\cup\\{M_{i}:i<$

$\delta\\}|<\mu$ then there is $M$ which is $\leq_{{\mathfrak{k}}}$-prime over
$\bar{M}$, i.e.

  1. $(*)\quad$ if $N\in K_{{\mathfrak{k}}}$ and $i<\delta\Rightarrow M_{i}\leq_{{\mathfrak{k}}}N$ then there is a $\leq_{{\mathfrak{k}}}$-embedding

of $M$ into $M$ over $\cup\\{|M_{i}|:i<\delta\\}$.

Ax(IV)(c) If $I$ is $\kappa$-directed and $\bar{M}=\langle M_{s}:s\in
I\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing and $N_{1},N_{2}$

are $\leq_{{\mathfrak{s}}}$-upper bounds of $\bar{M}$ then for some
$(N^{\prime}_{2},f)$ we have

$N_{2}\leq_{{\mathfrak{k}}}N^{\prime}_{2}$ and $f$ is a
$\leq_{{\mathfrak{k}}}$-embedding of $N_{1}$ into $N_{2}$

which is the identity on $M_{s}$ for every $s\in I$

(this is a weak form of uniqueness)

Ax(IV)(d) If $I$ is a $\kappa$-directed partial order, $\bar{M}=\langle
M_{s}:s\in I\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing,

$s\in I\Rightarrow M_{s}\leq_{{\mathfrak{k}}}M$ and $|M|=\cup|M_{s}|:s\in
I\\}$, then

$\bigwedge\limits_{s}M_{s}\leq_{{\mathfrak{k}}}N\Rightarrow
M\leq_{{\mathfrak{k}}}N$.

Ax(IV)(e) Like Ax(IV)(c) but $I$ is just directed.

Ax(IV)(f) If $I$ is directed and $\bar{M}=\langle M_{s}:s\in I\rangle$ is
$\leq_{{\mathfrak{k}}}$-increasing then there is

$M$ which is a $\leq_{{\mathfrak{k}}}$-prime over $\bar{M}$, defined as in
Ax(III)(f).

######  Claim 5.5.

Assume131313By 5.1 no need to say this ${\mathfrak{k}}$ is a d.a.e.c.

1) Ax(III)(d) implies Ax(III)(c).

2) Ax(III)(e) implies Ax(III)(c) and it implies Ax(III)(b).

3) Ax(IV)(d) implies Ax(IV)(c).

4) Ax(IV)(e) implies Ax(IV)(c) and implies Ax(III)(b).

5) In all the axioms in Definition 5.4 it is necessary that
$|\cup\\{M_{s}:s\in I\\}|<\mu_{{\mathfrak{k}}}$.

6) Ax(IV)(b) implies that ${\mathfrak{k}}$ has amalgamation.

######  Definition 5.6.

We say $\langle M_{i}:i<\alpha\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing
$(\geq\kappa)$-continuous when it is $\leq_{{\mathfrak{k}}}$-increasing and
$\delta<\alpha$ and ${\rm
cf}(\delta)\geq\kappa\Rightarrow|M_{i}|=\cup\\{|M_{j}|:j<\delta\\}$.

As an exercise we consider directed systems with mappings.

######  Definition 5.7.

1) We say that $\bar{M}=\langle M_{t},h_{s,t}:s\leq_{I}t\rangle$ is a
$\leq_{{\mathfrak{k}}}$-directed system when $I$ is a directed partial order
and if $t_{0}\leq_{I}t_{2}\leq_{I}t_{2}$ then
$h_{t_{2},t_{0}}=h_{t_{2},t_{1}}\circ h_{t_{1},t_{0}}$.

1A) We say that $\bar{M}=\langle M_{t},h_{s,t}:s\leq_{I}t\rangle$ is a
$\leq_{{\mathfrak{k}}}-\theta$-directed system when in addition $I$ is
$\theta$-directed.

2) We omit $h_{s,t}$ when $s\leq_{I}t\Rightarrow h_{s,t}={\rm id}_{M_{s}}$ and
write $\bar{M}=\langle M_{t}:t\in I\rangle$.

3) We say $(M,\bar{h})$ is a $\leq_{{\mathfrak{k}}}$-limit of $\bar{M}$ when
$\bar{h}=\langle h_{s}:s\in I\rangle,h_{s}$ is a
$\leq_{{\mathfrak{k}}}$-embedding of $M_{s}$ into $M_{s}$ and
$s\leq_{I}t\Rightarrow h_{s}=h_{t}\circ h_{t,s}$.

4) We say $\bar{M}=\langle M_{\alpha}:\alpha<\alpha^{*}\rangle$ is
$\leq_{{\mathfrak{k}}}$-semi-continuous when : (see Ax(III)(f) in 5.4)

  1. $(a)$

$\bar{M}$ is $\leq_{{\mathfrak{k}}}$-increasing

  2. $(b)$

if $\alpha<\alpha^{*}$ has cofinality $\geq\kappa$ then
$M_{\alpha}=\cup\\{M_{\beta}:\beta<\alpha\\}$

  3. $(c)$

if $\alpha<\alpha^{*}$ has cofinality $<\kappa$ then $M_{\delta}$ is
$\leq_{{\mathfrak{k}}}$-prime over $\bar{M}\restriction\alpha$.

######  Observation 5.8.

[${{\mathfrak{k}}}$ is an d.a.e.c.]

1) If $\bar{M}=\langle M_{t},h_{s,t}:s\leq_{I}t\rangle$ is a
$\leq_{{\mathfrak{k}}}$-directed system, then we can find a
$\leq_{{\mathfrak{k}}}$-directed system $\langle M^{\prime}_{t}:t\in I\rangle$
(so $s\leq_{I}t\Rightarrow M^{\prime}_{s}\leq_{{\mathfrak{k}}}M^{\prime}_{t}$)
and $\bar{g}=\langle g_{t}:t\in I\rangle$ such that:

  1. $(a)$

$g_{t}$ is an isomorphism from $M_{t}$ onto $M^{\prime}_{t}$

  2. $(b)$

if $s\leq_{I}t$ then $g_{s}=g_{t}\circ h_{s,t}$.

2) So in the axioms (III)(a),(b)(IV)(a) from Definition 5.2 as well as those
of 5.4 we can use $\leq_{{\mathfrak{k}}}$-directed system $\langle
M_{s},h_{s,t}:s\leq_{I}t\rangle$ with $I$ as there.

3) If ${\mathfrak{k}}$ is an ess-$(\mu,\lambda)$-a.e.c., see §1 then
${\mathfrak{k}}$ is a $(\mu,\lambda,\aleph_{0})$-d.a.e.c. and satisfies all
the axioms from 5.4.

4) If $(M,\bar{h})$ is prime over $\bar{M}=\langle
M_{t},h_{s,t}:s\leq_{I}t\rangle$ and $\chi=\Sigma\\{\|M_{t}\|:t\in I\\}$ then
$\|M\|\leq\chi^{<\kappa}$.

###### Proof..

Straightforward, e.g. we can use “${{\mathfrak{k}}}$ has
$(\chi^{<\kappa})$-LST”, i.e. Observation 5.9. ∎

More serious is proving the LST theorem in our content (recall that in the
axioms, see Ax(VI), we demand it only down to $\lambda$).

######  Claim 5.9.

[${\mathfrak{k}}$ is a $(\mu,\lambda,\kappa)-2$-d.a.e.c., see Definition 5.4.]

If
$\lambda_{{\mathfrak{k}}}\leq\chi=\chi^{<\kappa}<\mu_{{\mathfrak{k}}},A\subseteq
N\in{\mathfrak{k}}$ and $|A|\leq\chi\leq\|N\|$ then there is
$M\leq_{{\mathfrak{k}}}N$ of cardinality $\chi$ such that $\|M\|=\chi$.

###### Proof..

Let $\langle u_{\alpha}:\alpha<\alpha(*)\rangle$ list
$[A]^{<\kappa({{\mathfrak{k}}})}$, let $I$ be the following partial order:

  1. $(*)_{1}$

$(\alpha)\quad$ set of elements is $\\{\alpha<\chi$: for no $\beta<\alpha$ do
we have $u_{\alpha}\subseteq u_{\beta}\\}$

  2. $(\beta)\quad\alpha\leq_{I}\beta$ iff $u_{\alpha}\subseteq u_{\beta}$ (hence $\alpha\leq\beta$).

Easily

  1. $(*)_{2}$

$(a)\quad I$ is $\kappa$-directed

  2. $(b)\quad$ for every $\alpha<\alpha(*)$ for some $\beta<\alpha(*)$ we have $u_{\alpha}\subseteq u_{\beta}\wedge\beta\in I$

  3. $(c)\quad\cup\\{u_{\alpha}:\alpha\in I\\}=A$.

Now we choose $M_{\alpha}$ by induction on $\alpha<\chi$ such that

  1. $(*)_{3}$

$(a)\quad M_{\alpha}\leq_{{\mathfrak{k}}}N$

  2. $(b)\quad\|M_{\alpha}\|=\lambda_{{\mathfrak{k}}}$

  3. $(c)\quad M_{\alpha}$ include $\cup\\{M_{\beta}:\beta<_{I}\alpha\\}\cup u_{\alpha}$.

Note that $|\\{\beta\in I:\beta<_{I}\alpha\\}|\leq|\\{u:u\subseteq
u_{\alpha}\\}|=2^{|u_{\alpha}|}\leq
2^{<\kappa({{\mathfrak{k}}})}\leq\lambda_{{\mathfrak{k}}}$ and by the
induction hypothesis
$\beta<\alpha\Rightarrow\|M_{\beta}\|\leq\lambda_{{\mathfrak{k}}}$ and recall
$|u_{\alpha}|<\kappa({{\mathfrak{k}}})\leq\lambda_{{\mathfrak{k}}}$ hence the
set $\cup\\{M_{\beta}:\beta<\alpha\\}\cup u_{\alpha}$ is a subset of $N$ of
cardinality $\leq\lambda$ hence by Ax(VI) there is $M_{\alpha}$ as required.

Having chosen $\langle M_{\alpha}:\alpha\in I\rangle$ clearly by Ax(V) it is a
$\leq_{{\mathfrak{k}}}$-directed system hence by Ax(III)(d),
$M=\cup\\{M_{\alpha}:\alpha\in I\\}$ is well defined with universe
$\cup\\{|M_{\alpha}|:\alpha\in I\\}$ and by Ax(IV)(d) we have
$M\leq_{{\mathfrak{k}}}N$.

Clearly $\|M\|\leq\Sigma\\{\|M_{\alpha}\|:\alpha\in
I\\}\leq|I|\cdot\lambda_{{\mathfrak{k}}}=\chi$, and by $(*)_{2}(c)+(*)_{3}(c)$
we have
$A\subseteq\cup\\{u_{\alpha}:\alpha<\chi\\}=\cup\\{u_{\alpha}:\alpha\in
I\\}\subseteq\cup\\{|M_{\alpha}|:\alpha\in I\\}=M$ and so $M$ is as required.
∎

######  Notation 5.10.

1) For $\chi\in[\lambda_{{\mathfrak{k}}},\mu_{{\mathfrak{k}}})$ let
$K_{\chi}=K^{{\mathfrak{k}}}_{\chi}=\\{M\in K:\|M\|=\chi\\}$ and
$K_{<\chi}=\bigcup\limits_{\mu<\chi}K_{\mu}$.

2) ${{\mathfrak{k}}}_{\chi}=(K_{\chi},\leq_{{\mathfrak{k}}}\restriction
K_{\chi})$.

3) If
$\lambda_{{\mathfrak{k}}}\leq\lambda_{1}<\mu_{1}\leq\mu_{{\mathfrak{k}}},\lambda_{1}=\lambda^{<\kappa}_{1}$
and $(\forall\alpha<\mu_{1})(|\alpha|^{<\kappa}<\mu_{1})$, then we define
$K_{[\lambda_{1},\mu_{1}]}=K^{{\mathfrak{s}}}_{[\lambda_{1},\mu_{1})}$ and
${\mathfrak{k}}_{1}={\mathfrak{k}}_{[\lambda_{1},\mu_{1}]}$ similarly, i.e.
$K_{{\mathfrak{k}}}=\\{M\in
K_{{\mathfrak{k}}}:\|M\|\in[\lambda_{1},\mu_{1})\\}$ and
$\leq_{{\mathfrak{k}}_{1}}=\leq_{{\mathfrak{k}}}{\restriction}K_{{\mathfrak{k}}_{1}}$
and
$\lambda_{{\mathfrak{k}}_{1}}=\lambda_{1},\mu_{{\mathfrak{k}}_{1}}=\mu_{1},\kappa_{{\mathfrak{k}}_{1}}=\kappa_{{\mathfrak{k}}}$.

######  Definition 5.11.

The embedding $f:N\rightarrow M$ is a ${{\mathfrak{k}}}$-embedding or a
$\leq_{{\mathfrak{k}}}$-embedding if its range is the universe of a model
$N^{\prime}\leq_{{\mathfrak{k}}}M$, (so $f:N\rightarrow N^{\prime}$ is an
isomorphism onto).

######  Claim 5.12.

[${\mathfrak{k}}$ is a 2-d.a.e.c.]

1) For every $N\in K$ there is a $\kappa_{{\mathfrak{k}}}$-directed partial
order $I$ of cardinality $\leq\|N\|$ and $\bar{M}=\langle M_{t}:t\in I\rangle$
such that $t\in I\Rightarrow M_{t}\leq_{{\mathfrak{k}}}N,\|M_{t}\|\leq{\rm
LST}({{\mathfrak{k}}}),I\models s<t\Rightarrow
M_{s}\leq_{{\mathfrak{k}}}M_{t}$ and $N=\bigcup\limits_{t\in I}M_{t}$.

2) For every $N_{1}\leq_{{\mathfrak{k}}}N_{2}$ we can find $\langle
M^{\ell}_{t}:t\in I^{*}\rangle$ as in part (1) for $N_{\ell}$ such that
$I_{1}\subseteq I_{2}$ and $t\in I_{1}\Rightarrow M^{2}_{t}=M^{1}_{t}$.

###### Proof..

1) As in the proof of 5.9.

2) Similarly. ∎

######  Claim 5.13.

Assume
$\lambda_{{\mathfrak{k}}}\leq\lambda_{1}=\lambda^{<\kappa}_{1}<\mu_{1}\leq\mu_{{\mathfrak{k}}}$
and $(\forall\alpha<\mu_{1})(|\alpha|^{<\kappa}<\mu_{1})$.

1) Then ${\mathfrak{k}}^{*}_{1}:={\mathfrak{k}}_{[\lambda_{1},\mu_{1})}$ as
defined in 5.10(3) is a
$(\lambda_{1},\mu_{1},\kappa_{{\mathfrak{k}}})$-d.a.e.c.

2) For each of the folllowing axioms if ${\mathfrak{k}}$ satisfies it then so
does ${\mathfrak{k}}_{1}$: Ax(III)(d),(IV)(b),(IV)(c),(IV)(d).

3) If if addition ${\mathfrak{k}}$ satisfies Ax(III)(d),(IV)(d), this part (2)
apply to all the axioms in 5.2, 5.4.

######  Claim 5.14.

1) If ${\mathfrak{k}}$ satisfies Ax(IV)(e) then ${\mathfrak{k}}$ satisfies
Ax(III)(e) provided that $\mu_{{\mathfrak{k}}}$ is regular or at least the
relevant $I$ has cardinality $<{\rm cf}(\mu_{{\mathfrak{k}}})$.

2) If Ax(III)(d),(IV)(d) we can waive $\mu_{{\mathfrak{k}}}$ is regular.

###### Proof..

We prove this by induction on $|I|$.

Case 1: $I$ is finite.

So there is $t^{*}\in I$ such that $t\in I\Rightarrow t\leq_{I}t^{*}$, so this
is trivial.

Case 2: $I$ is countable.

So we can find a sequence $\langle t_{n}:n<\omega\rangle$ such that $t_{n}\in
I,t_{n}\leq_{I}t_{n+1}$ and $s\in
I\Rightarrow\bigvee\limits_{n<\omega}s\leq_{I}t_{n}$. Now we can apply the
axiom to $\langle M_{t_{n}},h_{t_{n,t_{m}}}:m<n<\omega\rangle$.

Case 3: $I$ uncountable.

First, we can find an increasing continuous sequence $\langle
I_{\alpha}:\alpha<|I|\rangle$ such that $I_{\alpha}\subseteq I$ is directed of
cardinality $\leq|\alpha|+\aleph_{0}$ and let
$I_{|I|}=I=\cup\\{I_{\alpha}:\alpha<|I|\\}$.

Second, by the induction hypothesis for each $\alpha<|I|$ we choose
$N_{\alpha},\bar{h}^{\alpha}=\langle h_{\alpha,t}:t\in I_{\alpha}\rangle$ such
that:

  1. $(a)$

$N_{\alpha}\in{{\mathfrak{k}}}^{{\mathfrak{s}}}_{\leq\chi}$

  2. $(b)$

$h_{\alpha,t}$ is a $\leq_{{\mathfrak{k}}}$-embedding of $M_{t}$ into
$N_{\alpha}$

  3. $(c)$

if $s<_{I}t$ are in $I_{\alpha}$ then $h_{\alpha,s}=h_{\alpha,t}\circ h_{t,s}$

  4. $(d)$

if $\beta<\alpha$ then $N_{\beta}\leq_{{\mathfrak{k}}}N_{\alpha}$ and $t\in
I_{\beta}\Rightarrow h_{\alpha,t}=h_{\beta,t}$.

For $\alpha=0$ use the induction hypothesis.

For $\alpha$ a limit ordinal by Ax(III)(a) there is $N_{\alpha}$ s required as
$I_{\alpha}=\cup\\{I_{\beta}:\beta<\alpha\\}$ there are no new $h_{t}$’s; well
we have to check $\Sigma\\{\|N_{\beta}\|:\beta<\alpha\\}<\mu_{{\mathfrak{k}}}$
but as we assume $\mu_{{\mathfrak{k}}}$ is regular this holds.

For $\alpha=\beta+1$, by the induction hypothesis there is
$(N^{\prime}_{\alpha},\bar{g}^{\alpha})$ which is a limit of $\langle
M_{s},h_{s,t}:s\leq_{I_{\alpha}}t\rangle$. Now apply Ax(IV)(e); well is the
directed system version with $\langle
M_{s},h_{s,t}:s\leq_{I_{\beta}}t\rangle,(N^{\prime}_{\alpha},\bar{g}_{\alpha}),(N_{\beta},\langle
h_{s}:s\in I_{\beta}\rangle$ here standing for $\bar{M},N_{1},N_{2}$ there.

So there are $N_{\alpha},f^{\alpha}_{s}(s\in I_{\beta})$ such that
$N_{\beta}\leq_{{\mathfrak{k}}}N_{\alpha}$ and $s\in I_{\beta}\Rightarrow
f^{\alpha}_{s}\cdot g_{s}=h_{s}$. Lastly, for $s\in I_{\alpha}\backslash
I_{\beta}$ we choose $h_{s}=f^{\alpha}_{s}\circ g_{s}$, so we are clearly
done.

2) Easy by 5.9 or 5.13. ∎

###  5(B). Basic Notions

As in [She09c, §1], we now recall the definition of orbital types (note that
it is natural to look at types only over models which are amalgamation bases
recalling Ax(IV)(b) implies every $M\in K_{{\mathfrak{k}}}$ is).

######  Definition 5.15.

1) For $\chi\in[\lambda_{{\mathfrak{k}}},\mu_{{\mathfrak{k}}})$ and $M\in
K_{\chi}$ we define ${{\mathscr{S}}}(M)$ as ${\rm
ortp}(a,M,N):M\leq_{{\mathfrak{k}}}N\in K_{\chi}\text{ and }a\in N\\}$ where
${\rm ortp}(a,M,N)=(M,N,a)/{{\mathscr{E}}}_{M}$ where ${{\mathscr{E}}}_{M}$ is
the transitive closure of ${{\mathscr{E}}}^{{\rm at}}_{M}$, and the two-place
relation ${{\mathscr{E}}}^{{\rm at}}_{M}$ is defined by:

$\begin{array}[]{clcr}(M,N_{1},a_{1}){{\mathscr{E}}}^{{\rm
at}}_{M}&(M,N_{2},a_{2})\text{{\text@underline{iff}}\,
}M\leq_{{\mathfrak{k}}}N_{\ell},a_{\ell}\in
N_{\ell},\|M\|\leq\|N_{\ell}\|=\|M\|^{<\kappa}\\\ &\text{ for }\ell=1,2\text{
and }\text{ there is }N\in K_{\chi}\text{ and
}\leq_{{\mathfrak{k}}}\text{-embeddings}\\\ &f_{\ell}:N_{\ell}\rightarrow
N\text{ for }\ell=1,2\text{ such that:}\\\ &f_{1}\restriction M={\rm
id}_{M}=f_{2}\restriction M\text{ and }f_{1}(a_{1})=f_{2}(a_{2}).\end{array}$

$(\text{of course
}M\leq_{{\mathfrak{k}}}N_{1},M\leq_{{\mathfrak{k}}}N_{2}\text{ and }a_{1}\in
N_{1},a_{2}\in N_{2})$

2) We say “$a$ realizes $p$ in $N$” when $a\in N,p\in{{\mathscr{S}}}(M)$ and
letting $\chi=\|M\|^{<\kappa}$ for some $N^{\prime}\in K_{\chi}$ we have
$M\leq_{{\mathfrak{k}}}N^{\prime}\leq_{{\mathfrak{k}}}N$ and $a\in N^{\prime}$
and $p={\rm ortp}(a,M,N^{\prime})$; so $M,N^{\prime}\in K_{\chi}$ but possibly
$N\notin K_{\chi}$.

3) We say “$a_{2}$ strongly 141414note that ${{\mathscr{E}}}^{{\rm at}}_{M}$
is not an equivalence relation and certainly in general is not
${\mathscr{E}}_{M}$ realizes $(M,N^{1},a_{1})/{{\mathscr{E}}}^{\text{at}}_{M}$
in $N$” when for some $N^{2}$ we have
$M\leq_{{\mathfrak{k}}}N^{2}\leq_{{\mathfrak{k}}}N$ and $a_{2}\in N^{2}$ and
$(M,N^{1},a_{1})\,{{\mathscr{E}}}^{{\rm at}}_{M}\,(M,N^{2},a_{2})$.

4) We say $M_{0}$ is a $\leq_{{\mathfrak{k}}[\chi_{0},\chi_{1})}$-amalgamation
base if this holds in ${{\mathfrak{k}}}_{[\chi_{0},\chi_{1})}$, see below.

4A) We say $M_{0}\in{{\mathfrak{k}}}$ is an amalgamation base or
$\leq_{{\mathfrak{k}}}$-amalgamation base when : for every
$M_{1},M_{2}\in{{\mathfrak{k}}}$ and $\leq_{{\mathfrak{k}}}$-embeddings
$f_{\ell}:M_{0}\rightarrow M_{\ell}$ (for $\ell=1,2$) there is
$M_{3}\in{{\mathfrak{k}}}_{\lambda}$ and $\leq_{{\mathfrak{k}}}$-embeddings
$g_{\ell}:M_{\ell}\rightarrow M_{3}$ (for $\ell=1,2$) such that $g_{1}\circ
f_{1}=g_{2}\circ f_{2}$.

5) We say ${{\mathfrak{k}}}$ is stable in $\chi$ when :

  1. $(a)$

$\lambda_{{\mathfrak{k}}}\leq\chi<\mu_{{\mathfrak{k}}}$

  2. $(b)$

$M\in K_{\chi}\Rightarrow|{{\mathscr{S}}}(M)|\leq\chi$

  3. $(c)$

$\chi\in{\rm Car}_{{\mathfrak{k}}}$ which means $\chi=\chi^{<\kappa}$ or the
conclusion of 5.9 holds

  4. $(d)$

${\mathfrak{k}}_{\chi}$ has amalgamation.

6) We say $p=q\restriction M$ if
$p\in{{\mathscr{S}}}(M),q\in{{\mathscr{S}}}(N),M\leq_{{\mathfrak{k}}}N$ and
for some $N^{+},N\leq_{{\mathfrak{k}}}N^{+}$ and $a\in N^{+}$ we have $p={\rm
ortp}(a,M,N^{+}),q={\rm ortp}(a,N,N^{+})$; note that $p\restriction M$ is well
defined if $M\leq_{{\mathfrak{k}}}N,p\in{{\mathscr{S}}}(N)$.

7) For finite $m$, for $M\leq_{{\mathfrak{k}}}N,\bar{a}\in{}^{m}N$ we can
define ${\rm ortp}(\bar{a},N,N)$ and ${{\mathscr{S}}}^{m}(M)$ similarly and
${{\mathscr{S}}}^{<\omega}(M)=\bigcup\limits_{m<\omega}{{\mathscr{S}}}^{m}(M)$,
(but we shall not use this in any essential way, hence we choose
${{\mathscr{S}}}(M)={{\mathscr{S}}}^{1}(M)$.)

######  Definition 5.16.

1) We say $N$ is $\lambda$-universal above or over $M$ if for every
$M^{\prime},M\leq_{{\mathfrak{k}}}M^{\prime}\in K^{{\mathfrak{k}}}_{\lambda}$,
there is a $\leq_{{\mathfrak{k}}}$-embedding of $M^{\prime}$ into $N$ over
$M$. If we omit $\lambda$ we mean ${\rm
rnd}_{{\mathfrak{k}}}(\lambda)=\min({\rm
Car})_{{\mathfrak{k}}}\backslash\lambda$) so
$\leq\|N\|^{<\kappa({{\mathfrak{k}}})}$; clearly this implies that $M$ is a
$\leq_{{\mathfrak{k}}_{[\chi_{0},\chi_{1}]}}$-amalgamation base where
$\chi_{0}=\|M\|,\chi_{1}=(\|N\|^{<\kappa})^{+}$.

2) $K^{3}_{{\mathfrak{k}}}=\\{(M,N,a):M\leq_{{\mathfrak{k}}}N,a\in N\backslash
M$ and $M,N\in K_{{\mathfrak{k}}}\\}$, with the partial order
$\leq=\leq_{{\mathfrak{k}}}$ defined by
$(M,N,a)\leq(M^{\prime},N^{\prime},a^{\prime})$ iff
$a=a^{\prime},M\leq_{{\mathfrak{k}}}M^{\prime}$ and
$N\leq_{{\mathfrak{k}}}N^{\prime}$. We say $(M,N,a)$ is minimal if
$(M,N,a)\leq(M^{\prime},N_{\ell},a)\in K^{3}_{{\mathfrak{k}}}$ for $\ell=1,2$
implies ${\rm ortp}(a,M^{\prime},N_{1})={\rm ortp}(a,M^{\prime},N_{2})$
moreover, $(M^{\prime},N_{1},a){{\mathscr{E}}}^{{\rm
at}}_{\lambda}(M^{\prime},N_{2},a)$, (not needed if every $M^{\prime}\in
K_{\lambda}$ is an amalgamation basis).

2A) $K^{3,{\mathfrak{k}}}_{\lambda}$ is defined similarly using
${\mathfrak{k}}_{[\lambda,{\rm rnd}(\lambda)]}$.

Generalizing superlimit, we have more than one reasonable choice.

######  Definition 5.17.

1) For $\ell=1,2$ we say $M^{*}\in K^{{\mathfrak{k}}}_{\lambda}$ is
superlimitℓ or $(\lambda,\geq\kappa)$-superlimitℓ when clause (c) of 5.15(5)
and:

  1. $(a)$

it is universal, (i.e., every $M\in K^{{\mathfrak{k}}}_{\lambda}$ can be
properly $\leq_{{\mathfrak{k}}}$-embedded into $M^{*}$), and

  2. $(b)$

Case 1: $\ell=1$ if $\langle M_{i}:i\leq\delta\rangle$ is
$\leq_{{\mathfrak{k}}}$-increasing, ${\rm
cf}(\delta)\geq\kappa,\delta<\lambda^{+}$ and $i<\delta\Rightarrow M_{i}\cong
M^{*}$ then $M_{\delta}\cong M^{*}$

  3. Case 2: $\ell=2$ if $I$ is a $(<\kappa)$-directed partial order of cardinality $\leq\chi$, $\langle M_{t}:t\in I\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing and $t\in I\Rightarrow M_{t}\cong M^{*}$ then $\cup\\{M_{t}:t\in I\\}\cong M^{*}$.

2) $M$ is $\lambda$-saturated above $\mu$ if $\|M\|\geq\lambda>\mu\geq{\rm
LST}({{\mathfrak{k}}})$ and:
$N\leq_{{\mathfrak{k}}}M,\mu\leq\|N\|<\lambda,p\in{{\mathscr{S}}}_{{\mathfrak{k}}}(N)$
implies $p$ is strongly realized in $M$. Let “$M$ is $\lambda^{+}$-saturated”
mean that “$M$ is $\lambda^{+}$-saturated above $\lambda$” and
$K(\lambda^{+}$-saturated) $=\\{M\in K:M$ is $\lambda^{+}$-saturated$\\}$ and
“$M$ is saturated $\ldots$” mean “$M$ is $\|M\|$-saturated $\ldots$”.

######  Definition 5.18.

1) We say $N$ is $(\lambda,\sigma)$-brimmed over $M$ if we can find a sequence
$\langle M_{i}:i<\sigma\rangle$ which is $\leq_{{\mathfrak{k}}}$-increasing
semi-continuous, $M_{i}\in K_{\lambda},M_{0}=M,M_{i+1}$ is
$\leq_{{\mathfrak{k}}}$-universal over $M_{i}$ and
$\bigcup\limits_{i<\sigma}M_{i}=N$. We say $N$ is $(\lambda,\sigma)$-brimmed
over $A$ if $A\subseteq N\in K_{\lambda}$ and we can find $\langle
M_{i}:i<\sigma\rangle$ as in part (1) such that $A\subseteq M_{0}$; if
$A=\emptyset$ we may omit “over $A$”.

2) We say $N$ is $(\lambda,*)$-brimmed over $M$ if for some
$\sigma\in[\kappa,\lambda),N$ is $(\lambda,\sigma)$-brimmed over $M$. We say
$N$ is $(\lambda,*)$-brimmed if for some $M,N$ is $(\lambda,*)$-brimmed over
$M$.

3) If $\alpha<\lambda^{+}$ let “$N$ is $(\lambda,\alpha)$-brimmed over $M$”
mean $M\leq_{{\mathfrak{k}}}N$ are from $K_{\lambda}$ and ${\rm
cf}(\alpha)\geq\kappa\Rightarrow N$ is $(\lambda,{\rm cf}(\alpha))$-brimmed
over $M$.

Recall

######  Claim 5.19.

1) If ${{\mathfrak{k}}}$ is a $(\mu,\lambda,\kappa)$-d.a.e.c. with
amalgamation, stable in $\chi$ and $\sigma={\rm cf}(\sigma)$ so
$\chi\in[\lambda,\mu)$, then for every $M\in K^{{\mathfrak{k}}}_{\chi}$ there
is $N\in K^{{\mathfrak{k}}}_{\lambda}$ universal over $M$ which is
$(\chi,\sigma)$-brimmed over $M$ (hence is $S^{\chi}_{\sigma}$-limit, see
[She09a], not used).

2) If $N_{\ell}$ is $(\chi,\theta)$-brimmed over $M$ for $\ell=1,2$, and
$\kappa\leq\theta={\rm cf}(\theta)\leq\chi^{+}$ then $N_{1},N_{2}$ are
isomorphic over $M$.

3) If $M_{2}$ is $(\chi,\theta)$-brimmed over $M$ and
$M_{0}\leq_{{\mathfrak{s}}}M_{1}$ then $M_{2}$ is $(\chi,\theta)$-brimmed over
$M_{0}$.

###### Proof..

Straightforward for part (1); recall clause (c) of Definition 5.19(5).

2),3) As in [She09c]. ∎

$*\qquad*\qquad*$

###  5(C). Liftings

Here we deal with lifting, there are two aspects. First, if
${\mathfrak{k}}^{1},{\mathfrak{k}}^{1}$ agree in $\lambda$ they agree in every
higher cardinal. Second, given ${\mathfrak{k}}$ we can find
${\mathfrak{k}}_{1}$ with
$\mu_{{\mathfrak{k}}_{1}}=\infty,({\mathfrak{k}}_{1})_{\lambda}={\mathfrak{k}}_{\lambda}$.

######  Theorem 5.20.

1) If ${{\mathfrak{k}}}^{\ell}$ is a $(\mu,\lambda,\kappa)$-a.e.c. for
$\ell=1,2$ and ${{\mathfrak{k}}}^{1}_{\lambda}={{\mathfrak{k}}}^{2}_{\lambda}$
then ${{\mathfrak{k}}}^{1}={\mathfrak{k}}^{2}$.

2) If ${{\mathfrak{k}}}_{\ell}$ is a $(\mu_{\ell},\lambda,\kappa)$-d.a.e.c.
for $\ell=1,2$ and ${\mathfrak{k}}^{1}$ satisfies Ax(IV)(d) and
$\mu_{1}\leq\mu_{2}$ and
${{\mathfrak{k}}}^{1}_{\lambda}={\mathfrak{k}}^{2}_{\lambda}$ then
${\mathfrak{k}}_{1}={\mathfrak{k}}_{2}[\lambda,\mu_{1})$.

###### Proof..

By 5.12. ∎

######  Theorem 5.21.

The lifting-up Theorem

1) If ${{\mathfrak{k}}}_{\lambda}$ is a
$(\lambda^{+},\lambda,\kappa)$-d.a.e.c.± then the pair
$(K^{\prime},\leq_{{{\mathfrak{k}}}^{\prime}})$ defined below is an
$(\infty,\lambda,\kappa)$-d.a.e.c.+ where we define

  1. $(A)$

$K^{\prime}$ is the class of $M$ such that $M$ is a
$\tau_{{{\mathfrak{k}}}_{\lambda}}$-model, and for some $I$ and $\bar{M}$ we
have

    1. $(a)$

$I$ is a $\kappa$-directed partial order

    2. $(b)$

$\bar{M}=\langle M_{s}:s\in I\rangle$

    3. $(c)$

$M_{s}\in K_{\lambda}$

    4. $(d)$

$I\models s<t\Rightarrow M_{s}\leq_{{{\mathfrak{k}}}_{\lambda}}M_{t}$

    5. $(e)$

if $J\subseteq I$ has cardinality $\leq\lambda$ and is $\kappa$-directed and
$M_{J}$ is the ${\mathfrak{k}}$-union of $\langle M_{t}:t\in J\rangle$, see
Definition 5.4, then $M_{J}$ is a submodel of $M$

    6. $(f)$

$M=\cup\\{M_{J}:J\subseteq I$ is $\kappa$-directed of cardinality
$\leq\lambda\\}$, i.e. both for the universe and for the relations and
functions.

  2. $(A)^{\prime}$

we call such $\langle M_{s}:s\in I\rangle$ a witness for $M\in K^{\prime}$, we
call it reasonable if $|I|\leq\|M\|^{<\kappa}$

  3. $(B)$

$M\leq_{{{\mathfrak{k}}}^{\prime}}N$ iff for some $I,J,\bar{M}$ we have

    1. $(a)$

$J$ is a $\kappa$-directed partial order

    2. $(b)$

$I\subseteq J$ is $\kappa$-directed

    3. $(c)$

$\bar{M}=\langle M_{s}:s\in J\rangle$ and is a
$\leq_{{{\mathfrak{k}}}_{\lambda}}$-increasing

    4. $(d)$

$\langle M_{s}:s\in J\rangle$ is a witness for $N\in K^{\prime}$

    5. $(e)$

$\langle M_{s}:s\in I\rangle$ is a witness for $M\in K^{\prime}$.

  4. $(B)^{\prime}$

We call such $I,\langle M_{s}:s\in J\rangle$ witnesses for
$M\leq_{{{\mathfrak{k}}}^{\prime}}N$ or say $(I,J,\langle M_{s}:s\in
J\rangle)$ witness $M\leq_{{{\mathfrak{k}}}^{\prime}}N$.

2) For the other axioms we have implications.

###### Proof..

The proof of part (2) is straightforward so we concentrate on part (1). So let
us check the axioms one by one.

AxO(a),(b),(c) and (d): $K^{\prime}$ is a class of $\tau$-models,
$\leq_{{{\mathfrak{k}}}^{\prime}}$ a two-place relation on $K,K^{\prime}$ and
$\leq_{{{\mathfrak{k}}}^{\prime}}$ are closed under isomorphisms and $M\in
K^{\prime}\Rightarrow\|M\|\geq\lambda$, etc.

[Why? trivially.]

AxI(a): If $M\leq_{{{\mathfrak{k}}}^{\prime}}N$ then $M\subseteq N$.

[Why? We use smoothness for $\kappa$-directed unions, i.e. Ax(IV)(x).]

AxII(a),(b),(c):

We prove the first, the others are easier.

Ax II(a):
$M_{0}\leq_{{{\mathfrak{k}}}^{\prime}}M_{1}\leq_{{{\mathfrak{k}}}^{\prime}}M_{2}$
implies $M_{0}\leq_{{{\mathfrak{k}}}^{\prime}}M_{2}$ and $M\in
K^{\prime}\Rightarrow M\leq_{{{\mathfrak{k}}}^{\prime}}M$.

[Why? The second phrase is trivial. For the first phrase let for
$\ell\in\\{1,2\\}$ the $\kappa$-directed partial orders $I_{\ell}\subseteq
J_{\ell}$ and $\bar{M}^{\ell}=\langle M^{\ell}_{s}:s\in J_{\ell}\rangle$
witness $M_{\ell-1}\leq_{{{\mathfrak{k}}}^{\prime}}M_{\ell}$.

We first observe

  1. $\boxdot$

if $I$ is a $\kappa$-directed partial order, $\langle M^{\ell}_{t}:t\in
I\rangle$ is a $\leq_{{{\mathfrak{k}}}_{\lambda}}$-system witnessing
$M_{\ell}\in K^{\prime}$ for $\ell=1,2$ and $t\in I\Rightarrow
M^{1}_{t}\leq_{{{\mathfrak{k}}}_{\lambda}}M^{2}_{t}$ then
$M_{1}\leq_{{\mathfrak{k}}}M_{2}$.

[Why? Let $I_{1}$ be the partial order with set of elements $I\times\\{1\\}$
ordered by $(s,1)\leq_{I_{1}}(t,1)\Leftrightarrow s\leq_{I}t$. Let $I_{2}$ be
the partial order with set of elements $I\times\\{1,2\\}$ ordered by
$(s_{1},\ell_{1})\leq_{I_{2}}(s_{2},\ell_{2})\Leftrightarrow
s_{1}\leq_{I}s_{2}\wedge\ell_{1}\leq\ell_{2}$. Clearly $I_{1}\subseteq I_{2}$
are both $\kappa$-directed.

Let $M_{(s,1)}=M^{1}_{s},M_{(s,2)}=M^{2}_{s}$, so clearly $\bar{M}=\langle
M_{t}:t\in I_{\ell}\rangle$ is a $\leq_{{\mathfrak{k}}}-\kappa$-directed
system witnessing $M_{\ell}\in K^{\prime}$ for $\ell=1,2$ and
$(I_{1},I_{2},\bar{M})$ witness $M_{1}\leq_{{{\mathfrak{k}}}^{\prime}}M_{2}$,
so we are done.]

Without loss of generality $J_{1},J_{2}$ are pairwise disjoint. Let
$\chi=(|J_{1}|+|J_{2}|)^{<\kappa}$ so $\lambda\leq\chi<\mu$ and let

$\begin{array}[]{clcr}{{\mathscr{U}}}:=\\{u:&u\subseteq J_{1}\cup J_{2}\text{
has cardinality }\leq\lambda\text{ and }u\cap I_{\ell}\\\ &\text{is
}\kappa\text{-directed under }\leq_{I_{\ell}}\text{ for }\ell=1,2\text{ and
}u\cap J_{\ell}\\\ &\text{is }\kappa\text{-directed under
}\leq_{J_{\ell}}\text{ for }\ell=1,2\text{ and}\\\ &\cup\\{|M^{2}_{t}|:t\in
I_{2}\\}=\cup\\{|M^{1}_{t}:t\in J_{1}\\}\\}.\end{array}$

Let $\langle u_{\alpha}:\alpha<\alpha^{*}\rangle$ list ${{\mathscr{U}}}$, and
we define a partial order $I$:

  1. $(a)^{\prime}$

its set of elements is $\\{\alpha<\alpha^{*}$: for no $\beta<\alpha$ do we
have $u_{\alpha}\subseteq u_{\alpha}\\}$

  2. $(b)^{\prime}$

$\alpha\leq_{I}\beta$ iff $u_{\alpha}\subseteq u_{\beta}\wedge\alpha\in
I\wedge\beta\in I$.

Note that the set $I$ may have ${\rm
card}(\sum\limits_{i<\delta}\|M_{i}\|)^{\lambda}$ which may be
$>\mu_{{\mathfrak{k}}}$.

As in the proof of 5.9, $I$ is $\kappa$-directed.

For $\ell=0,1,2$ and $\alpha\in I$ let $M_{\ell,\alpha}$ be

  1. $(a)$

$\leq_{{\mathfrak{k}}}$-union of $\langle M^{\ell}_{t}:t\in u_{\alpha}\cap
I_{1}\rangle$ if $\ell=0$

  2. $(b)$

the $\leq_{{\mathfrak{k}}}$-union of the
$\leq_{{\mathfrak{k}}_{\lambda}}$-directed system $\langle M^{1}_{t}:t\in
J_{1}\rangle$, equivalently the $\leq_{{{\mathfrak{k}}}_{\lambda}}$-directed
system of $\langle M^{2}_{t}:t\in I_{2}\rangle$, when $\ell=1$

  3. $(c)$

the $\leq_{{\mathfrak{k}}}$-union of the
$\leq_{{{\mathfrak{k}}}_{\lambda}}$-directed system $\langle M^{2}_{t}:t\in
J_{2}\rangle$ when $\ell=2$.

Now

  1. $(*)_{1}$

if $\ell=0,1,2$ and $\alpha\leq_{I}\beta$ then
$M^{\ell}_{\alpha}\leq_{{{\mathfrak{k}}}_{\lambda}}M^{\ell}_{\beta}$

  2. $(*)_{2}$

if $\alpha\in I$ then
$M^{0}_{\alpha}\leq_{{{\mathfrak{k}}}_{\lambda}}M^{1}_{\alpha}\leq_{{{\mathfrak{k}}}_{\lambda}}M^{1}_{\alpha}$

  3. $(*)_{3}$

$\langle M_{\ell,\alpha}:\alpha\in I\rangle$ is a witness for $M_{\ell}\in
K^{\prime}$

  4. $(*)_{4}$

$M_{0,\alpha}\leq_{{{\mathfrak{k}}}_{\lambda}}M_{2,\alpha}$ for $\alpha\in I$.

Together by $\boxdot$ we get that $M_{0}\leq_{{\mathfrak{k}}^{\prime}}M_{2}$
as required.

Ax III(a): In general.

Let $(I_{i,j},J_{i,j},\bar{M}^{i,j})$ witness
$M_{i}\leq_{{{\mathfrak{k}}}^{\prime}}M_{j}$ when $i\leq j<\delta$ and without
loss of generality $\langle J_{i,j}:i<j<\delta\rangle$ are pairwise disjoint.
Let ${{\mathscr{U}}}$ be the family of $u$ such that for some
$v\in[\delta]^{\leq\lambda}$,

  1. $(a)$

$v\subseteq\delta$ has cardinality $\leq\lambda$ and has order type of
cofinality $\geq\kappa$

  2. $(b)$

$u\subseteq\cup\\{J_{i,j}:i<j$ are from $v\\}$ has cardinality $\leq\lambda$
and

  3. $(b)$

for $i<j$ from $v$ the set $u\cap J_{i,j}$ is $\kappa$-directed under
$\leq_{J_{i,j}}$ and $u\cap I_{i,j}$ is $\kappa$-directed under
$\leq_{I_{i,j}}$

  4. $(c)$

if $i_{0}\leq i_{1}\leq i_{2}$ then $\cup\\{M^{i(0),i(1)}_{t,s}:s\in u\cap
J_{i(0),i(1)}\\}=\cup\\{M^{i(1),i(2)}_{s}:s\in u\cap I_{i(1),i(2)}\\}$

  5. $(d)$

if $i(0)\leq i(1)\leq i(2)$ are from $v$ then $\cup\\{M^{i(0),i(1)}_{t,s}:s\in
u\cap J_{i(0),i(1)}\\}=\cup\\{M^{i(1),i(2)}_{s}:s\in u\cap I_{i(1),i(2)}\\}$

  6. $(e)$

if $i(0)\leq k(0)\leq j(1)$ and $i(1)\leq j(1)$ are from $u$ then
$\cup\\{M^{i(0),j(0)}_{s}:s\in u\cap
J^{i(0),j(0)}_{s}\\}=\cup\\{M^{i(1),j(1)}_{s}:s\in u\cap J_{i(1),j(1)}\\}$.

Let the rest of the proof be as before.

Ax(IV)(a):

Similar, but ${\mathscr{U}}=\\{u\subseteq I:u$ has cardinality $\leq\lambda$
and is $\kappa$-directed$\\}$.

Ax(III)(d):

Assuming ${\mathfrak{k}}$ satisfies Ax(III)(d). Similar.

Ax(IV)(d):

Assuming ${\mathfrak{k}}$ satisfiefs Ax(IV)(d). Similar.

Axiom V: Assume $N_{0}\leq_{{{\mathfrak{k}}}^{\prime}}M$ and
$N_{1}\leq_{{{\mathfrak{k}}}^{\prime}}M$.

If $N_{0}\subseteq N_{1}$, then $N_{0}\leq_{{{\mathfrak{k}}}^{\prime}}N_{1}$.

[Why? Let $(I_{\ell},J_{\ell},\langle M^{\ell}_{s}:s\in J_{\ell}\rangle)$
witness $N_{\ell}\leq_{{\mathfrak{k}}}M$ for $\ell=0,1$; without loss of
generality $J_{0},J_{1}$ are disjoint.

Let

$\begin{array}[]{clcr}{\mathscr{U}}:=\\{u\subseteq J_{0}\cup
J_{1}:&|u|\leq\lambda\text{ and }u\cap J_{\ell}\text{ is
}\kappa\text{-directed}\\\ &\text{ and }u\cap I_{\ell}\text{ is
}\kappa\text{-directed for }\ell=0,1\text{ and}\\\ &\cup\\{|M^{0}_{s}|:s\in
u\cap J_{0}\\}=\cup\\{|M^{0}_{s}|:s\in u\in J_{1}\\}\\}.\end{array}$

For $u\in{\mathscr{U}}$ let

  1. $\bullet$

$M_{u}=M{\restriction}\cup\\{(M^{\ell}_{s}:s\in u\cap J_{\ell}\\}$ for $i=0,1$

  2. $\bullet$

$N_{\ell,u}=N_{\ell}{\restriction}\\{(M^{\ell}_{s}):s\in u\cap I_{\ell}\\}$.

Let

  1. $(*)$

$(a)\quad({\mathscr{U}},\subseteq)$ is $\kappa$-directed

  2. $(b)\quad N_{\ell,u}\leq_{{\mathfrak{k}}}M$

  3. $(c)\quad M_{\ell,u}\leq_{{\mathfrak{k}}}M_{\ell,v}$ when $u\subseteq v$ are from ${\mathscr{U}}$ and $\ell=0,1$

  4. $(d)\quad M_{0,u}\leq_{{\mathfrak{k}}}M_{1,u}$

  5. $(e)\quad N_{|}ell=\cup\\{N_{\ell,u}:u\in{\mathscr{U}}\\}$; as in ?

By $\boxdot$ above we are done.

Axiom VI: LST$({{\mathfrak{k}}}^{\prime})=\lambda$.

[Why? Let $M\in K^{\prime},A\subseteq M,|A|+\lambda\leq\chi<\|M\|$ and let
$\langle M_{s}:s\in I\rangle$ witness $M\in K^{\prime}$; without loss of
generality $|A|=\chi^{<\kappa}$. Now choose a directed $I\subseteq J$ of
cardinality $\leq|A|=\chi^{<\kappa}$ such that $A\subseteq
M^{\prime}=:\bigcup\limits_{s\in I}M_{s}$ and so $(I,J,\langle M_{s}:s\in
J\rangle)$ witnesses $M^{\prime}\leq_{{{\mathfrak{k}}}^{\prime}}M$, so as
$A\subseteq M^{\prime}$ and $\|M^{\prime}\|\leq|A|+\mu$ we are done.] ∎

Also if two such d.a.e.c.’s have some cardinal in common then we can put them
together.

######  Claim 5.22.

Let $\iota\in\\{1,2,3\\}$ and assume $\lambda_{1}<\lambda_{2}<\lambda_{3}$ and

  1. $(a)$

${{\mathfrak{k}}}^{1}$ is an $(\lambda^{+}_{2},\lambda,\kappa)-2$-d.a.e.c.,
$K^{1}=K^{1}_{\geq\lambda}$

  2. $(b)$

${{\mathfrak{k}}}^{2}$ is a $(\lambda_{3},\lambda_{2},\kappa)-\iota$-d.a.e.c.

  3. $(c)$

$K^{{\mathfrak{k}}^{1}}_{\lambda_{2}}=K^{{\mathfrak{k}}^{2}}_{\lambda_{2}}$
and
$\leq_{{{\mathfrak{k}}}^{2}}{\restriction}K^{{\mathfrak{k}}^{2}}_{\lambda_{2}}=\leq_{{{\mathfrak{k}}}^{1}}\restriction
K^{{\mathfrak{k}}^{1}}_{\lambda_{2}}$

  4. $(d)$

we define ${{\mathfrak{k}}}$ as follows:
$K_{{\mathfrak{k}}}=K_{{\mathfrak{k}}}\cup
K_{{\mathfrak{k}}_{2}},M\leq_{{\mathfrak{k}}}N$ iff
$M\leq_{{{\mathfrak{k}}}^{1}}N$ or $M\leq_{{{\mathfrak{k}}}^{2}}N$ or for some
$M^{\prime},M\leq_{{{\mathfrak{k}}}^{1}}M^{\prime}\leq_{{{\mathfrak{k}}}^{2}}N$.

Then ${{\mathfrak{k}}}$ is an
$(\lambda_{3},\lambda_{1},\kappa)-\iota$-d.a.e.c.

###### Proof..

Straightforward. E.g.

Ax(III)(d): So $\langle M_{s}:s\in I\rangle$ is
$\leq_{{\mathfrak{s}}}-\kappa$-directed system.

If $\|M_{s}\|\geq\lambda_{2}$ for some $\lambda$, use $\langle M_{s}:s\leq
t\in I\rangle$ and clause (b) of the assumption. If $\cup\\{M_{s}:s\in I\\}$
has cardinality $\leq\lambda_{2}$ use clause (a) in the assumption. If neither
one of them holds, recall $\lambda_{2}=\lambda^{<\kappa}_{2}$ by clause (b) of
the assumption, and let

$\\\ cU=\\{u\subseteq I:|u|\ le\lambda_{2},u\text{ is }\kappa\text{-directed
(in I), and }\cup\\{M_{s}:s\in u\\}\text{ has cardinality }\lambda\\}.$

Easily $({\mathscr{U}},\subseteq)$ is $\lambda_{2}$-directed, for $u\in J$ let
$M_{u}$ be the $\leq_{{\mathfrak{s}}}$-union of $\langle M_{s}:s\in u\rangle$.
Now by clause (a) of the assumption

  1. $(*)_{1}$

$M_{u}\in K^{{\mathfrak{k}}^{1}}_{\lambda_{2}}=K^{k^{*}}_{\lambda_{2}}$

  2. $(*)_{2}$

if $u_{1}\subseteq v$ are from ${\mathscr{U}}$ then
$M_{u}\leq_{{\mathfrak{k}}^{1}}M_{v},M_{u}\leq_{{\mathfrak{k}}^{2}}M_{v}$.

Now use clause (b) of the assumption.

Axiom V: We shall use freely

  1. $(*)$

${{\mathfrak{k}}}^{2}_{\lambda}={{\mathfrak{k}}}^{2}$ and
${{\mathfrak{k}}}^{1}_{\lambda}={{\mathfrak{k}}}^{1}$.

So assume
$N_{0}\leq_{{\mathfrak{k}}}M,N_{1}\leq_{{\mathfrak{k}}}M,N_{0}\subseteq
N_{1}$.

Now if $\|N_{0}\|\geq\lambda_{1}$ use assumption (b), so we can assume
$\|N_{0}\|<\lambda_{1}$. If $\|M\|\leq\lambda_{1}$ we can use assumption (a)
so we can assume $\|M\|>\lambda_{1}$ and by the definition of
$\leq_{{\mathfrak{k}}}$ there is $M^{\prime}_{0}\in
K^{{\mathfrak{k}}^{1}}_{\lambda_{1}}=K^{{\mathfrak{k}}^{2}}_{\lambda_{1}}$
such that
$N_{0}\leq_{{\mathfrak{k}}^{1}}M^{\prime}_{0}\leq_{{\mathfrak{k}}^{2}}M$.
First assume $\|N_{1}\|\leq\lambda_{1}$, so we can find $M^{\prime}_{1}\in
K^{{\mathfrak{k}}^{1}}_{\lambda_{1}}$ such that
$N_{1}\leq_{{{\mathfrak{k}}}^{1}}M^{\prime}_{1}\leq_{{{\mathfrak{k}}}^{2}}M$
(why? if $N_{1}\in K^{{\mathfrak{k}}^{1}}_{<\lambda_{1}}$, by the definition
of $\leq_{{\mathfrak{k}}}$ and if $N_{1}\in
K^{{\mathfrak{k}}^{1}}_{\lambda_{1}}$ just choose $M^{\prime}_{1}=N_{1}$). Now
we can by assumption (b) find $M^{\prime\prime}\in
K^{{\mathfrak{k}}^{1}}_{\lambda_{1}}$ such that $M^{\prime}_{0}\cup
M^{\prime}_{1}\subseteq M^{\prime\prime}\leq_{{\mathfrak{k}}}M$, hence by
assumption (b) (i.e. AxV for ${{\mathfrak{k}}}^{2}$) we have
$M^{\prime}_{0}\leq_{{\mathfrak{k}}}M^{\prime\prime},M^{\prime}_{1}\leq_{{\mathfrak{k}}}M^{\prime\prime}$.
As
$N_{0}\leq_{{\mathfrak{k}}}M^{\prime}_{0}\leq_{{\mathfrak{k}}}M^{\prime\prime}\in
K^{{\mathfrak{k}}}_{\leq\lambda_{1}}$ by assumption (a) we have
$N_{0}\leq_{{\mathfrak{k}}}M^{\prime\prime}$, and similarly we have
$N_{1}\leq_{{\mathfrak{k}}}M^{\prime\prime}$. So $N_{0}\subseteq
N_{1},N_{0}\leq_{{\mathfrak{k}}}M^{\prime\prime},N_{1}\leq_{{\mathfrak{k}}}M^{\prime}$
so by assumption (b) we have $N_{0}\leq_{{\mathfrak{k}}}N_{1}$.

We are left with the case $\|N_{1}\|>\lambda$, by assumption (b) there is
$N^{\prime}_{1}\in K_{\lambda_{1}}$ such that $N_{0}\subseteq
N^{\prime}_{1}\leq_{{\mathfrak{k}}^{2}}N_{2}$. By assumption (b) we have
$N^{\prime}_{1}\leq_{{\mathfrak{k}}}M$, so by the previous paragraph we get
$N_{0}\leq_{{\mathfrak{k}}}N^{\prime}_{1}$, together with the previous
sentence we have
$N_{0}\leq_{{{\mathfrak{k}}}^{1}}N^{\prime}_{1}\leq_{{{\mathfrak{k}}}^{2}}N_{1}$
so by the definition of $\leq_{{\mathfrak{k}}}$ we are done. ∎

######  Definition 5.23.

If $M\in K_{\chi}$ is $(\chi,\geq\kappa)$-superlimit1 let
$K^{[M]}_{\chi}=\\{N\in K_{\chi}:N\cong
M\\},{{\mathfrak{K}}}^{[M]}_{\chi}=(K^{[M]}_{\chi},\leq_{{\mathfrak{k}}}\restriction
K^{[M]}_{\chi})$ and ${{\mathfrak{k}}}^{[M]}$ is the
${{\mathfrak{k}}}^{\prime}$ we get in 5.21(1) for
${{\mathfrak{k}}}^{\prime}={{\mathfrak{k}}}^{[M]}_{\chi}$.

######  Claim 5.24.

1) If ${{\mathfrak{k}}}$ is an $(\mu,\lambda,\kappa)$-a.e.c.,
$\lambda\leq\chi<\mu,M\in K_{\chi}$ is $(\chi,\geq\kappa)$-superlimit1 then
${{\mathfrak{k}}}^{[M]}_{\chi}$ is a $(\chi^{+},\chi,\kappa)$-d.a.e.c.

2) If in addition ${{\mathfrak{k}}}$ is a $(\mu,\lambda,\kappa)$-d.a.e.c.±
then ${{\mathfrak{k}}}^{[M]}_{\chi}$ is a $(\chi^{+},\chi,\kappa)$-d.a.e.c.±.

##  6\. II pr frames

######  Definition 6.1.

For $\iota=1,2,3,4$. We say that ${{\mathfrak{s}}}$ is a good
$(\mu,\lambda,\kappa)-\iota$-frame when ${{\mathfrak{s}}}$ consists of the
following objects satisfying the following condition: $\mu,\lambda,\kappa$ (so
we may write
$\mu_{{\mathfrak{s}}},\lambda_{{\mathfrak{s}}},\kappa_{{\mathfrak{s}}}$ but we
usually ignore them defining ${{\mathfrak{s}}}$) and

  1. $(A)$

${{\mathfrak{k}}}={{\mathfrak{k}}}_{{\mathfrak{s}}}$ is a
$(\mu,\lambda,\kappa)-6$-d.a.e.c., so we may write ${{\mathfrak{s}}}$ instead
of ${{\mathfrak{k}}}$, e.g. $\leq_{{\mathfrak{s}}}$-increasing, etc. and
$\chi\in[\lambda,\mu)\Rightarrow{\rm LST}(\chi^{<\kappa})$

  2. $(B)$

${{\mathfrak{k}}}$ has a $(\lambda,\geq\kappa)$-superlimit model $M^{*}$ which
151515follows by (C) in fact is not $<_{{\mathfrak{k}}}$-maximal, i.e.,

    1. $(a)$

$M^{*}\in K^{{\mathfrak{s}}}_{\lambda}$

    2. $(b)$

if $M_{1}\in K^{{\mathfrak{s}}}_{\lambda}$ then for some
$M_{2},M_{1}<_{{\mathfrak{s}}}M_{2}\in K^{{\mathfrak{s}}}_{\lambda}$ and
$M_{2}$ is isomorphic to $M^{*}$

    3. $(c)$

if $\langle M_{i}:i<\delta\rangle$ is $\leq_{{\mathfrak{s}}}$-increasing,
$i<\delta\Rightarrow M_{i}\cong M$ and ${\rm
cf}(\delta)\geq\kappa,\delta<\lambda^{+}$ then $\cup\\{M_{i}:i<\delta\\}$ is
isomorphic to $M^{*}$

  3. $(C)$

${{\mathfrak{k}}}$ has the amalgamation property, the JEP (joint embedding
property), and has no $\leq_{{\mathfrak{k}}}$-maximal member; if of $\iota\geq
2,{{\mathfrak{k}}}$ has primes- and if $\iota\geq 4,{{\mathfrak{k}}}$ has
primes+

  4. $(D)$

$(a)\quad{{\mathscr{S}}}^{{\rm bs}}={{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}$ (the class of basic types for
${{\mathfrak{k}}}_{{\mathfrak{s}}}$) is included in

$\bigcup\\{{{\mathscr{S}}}(M):M\in K_{{\mathfrak{s}}}\\}$ and is closed under
isomorphisms including

automorphisms; for $M\in K_{\lambda}$ let ${{\mathscr{S}}}^{{\rm
bs}}(M)={{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}\cap{{\mathscr{S}}}(M)$;

no harm in allowing types of finite sequences.

  5. $(b)\quad$ if $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M)$, then $p$ is non-algebraic (i.e., not realized by any

$a\in M$).

  6. $(c)\quad$ (density)

if $M\leq_{{\mathfrak{k}}}N$ are from $K_{{\mathfrak{s}}}$ and $M\neq N$, then
for some $a\in N\backslash M$

we have ${\rm ortp}(a,M,N)\in{{\mathscr{S}}}^{{\rm bs}}$

[intention: examples are: minimal types in [She01], regular types

for superstable theories]

  7. $(d)\quad$ bs-stability

${{\mathscr{S}}}^{{\rm bs}}(M)$ has cardinality $\leq\|M\|^{<\kappa}$ for
$M\in K_{{\mathfrak{s}}}$.

  8. $(E)$

$(a)\quad\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits=\mathop{\hbox{\hbox
to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits_{\textstyle{\mathfrak{s}}}$
is a four place relation called nonforking with $\mathop{\hbox{\hbox
to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits(M_{0},M_{1},a,M_{3})$
implying $M_{0}\leq_{{\mathfrak{k}}}M_{1}\leq_{{\mathfrak{k}}}M_{3}$ are from
$K_{{\mathfrak{s}}},a\in M_{3}\backslash M_{1}$ and ${\rm
ortp}(a,M_{0},M_{3})\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M_{0})$ and
${\rm ortp}(a,M_{1},M_{3})\in{{\mathscr{S}}}^{{\rm bs}}(M_{1})$. Also
$\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits$
is preserved under isomorphisms.

We also write $M_{1}\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits_{\textstyle
M_{0}}^{\textstyle M_{3}}a$ and demand: if
$M_{0}=M_{1}\leq_{{\mathfrak{k}}}M_{3}$ both in $K_{\lambda}$ then:
$\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits(M_{0},M_{1},a,M_{3})$
is equivalent to “${\rm ortp}(a,M_{0},M_{3})\in{{\mathscr{S}}}^{{\rm
bs}}(M_{0})$”. Also we may state $M_{1}\mathop{\hbox{\hbox
to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits_{\textstyle
M_{0}}^{\textstyle M_{3}}a$ “${\rm ortp}(a,M_{1},M_{3})$ does not fork over
$M_{0}$ (inside $M_{3}$)” (this is justified by clause (b) below).

[Explanation: The intention is to axiomatize non-forking of types, but we
allow dealing with basic types only. Note that in [She01] we know something on
minimal types but other types are something else.]

  9. $(b)\quad$ (monotonicity):

if
$M_{0}\leq_{{\mathfrak{k}}}M^{\prime}_{0}\leq_{{\mathfrak{k}}}M^{\prime}_{1}\leq_{{\mathfrak{k}}}M_{1}\leq_{{\mathfrak{k}}}M_{3}\leq_{{\mathfrak{k}}}M^{\prime}_{3},M_{1}\cup\\{a\\}\subseteq
M^{\prime\prime}_{3}\leq_{{\mathfrak{k}}}M^{\prime}_{3}$ all of them in
$K_{\lambda}$, then $\mathop{\hbox{\hbox to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits(M_{0},M_{1},a,M_{3})\Rightarrow\mathop{\hbox{\hbox
to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits(M^{\prime}_{0},M^{\prime}_{1},a,M^{\prime}_{3})\Leftrightarrow\mathop{\hbox{\hbox
to9.99998pt{\hfil\vrule
width=0.3pt,depth=10.00012pt,height=0.0pt\hfil}\hbox{$\textstyle\bigcup$}}}\limits(M^{\prime}_{0},M^{\prime}_{1},a,M^{\prime\prime}_{3})$,
so it is legitimate to just say “${\rm ortp}(a,M_{1},M_{3})$ does not fork
over $M_{0}$”.

[Explanation: non-forking is preserved by decreasing the type, increasing the
basis (= the set over which it does not fork) and increasing or decreasing the
model inside which all this occurs. The same holds for stable theories only
here we restrict ourselves to “legitimate” types.]

  10. $(c)\quad$ (local character):

Case 1: $\iota=1,2,3$.

If $\langle M_{i}:i\leq\delta\rangle$ is $\leq_{{\mathfrak{s}}}$-semi-
continuous and $p\in{{\mathscr{S}}}^{{\rm bs}}(M_{\delta})$ and ${\rm
cf}(\delta)\geq\kappa$ then for every $\alpha<\delta$ large enough, $p$ does
not fork over $M_{\alpha}$.

Case 2: $\iota=4$.

If $I$ is a $\kappa$-directed partial order and $\bar{M}=\langle M_{t}:t\in
I\rangle$ is a $\leq_{{\mathfrak{s}}}$-directed system and $M$ is its
$\leq_{{\mathfrak{k}}}$-union and $M\leq_{{\mathfrak{s}}}N$ and ${\rm
ortp}(a,M,N)\in{{\mathscr{S}}}^{{\rm bs}}(M_{\delta})$ then for every $s\in I$
large enough ${\rm ortp}(a,M,N)$ does not fork over $M_{s}$.

[Explanation: This is a replacement for $\kappa\geq\kappa_{r}(T)$; if
$p\in{{\mathscr{S}}}(A)$ then there is a $B\subseteq A$ of cardinality
$<\kappa$ such that $p$ does not fork over $A$. The case $\iota=2$? is a very
strong demand even for stable first order theories.] It means dimensional
continuity, i.e. $M_{\delta}$ is minimal over
$\cup\\{M_{\alpha}:\alpha<\delta\\}$ and $\kappa$-saturated models.]

  11. $(d)\quad$ (transitivity):

if
$M_{0}\leq_{{\mathfrak{k}}}M^{\prime}_{0}\leq_{{\mathfrak{k}}}M^{\prime\prime}_{0}\leq_{{\mathfrak{k}}}M_{3}$
and $a\in M_{3}$ and ${\rm ortp}(a,M^{\prime\prime}_{0},M_{3})$ does not fork
over $M^{\prime}_{0}$ and ${\rm ortp}(a,M^{\prime}_{0},M_{3})$ does not fork
over $M_{0}$ (all models are in $K_{\lambda}$, of course, and necessarily the
three relevant types are in ${{\mathscr{S}}}^{{\rm bs}}$), then ${\rm
ortp}(a,M^{\prime\prime}_{0},M_{3})$ does not fork over $M_{0}$

  12. $(e)\quad$ uniqueness:

if $p,q\in{{\mathscr{S}}}^{{\rm bs}}(M_{1})$ do not fork over
$M_{0}\leq_{{\mathfrak{k}}}M_{1}$ (all in $K_{\mathfrak{s}}$) and

$p\restriction M_{0}=q\restriction M_{0}$ then $p=q$

  13. $(f)\quad$ symmetry:

Case 1: $\ell\geq 3$.

If $M_{0}\leq_{{\mathfrak{s}}}M_{\ell}\leq_{{\mathfrak{s}}}M_{3}$ and
$(M_{0},M_{\ell},a_{\ell})\in K^{3,{\rm pr}}_{{\mathfrak{s}}}$, see clause (j)
below for $\ell=1,2$ then ${\rm ortp}_{{\mathfrak{s}}}(a_{2},M_{1},M_{3})$
does not fork over $M_{0}$ iff ${\rm
ortp}_{{\mathfrak{s}}}(a_{1},M_{2},M_{3})$ does not fork over $M_{0}$.

Case 2: $\iota=1,2$.

If $M_{0}\leq_{{\mathfrak{k}}}M_{3}$ are in ${{\mathfrak{k}}}_{\lambda}$ and
for $\ell=1,2$ we have $a_{\ell}\in M_{3}$ and ${\rm
ortp}(a_{\ell},M_{0},M_{3})\in{{\mathscr{S}}}^{{\rm bs}}(M_{0})$, then the
following are equivalent:

    1. $(\alpha)\quad$ there are $M_{1},M^{\prime}_{3}$ in $K_{{\mathfrak{s}}}$ such that $M_{0}\leq_{{\mathfrak{k}}}M_{1}\leq_{{\mathfrak{K}}}M^{\prime}_{3}$,

$a_{1}\in M_{1},M_{3}\leq_{{\mathfrak{k}}}M^{\prime}_{3}$ and ${\rm
ortp}(a_{2},M_{1},M^{\prime}_{3})$ does not fork over $M_{0}$

    2. $(\beta)\quad$ there are $M_{2},M^{\prime}_{3}$ in $K_{\lambda}$ such that $M_{0}\leq_{{\mathfrak{k}}}M_{2}\leq_{{\mathfrak{k}}}M^{\prime}_{3}$,

$a_{2}\in M_{2},M_{3}\leq_{{\mathfrak{k}}}M^{\prime}_{3}$ and ${\rm
ortp}(a_{1},M_{2},M^{\prime}_{3})$ does not fork over $M_{0}$.

[Explanation: this is a replacement to “${\rm
ortp}(a_{1},M_{0}\cup\\{a_{2}\\},M_{3})$ forks over $M_{0}$ iff ${\rm
ortp}(a_{2},M_{0}\cup\\{a_{1}\\},M_{3})$ forks over $M_{0}$”; which is not
well defined in out context]

  14. $(g)\quad$ [existence] if $M\leq_{{\mathfrak{s}}}N,p\in{{\mathscr{S}}}^{{\rm bs}}(M)$ then there is $q\in{{\mathscr{S}}}^{{\rm bs}}(N)$

a non-forking extension of $p$

  15. $(h)\quad$ [continuity] Case 1: $\iota\geq 1$.

If $\langle M_{\alpha}:\alpha\leq\delta\rangle$ is
$\leq_{{\mathfrak{s}}}$-increasing, $\leq_{{\mathfrak{s}}}$-semi-continuity,
$M_{\delta}=\bigcup\limits_{\alpha<\delta}M_{\alpha}$ which holds if ${\rm
cf}(\delta)\geq\kappa$ and $p\in{{\mathscr{S}}}(M_{\delta})$ and
$p\restriction M_{\alpha}$ does not fork over $M_{0}$ for $\alpha<\delta$ then
$p\in{{\mathscr{S}}}^{{\rm bs}}(M_{\delta})$ and it does not fork over $M_{0}$

  16. Case 2: $\iota=4$.

Similarly for $\bar{M}=\langle M_{t}:t\in I\rangle,I$ directed,
$M=\cup\\{M_{t}:t\in I\\}$ is a $\leq_{{\mathfrak{s}}}$-upper bound of
$\bar{M}$

  17. $(j)\quad{{\mathfrak{s}}}$ has $K^{3,{\rm pr}}_{{\mathfrak{s}}}$-primes, see 6.8

  18. $(k)\quad$ Case 1: $\iota\geq 1$.

If $p\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(N)$ then $p$ does not fork
over $M$ for some $M\leq_{{\mathfrak{s}}}N$ from $K_{\lambda}$

  19. Case 2: $\iota=3,4$.

If $M_{\ell}(\ell\leq 3),a_{\ell},p_{\ell}(\ell=1,2)$ are as in (E)(f).

######  Discussion 6.2.

Consider using: semi-continuous + ${\rm cf}(\delta)\geq\kappa$ for
$(E)(c),(E)(x):{\rm cf}(\delta)\geq\kappa$ stable only if
$\chi=\chi^{<\kappa}$.

######  Claim 6.3.

1) If $\langle M_{i}:i<\delta\rangle$ is $\leq_{{\mathfrak{k}}}$-increasing,
$(\Sigma\\{\|M_{i}\|:i<\delta\\})<\mu$ and $p_{i}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M_{i})$ does not fork over $M_{0}$ for $i<\delta$ and
$[i<j\Rightarrow p_{j}\restriction M_{i}=p_{i}]$ then :

  1. $(a)$

we can find $M_{\delta}$ such that $i<\delta\Rightarrow
M_{i}\leq_{\mathfrak{k}}M_{\delta}$

  2. $(b)$

for any such $M_{\delta}$, we can find
$p\in{{\mathscr{S}}}_{{\mathfrak{s}}}(M_{\delta})$ such that
$\bigwedge\limits_{i<\delta}p\restriction M_{i}=p_{i}$ and $p$ does not fork
over $M_{0}$

  3. $(c)$

$p_{\delta}$ is unique in clause (b)

  4. $(d)$

if $\ell\geq\kappa\wedge{\rm cf}(\delta)\geq\kappa$ we can add
$M=\cup\\{M_{\alpha}:\alpha<\delta\\}$.

2) Similarly for $\bar{M}=\langle M_{t}:t\in I\rangle,I$ directed.

###### Proof..

1) First choose $M_{\delta}$ by 6.1, Clause (A). Second choose
$p_{\delta}\in{{\mathscr{S}}}^{{\rm bs}}_{{\mathfrak{s}}}(M_{\delta})$, a non-
forking extension of $p_{0}$, exist by Ax(g) of (E) of 6.1. Now
$p_{\delta}\restriction M_{i}\in{{\mathscr{S}}}^{{\rm
bs}}_{{\mathfrak{s}}}(M_{i})$ does not fork over $M_{0}$ by (b) of (E) of 6.1