thlabel [name=notes,title=Notes throughout the document (red text),columns=1]

###### Proof.

###### Bilinear spaces over a fixed field are simple unstable

###### Abstract.

We study the model theory of vector spaces with a bilinear form over a fixed
field. For finite fields this can be, and has been, done in the classical
framework of full first-order logic. For infinite fields we need different
logical frameworks. First we take a category-theoretic approach, which
requires very little set-up. We show that linear independence forms a simple
unstable independence relation. With some more work we then show that we can
also work in the framework of positive logic, which is much more powerful than
the category-theoretic approach and much closer to the classical framework of
full first-order logic. We fully characterise the existentially closed models
of the arising positive theory. Using the independence relation from before we
conclude that the theory is simple unstable, in the sense that dividing has
local character but there are many distinct types. We also provide positive
version of what is commonly known as the Ryll-Nardzewski theorem for
$\omega$-categorical theories in full first-order logic, from which we
conclude that bilinear spaces over a countable field are $\omega$-categorical.

###### Key words and phrases:

bilinear form; simple theory; positive logic; independence relation

This project was supported by EPSRC grant EP/W522314/1.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 The independence relation
  4. 4 Bilinear spaces in positive logic
  5. 5 $\omega$-Categoricity in positive logic
  6. 6 Comparison to different approaches

##  1\. Introduction

Vector spaces with a bilinear form, or _bilinear spaces_ as we will call them,
appear in many different places in mathematics. Examples include inner product
spaces (such as Hilbert spaces) and symplectic spaces arising from symplectic
geometry. The model theory of bilinear spaces has been studied using various
approaches. One approach is to study $K$-bilinear spaces over some fixed
finite field $K$ [KLM89, CH03]. As $K$ is finite, its elements can simply be
named in the signature (e.g. as constants). The arising theory turns out to be
simple unstable. In [Gra99] another approach is taken to study bilinear spaces
over infinite fields. Their set-up is to consider a bilinear space as a two-
sorted structure, with one vector space sort $V$ and one field sort $K$. They
prove that the arising theory is non-simple. Later it was shown in [CR16] that
the theory is NSOP1 (over algebraically closed fields).

When studying the model theory of vector spaces we generally take a one-sorted
approach, where scalar multiplication is coded by introducing a unary function
symbol for each field element. The arising theory is well known to be very
well-behaved (i.e. stable), regardless of the field. In contrast, when taking
a two-sorted approach the arising theory will be at least as complicated as
the theory of the field sort. So by making the field part of the language, and
thus fixing it, it no longer complicates the arising theory. The main idea of
this paper is to do something similar for bilinear spaces, namely fix the
field and then study its model theory.

The main problem with the classical first-order approach to vector spaces with
a bilinear form $[\cdot,\cdot]$ over some infinite field $K$, such as the
approach in [Gra99], is the strength of compactness. If $[x,y]=\lambda$ is
definable for every $\lambda\in K$, as it should be in any reasonable
signature, then the set $\\{[x,y]\neq\lambda:\lambda\in K\\}$ has a
realisation. This means that either the field needs to vary with the models,
as in the approach in [Gra99], or there are models where the bilinear form
$[\cdot,\cdot]$ is incomplete. In this paper we sidestep these issues by
working in different logical frameworks. We first take a category-theoretic
approach. This is a very general framework that requires very little set-up.
Then we consider the more powerful setting of positive logic, which is much
closer to the classical framework of full first-order logic, but still allows
us to fix the field $K$ to be any field we like.

Independence relations are a central tool in determining where a theory
belongs in the stability hierarchy. General theory for simple independence
relations has been developed for positive logic [Pil00, BY03b, BY03c] and for
the category-theoretic approach [Kam20]. These results are roughly of the form
“a given theory/category can only have one nice enough independence relation
(the _canonical independence relation_), and that relation determines its
place in the stability hierarchy”. In [HK21] such tools are used to study the
positive theory of exponential fields. In this paper we will, in a similar
way, employ these tools to show that linear independence is the canonical
independence relation in bilinear spaces over a fixed field, and that this
implies simplicity and non-stability.

Main results. We study the category $\mathbf{Bil}_{K}$ of $K$-bilinear spaces
with bilinear monomorphisms (injective linear maps that respect the bilinear
form). We write $\mathbf{Bil}^{\textup{s}}_{K}$ and
$\mathbf{Bil}^{\textup{a}}_{K}$ for the full subcategories of symmetric and
alternating $K$-bilinear spaces respectively, and we write
$\mathbf{Bil}^{*}_{K}$ when we mean any of these three categories. To study
the model theory using a category-theoretic approach we use the framework of
_abstract elementary categories_ , or _AECats_ , from [Kam20], see section 2.2
for the relevant details.

######  Theorem 1.1.

thm:category-of-bilinear-spaces-is-simple-aecat The category
$\mathbf{Bil}^{*}_{K}$ is an AECat with the amalgamation property that has a
canonical simple unstable independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ given by linear
independence.

We then move to positive logic and define a theory $T_{K}$ for $K$-bilinear
spaces, again with the notation $T_{K}^{s}$ and $T_{K}^{a}$ for the symmetric
and alternating cases respectively and $T_{K}^{*}$ for any of these theories.
The bilinear form is coded by binary relation symbols of the form
$[x,y]=\lambda$ for every $\lambda\in K$. The important part is that
$[x,y]\neq\lambda$ will not be positively definable when $K$ is infinite,
resulting in the fact that the e.c. (existentially closed) models are
$K$-bilinear spaces. In fact, we fully characterise the e.c. models.

######  Theorem 1.2.

thm:ec-models The following are equivalent for an $\mathcal{L}_{K}$-structure
$V$:

  1. (i)

$V$ is an e.c. model of $T_{K}^{*}$,

  2. (ii)

$V$ is a finitely injective $K$-bilinear space,

  3. (iii)

$V$ is an infinite dimensional non-degenerate finitely injective $K$-bilinear
space.

In (ii) and (iii) we restrict to symmetric/alternating $K$-bilinear spaces for
$T_{K}^{s}$ and $T_{K}^{a}$ respectively.

We then prove some results about $T_{K}^{*}$ that are useful for technical
reasons. We prove that the quantifier-free part of a type determines the
entire type (thm:quantifier-free-type-determines-type), but $T_{K}^{*}$ has
positive quantifier elimination precisely when $K$ is finite (thm:quantifier-
elimination). If $K$ is finite then $T_{K}^{*}$ is Boolean (every full first-
order formula is equivalent to a positive formula), so we are essentially back
in the classical first-order case mentioned earlier in this introduction.
However, when $K$ is infinite then $T_{K}^{*}$ cannot be Boolean (not even
_Hausdorff_ , see def:hausdorff), but we still have that equality of types is
type-definable (i.e.  _semi-Hausdorff_ , see def:hausdorff). Putting
everything together we can cast thm:category-of-bilinear-spaces-is-simple-
aecat in the setting of positive logic.

######  Theorem 1.3.

thm:positive-theory-simple-unstable The theory $T_{K}^{*}$ is simple unstable,
and non-dividing coincides with linear independence. That is, for every
$\bar{a}$, $\bar{b}$ and $C$ in some e.c. model $M$ we have
$\bar{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{M}\bar{b}$
iff $\operatorname{tp}(\bar{a}/C\bar{b})$ does not divide over $C$.

In the above $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ is the relation
given by linear independence, see def:linear-independence.

We also extend [Hay19, Theorem 6.5] to thm:omega-categoricity, resulting in a
characterisation of $\omega$-categorical positive theories (i.e. theories
where there is exactly one countable e.c. model, up to isomorphism). This
includes a positive variant of being an isolated type, which then allows us to
conclude that $T_{K}^{*}$ is $\omega$-categorical (for countable $K$), see
cor:bilinear-spaces-omega-categorical.

Overview. We start with some preliminaries in section 2 about bilinear spaces,
the category-theoretic framework of AECats and positive logic. Then in section
3 we establish the properties that make linear independence a simple unstable
independence relation in bilinear spaces (over a fixed field). In section 4 we
study the positive theory of bilinear spaces over a fixed field. In the stand-
alone section 5 we prove a theorem characterising $\omega$-categorical
theories. Finally, in section 6 we discuss two other model-theoretic
approaches to certain bilinear spaces, and compare them to our approach:
Hilbert spaces in continuous logic and the approach of [Gra99] to bilinear
spaces over an infinite field.

Acknowledgements. I would like to thank Jan Dobrowolski, Jonathan Kirby and
Rosario Mennuni for their feedback on earlier versions of this paper.

##  2\. Preliminaries

We discuss some preliminaries about bilinear spaces and the logical frameworks
of AECats and positive logic. All claims made in this section are well-known
and are either trivial or can be found in the references given at the start of
each subsection.

Lowercase letters such as $a,b,c$ and $x,y,z$ will generally denote single
elements or variables. We write $\bar{a}$ to mean a (possibly empty, possibly
infinite) tuple. The exact length of a tuple often does not matter, so we
write $\bar{a}\in A$ instead of $\bar{a}\in A^{n}$. We write unions in
juxtaposition, so $AB$ means $A\cup B$.

###  2.1. Bilinear spaces

Throughout this subsection we fix a field $K$. We will drop the $K$ from any
names and terms, e.g. we write “vector space” instead of “$K$-vector space”.

######  Definition 2.1.

def:bilinear-space Let $V$ be a vector space. A _bilinear form_ is a map
$[\cdot,\cdot]:V\times V\to K$ that is linear in each argument. That is, for
all $x,y,z\in V$ and $\lambda\in K$:

  * •

$[x,y+z]=[x,y]+[x,z]$ and $[x,\lambda y]=\lambda[x,y]$,

  * •

$[x+y,z]=[x,z]+[y,z]$ and $[\lambda x,y]=\lambda[x,y]$.

A _bilinear space_ is a vector space equipped with a bilinear form.

For any $A\subseteq V$ we write $\langle A\rangle$ for the linear span of $A$.
If $V$ is a bilinear space we naturally view $\langle A\rangle$ as a bilinear
space by restricting the bilinear form.

######  Definition 2.2.

def:symmetric-alternating Let $V$ be a bilinear space, we call $V$ or the
bilinear form on $V$:

  * •

_symmetric_ if $[x,y]=[y,x]$ for all $x,y\in V$;

  * •

_alternating_ if $[x,x]=0$ for all $x\in V$.

Note that in an alternating bilinear space $V$ we always have $[x,y]=-[y,x]$
for all $x,y\in V$, this is easily seen by expanding $[x+y,x+y]$.

######  Example 2.3.

ex:bilinear-spaces We give some common examples of bilinear spaces.

  1. (i)

For any field $K$ we can consider the $n$-dimensional vector space $K^{n}$
with the dot product as bilinear form, which yields a symmetric $K$-bilinear
space.

  2. (ii)

A real Hilbert space $H$ is a real vector space with an inner product such
that the associated metric makes it into a complete space. In particular, $H$
together with the inner product is a symmetric real bilinear space.

  3. (iii)

The standard symplectic space $\mathbb{R}^{2n}$: let
$x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ be the standard basis, then the
bilinear form is determined by $[x_{i},y_{i}]=-[y_{i},x_{i}]=1$ and $0$ for
any other combination of basis vectors.

We will use the following straightforward fact implicitly throughout this
paper.

######  Fact 2.4.

fact:bilinear-data-of-basis-determines-bilinear-form Let $V$ be a vector space
with basis $B$. To specify a bilinear form on $V$ it suffices to specify the
value for $[a,b]$ for every pair $a,b\in B$ (extend linearly in each
argument). In fact, any bilinear form is uniquely determined by the values
$[a,b]$ for all $a,b\in B$. It is symmetric precisely when $[a,b]=[b,a]$ for
all $a,b\in B$ and it is alternating precisely when $[a,a]=0$ and
$[a,b]=-[b,a]$ for all $a,b\in B$.

######  Definition 2.5.

def:non-degenerate We call a bilinear space $V$ _non-degenerate_ if $[x,y]=0$
for all $y\in V$ implies $x=0$ and, symmetrically, $[x,y]=0$ for all $x\in V$
implies $y=0$.

All examples in ex:bilinear-spaces are non-degenerate. One might normally only
be interested in non-degenerate spaces. However, we will not worry about this
so much as every degenerate space embeds into a non-degenerate one. This last
claim follows from thm:ec-models and fact:complete-to-ec-model, but we give a
sketch of a direct proof here. Start with some bilinear space $V_{0}$. We
extend $V_{0}$ to $V_{1}$ by adding a new basis vector $y_{x}$ for each $x\in
V_{0}$ and extend the bilinear form so that $x$ and $y_{x}$ have non-zero
bilinear product. We repeat this process $\omega$ times and then take the
union of the resulting chain of bilinear spaces.

######  Definition 2.6.

def:category-of-vector-spaces Let $K$ be a field. We write $\mathbf{Vec}_{K}$
for the category of $K$-vector spaces with injective linear maps as arrows.

######  Definition 2.7.

def:bilinear-homomorphism A _bilinear homomorphism_ is a linear map between
bilinear spaces that respects the bilinear forms. That is, a linear map
$f:V\to W$ where $V$ and $W$ are bilinear spaces, such that
$[x,y]=[f(x),f(y)]$ for all $x,y\in V$. A _bilinear monomorphism_ is an
injective bilinear homomorphism.

######  Definition 2.8.

def:category-of-bilinear-spaces Let $K$ be a field. We write
$\mathbf{Bil}_{K}$ for the category of $K$-bilinear spaces with bilinear
monomorphisms. Furthermore, we write $\mathbf{Bil}^{\textup{s}}_{K}$ and
$\mathbf{Bil}^{\textup{a}}_{K}$ for the full subcategory of symmetric and
alternating $K$-bilinear spaces respectively. We will write
$\mathbf{Bil}^{*}_{K}$ if something applies to all of $\mathbf{Bil}_{K}$,
$\mathbf{Bil}^{\textup{s}}_{K}$ and $\mathbf{Bil}^{\textup{a}}_{K}$.

###  2.2. Abstract Elementary Categories (AECats)

We first recall the basic notions concerning accessible categories. A great
reference for this is [AR94].

######  Definition 2.9.

def:lambda-presentable Let $\mathcal{C}$ be a category and let $\lambda$ be a
regular cardinal. An object $X$ in $\mathcal{C}$ is called _$\lambda$
-presentable_ if whenever $Y=\operatorname{colim}_{i\in I}Y_{i}$ is a
$\lambda$-directed colimit then every arrow $X\to Y$ factors essentially
uniquely as $X\to Y_{i}\to Y$ for some $i\in I$. Equivalently:
$\operatorname{Hom}(X,-)$ preserves $\lambda$-directed colimits.

######  Definition 2.10.

def:accessible-category A category $\mathcal{C}$ is called _$\lambda$
-accessible_ if:

  1. (i)

$\mathcal{C}$ has $\lambda$-directed colimits;

  2. (ii)

there is a set $\mathcal{A}$ of $\lambda$-presentable objects, such that every
object in $\mathcal{C}$ can be written as a $\lambda$-directed colimit of
objects in $\mathcal{A}$.

In the case where $\lambda=\omega$ we say that $\mathcal{C}$ is finitely
accessible. A category is called _accessible_ if it is $\lambda$-accessible
for some $\lambda$.

In [Kam20, Definition 2.5] an AECat is defined as a pair of categories
$(\mathcal{C},\mathcal{M})$ satisfying some properties. We will only be
interested in the case where $\mathcal{C}=\mathcal{M}$, so when we say
“$\mathcal{C}$ is an AECat” we will actually mean “$(\mathcal{C},\mathcal{C})$
is an AECat”. This also allows us to simplify the definition.

######  Definition 2.11.

def:aecat An accessible category $\mathcal{C}$ is called an _AECat_ if it has
all directed colimits and all arrows are monomorphisms.

######  Definition 2.12.

def:ap A category $\mathcal{C}$ is said to have the _amalgamation property_ if
any span of arrows $Y_{1}\leftarrow X\to Y_{2}$ can be completed to a
commuting square.

There are many examples of AECats with the amalgamation property, such as the
motivating example: any category of models of some first-order theory with
elementary embeddings. In this paper we will just be interested in
$\mathbf{Vec}_{K}$ and $\mathbf{Bil}_{K}^{*}$. Note that the arrows are
injective maps in both cases, because all arrows in an AECat have to be
monomorphisms. The following is straightforward to check.

######  Fact 2.13.

fact:categories-are-aecats For any $K$, the categories $\mathbf{Vec}_{K}$ and
$\mathbf{Bil}^{*}_{K}$ are finitely accessible. An object $V$ in any of these
categories is $\lambda$-presentable precisely when $\dim(V)<\lambda$. In
particular, each of these categories is an AECat.

It is easily seen that $\mathbf{Vec}_{K}$ has the amalgamation property.
Later, we will see that $\mathbf{Bil}^{*}_{K}$ also has the amalgamation
property (prop:independent-amalgamation). This is mainly relevant for the
notion of Galois type, which we define below. The amalgamation property
ensures that “having the same Galois type” is indeed transitive and thus an
equivalence relation. Once again, we simplify the definition for our specific
case.

######  Definition 2.14.

def:galois-type Let $\mathcal{C}$ be one of $\mathbf{Vec}_{K}$ or
$\mathbf{Bil}^{*}_{K}$. Let $V$ and $V^{\prime}$ be objects in $\mathcal{C}$,
$\bar{a}\in V$, $\bar{a}^{\prime}\in V^{\prime}$ and let $B\subseteq
V,V^{\prime}$ be a shared subset. Then we say that $\bar{a}$ and
$\bar{a}^{\prime}$ have the same _Galois type over $B$_, and we write

$\operatorname{gtp}(\bar{a}/B;V)=\operatorname{gtp}(\bar{a}^{\prime}/B;V^{\prime}),$

if there are arrows $V\xrightarrow{f}W\xleftarrow{g}V^{\prime}$ in our
category $\mathcal{C}$ that agree on $B$ and are such that
$f(\bar{a})=g(\bar{a}^{\prime})$. If $B=\emptyset$ we drop it from the
notation altogether.

We can make sense of arbitrary elements and subsets of the objects in
$\mathbf{Vec}_{K}$ and $\mathbf{Bil}_{K}^{*}$ because any tuple $\bar{a}\in V$
uniquely determines a subspace $\langle\bar{a}\rangle\subseteq V$, which is
then again an object in our category. Furthermore, as $\bar{a}$ generates
$\langle\bar{a}\rangle$ arrows with domain $\bar{a}$ are in one-to-one
correspondence with arrows with domain $\langle\bar{a}\rangle$.

###  2.3. Positive logic

For a more extensive treatment we refer to [BY03a, PY18].

######  Definition 2.15.

def:positive-syntax Fix a signature $\mathcal{L}$. A _positive existential
formula_ in $\mathcal{L}$ is one that is built using atomic formulas and
$\wedge$, $\vee$, $\top$, $\bot$ and $\exists$. An _h-inductive sentence_ is a
sentence of the form $\forall\bar{x}(\varphi(\bar{x})\to\psi(\bar{x}))$, where
$\varphi(\bar{x})$ and $\psi(\bar{x})$ are positive existential formulas. A
_positive theory_ is a set of h-inductive sentences.

######  Convention 2.16.

conv:positive-is-standard Whenever we say “formula” or “theory” we will mean
“positive existential formula” and “positive theory” respectively, unless
explicitly stated otherwise. This also means that every formula and theory we
consider will be implicitly assumed to be positive (existential).

In full first-order logic we consider elementary embeddings because they
preserve and reflect truth of all first-order formulas. We do not have
negation in positive logic, so there is a difference between preserving and
reflecting truth of formulas.

######  Definition 2.17.

def:homomorphism-immersion A function $f:M\to N$ between
$\mathcal{L}$-structures is called a _homomorphism_ if it preserves truth of
formulas. That is, if for every $\varphi(\bar{x})$ and every $\bar{a}\in M$ we
have

$M\models\varphi(\bar{a})\implies N\models\varphi(f(\bar{a})).$

We call $f$ an _immersion_ if the converse implication also holds.

######  Definition 2.18.

def:ec-model We call a model $M$ of $T$ an _existentially closed model_ or an
_e.c. model_ if the following equivalent conditions hold:

  1. (i)

every homomorphism $f:M\to N$ with $N\models T$ is an immersion;

  2. (ii)

for every $\bar{a}\in M$ and $\varphi(\bar{x})$ such that
$M\not\models\varphi(\bar{a})$ there is $\psi(\bar{x})$ with
$T\models\neg\exists\bar{x}(\varphi(\bar{x})\wedge\psi(\bar{x}))$ and
$M\models\psi(\bar{a})$.

Positive model theory generally studies the e.c. models of a theory. Every
model can be completed to an e.c. model anyway, see the fact below.

######  Fact 2.19.

fact:complete-to-ec-model Let $M$ be a model of some theory $T$ then there is
a homomorphism $f:M\to N$ such that $N$ is an e.c. model of $T$.

######  Definition 2.20.

def:type Let $M$ be an e.c. model, $B\subseteq M$ and $\bar{a}\in M$. Then the
_type of $\bar{a}$ over $B$_ is defined as:

$\operatorname{tp}(\bar{a}/B)=\\{\varphi(\bar{x})\text{ with parameters in
}B:M\models\varphi(\bar{a})\\}.$

In other words, it is a maximal consistent set of formulas with parameters in
$B$. A _partial type_ over $B$ is just any consistent set of formulas over
$B$.

######  Definition 2.21.

def:saturation We call an e.c. model $M$ of some theory $T$ _$\kappa$
-saturated_ if any partial type with $<\kappa$ variables and $<\kappa$
parameters from $M$ that is finitely satisfiable in $M$ has a realisation in
$M$.

Because e.c. models are generally not the same as just models of some theory
there can be h-inductive sentences that are true in all e.c. models, but fail
in some models. Such sentences can be added to the theory without changing the
class of e.c. models. It will be useful to have some notation for this.

######  Definition 2.22.

def:t-ec Let $T$ be a theory. The _Kaiser hull of $T$_ is defined as:

$T^{\textup{ec}}=\\{\chi\text{ an h-inductive sentence}:M\models\chi\text{ for
every e.c.\ model $M$ of $T$}\\}.$

The following definitions, except for being Boolean, are taken from [BY03c],
and are very useful for developing (neo)stability theory for positive logic.

######  Definition 2.23.

def:hausdorff Let $T$ be a positive theory. We call $T$:

  * •

_Boolean_ if every formula in full first-order logic is equivalent to some
positive existential formula modulo $T$, or equivalently: for every positive
existential formula $\varphi(\bar{x})$ there is a positive existential
$\psi(\bar{x})$ that is equivalent to $\neg\varphi(\bar{x})$ modulo $T$;

  * •

_Hausdorff_ if for any two distinct types $p(\bar{x})$ and $q(\bar{x})$ there
are $\varphi(\bar{x})\not\in p(\bar{x})$ and $\psi(\bar{x})\not\in q(\bar{x})$
such that
$T^{\textup{ec}}\models\forall\bar{x}(\varphi(\bar{x})\vee\psi(\bar{x}))$;

  * •

_semi-Hausdorff_ if equality of types is type-definable, so there is a partial
type $\Omega(\bar{x},\bar{y})$ such that for any $\bar{a},\bar{b}$ in some
e.c. model $M$ we have $\operatorname{tp}(\bar{a})=\operatorname{tp}(\bar{b})$
if and only if $M\models\Omega(\bar{a},\bar{b})$;

The reason for the name Hausdorff is that this corresponds to the type spaces
being Hausdorff, where formulas correspond to closed sets.

######  Fact 2.24.

fact:boolean-implies-hausdorff-implies-semi-hausdorff Boolean implies
Hausdorff implies semi-Hausdorff.

The reader that is familiar with the above terminology might be missing one
term: _thickness_ , which asserts that being an indiscernible sequence is
type-definable and which is again weaker than being semi-Hausdorff. However,
we will have no use for that notion here, so we leave it at an honourable
mention.

Boolean theories are essentially the classical full first-order theories.
Through a process called _positive Morleyisation_ [PY18, section 2.3] we can
view any theory in full first-order logic as a Boolean positive theory, and we
will implicitly do so.

The following fact is useful for proving or disproving that a theory is
Hausdorff.

######  Fact 2.25 ([PY18, Theorem 8]).

fact:hausdorff-iff-aph The following are equivalent for a theory $T$:

  1. (i)

$T$ is Hausdorff;

  2. (ii)

any model of $T^{\textup{ec}}$ is an amalgamation base, so any span of
homomorphisms $M_{1}\leftarrow M\to M_{2}$ between models of $T$ (not
necessarily e.c.) with $M\models T^{\textup{ec}}$ can be amalgamated to
$M_{1}\to N\leftarrow M_{2}$ where $N\models T$.

The following is the positive analogue of being complete in full first-order
logic.

######  Definition 2.26.

def:jep A theory $T$ has the _joint embedding property_ , or _JEP_ , if for
any two e.c. models $M_{1}$ and $M_{2}$ there is $N\models T$ with
homomorphisms $M_{1}\to N\leftarrow M_{2}$.

##  3\. The independence relation

Throughout this section we again fix a field $K$ and drop it from any names
and terms (except for the names of our categories).

The notion of linear independence can be cast in the form of a ternary
relation on subsets of vector spaces. We give two equivalent formulations.

######  Definition 3.1.

def:linear-independence Let $V$ be a vector space and let $A,B,C\subseteq V$.
We say that _$A$ is (linearly) independent from $B$ over $C$_, and write
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{V}_{C}B$, if the
following equivalent statements hold:

  1. (i)

$\langle AC\rangle\cap\langle BC\rangle=\langle C\rangle$;

  2. (ii)

given a basis $C_{0}$ for $\langle C\rangle$ and $A_{0}$ and $B_{0}$ such that
$A_{0}C_{0}$ is a basis for $\langle AC\rangle$ and $B_{0}C_{0}$ is a basis
for $\langle BC\rangle$, we have that $A_{0}B_{0}C_{0}$ is a linearly
independent set.

Notation: any of $A$, $B$ or $C$ can be replaced by a tuple enumerating them.
For example, if $\bar{a}$ enumerates $A$ then
$\bar{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ just
means $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$.

The relation $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ really is a
ternary relation on subobjects of objects in $\mathbf{Vec}_{K}$ (again using
the idea that an arbitrary subset $A$ is essentially the same as $\langle
A\rangle$). This independence relation is known to have many desirable
properties in $\mathbf{Vec}_{K}$. Most of these properties are immediate from
the definition, for the remainder see for example [Kim14, Fact 2.1.4].

######  Fact 3.2.

fact:linear-independence-facts The independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ has the
following properties in $\mathbf{Vec}_{K}$.

Invariance:

For any arrow $f:V\to W$ we have $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ iff
$f(A)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{f(C)}^{W}f(B)$.

Monotonicity:

If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ and
$B^{\prime}\subseteq B$ then $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{C}^{V}B^{\prime}$.

Base-Monotonicity:

If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ and
$C\subseteq C^{\prime}\subseteq B$ then
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{C^{\prime}}^{V}B$.

Transitivity:

If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{B}^{V}C$ and
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}D$ with
$B\subseteq C$ then $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{B}^{V}D$

Symmetry:

If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ then
$B\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}A$.

Existence:

We always have $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}C$.

Finite Character:

If $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{C}^{V}B^{\prime}$ for all finite $B^{\prime}\subseteq B$ then
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$.

Local Character:

For any $A,B\subseteq V$ there is $B^{\prime}\subseteq B$ with $\dim(\langle
B^{\prime}\rangle)\leq\dim(\langle A\rangle)$ such that
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{B^{\prime}}^{V}B$.

Extension:

If $\bar{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ then
for any $D\subseteq V$ there is an extension $V\subseteq W$ with some
$\bar{a}^{\prime}$ in $W$ such that
$\operatorname{gtp}(\bar{a}/BC;V)=\operatorname{gtp}(\bar{a}^{\prime}/BC;W)$
and $\bar{a}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{W}BD$.

Stationarity:

If $\operatorname{gtp}(\bar{a}/C;V)=\operatorname{gtp}(\bar{a}^{\prime}/C;V)$
then $\bar{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ and
$\bar{a}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$ implies
$\operatorname{gtp}(\bar{a}/BC;V)=\operatorname{gtp}(\bar{a}^{\prime}/BC;V)$.

######  Definition 3.3.

def:stable-independence-relation An independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ satisfying the
properties in fact:linear-independence-facts is called a _stable independence
relation_.

The fact that $\mathbf{Vec}_{K}$ has a stable independence relation means that
it is model-theoretically very well-behaved. The situation in
$\mathbf{Bil}^{*}_{K}$ turns out to be a little bit more complicated: we lose
the Stationarity property.

######  Proposition 3.4.

prop:failure-of-stationarity The Stationarity property fails for
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ over every $C$
in $\mathbf{Bil}^{*}_{K}$. That is, for any $C$ in $\mathbf{Bil}^{*}_{K}$
there is an extension $C\subseteq V$ with $a,a^{\prime},b\in V$ such that
$\operatorname{gtp}(a/C;V)=\operatorname{gtp}(a^{\prime}/C;V)$,
$a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}b$ and
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}b$, while
$\operatorname{gtp}(a/Cb;V)\neq\operatorname{gtp}(a^{\prime}/Cb;V)$.

###### Proof.

Let $C$ be a bilinear space and introduce new linearly independent vectors
$a,a^{\prime},b$ and set $V=\langle Caa^{\prime}b\rangle$. By construction we
then have $a\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}b$ and
$a^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}b$. We
make $V$ into a bilinear space by setting $[x,c]=[c,x]=0$ for
$x\in\\{a,a^{\prime},b\\}$ and $c\in C$. We set
$[a^{\prime},b]=[b,a^{\prime}]=1$ (in the alternating case we set
$[b,a^{\prime}]=-1$), for the remainder of the pairs in $\\{a,a^{\prime},b\\}$
we take their bilinear product to be $0$.

Define $f:\langle Ca\rangle\to V$ to be the identity on $C$ and
$f(a)=a^{\prime}$, and extend linearly. Then $f$ is a bilinear monomorphism.
So we can amalgamate $V\supseteq\langle Ca\rangle\xrightarrow{f}V$ using
prop:independent-amalgamation to find $V\xrightarrow{g}W\xleftarrow{h}V$ such
that $g|_{\langle Ca\rangle}=hf$. So $g$ and $h$ agree on $C$ and
$g(a)=h(a^{\prime})$, and thus
$\operatorname{gtp}(a/C;V)=\operatorname{gtp}(a^{\prime}/C;V)$.

Suppose for a contradiction that
$\operatorname{gtp}(a/Cb;V)=\operatorname{gtp}(a^{\prime}/Cb;V)$, then there
are $V\xrightarrow{g}W\xleftarrow{h}V$ that agree on $Cb$ and
$g(a)=h(a^{\prime})$. But then
$0=[a,b]=[g(a),g(b)]=[h(a^{\prime}),h(b)]=[a^{\prime},b]=1$. ∎

The independence relation on $\mathbf{Bil}_{K}^{*}$ is still reasonably nice.
We just have to replace Stationarity with some weaker property, namely
3-amalgamation. Once again, we give a simplified definition for our situation.

######  Definition 3.5.

def:3-amalgamation An independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ has
3-amalgamation if the following holds. Suppose that we have a commuting
diagram as below, but without the dashed arrows and without $W$. We view all
the arrows as inclusions. Suppose furthermore that
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{1}}B$,
$B\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{3}}C$ and
$C\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{2}}A$.
Then we can find the dashed arrows and $W$, such that
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{W}V_{3}$ and
the resulting diagram commutes.

${V_{2}}$${W}$${A}$${V_{1}}$${C}$${V_{3}}$${D}$${B}$

######  Definition 3.6.

def:simple-independence-relation An independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ that satisfies
the properties in fact:linear-independence-facts, except possibly
Stationarity, and also satisfies 3-amalgamation is called a _simple
independence relation_.

The possible failure of Stationarity is precisely what distinguishes a simple
independence relation from a stable one. This is because we get 3-amalgamation
from Stationarity, modulo the rest of the properties.

######  Fact 3.7 ([Kam20, Proposition 6.16]).

fact:stationarity-implies-3-amalgamation Any stable independence relation also
satisfies 3-amalgamation. So every stable independence relation is also
simple.

Simple independence relations are always canonical, in the sense that there
can only be one on a given AECat, see the fact below. So
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$, being stable
and thus in particular simple, is the canonical independence relation on
$\mathbf{Vec}_{K}$. Even though the formulation of the properties is slightly
different from the formulation in [Kam20], it is an easy exercise to see that
they are equivalent.

######  Fact 3.8 ([Kam20, Theorem 1.1]).

fact:canonicity-of-independence Let $\mathcal{C}$ be an AECat with the
amalgamation property. If $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ and
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\prime}$ are
simple independence relations on $\mathcal{C}$ then
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}=\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}^{\prime}$.

The remainder of this section is now devoted to proving that
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ is a simple
independence relation on $\mathbf{Bil}^{*}_{K}$, and is thus also the
canonical independence relation.

######  Proposition 3.9 (Independent amalgamation).

prop:independent-amalgamation Let $V,W_{1},W_{2}$ be bilinear spaces. Let
$W_{1}\xleftarrow{f_{1}}V\xrightarrow{f_{2}}W_{2}$ be bilinear monomorphisms.
Then there is a bilinear space $U$ and bilinear monomorphisms
$W_{1}\xrightarrow{g_{1}}U\xleftarrow{g_{2}}W_{2}$, such that
$g_{1}f_{1}=g_{2}f_{2}$ and
$g_{1}(W_{1})\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{g_{1}f_{1}(V)}^{U}g_{2}(W_{2})$. Furthermore, if $V,W_{1},W_{2}$
are symmetric/alternating then we can choose $U$ to be symmetric/alternating.

###### Proof.

For convenience we may assume $V\subseteq W_{1},W_{2}$ and $W_{1}\cap
W_{2}=V$. So we can take $U=\langle W_{1}W_{2}\rangle$ and take $g_{1}$ and
$g_{2}$ to be the relevant embeddings. So we have
$W_{1}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{V}^{U}W_{2}$ and
we are left to extend the bilinear form to all of $U$. Let
$V^{\prime}\subseteq V$ be a basis for $V$ and for $i\in\\{1,2\\}$ let
$W^{\prime}_{i}\subseteq W_{i}$ be such that $V^{\prime}W^{\prime}_{i}$ is a
basis for $W_{i}$. Since $W_{1}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{V}^{U}W_{2}$ we
have that $V^{\prime}W^{\prime}_{1}W^{\prime}_{2}$ is a linearly independent
set, and hence a basis for $U$. So we can set $[w_{1},w_{2}]=[w_{2},w_{1}]=0$
for all $w_{1}\in W_{1}^{\prime}$ and $w_{2}\in W_{2}^{\prime}$, and extend
linearly in each argument. The final claim about the symmetric/alternating
property then immediately follows. ∎

######  Proposition 3.10.

prop:extension The independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ on
$\mathbf{Bil}^{*}_{K}$ satisfies Extension.

###### Proof.

Let $\bar{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{V}B$. We
will prove that there is some extension $V\subseteq W$ and a tuple
$\bar{a}^{\prime}$ in $W$ such that
$\bar{a}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{W}V$ and
$\operatorname{gtp}(\bar{a}/BC;V)=\operatorname{gtp}(\bar{a}^{\prime}/BC;W)$.
Consider the span of inclusions $\langle BC\bar{a}\rangle\supseteq\langle
BC\rangle\subseteq V$ and use prop:independent-amalgamation to find $\langle
BC\bar{a}\rangle\xrightarrow{f}W\xleftarrow{g}V$ completing this span to a
commuting square. We may assume $g$ is an inclusion, so we have our extension
$V\subseteq W$. Set $\bar{a}^{\prime}=f(\bar{a})$, then
$\operatorname{gtp}(\bar{a}^{\prime}/BC;W)=\operatorname{gtp}(\bar{a}/BC;\langle
BC\bar{a}\rangle)=\operatorname{gtp}(\bar{a}/BC;V)$. It follows that
$\bar{a}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{W}\langle
BC\rangle$ and applying Monotonicity to $f(\langle
BC\bar{a}\rangle)\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{\langle
BC\rangle}^{W}V$ yields
$\bar{a}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{\langle
BC\rangle}^{W}V$. So by Transitivity we get
$\bar{a}^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{W}V$, as
required. ∎

We prove a stronger version of 3-amalgamation, where the resulting cube
satisfies two extra instances of independence.

######  Theorem 3.11 (3-amalgamation).

thm:3-amalgamation Suppose that we have a commuting diagram as below of
bilinear monomorphisms (which we view as inclusions), but without the dashed
arrows and without $W$. Suppose furthermore that
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{1}}B$,
$B\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{3}}C$ and
$C\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{2}}A$.
Then there is a bilinear space $W$, together with the dashed bilinear
monomorphisms, such that $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{W}V_{3}$,
$B\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{W}V_{2}$ and
$C\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{W}V_{1}$ and
the resulting diagram commutes.

${V_{2}}$${W}$${A}$${V_{1}}$${C}$${V_{3}}$${D}$${B}$

Furthermore, if $V_{1},V_{2},V_{3}$ are all symmetric/alternating then we can
choose $W$ to be symmetric/alternating.

###### Proof.

First pick a basis $D^{\prime}$ of $D$ and let
$A^{\prime},B^{\prime},C^{\prime}$ be such that $A^{\prime}D^{\prime}$,
$B^{\prime}D^{\prime}$ and $C^{\prime}D^{\prime}$ are bases of $AD$, $BD$ and
$CD$ respectively. Because $A\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{1}}B$ we
have that $A^{\prime}B^{\prime}D^{\prime}$ is a linearly independent set, so
we extend it to a basis for $V_{1}$. So let $V^{\prime}_{1}$ such that
$A^{\prime}B^{\prime}D^{\prime}V^{\prime}_{1}$ is a basis for $V_{1}$.
Similarly we find $V^{\prime}_{2}$ and $V^{\prime}_{3}$. We may assume that
$V^{\prime}_{1},V^{\prime}_{2},V^{\prime}_{3}$ are such that
$D^{\prime}A^{\prime}B^{\prime}C^{\prime}V^{\prime}_{1}V^{\prime}_{2}V^{\prime}_{3}$
is a basis for $W=\langle
D^{\prime}A^{\prime}B^{\prime}C^{\prime}V^{\prime}_{1}V^{\prime}_{2}V^{\prime}_{3}\rangle$.
This induces canonical inclusions $V_{i}\subseteq W$. So in particular
$V_{1}\cap V_{2}=A$, $V_{1}\cap V_{3}=B$ and $V_{2}\cap V_{3}=C$, as subspaces
of $W$. From this the independence relations in the conclusions easily follow.
For example: $A\cap V_{3}=A\cap V_{1}\cap V_{3}=A\cap B=D$ hence
$A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{W}V_{3}$,
where we used $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{D}^{V_{1}}B$ in
the final equality.

We are left to define a bilinear form on $W$. For any $e,e^{\prime}\in
D^{\prime}A^{\prime}B^{\prime}C^{\prime}V^{\prime}_{1}V^{\prime}_{2}V^{\prime}_{3}$
with $e,e^{\prime}\in V_{i}$ for some $1\leq i\leq 3$ the choice for
$[e,e^{\prime}]$ is forced, and these choices are compatible by commutativity
of the original diagram. For the remainder of the pairs $e,e^{\prime}$ we set
$[e,e^{\prime}]=0$ and extend linearly in each argument. The final claim about
the symmetric/alternating property then immediately follows. ∎

######  thm:category-of-bilinear-spaces-is-simple-aecat, repeated.

The category $\mathbf{Bil}^{*}_{K}$ is an AECat with the amalgamation property
that has a canonical simple unstable independence relation
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ given by linear
independence.

###### Proof.

From fact:categories-are-aecats we know that $\mathbf{Bil}^{*}_{K}$ is an
AECat. We get the amalgamation property from prop:independent-amalgamation.
That $\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ then forms a
simple independence relation comes down to checking all the required
properties. We only need to verify Invariance, Extension and 3-amalgamation,
as the remaining properties do not depend on the category we are working in,
so we get them directly from fact:linear-independence-facts. As any bilinear
monomorphism is an injective linear map, Invariance is immediate. The two
remaining properties are exactly prop:extension and thm:3-amalgamation. In
prop:failure-of-stationarity we saw that Stationarity fails for
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ on
$\mathbf{Bil}^{*}_{K}$, and so $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ is not stable.
Finally, canonicity follows from fact:canonicity-of-independence. ∎

##  4\. Bilinear spaces in positive logic

We define and study the positive theory of $K$-bilinear spaces.

######  Definition 4.1.

def:signature-of-bilinear-spaces Let $K$ be some field. We define the
signature $\mathcal{L}_{K}$ as follows: $\mathcal{L}_{K}$ includes the
standard signature for $K$-vector spaces, with a function symbol for scalar
multiplication for each $\lambda\in K$ and a symbol $\neq$ for inequality.
Furthermore, for every $\lambda\in K$ we have a binary relation symbol
$[x,y]=\lambda$, which will express that the bilinear product of $x$ and $y$
is $\lambda$.

We naturally view every $K$-bilinear space as an $\mathcal{L}_{K}$-structure.

######  Definition 4.2.

def:theories-of-bilinear-spaces Let $K$ be some field. We define the following
$\mathcal{L}_{K}$-theories:

  * •

$T_{K}$ is the common h-inductive theory of all $K$-bilinear spaces,

  * •

$T_{K}^{s}$ is the common h-inductive theory of all symmetric $K$-bilinear
spaces,

  * •

$T_{K}^{a}$ is the common h-inductive theory of all alternating $K$-bilinear
spaces.

We write $T_{K}^{*}$ if something applies to all of $T_{K}$, $T_{K}^{s}$ and
$T_{K}^{a}$.

Any positive existential formula $\varphi(\bar{x})$ is equivalent to one of
the form

$\bigvee_{i=1}^{n}\exists\bar{y}_{i}\psi_{i}(\bar{x},\bar{y}_{i}),$

where each $\psi_{i}(\bar{x},\bar{y}_{i})$ is a conjunction of atomic
formulas. We will implicitly use this fact in what follows by assuming all
formulas are of this form, and we recall some relevant terminology.

######  Definition 4.3.

def:regular-formula A _regular formula_ (sometimes also called _pp-formula_)
is one of the form $\exists\bar{y}\psi(\bar{x},\bar{y})$ where
$\psi(\bar{x},\bar{y})$ is a conjunction of atomic formulas.

######  Lemma 4.4.

lem:negation-of-bilinear-value Let $\lambda\in K$ and let $\psi(x,y)$ be a
regular formula that contains no linear equations such that
$T_{K}^{*}\models\neg\exists xy(\psi(x,y)\wedge[x,y]=\lambda)$. Then there is
some $\lambda^{\prime}\in K$ such that $T_{K}^{*}\models\forall
xy(\psi(x,y)\to[x,y]=\lambda^{\prime})$. Similarly, for a regular formula
$\psi(x)$ that contains no linear equations such that
$T_{K}^{*}\models\neg\exists xy(\psi(x)\wedge[x,x]=\lambda)$ there is some
$\lambda^{\prime}\in K$ such that $T_{K}^{*}\models\forall
x(\psi(x)\to[x,x]=\lambda^{\prime})$.

###### Proof.

The proof below is for the case of $T_{K}$ and a formula $\psi(x,y)$ in two
variables, at the end we discuss how to make the same proof work for a single
variable and in $T_{K}^{s}$ and $T_{K}^{a}$. We can write $\psi(x,y)$ as

$\exists\bar{z}(\chi(x,y,\bar{z})\wedge\bigwedge_{i\in
I}[v_{i},w_{i}]=\mu_{i}),$

where $\chi(x,y,\bar{z})$ is a conjunction of linear inequalities (as
$\psi(x,y)$ contains no linear equations) and each $v_{i}$ and $w_{i}$ is a
linear combination of $x$, $y$ and $\bar{z}$.

We now claim that there is some $\lambda^{\prime}$ such that for any bilinear
space $V$ and any $c,d\in V$ such that $V\models\psi(c,d)$ we have
$V\models[c,d]=\lambda^{\prime}$. This is enough, because then $\forall
xy(\psi(x,y)\to[x,y]=\lambda^{\prime})$ is in $T_{K}$, as required.

To prove the claim we argue by contradiction. Suppose there are distinct
$\lambda_{1}$ and $\lambda_{2}$ and bilinear spaces $V_{1}$ and $V_{2}$ with
$c_{1},d_{1}\in V_{1}$ and $c_{2},d_{2}\in V_{2}$ such that
$V_{1}\models\psi(c_{1},d_{1})\wedge[c_{1},d_{1}]=\lambda_{1}$ and
$V_{2}\models\psi(c_{2},d_{2})\wedge[c_{2},d_{2}]=\lambda_{2}$. We rename the
variables appearing in the quantifier-free part of $\psi$ as follows: $x$
becomes $z_{1}$, $y$ becomes $z_{2}$ and then we can enumerate $\bar{z}$ as
$z_{3},\ldots,z_{n}$ for some $n$. We introduce a variable $u_{ij}$ all $1\leq
i,j\leq n$, so letting $u_{ij}$ represent $[z_{i},z_{j}]$ we can view
$\bigwedge_{i\in I}[v_{i},w_{i}]=\mu_{i}$ as a system $S$ of linear equations
in variables $(u_{ij})_{1\leq i,j\leq n}$. This determines an affine space

$A=\\{(\alpha_{ij})_{1\leq i,j\leq n}\in K^{n^{2}}:(\alpha_{ij})_{1\leq
i,j\leq n}\text{ is a solution to }S\\}.$

By our assumption about the existence of $V_{1}$ and $V_{2}$ we know that
there must be solutions $(\alpha_{ij})_{1\leq i,j\leq n}$ and
$(\beta_{ij})_{1\leq i,j\leq n}$ in $A$ with $\alpha_{1,2}=\lambda_{1}$ and
$\beta_{1,2}=\lambda_{2}$. Using basic linear algebra and the fact that
$\lambda_{1}$ and $\lambda_{2}$ are distinct, we find a solution
$(\gamma_{ij})_{1\leq i,j\leq n}$ with $\gamma_{1,2}=\lambda$. Let $V$ be a
vector space with basis $e_{1},\ldots,e_{n}$ and make it into a bilinear space
by setting $[e_{i},e_{j}]=\gamma_{ij}$ for all $1\leq i,j\leq n$. We now have
$V\models\psi(e_{1},e_{2})$, where the existential quantifier over $\bar{z}$
is satisfied by $e_{3},\ldots,e_{n}$. The $\chi$ part is then satisfied
because it only contains linear inequalities and $e_{1},\ldots,e_{n}$ are
linearly independent, while the part $\bigwedge_{i\in I}[v_{i},w_{i}]=\mu_{i}$
is satisfied by our choice of $[e_{i},e_{j}]$. At the same time we have
$V\models[e_{1},e_{2}]=\lambda$, but this contradicts $T_{K}\models\neg\exists
xy(\psi(x,y)\wedge[x,y]=\lambda)$.

The case where $\psi(x)$ has only one variable is a simpler version of the
argument above. We replace any occurrence of the $y$ variable by the $x$
variable, and we will just have that $c_{1}=d_{1}$, $c_{2}=d_{2}$ and
$e_{1}=e_{2}$.

To make the above argument work for $T_{K}^{s}$ we restrict ourselves to
symmetric bilinear spaces in the entire argument. To make sure the $V$ we
construct is also symmetric, we can just add equations $u_{ij}=u_{ji}$ to our
system $S$ of linear equations. Since the $V_{1}$ and $V_{2}$ we find are then
assumed to be symmetric bilinear spaces, they still yield solutions to $S$. A
similar trick works for $T_{K}^{a}$. ∎

######  Lemma 4.5.

lem:ec-is-bilinear Every e.c. model of $T_{K}^{*}$ is a $K$-bilinear space.

###### Proof.

Let $M$ be an e.c. model of $T_{K}^{*}$. Clearly $T_{K}^{*}$ will already
specify that $M$ is a $K$-vector space. We need to prove that the binary
relations $[x,y]=\lambda$ encode a bilinear form on $M$. For that we only need
to check that for every $a,b\in M$ there is some $\lambda\in K$ such that
$M\models[a,b]=\lambda$. Then $T_{K}^{*}$ will guarantee that this $\lambda$
is unique and that the binary function $M\times M\to K$ encoded by the
relation symbols in $\mathcal{L}_{K}$ is actually a bilinear form. In fact, it
is enough to check this for linearly independent $a$ and $b$, and when $a=b$.
Then we can take any basis $B$ of $M$ and since $[\cdot,\cdot]$ will then be
defined on any $a,b\in B$ it extends to all of $M$, which is forced by
$T_{K}^{*}$.

If $M\models[a,b]=0$ then we are done. So suppose that $M\not\models[a,b]=0$.
Then as $M$ is e.c. there must be some $\varphi(x,y)$ such that
$T_{K}^{*}\models\neg\exists xy(\varphi(x,y)\wedge[x,y]=0)$ and
$M\models\varphi(a,b)$. We can write $\varphi(x,y)$ as a disjunction of
regular formulas. Let $\psi(x,y)$ be a disjunct so that $M\models\psi(a,b)$,
then we have $T_{K}^{*}\models\neg\exists xy(\psi(x,y)\wedge[x,y]=0)$. We can
write $\psi(x,y)$ as $\exists\bar{z}\chi(x,y,\bar{z})$, where $\chi(\bar{z})$
is a conjunction of atomic formulas. We may assume that $\chi(x,y,\bar{z})$
does not contain any linear equations. This is because we assumed $a$ and $b$
to be linearly independent, or $a=b$ and then we just drop the $y$ variable,
and any $z\in\bar{z}$ that is linearly dependent on the remainder of the
variables can be eliminated from the quantifier by replacing it by the
appropriate linear combination of variables. By lem:negation-of-bilinear-value
there is then a $\lambda\in K$ such that $T_{K}^{*}\models\forall
xy(\psi(x,y)\to[x,y]=\lambda)$. We thus have $M\models[a,b]=\lambda$, as
required. ∎

The following is the analogue of being complete for first-order theories, and
in fact together with prop:boolean-or-semi-hausdorff-not-hausdorff this
implies that $T_{K}^{*}$ is complete for finite $K$.

######  Corollary 4.6.

cor:jep The theory $T_{K}^{*}$ has JEP.

###### Proof.

Use that e.c. models are $K$-bilinear spaces (lem:ec-is-bilinear) to
amalgamate (prop:independent-amalgamation) over the trivial bilinear space. ∎

######  Proposition 4.7.

prop:formula-for-linear-independence For every $n\geq 1$ there is a formula
$\theta_{n}(x_{1},\ldots,x_{n})$ such that for every $a_{1},\ldots,a_{n}$ in
any e.c. model $M$ of $T_{K}^{*}$ we have that
$M\models\theta_{n}(a_{1},\ldots,a_{n})$ iff $a_{1},\ldots,a_{n}$ are linearly
independent.

###### Proof.

We will build the formula $\theta_{n}(x_{1},\ldots,x_{n})$ by induction on
$n$. For $\theta_{1}(x_{1})$ there is nothing to do. That is, we can just take
$x_{1}=x_{1}$. Having built $\theta_{n}(x_{1},\ldots,x_{n})$ we define
$\theta_{n+1}(x_{1},\ldots,x_{n},x_{n+1})$ as

$\theta_{n}(x_{1},\ldots,x_{n})\wedge\exists
yz\left(\bigwedge_{i=1}^{n}([y,x_{i}]=1\wedge[z,x_{i}]=1)\wedge[y,x_{n+1}]=1\wedge[z,x_{n+1}]=0\right).$

Let $M$ be an e.c. model of $T_{K}^{*}$ and let $a_{1},\ldots,a_{n+1}\in M$.
Suppose that $M\models\theta_{n+1}(a_{1},\ldots,a_{n+1})$. Then by the
inductive hypothesis $a_{1},\ldots,a_{n}$ are linearly independent. So it
remains to be shown that $a_{n+1}$ is not a linear combination of
$a_{1},\ldots,a_{n}$. Suppose for a contradiction that
$a_{n+1}=\lambda_{1}a_{1}+\ldots+\lambda_{n}a_{n}$. Let $b$ and $c$ be
realisations for the $y$ and $z$ variables respectively. Then as $[b,a_{i}]=1$
for all $1\leq i\leq n$ we must have that
$1=[b,a_{n+1}]=\lambda_{1}+\ldots+\lambda_{n}$. By similar reasoning we also
get $0=[c,a_{n+1}]=\lambda_{1}+\ldots+\lambda_{n}$, and thus $1=0$. So we
arrive at a contradiction and conclude that $a_{1},\ldots,a_{n+1}$ must indeed
be linearly independent.

Conversely, suppose that $a_{1},\ldots,a_{n+1}$ are linearly independent. Then
by the induction hypothesis $M\models\theta_{n}(a_{1},\ldots,a_{n})$. Extend
$a_{1},\ldots,a_{n+1}$ to a basis $B$ of $M$ and add two new basis vectors $b$
and $c$. Let $V=\langle Bbc\rangle$, so it extends $M$. We extend the bilinear
form on $M$ to $V$ as follows. We set $[b,a_{i}]=[c,a_{i}]=1$ for all $1\leq
i\leq n$, and we set $[b,a_{n+1}]=1$ and $[c,a_{n+1}]=0$. We set the values
for $[a_{i},b]$ and $[a_{i},c]$ for $1\leq i\leq n+1$ according to whether we
work in $T_{K}^{s}$ or $T_{K}^{a}$ (in the case of just $T_{K}$ it does not
matter). Then $V\models\theta_{n+1}(a_{1},\ldots,a_{n+1})$ because $b$ and $c$
are realisations for the $y$ and $z$ variables. So by existential closedness
$M\models\theta_{n+1}(a_{1},\ldots,a_{n+1})$. ∎

We can now give a full characterisation of the e.c. models of $T_{K}^{*}$,
which will use the following definition (borrowing some category-theoretic
terminology).

######  Definition 4.8.

def:finitely-injective We call a $K$-bilinear space $V$ _finitely injective_
if for any finite dimensional $K$-bilinear spaces $A\subseteq B$ and any
bilinear monomorphism $f:A\to V$ we can find a bilinear monomorphism $g:B\to
V$ such that $g$ extends $f$. In the symmetric/alternating case we require all
the spaces involved to be symmetric/alternating.

######  thm:ec-models, repeated.

The following are equivalent for an $\mathcal{L}_{K}$-structure $V$:

  1. (i)

$V$ is an e.c. model of $T_{K}^{*}$,

  2. (ii)

$V$ is a finitely injective $K$-bilinear space,

  3. (iii)

$V$ is an infinite dimensional non-degenerate finitely injective $K$-bilinear
space.

In (ii) and (iii) we restrict to symmetric/alternating $K$-bilinear spaces for
$T_{K}^{s}$ and $T_{K}^{a}$ respectively.

###### Proof.

(ii) $\Leftrightarrow$ (iii). Let $V$ be a finitely injective $K$-bilinear
space. Then $V$ is infinite dimensional, as we can embed spaces of arbitrarily
large (finite) dimension in it. For any $a\in V$ let $A=\langle a\rangle$ and
define $B=\langle ab\rangle$ where $b$ is some new vector linearly independent
from $a$. Make $B$ into a $K$-bilinear space by setting $[a,b]=1$ and also
$[b,a]=\pm 1$ (the exact value depends on whether we are in the symmetric or
alternating case). We can then extend the embedding $A\subseteq V$ to some
bilinear monomorphism $g:B\to V$, and so we have $[a,g(b)]\neq 0$ and
$[g(b),a]\neq 0$. Since $a$ was arbitrary we conclude that $V$ is non-
degenerate.

(i) $\implies$ (ii). By lem:ec-is-bilinear we already know that every e.c.
model is in fact a $K$-bilinear space. Let $A\subseteq B$ be finite
dimensional $K$-bilinear spaces and let $f:A\to V$ be a bilinear monomorphism.
Let $\bar{a}$ be a basis for $A$ and extend it to a basis $\bar{a}\bar{b}$ for
$B$. Let $\bar{x}$ and $\bar{y}$ be variables matching $\bar{a}$ and $\bar{b}$
respectively. Let $\chi(\bar{x},\bar{y})$ be a conjunction of binary relation
symbols $[\cdot,\cdot]=\lambda$ capturing all the bilinear products in
$\bar{a}\bar{b}$. Amalgamate $V\xleftarrow{f}A\subseteq B$ to
$V\xrightarrow{h}W\supseteq B$, using prop:independent-amalgamation. We may
assume that $W$ is an e.c. model (otherwise complete it to one). Consider the
formula $\varphi(\bar{x},\bar{y})$ given by
$\theta_{n}(\bar{x},\bar{y})\wedge\chi(\bar{x},\bar{y})$, where $\theta_{n}$
is the formula from prop:formula-for-linear-independence capturing linear
independence. Then $W\models\varphi(\bar{a},\bar{b})$ and thus
$W\models\exists\bar{y}\varphi(\bar{a},\bar{y})$. As $\bar{a}=hf(\bar{a})$ and
$h:V\to W$ is an immersion we have
$V\models\exists\bar{y}\varphi(f(\bar{a}),\bar{y})$. We thus find $\bar{c}\in
V$ with $V\models\varphi(f(\bar{a}),\bar{c})$. We can now extend $f$ to
$g:B\to V$ by setting $g(\bar{b})=\bar{c}$ and extend linearly.

(ii) $\implies$ (i). Let $V\subseteq W$ be an extension and
$W\models\varphi(\bar{a})$ for some $\bar{a}\in V$. We may assume that $W$ is
an e.c. model (otherwise complete it to one), so in particular it is a
$K$-bilinear space. Write $\varphi(\bar{x})$ as
$\exists\bar{y}\psi(\bar{x},\bar{y})$, where $\psi(\bar{x},\bar{y})$ is
quantifier-free. Let $\bar{b}\in W$ be such that
$W\models\psi(\bar{a},\bar{b})$. Define $A=\langle\bar{a}\rangle$ and
$B=\langle\bar{a}\bar{b}\rangle$. So $A\subseteq V$ and $A\subseteq B$ are
finite dimensional $K$-bilinear spaces. As $V$ is finitely injective we find
$g:B\to V$ extending the inclusion $A\subseteq V$. As $\psi$ is quantifier-
free we have $B\models\psi(\bar{a},\bar{b})$ and thus
$V\models\psi(\bar{a},g(\bar{b}))$. So $V\models\varphi(\bar{a})$, as
required. ∎

######  Theorem 4.9.

thm:quantifier-free-type-determines-type In $T_{K}^{*}$ all types are
determined by their quantifier-free part.

###### Proof.

Let $\bar{a}$ be some tuple in some e.c. model $M$ and let $\bar{b}$ be a
tuple in some e.c. model $N$, such that
$\operatorname{qftp}(\bar{a})=\operatorname{qftp}(\bar{b})$. Define a function
$f:\langle\bar{a}\rangle\to N$ by $f(\bar{a})=\bar{b}$ and extending linearly.
Then $f$ is a bilinear monomorphism because
$\operatorname{qftp}(\bar{a})=\operatorname{qftp}(\bar{b})$. So we can
amalgamate, using prop:independent-amalgamation, to find bilinear
monomorphisms $M\xrightarrow{g}V\xleftarrow{h}N$ such that
$g(\bar{a})=h(\bar{b})$. As $M$ and $N$ are e.c. models, $g$ and $h$ are
immersions, so $\operatorname{tp}(\bar{a})=\operatorname{tp}(\bar{b})$
follows. ∎

######  Corollary 4.10.

cor:all-types-isolated Modulo $(T_{K}^{*})^{\textup{ec}}$ every type in
finitely many variables without parameters is equivalent to a formula. If $K$
is finite, this formula can be taken to be quantifier-free.

###### Proof.

Let $\bar{a}$ be a finite tuple in an e.c. model $M$. By thm:quantifier-free-
type-determines-type it is enough to construct a formula $\varphi(\bar{x})$
that is equivalent to $\operatorname{qftp}(\bar{a})$ (modulo
$(T_{K}^{*})^{\textup{ec}}$). Let $\bar{a}^{\prime}$ be a maximal linearly
independent subtuple of $\bar{a}$. Let $\psi(\bar{x})$ be a conjunction of
linear equations capturing how the remainder of $\bar{a}$ depends on
$\bar{a}^{\prime}$. Let $\chi(\bar{x})$ be a conjunction of binary relation
symbols $[\cdot,\cdot]=\lambda$ capturing the bilinear products in
$\bar{a}^{\prime}$. Then the formula
$\theta_{n}(\bar{x}^{\prime})\wedge\psi(\bar{x})\wedge\chi(\bar{x}^{\prime})$
is the $\varphi(\bar{x})$ we are looking for, where $\theta_{n}$ is the
formula from prop:formula-for-linear-independence capturing linear
independence. The final claim follows from the fact that linear independence
over a finite field can be expressed by a quantifier-free formula, so we can
replace $\theta_{n}$ by this quantifier-free formula. ∎

######  Remark 4.11.

rem:quantifier-free-type-vs-quantifier-elimination In full first-order logic
we have that if all types are determined by their quantifier-free part, then
the theory has quantifier elimination, see for example [Gra99, Proposition
9.2.8]. In positive logic this is no longer true. As we will see below,
$T_{K}^{*}$ for infinite $K$ is an example. By thm:quantifier-free-type-
determines-type we do have that in $T_{K}^{*}$ every type is determined by its
quantifier-free part. However, thm:quantifier-elimination shows that we do not
have quantifier elimination.

######  Definition 4.12.

def:positive-quantifier-elimination We say that a theory $T$ has _positive
quantifier elimination_ if for every formula $\varphi(\bar{x})$ there is some
quantifier-free formula $\psi(\bar{x})$ that is equivalent to
$\varphi(\bar{x})$ modulo $T^{\textup{ec}}$.

######  Theorem 4.13.

thm:quantifier-elimination The theory $T_{K}^{*}$ has positive quantifier
elimination iff $K$ is finite.

###### Proof.

We first prove the right to left direction. The quantifier-free type of any
$n$-tuple is fully determined by any linear dependencies and the bilinear
products in that tuple. As $K$ is finite, there are only finitely many
possibilities for this for a fixed $n$. So there are only finitely many
quantifier-free $n$-types, and hence finitely many $n$-types by
thm:quantifier-free-type-determines-type.

Let now $\varphi(\bar{x})$ be some formula and write $[\varphi(\bar{x})]$ for
the set of all types that contain $\varphi(\bar{x})$. By the above discussion
$[\varphi(\bar{x})]$ is finite and, by cor:all-types-isolated, for each
$p\in[\varphi(\bar{x})]$ there is some quantifier-free $\chi_{p}(\bar{x})$
that is equivalent to $p$. It then immediately follows that $\varphi(\bar{x})$
is equivalent to $\bigvee_{p\in[\varphi(\bar{x})]}\chi_{p}(\bar{x})$, modulo
$(T_{K}^{*})^{\textup{ec}}$.

We now prove the contrapositive of the converse. So let $K$ be infinite, we
prove that for any $n\geq 2$ the formula $\theta_{n}$ is from prop:formula-
for-linear-independence is not equivalent to a quantifier-free formula, modulo
$(T_{K}^{*})^{\textup{ec}}$. We give a proof for $T_{K}$ and at the end of the
proof we describe how to make the proof work for $T_{K}^{s}$ and $T_{K}^{a}$.
Let $n\geq 2$ and suppose that $\theta_{n}(x_{1},\ldots,x_{n})$ is equivalent
(modulo $T_{K}^{\textup{ec}}$) to some quantifier free formula
$\psi(x_{1},\ldots,x_{n})$. We can write $\psi$ as a disjunction
$\bigvee_{\ell=1}^{k}\psi_{\ell}(x_{1},\ldots,x_{n})$, where each
$\psi_{\ell}$ is a conjunction of atomic formulas.

Let $1\leq\ell\leq k$. For each $1\leq i,j\leq n$ introduce a variable
$u_{ij}$ that will represent $[x_{i},x_{j}]$. The atomic formulas in
$\psi_{\ell}$ of the form $[t,s]=\lambda$, where $\lambda\in K$ and $t$ and
$s$ are linear combinations of $x_{1},\ldots,x_{n}$, will then determine a
linear system of equations $S_{\ell}$ in variables $(u_{ij})_{1\leq i,j\leq
n}$. We define the affine space

$A_{\ell}=\\{(\alpha_{ij})_{1\leq i,j\leq n}\in K^{n^{2}}:(\alpha_{ij})_{1\leq
i,j\leq n}\text{ is a solution to }S_{\ell}\\}.$

We distinguish two cases.

  1. (1)

There is some $\ell$ such that $A_{\ell}$ is the entire space $K^{n^{2}}$.
This means that $\psi_{\ell}$ is (equivalent to) some formula purely in the
language of vector spaces. Then as linear independence is not definable in
vector spaces over an infinite field, there is some vector space $V$ with
$v_{1},\ldots,v_{n}\in V$ such that $V\models\psi_{\ell}(v_{1},\ldots,v_{n})$
while $v_{1},\ldots,v_{n}$ are not linearly independent. We make $V$ into a
bilinear space (the choice of the bilinear form does not matter) and then
extend it to an e.c. model $M$. This process does not invalidate the truth of
$\psi_{\ell}(v_{1},\ldots,v_{n})$. So
$M\models\psi_{\ell}(v_{1},\ldots,v_{n})$ and hence
$M\models\psi(v_{1},\ldots,v_{n})$. However, we do not have
$M\models\theta_{n}(v_{1},\ldots,v_{n})$, so $\theta_{n}$ cannot be equivalent
to $\psi$ in all e.c. models.

  2. (2)

Every $A_{\ell}$ is a proper subspace of $K^{n^{2}}$. As $K$ is infinite,
$A=\bigcup_{1\leq\ell\leq k}A_{\ell}$ is still a proper subset of $K^{n^{2}}$.
Pick some $(\alpha_{ij})_{1\leq i,j\leq n}\in K^{n^{2}}-A$. Let $V=\langle
a_{1},\ldots,a_{n}\rangle$ be an $n$-dimensional vector space with basis
$a_{1},\ldots,a_{n}$. Make $V$ into a bilinear space by setting
$[a_{i},a_{j}]=\alpha_{ij}$ for all $1\leq i,j\leq n$. Extend $V$ to some e.c.
model $M\supseteq V$. We now have $M\models\theta_{n}(a_{1},\ldots,a_{n})$, as
$a_{1},\ldots,a_{n}$ are linearly independent. However, for every
$1\leq\ell\leq k$ we have that $([a_{i},a_{j}]_{ij})_{1\leq i,j\leq
n}=(\alpha_{ij})_{1\leq i,j\leq n}$ is not a solution to $S_{\ell}$, so
$M\not\models\psi_{\ell}(a_{1},\ldots,a_{n})$. Hence
$M\not\models\psi(x_{1},\ldots,x_{n})$, which contradicts that
$\theta_{n}(x_{1},\ldots,x_{n})$ and $\psi(x_{1},\ldots,x_{n})$ are equivalent
in all e.c. models.

A similar proof works for $T_{K}^{s}$, we just restrict the variables
$(u_{ij})_{1\leq i,j\leq n}$ and the corresponding systems $S_{\ell}$ and
affine subspaces $A_{\ell}$ to those $i$ and $j$ with $i\leq j$. This suffices
as they will encode all the necessary information for a symmetric bilinear
form. Similarly, for $T_{K}^{a}$ we restrict things to those $i$ and $j$ where
$i<j$. ∎

######  Proposition 4.14.

prop:boolean-or-semi-hausdorff-not-hausdorff If $K$ is finite then $T_{K}^{*}$
is Boolean. If $K$ is infinite then $T_{K}^{*}$ is semi-Hausdorff, but not
Hausdorff.

###### Proof.

Suppose that $K$ is finite. Then $[x,y]\neq\lambda$ is equivalent to
$\bigvee_{\lambda^{\prime}\neq\lambda}[x,y]=\lambda^{\prime}$. So for every
atomic formula $\chi(\bar{x})$ there is a positive (quantifier-free) formula
that is equivalent to $\neg\chi(\bar{x})$. The same is then true for
quantifier-free formulas. We conclude that $T_{K}^{*}$ is Boolean by positive
quantifier elimination (thm:quantifier-elimination).

We now move on to the case where $K$ is infinite. We first prove that
$T_{K}^{*}$ is semi-Hausdorff. By thm:quantifier-free-type-determines-type we
only need to prove that having the same quantifier-free type is type-
definable. This is clearly true when we restrict to just the language of
$K$-vector spaces. So it suffices to show that $[x,y]=[x^{\prime},y^{\prime}]$
is definable. Consider the formula $\varphi(x,y,x^{\prime},y^{\prime})$ given
by:

$\exists
zz^{\prime}([x,y-z]=0\wedge[x-z^{\prime},z]=0\wedge[z^{\prime},z-y^{\prime}]=0\wedge[z^{\prime}-x^{\prime},y^{\prime}]=0).$

Let $M$ be an e.c. model and let $a,b,a^{\prime},b^{\prime}\in M$ with
$M\models\varphi(a,b,a^{\prime},b^{\prime})$. Let $c,c^{\prime}\in M$ be such
that
$[a,b-c]=[a-c^{\prime},c]=[c^{\prime},c-b^{\prime}]=[c^{\prime}-a^{\prime},b^{\prime}]=0$.
We thus get
$[a,b]-[a,c]=[a,c]-[c^{\prime},c]=[c^{\prime},c]-[c^{\prime},b^{\prime}]=[c^{\prime},b^{\prime}]-[a^{\prime},b^{\prime}]=0$,
and so
$[a,b]=[a,c]=[c^{\prime},c]=[c^{\prime},b^{\prime}]=[a^{\prime},b^{\prime}]$.
So $\varphi(x,y,x^{\prime},y^{\prime})$ does indeed imply
$[x,y]=[x^{\prime},y^{\prime}]$.

Conversely, let $a,b,a^{\prime},b^{\prime}\in M$ such that
$[a,b]=[a^{\prime},b^{\prime}]=\lambda$. Define $A=\langle
aba^{\prime}b^{\prime}\rangle$ and introduce two new linearly independent
vectors $c,c^{\prime}$ and form $B=\langle
aba^{\prime}b^{\prime}cc^{\prime}\rangle$. Make $B$ into a $K$-bilinear space
by setting $[a,c]=[c^{\prime},c]=[c^{\prime},b^{\prime}]=\lambda$. Pick
anything for the remainder of the bilinear products (respecting the form
having to be symmetric/alternating). As $M$ is e.c. it is finitely injective
(thm:ec-models), so the inclusion $A\subseteq M$ extends to a bilinear
monomorphism $g:B\to M$. Then $g(c)$ and $g(c^{\prime})$ are realisations for
$z$ and $z^{\prime}$ respectively in $\varphi$, so
$M\models\varphi(a,b,a^{\prime},b^{\prime})$.

Now we prove that $T_{K}^{*}$ is not Hausdorff. Let $M$ be an e.c. model of
$T_{K}^{*}$. Using the assumption that $K$ is infinite, we can use compactness
for full first-order logic to find an elementary extension $N$ of $M$ with
linearly independent $a,b\in N$ such that $N\not\models[a,b]=\lambda$ for all
$\lambda\in K$. We claim that there are extensions $N_{1}$ and $N_{2}$ of $N$,
both models of $T_{K}^{*}$, such that $N_{1}\models[a,b]=0$ and
$N_{2}\models[a,b]=1$. This shows that $T_{K}^{*}$ is not Hausdorff by
fact:hausdorff-iff-aph, as $N_{1}\supseteq N\subseteq N_{2}$ cannot be
amalgamated.

Suppose for a contradiction that one of these extensions, say $N_{1}$, does
not exist. Then by compactness there is a finite conjunction
$\varphi(a,b,\bar{c})$ of atomic formulas that are true in $N$ such that
$T_{K}^{*}\models\neg\exists
xy(\exists\bar{z}\varphi(x,y,\bar{z})\wedge[x,y]=0)$. We may assume $\varphi$
contains no linear equations since $a$ and $b$ are linearly independent and we
can replace any $c\in\bar{c}$ that is linearly dependent on $a$, $b$ and the
remainder of $\bar{c}$ by replacing it by the appropriate linear combination.
We can thus apply lem:negation-of-bilinear-value to obtain some $\lambda\in K$
such that $T_{K}^{*}\models\forall
xy(\exists\bar{z}\varphi(x,y,\bar{z})\to[x,y]=\lambda)$. However, this would
imply that $N\models[a,b]=\lambda$, a contradiction. ∎

We close out this section by recalling the definitions of simplicity (in the
sense of [BY03b]) and stability for positive theories, and prove that
$T_{K}^{*}$ is simple unstable.111Having JEP allows us to work in a monster
model, but we have so far not used monster models and it seems unnecessary to
introduce them just for the final result.

######  Definition 4.15.

def:dividing Let $M$ be some e.c. model and let $\bar{a},\bar{b}\in M$,
$C\subseteq M$. We say that a type
$p(x,\bar{b})=\operatorname{tp}(\bar{a}/C\bar{b})$ _divides over $C$_ if there
is an extension $M\subseteq N$ with a $C$-indiscernible sequence
$(\bar{b}_{i})_{i<\omega}$ in $N$ such that
$\bigcup_{i<\omega}p(x,\bar{b}_{i})$ is inconsistent.

######  Definition 4.16.

def:simplicity A theory $T$ is called _simple_ if dividing has _local
character_. That is, there is some cardinal $\mu$ such that for any finite
$\bar{a}$ in any e.c. model $M$ and any $B\subseteq M$ there is
$B_{0}\subseteq B$ with $|B_{0}|\leq\mu$ and $\operatorname{tp}(\bar{a}/B)$
does not divide over $B_{0}$.

######  Definition 4.17.

def:stability A theory $T$ is called _stable_ if there is some cardinal $\mu$
such that for all $A\subseteq M$, where $M$ is an e.c. model and $|A|\leq\mu$,
there are at most $\mu$ different types over $A$ (possibly realised in
extensions of $M$).

The following fact is one half of a Kim-Pillay style theorem for positive
logic, which allows us to characterise simple positive theories based on the
existence of a simple independence relation. We simplified the statement for
our setting and only mentioned one half as that is what we need, the original
theorem is much stronger.

######  Fact 4.18 ([BY03b, Theorem 1.51]).

fact:kim-pillay-positive-logic Let $T$ be a positive theory. If there is a
simple independence relation $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ on subsets of
the e.c. models of $T$ then $T$ is simple and
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ coincides with
non-dividing.

######  thm:positive-theory-simple-unstable, repeated.

The theory $T_{K}^{*}$ is simple unstable, and non-dividing coincides with
linear independence. That is, for every $\bar{a}$, $\bar{b}$ and $C$ in some
e.c. model $M$ we have $\bar{a}\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}_{C}^{M}\bar{b}$
iff $\operatorname{tp}(\bar{a}/C\bar{b})$ does not divide over $C$.

###### Proof.

We get simplicity and the claim about $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ coinciding with
non-dividing directly from applying fact:kim-pillay-positive-logic to
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$, as all the
necessary properties have been verified in section 3.

We could prove non-stability using the fact that Stationarity fails, as in
prop:failure-of-stationarity, because in stable positive theories we must have
Stationarity over certain sets [BY03b, Theorem 2.8]. However, we will give a
direct proof for def:stability. Let $\mu$ be any infinite cardinal, and let
$M$ be an e.c. model with at least $\mu$ many linearly independent vectors
$(a_{i})_{i<\mu}$. Then for any $\chi:\mu\to\\{0,1\\}$ we define the partial
type $\Sigma_{\chi}(x)=\\{[x,a_{i}]=\chi(i):i<\mu\\}$. Then each
$\Sigma_{\chi}(x)$ can be extended to a type $p_{\chi}(x)$ over
$(a_{i})_{i<\mu}$ (i.e. it will have a realisation in some extension of $M$).
This yields an injection from $2^{\mu}$ into the space of types over
$(a_{i})_{i<\mu}$, so there are more than $\mu$ many types over
$(a_{i})_{i<\mu}$. ∎

We named real Hilbert spaces as an example of bilinear spaces (ex:bilinear-
spaces(ii)). Complex Hilbert spaces are not bilinear spaces, because they are
required to be conjugate symmetric, that is $[x,y]=\overline{[y,x]}$. Together
with linearity in the first argument this means that the form is conjugate
linear in the second argument: $[x,\alpha y+\beta
z]=\bar{a}[x,y]+\bar{\beta}[x,z]$. This is an example of a _Hermitian space_ ,
which is a generalisation of bilinear spaces. Even more general are
_sesquilinear spaces_. It seems likely that similar techniques can be used to
study such spaces. We thank Jan Dobrowolski for asking the following question.

######  Question 4.19.

q:sesquilinear-forms Can we use positive logic to study sesquilinear or
Hermitian spaces over a fixed base field (or even: division ring)? Is the
arising theory still simple?

##  5\. $\omega$-Categoricity in positive logic

In this section we provide a positive version of what is commonly known as the
Ryll-Nardzewski theorem in full first-order logic, extending [Hay19, Theorem
6.5]. This will then be used to conclude that, for countable $K$, the theory
$T_{K}^{*}$ of (symmetric/alternating) $K$-bilinear spaces is
$\omega$-categorical, see cor:bilinear-spaces-omega-categorical.

######  Definition 5.1.

def:categoricity Let $\kappa$ be a cardinal. A theory $T$ is called _$\kappa$
-categorical_ if it has only one e.c. model of cardinality $\kappa$, up to
isomorphism.

The following definition is taken from [Hay19]. More precisely, [Hay19, page
844] gives a topological definition of what it means for a set of formulas to
be supported. We translate that to a logical property, where we restrict our
attention to types (remember, for us these are maximal types, in contrast to
[Hay19]). This results in def:supported-type(ii). We also provide an
equivalent property, which is similar to the usual notion of isolated type. In
fact, in full first-order logic supported is the same as isolated. In positive
logic the latter would not be a good term, as it no longer corresponds to a
type being an isolated point in the type space.

######  Definition 5.2.

def:supported-type A type $p(\bar{x})$ in finitely many variables is called
_supported_ if there is a formula $\varphi(\bar{x})\in p(\bar{x})$ such that
the following equivalent conditions hold:

  1. (i)

for all $\chi(\bar{x})\in p(\bar{x})$ we have
$T^{\textup{ec}}\models\forall\bar{x}(\varphi(\bar{x})\to\chi(\bar{x}))$,

  2. (ii)

for all $\psi(\bar{x})\not\in p(\bar{x})$ we have
$T\models\neg\exists\bar{x}(\varphi(\bar{x})\wedge\psi(\bar{x}))$.

In this case we call $\varphi(\bar{x})$ the _support_ of $p(\bar{x})$.

######  Lemma 5.3.

lem:supported-type The conditions in def:supported-type are indeed equivalent.

###### Proof.

(i) $\Rightarrow$ (ii) Let $\psi(\bar{x})\not\in p(\bar{x})$ and assume for a
contradiction that there is some model $M$ of $T$ with $\bar{a}\in M$ such
that $M\models\varphi(\bar{a})\wedge\psi(\bar{a})$. We may assume $M$ to be
e.c. Write $q(\bar{x})=\operatorname{tp}(\bar{a})$, so we have
$\varphi(\bar{x})\in q(\bar{x})$ and hence $p(\bar{x})\subseteq q(\bar{x})$ by
our assumption on $\varphi$. By maximality of types we then have
$p(\bar{x})=q(\bar{x})$, and hence $\psi(\bar{x})\in q(\bar{x})=p(\bar{x})$,
which is a contradiction.

(ii) $\Rightarrow$ (i) Let $\chi(\bar{x})\in p(\bar{x})$. If
$T^{\textup{ec}}\not\models\forall\bar{x}(\varphi(\bar{x})\to\chi(\bar{x}))$
then by definition of $T^{\textup{ec}}$ there must be an e.c. model $M$ with
$a\in M$ such that $M\models\varphi(\bar{a})$ and $M\not\models\chi(\bar{a})$.
So there is a negation $\psi(\bar{x})$ of $\chi(\bar{x})$ such that
$M\models\psi(\bar{a})$. As $\chi(\bar{x})\in p(\bar{x})$ we must have
$\psi(\bar{x})\not\in p(\bar{x})$, so by our assumption on $\varphi(\bar{x})$
we have $T\models\neg\exists\bar{x}(\varphi(\bar{x})\wedge\psi(\bar{x}))$.
However, this contradicts $M\models\varphi(\bar{a})\wedge\psi(\bar{a})$. ∎

######  Definition 5.4.

def:atomic-prime-model Let $M$ be an e.c. model. We call $M$ _atomic_ if it
only realises supported types. We call $M$ _prime_ if every e.c. model $N$ is
a continuation of $M$.

######  Fact 5.5 ([Hay19, Proposition 6.3]).

fact:prime-iff-countable-and-atomic Let $M$ be an e.c. model of a countable
theory $T$ with JEP. Then $M$ is prime if and only if it is countable and
atomic.

######  Lemma 5.6.

lem:supported-n-types-implies-omega-saturated If every $n$-type is supported
then every e.c. model is $\omega$-saturated.

###### Proof.

Let $M$ be an e.c. model and let $\Sigma(\bar{x},\bar{b})$ be finitely
satisfiable in $M$, where $\bar{x}$ and $\bar{b}\in M$ are finite. Then there
is a realisation $\bar{a}$ in some e.c. model $N$ that is a continuation of
$M$. Set $p(\bar{x},\bar{y})=\operatorname{tp}(\bar{a},\bar{b})$ and let
$\varphi(\bar{x},\bar{y})$ be the support of $p(\bar{x},\bar{y})$. Then
$N\models\exists\bar{x}\varphi(\bar{x},\bar{b})$ so because $M$ is e.c. we
find $\bar{a}^{\prime}\in M$ with $M\models\varphi(\bar{a}^{\prime},\bar{b})$.
As $\varphi$ supports $p$ we have that $M\models p(\bar{a^{\prime}},\bar{b})$
and hence $M\models\Sigma(\bar{a}^{\prime},\bar{b})$. ∎

######  Remark 5.7.

rem:finite-type-spaces thm:omega-categoricity below provides several
equivalent characterisations of being $\omega$-categorical for positive
theories. However, compared to the analogous theorem for full first-order
logic one important characterisation is missing: namely that the space of
$n$-types is finite, for every $n<\omega$. As pointed out in [Hay19, Example
6.6] this is simply no longer equivalent to being $\omega$-categorical in
positive logic. In fact, one easily sees that having finite type spaces is
equivalent to being $\omega$-categorical and Boolean, where being Boolean
follows because the complement of any positively definable set is positively
definable using a finite disjunction.

The counterexample from [Hay19] is quite simple, so we repeat it here.
Consider the theory $T$ with constants $\\{c_{i}\\}_{i<\omega}$, asserting
that $c_{i}\neq c_{j}$ for all $i\neq j$. Then $T$ has a unique e.c. model
consisting of only interpretations for the constants. We see that $T$ is
$\omega$-categorical, but each constant yields a different type, so we have
infinitely many $1$-types.

######  Theorem 5.8.

thm:omega-categoricity Let $T$ be a countable theory with JEP. Then the
following are equivalent:

  1. (i)

$T$ is $\omega$-categorical,

  2. (ii)

every $n$-type is supported,

  3. (iii)

all e.c. models are atomic,

  4. (iv)

all countable e.c. models are atomic,

  5. (v)

every e.c. model is $\omega$-saturated,

  6. (vi)

there is a saturated prime model.

###### Proof.

The equivalence between (i), (ii), (iii) and (iv) is [Hay19, Corollary 6.4 and
Theorem 6.5]. lem:supported-n-types-implies-omega-saturated yields (ii)
$\Rightarrow$ (v), while (v) $\Rightarrow$ (i) easily follows from back-and-
forth. We are left to prove that (vi) is equivalent to properties (i) to (v).

(vi) $\Rightarrow$ (iii) Let $M$ be prime and saturated. By fact:prime-iff-
countable-and-atomic $M$ is atomic. Let $p$ be a type that is realised in some
e.c. model $N$. By JEP and saturation $p$ is also realised in $M$ and is thus
supported. As $p$ and $N$ were arbitrary we conclude that indeed all e.c.
models are atomic.

(i)–(v) $\Rightarrow$ (vi) Let $M$ be the unique countable model. By (iii) $M$
is atomic and by (v) $M$ is $\omega$-saturated, and hence saturated. Finally,
by fact:prime-iff-countable-and-atomic $M$ is prime. ∎

######  Corollary 5.9.

cor:bilinear-spaces-omega-categorical Let $K$ be any field. Every e.c. model
of $T_{K}^{*}$ is $\omega$-saturated. If $K$ is at most countable then
$T_{K}^{*}$ is $\omega$-categorical.

###### Proof.

By cor:all-types-isolated every type in finitely many variables is supported
and cor:jep gives us JEP. So lem:supported-n-types-implies-omega-saturated and
thm:omega-categoricity apply. ∎

##  6\. Comparison to different approaches

We consider two other model-theoretic approaches to certain bilinear spaces:
Hilbert spaces and the two-sorted approach in full first-order logic. The
former is known to be stable, while the latter is known to be non-simple (but
is NSOP1). This is in contrast to our main results thm:category-of-bilinear-
spaces-is-simple-aecat and thm:positive-theory-simple-unstable, which claim
simplicity and non-stability. In each of these two cases we point out
precisely where the difference lies, in terms of the canonical independence
relation.

###  6.1. Hilbert spaces

From [BYBHU08, section 15] we know that Hilbert spaces (over the real
numbers), studied in the framework of continuous logic, are stable. The
canonical independence relation is given by orthogonality. So linear
independence cannot be a simple independence relation on the category of
Hilbert spaces, as it would have to coincide with orthogonality by canonicity.
It is then natural to ask: what property fails? The answer to this question
turns out to be 3-amalgamation, as we will show in ex:linear-independence-
fails-in-hilbert-spaces.

######  Example 6.1.

ex:linear-independence-fails-in-hilbert-spaces We write
$\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ for linear
independence (as in def:linear-independence), and we will show that
3-amalgamation fails for $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ in the category
of Hilbert spaces. We work in the Hilbert space $\mathbb{R}^{3}$ with the
usual inner product as bilinear form. Let $a=(1,0,0)$, $a^{\prime}=(0,1,0)$,
$b=(1,0,1)$, $c=(0,1,1)$ and $d=(\frac{1}{2},\frac{1}{2},2)$. Set $A=\langle
ad\rangle$, $A^{\prime}=\langle a^{\prime}d\rangle$, $B=\langle bd\rangle$,
$C=\langle cd\rangle$ and $D=\langle d\rangle$. So we have a commuting diagram
as below (ignoring the dashed arrows), where every arrow is an inclusion,
except for $f$, which is defined by $f(d)=d$ and $f(a)=a^{\prime}$, and then
extend linearly.

${\mathbb{R}^{3}}$${V}$${A}$${\mathbb{R}^{3}}$${C}$${\mathbb{R}^{3}}$${D}$${B}$$\scriptstyle{h}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{i}$

Furthermore, we have $A\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{D}^{\mathbb{R}^{3}}B$, $B\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{C}^{\mathbb{R}^{3}}C$ and
$A^{\prime}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{D}^{\mathbb{R}^{3}}C$. Noting that $A^{\prime}=f(A)$
3-amalgamation would give us the dashed arrows such that everything commutes.
We will view $i$ as a genuine inclusion and write $a^{*}=g(a)=h(a^{\prime})$.

Set $v=2(b+c-d)$, so in $\mathbb{R}^{3}$ this is just $(1,1,0)$. We calculate:

$\displaystyle[a^{*}-v,a^{*}-v]$ $\displaystyle=[a^{*},a^{*}-v]-[v,a^{*}-v]$
$\displaystyle=([a^{*},a^{*}]-[a^{*},v])-([v,a^{*}]-[v,v])$
$\displaystyle=(1-([a^{*},2b]+[a^{*},2c]-[a^{*},2d]))-([v,a^{*}]-[v,v])$
$\displaystyle=(1-(2+2-1))-([2b,a^{*}]+[2c,a^{*}]-[2d,a^{*}]-2)$
$\displaystyle=-2-(2+2-1-2)$ $\displaystyle=-3,$

here we have used the definition of $a^{*}$ and commutativity of the above
diagram multiple times for simplifications like
$[a^{*},2b]=[g(a),i(2b)]=[g(a),g(2b)]=[a,2b]=2$.

So we have found an element of $V$, namely $a^{*}-v$, such that
$[a^{*}-v,a^{*}-v]=-3$. This means that the form on $V$ is not positive
definite. So $V$ cannot be a Hilbert space and we conclude that 3-amalgamation
fails for linear independence $\mathop{\mathchoice{\displaystyle\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 1.42882pt}}$ in the category
of Hilbert spaces.

Of course, we could apply thm:3-amalgamation to the diagram in ex:linear-
independence-fails-in-hilbert-spaces to find $V$ together with the dashed
arrows. We will just get a bilinear form that is not positive definite. In
fact, we can give an explicit description of $V$ and $a^{*}$ (which completely
determines the diagram). We take $V=\mathbb{R}^{4}$, and let the bilinear form
be defined by
$[(x,y,z,w),(x^{\prime},y^{\prime},z^{\prime},w^{\prime})]=xx^{\prime}+yy^{\prime}+zz^{\prime}-ww^{\prime}$.
Now we can take
$a^{*}=(1\frac{1}{2},1\frac{1}{2},-\frac{1}{2},\frac{\sqrt{15}}{2})$. It is
then straightforward to verify that this does indeed form a solution to the
3-amalgamation problem.

###  6.2. Failure of simplicity when the field varies

Bilinear spaces over some infinite field $K$ have been studied in the
framework of full first-order logic by considering a two-sorted theory
$T_{\infty}^{K}$: one sort for the vector space and one for the field, which
has to be elementarily equivalent to $K$ in the language of rings. This comes
with two disadvantages: the theory is going to be at least as complicated as
the theory of the field and the field varies between different models of the
theory.

Even when we restrict ourselves to algebraically closed fields, for which we