MS-measurability via coordinatization

Mostafa Mirabi Department of Mathematics and Computer Science, Wesleyan University, 655 Exley Science Tower, 265 Church Street, Middletown, CT 06459 mmirabi@wesleyan.edu https://sites.google.com/site/mostafamirabi/

Abstract.

We define a notion of coordinatization for $\aleph_{0}$ -categorical structures which is, like Lie coordinatized structures in [2], a certain kind of expansion of a tree. We show that a structure which is coordinatized, in a certain strong sense, by $\aleph_{0}$ -categorical MS-measurable structures itself is MS-measurable.

Key words and phrases:

MS-measurable structures, asymptotic classes, coordinatization,

\aleph_{0}

-categorical theories

2010 Mathematics Subject Classification:

03C13 , 03C45, 03C15

1. Introduction

The notion of MS-measurable structures was introduced by Macpherson and Steinhorn in [9]. An MS-measurable structure is an infinite structure whose definable sets are equipped with an $\mathbb{N}$ -valued dimension and an $\mathbb{R}$ -valued measure satisfying natural properties (e.g., a variant of Fubini’s Theorem). All MS-measurable structures are supersimple of finite SU-rank. The motivating examples of MS-measurable structures are infinite pseudofinite fields. More generally, a non-principal ultraproduct of a so-called asymptotic class of finite structures is always MS-measurable. An asymptotic class means that we have, in some sense, tight estimates of the sizes of the definable sets in all structures in the class; this condition is essentially a generalization of the classical Lang–Weil estimates for varieties in the class of finite fields [1]. There are also MS-measurable structures which are not even pseudofinite, so cannot be obtained from asymptotic classes [4].

Coordinatization is a method of decomposing a model as a tree of geometries. Roughly speaking, it means that the structure can be constructed in a nice way from a tree, the skeleton of the structure, by gluing additional structures on some nodes of tree. By allowing the components to interact with each other, we may define different notions of coordinatization. In [2], for example, Cherlin and Hrushovski define a “Lie coordinatized” structure to be a structure with a treelike form of finite height which witnesses how a structure is built by a sequence of finite and affine covers; they then prove that a countable structure is smoothly approximated if and only if it is Lie coordinatizable. Elwes, in [5], has proven that any smoothly approximable structure is MS-measurable by showing that it arises from an approximating sequence of envelopes which forms an asymptotic class.

In [7], Hill has studied the construction of pseudo-finite structures using a certain notion of coordinatization. In particular, he has shown that a structure that is coordinatized by structures with the finite sub-model property itself has the finite sub-model property. In [7], Hill uses a notion of coordinatization with a lot of independence similar to what is found in [8], but does not come with size estimates. Also, in [6], Hill uses a different but related kind of independence to get size estimates. Insofar as there is some overlap between [6] and [7] which suggests that there is a nice intraction between MS-measurability and pseudo-finiteness, it is worthwhile to study MS-measurability of structures obtained as expansions of trees.

In this paper, we define a strong notion of coordinatization for $\aleph_{0}$ -categorical structures obtained as certain kinds of expansions of trees. We show that a structure which is coordinatized by one-dimensional $\aleph_{0}$ -categorical MS-measurable structures is itself MS-measurable. This gives new examples of MS-measurable structures. This paper can be counted as the first step towards a more general problem for the study of an approach to understanding the intersection of two model-theoretically natural classes of structures, countably categorical 1-based structures on one hand (see [3]), and MS-measurable structures (see [9]) on the other hand. In the long run, the goal is to develop a notion of coordinatization that captures the structures in that intersection. Here, we have worked out how we would expect that notion to look by working out the details of a certain strong notion of coordinatization. We expect that the ”right” notion will turn out to be a relaxation of the notion of coordinatization here.

The paper is organized as follows. In Subsection 1.1, we fix some notation and conventions which we use throughout the paper. In Section 2, we express some preliminaries and building blocks which will be used for the rest of the paper including the notions of tree plans, tree-closure, and expansions of trees. We provide all material which we need to define our notion of coordinatization. In Section 3, we define a certain notion of coordinatization and show how to construct structures coordinatized in this way. In Section 4, we recall the definition of MS-measurability and prove a criterion for MS-measurability of $\aleph_{0}$ -categorical structures. Then we prove the main theorem saying that one can lift MS-measurability to coordinatized structures.

1.1. Notation and conventions

Throughout this paper, we use calligraphic upper-case letters like $\mathcal{M}$ , $\mathcal{N}$ to denote infinite structures with universes $M$ and $N$ , respectively. We use simple upper-case letters like $A,B,C$ to denote finite structures and identify them with their universes. In general, our notation for such structures is standard (see [10]). By $\mathcal{M}\preceq\mathcal{N}$ we mean that $\mathcal{M}$ is an elementary substructure of $\mathcal{N}$ . By $A\subset_{\textrm{fin}}M$ we mean that $A$ is a finite subset of $M$ . We write $\operatorname{acl}^{\mathcal{M}}(A)$ and $\operatorname{dcl}^{\mathcal{M}}(A)$ to denote the algebraic closure and definable closure of $A$ in $\mathcal{M}$ , respectively. For $A,B\subseteq M$ and $c\in M$ , by $AB$ and $Ac$ we mean $A\cup B$ and $A\cup\{c\}$ , respectively. We use $\uplus$ to denote disjoint union. By $A^{<\omega}$ we mean the set of all finite tuples of members of $A$ . We write $\mathrm{Aut}(\mathcal{M}/A)$ to denote the set of all automorphisms of $\mathcal{M}$ that fix $A$ pointwise. A theory $T$ is algebraically trivial if for all $\mathcal{M}\vDash T$ , for all $A\subseteq M$ , $\operatorname{acl}^{\mathcal{M}}(A)=A$ . If $\varphi(\overline{x},\overline{y})$ is an $\mathcal{L}$ -formula and $\overline{b}\in M^{m}$ , we write $\varphi(\mathcal{M},\overline{b})$ to denote the set defined in $\mathcal{M}$ by the $\mathcal{L}(\overline{b})$ -formula $\varphi(\overline{x},\overline{b})$ , i.e. $\varphi(\mathcal{M},\overline{b})=\left\{\overline{a}\in M^{n}:\mathcal{M}\vDash\varphi(\overline{a},\overline{b})\right\}$ . By $\overline{a}\equiv_{C}\overline{b}$ we mean $\operatorname{tp}(\overline{a}/C)=\operatorname{tp}(\overline{b}/C)$ . For a countably infinite $\aleph_{0}$ -categorical structure $\mathcal{M}$ , $\overline{a}\in M^{n}$ and $C\subset_{\textrm{fin}}M$ , we remind the reader that $\operatorname{tp}(\overline{a}/C)$ is always isolated, by the Ryll-Nardzewski theorem. Also in such a structure,

i.

By $X=\operatorname{tp}(\overline{a}/C)$ , we mean $X$ is the set defined by some formula that isolates $\operatorname{tp}(\overline{a}/C)$ .
ii.

We will introduce notions of dimension and measure for definable sets. By $\dim(a/C)$ and $\mathrm{meas}(a/C)$ , we mean $\dim(X)$ and $\mathrm{meas}(X)$ where $X$ is the set defined by a formula which isolates $\operatorname{tp}(a/C)$ .
iii.

When we write $X=\operatorname{tp}(\overline{a}_{1}/C)\uplus\dots\uplus\operatorname{tp}(\overline{a}_{k}/C)$ , we mean that

$X=\varphi_{1}(\mathcal{M},\overline{c})\uplus\dots\uplus\varphi_{k}(\mathcal{M},\overline{c})$

where $\varphi_{i}(\overline{x},\overline{c})$ isolates $\operatorname{tp}(\overline{a}_{i}/C)$ for each $i\in\{1,\dots,k\}$ , and $\overline{c}$ enumerates $C$ .

2. Tree structures

In this section, we introduce some basic notions and techniques which are needed for the rest of the paper.

2.1. Tree plans

In order to define the notion of coordinatized structure in the next section, that is a kind of expansion of a tree, we first need the notion of tree plan.

Definition 2.1.1.

I.

The language of trees, $\mathcal{L}_{t}$ , has signature $\mathrm{sig}(\mathcal{L}_{t})=\left\{\leq,\varepsilon,\sqcap,\mathtt{pred}\right\}$ , where $\leq$ is a binary relation symbol, $\varepsilon$ is a constant symbol, and $\sqcap$ is a binary function symbol, and $\mathtt{pred}$ is a unary function symbol.
II.
A tree is an $\mathcal{L}_{t}$ -structure $\mathcal{A}$ such that:
- •
  
  $\leq^{\mathcal{A}}$ is a partial ordering of the universe in which every downset $\left\{a:a\leq^{\mathcal{A}}b\right\}$ ( $b\in A$ ) is well-ordered.
- •
  
  $\varepsilon^{\mathcal{A}}$ is the unique minimum element of $\leq^{\mathcal{A}}$ — the root of $\mathcal{A}$ .
- •
  
  For $a,a^{\prime}\in A$ , $a\sqcap^{\mathcal{A}}a^{\prime}$ is the unique maximum element of $A$ that is below both $a$ and $a^{\prime}$ — the meet of $a$ and $a^{\prime}$ .
- •
  
  For each $a\in A$ , $\mathtt{pred}^{\mathcal{A}}(a)$ is the maximum element among the elements that are strictly less than $a$ . It is called the predecessor of $a$ . Also, as a convention we let $\mathtt{pred}^{\mathcal{A}}(\varepsilon)=\varepsilon$ .
Note that the class of trees in this sense is not first-order axiomatizable.

It follows from Definition 2.1.1 that $\{a:a\leq b\}$ is actually finite for all $b$ , and so predecessors exist.

Note that $\omega^{<\omega}$ with function extension ordering has a natural tree structure, and when we write a tree $\Gamma\subseteq\omega^{<\omega}$ , we mean that $\Gamma$ is a substructure. In paticular, it contains $\left\langle\right\rangle$ and is closed under $\mathtt{pred}$ .

Definition 2.1.2.

I.

A tree plan is just a pair $(\Gamma,\lambda)$ , where $\Gamma\subset_{\textrm{fin}}\omega^{<\omega}$ is a finite tree (containing $\left\langle\right\rangle$ ) and $\lambda:\Gamma\to\{1,\infty\}$ is a function such that $\lambda(\left\langle\right\rangle)=1$ .

Under almost all circumstances it’s preferable to omit $\lambda$ from the notation and start from the declaration “ $\Gamma=(\Gamma,\lambda)$ is tree plan…”
II.

Given a tree plan $\Gamma=(\Gamma,\lambda)$ , we define

$I(\Gamma):=\lambda^{-1}[\infty]=\left\{\sigma\in\Gamma:\lambda(\sigma)=\infty\right\}.$

We call each member of $I(\Gamma)$ an infinity-node of $\Gamma$ .

For a given tree plan $\Gamma$ and a set $X$ (possibly infinite), we can define a tree, called $\Gamma(X)$ , by identifying every infinity-node of $\Gamma$ with the set $X$ . Roughly speaking, $\Gamma(X)$ is an extension of $\Gamma$ obtained by replacing each infinity-node with $|X|$ -many new nodes, and leaving other nodes to themselves.

Definition 2.1.3.

Let $\Gamma=(\Gamma,\lambda)$ be a tree plan. For any non-empty set $X$ , we define a tree $\Gamma(X)\subseteq\left(\omega\times\big{(}X\cup\{\star\}\big{)}\right)^{<\omega}$ , and a function $\pi_{X}:\Gamma(X)\to\Gamma$ as follows:

$\left\langle(i_{0},t_{0}),...,(i_{n},t_{n})\right\rangle\in\Gamma(X)$ if $\left\langle i_{0},...,i_{n}\right\rangle\in\Gamma$ and for each $k\leq n$ :

–

if $\lambda(\left\langle i_{0},...,i_{k}\right\rangle)=1$ , then $t_{k}=\star$ ;
–

if $\lambda(\left\langle i_{0},...,i_{k}\right\rangle)=\infty$ , then $t_{k}\in X$ .

We then define $\pi_{X}$ just by setting $\pi_{X}{\left(\left\langle(i_{0},t_{0}),...,(i_{n},t_{n})\right\rangle\right)}=\left\langle i_{0},...,i_{n}\right\rangle$ , and we define

I(X)=\left\{a\in\Gamma(X):\pi_{X}(a)\in I(\Gamma)\right\}=\left\{a\in\Gamma(X):\lambda(\pi_{X}(a))=\infty\right\}

which is the set of infinity-nodes of $\Gamma(X)$ . Note that for each $\sigma{{}^{\smallfrown}}i\in I(\Gamma)$ and $b\in\Gamma(X)$ such that $\pi_{X}(b)=\sigma$ , there is a bijection between $X$ and the set

\left\{a\in\Gamma(X):\mathtt{pred}(a)=b\,\,\,\wedge\,\,\,\pi_{X}(a)=\sigma{{}^{\smallfrown}}i\right\}.

Observation 2.1.4.

Let $\Gamma=(\Gamma,\lambda)$ be a tree plan. Then for any two sets $X$ and $Y$ , if $X\subseteq Y$ , then $\Gamma(X)\leq\Gamma(Y)$ and $\pi_{X}=\pi_{Y}{\upharpoonright}\Gamma(X)$ . In fact, it can be shown that if $X$ and $Y$ are both infinite and $X\subseteq Y$ , then $\Gamma(X)\preceq\Gamma(Y)$ .

In the following, for a subset $B$ of a tree we define the set of points below (and above) $B$ , and also define the height of the tree.

Definition 2.1.5.

Let $\Gamma$ be a tree plan and $\sigma\in\Gamma$ . Let $X$ be an arbitrary set and $B\subseteq\Gamma(X)$ .

(1)

We define ${\downarrow}B=\left\{a\in\Gamma(X):(\exists b\in B)\,\,a\leq b\right\}$ . We say $B$ is downward closed if $B={\downarrow}B$ .
(2)

We define ${\uparrow}B=\left\{a\in\Gamma(X):(\exists b\in B)\,\,b\leq a\right\}$ .
(3)
We define the height of $\sigma$ , $\mathsf{h}(\sigma)$ , to be the smallest number $k<\omega$ such that $\mathtt{pred}^{k}(\sigma)=\varepsilon$ . I.e.
- •
  
  $\mathsf{h}(\left\langle\right\rangle)=0$ ,
- •
  
  if $\sigma\neq\left\langle\right\rangle$ , then $\mathsf{h}(\sigma)=k$ if and only if $\mathtt{pred}^{k}(\sigma)=\varepsilon$ and $\mathtt{pred}^{k-1}(\sigma)\neq\varepsilon$ .
Also we define the height of the tree plan $\Gamma$ as follows:

$\mathsf{h}(\Gamma)=\max\left\{\mathsf{h}(\sigma):\sigma\in\Gamma\right\}.$

In the next definition we define a closure operator on a tree $\Gamma(X)$ . Roughly speaking, for a given $B\subseteq\Gamma(X)$ we define the tree-closure of $B$ in $\Gamma(X)$ to be the smallest subset of $\Gamma(X)$ containing $B$ that is definably closed. Note that a definably closed set will be downward closed as well, by applying $\mathtt{pred}$ function.

Definition 2.1.6.

Let $\Gamma=(\Gamma,\lambda)$ be a tree plan. For a set $X$ , we define tree-closure

\mathrm{tcl}=\mathrm{tcl}^{\Gamma(X)}:\mathcal{P}(\Gamma(X))\longrightarrow\mathcal{P}(\Gamma(X))

as follows:

	$\displaystyle\mathrm{tcl}_{0}(B)$	$\displaystyle=\Big{\{}\left\langle\right\rangle\Big{\}}\cup\Big{\{}a\in\Gamma(X)\,:\,(\exists b\in B)\,\,a\leq b\Big{\}}$
		$\displaystyle\vdots$
	$\displaystyle\mathrm{tcl}_{n+1}(B)$	$\displaystyle=\mathrm{tcl}_{n}(B)\cup\Big{\{}a\in\Gamma(X)\,:\,\lambda(\pi(a))=1\text{ and }\mathtt{pred}(a)\in\mathrm{tcl}_{n}(B)\Big{\}}$

We set $\mathrm{tcl}(B)=\bigcup_{n}\mathrm{tcl}_{n}(B)$ .

Using $\mathrm{tcl}$ , we formulate another definition which is useful in relating an element $a$ to $\mathrm{tcl}(B)$ that may not itself be a member of $\mathrm{tcl}(B)$ . For $a\in\Gamma(X)$ and $B\subseteq\Gamma(X)$ , we set

[a{\wedge}B;X]=\max\left\{e\in\mathrm{tcl}(B):e\leq a\right\}.

Note that $a\in\mathrm{tcl}(X)(B)$ if and only $[a{\wedge}B;X]=a$ .

From now on, we fix a tree plan $\Gamma=(\Gamma,\lambda)$ and a countably infinite set $S$ . Let

\mathrm{sig}(\mathcal{L}_{\Gamma})=\mathrm{sig}(\mathcal{L}_{t})\cup\left\{P_{\sigma}^{(1)}:\sigma\in\Gamma\right\},

where for each $\sigma\in\Gamma$ , by $P_{\sigma}^{(1)}$ we mean that $P_{\sigma}$ is a unary relation symbol. We consider $\Gamma(S)$ as an $\mathcal{L}_{\Gamma}$ -structure with $P_{\sigma}^{\Gamma(S)}=\left\{a:\pi(a)=\sigma\right\}$ for each $\sigma\in\Gamma$ .

When $S$ is clear from context, for brevity, we write:

•

$[a\wedge B]$ in place of $[a\wedge B;S]$ ;
•

$\mathrm{tcl}(B)$ in place of $\mathrm{tcl}^{\Gamma(S)}(B)$ ;
•

$\operatorname{acl}(B)$ in place of $\operatorname{acl}^{\Gamma(S)}(B)$ ;
•

$\operatorname{tp}_{\Gamma}(...)$ when we mean $\operatorname{tp}^{\Gamma(S)}(...)$ ;
•

$\pi$ in place of $\pi_{S}$ ;

Finally, we set

I(S)=\left\{a\in\Gamma(S):\lambda(\pi(a))=\infty\right\}

and for an expansion $\mathcal{M}$ of $\Gamma(S)$ , and each $a\in\Gamma(S)$ ,

M(a)=\left\{a^{\prime}\in\Gamma(S):\mathtt{pred}(a^{\prime})=\mathtt{pred}(a)\text{ and }\pi(a^{\prime})=\pi(a)\right\}.

These sets are called components of $\mathcal{M}$ . Note that if $\pi(a)=\pi(b)$ and $\mathtt{pred}(a)=\mathtt{pred}(b)$ , then $M(a)=M(b)$ .

Here is a lemma which will be used in Section 3, in particular in the proof of Corollary 3.2.4, showing that certain expansions of $\Gamma(S)$ are $\aleph_{0}$ -categorical and eliminate quantifiers.

Lemma 2.1.7.

$\Gamma(S)$ is ultra-homogeneous, and its theory, $Th(\Gamma(S))$ , is $\aleph_{0}$ -categorical.

Proof.

Let $\mathbf{K}(\Gamma)$ be the class obtained by closing $\left\{A\,:\,A\leq\Gamma(X)\text{ and }X\text{ is a finite set}\right\}$ under isomorphisms. It’s easily seen that $\Gamma(S)$ is the Fraïssé limit of $\mathbf{K}(\Gamma)$ . So $\Gamma(S)$ is an ultra-homogeneous $\mathcal{L}_{\Gamma}$ -structure, and its theory, $Th(\Gamma(S))$ , is $\aleph_{0}$ -categorical. ∎

3. Coordinatization

In this section, we introduce a strong notion of coordinatization. Then, by constructing an example, in Subsection 3.2 we show such coordinatized structures exist. Finally, we verify that the example works.

3.1. Nil-interaction coordinatized structures

For a tree plan $\Gamma$ and a countably infinite set $S$ , we define a nil-interaction coordinatized structure to be an $\aleph_{0}$ -categorical expansion of $\Gamma(S)$ in which, roughly speaking, there is no interaction between different components other than what $\Gamma(S)$ already supplies. This is analogous to what Cherlin and Hrushovski, in [2], call a Lie coordinatized structure in contrast to a Lie coordinatizable structure.

For the rest of this section, we take $\Gamma$ and $S$ as fixed. First, we make some notation for discussing $\aleph_{0}$ -categorical expansions of $\Gamma(S)$ in general.

Definition 3.1.1.

Let $\mathcal{M}$ be an expansion of $\Gamma(S)$ , and let $\sigma\in I(\Gamma)$ . For each $n<\omega$ , let

\mathcal{S}_{n}^{\sigma}=\left\{\operatorname{tp}^{\mathcal{M}}(b,\overline{a})\,\,\,:\,\,\begin{array}[]{l}\overline{a}=(a_{0},\dots,a_{n-1}),\\ a_{0}\in\pi^{-1}[\sigma],\\ b=\mathtt{pred}(a_{0})\text{ and }a_{1},\dots,a_{n-1}\in M(a_{0})\end{array}\right\},

and

\mathrm{sig}(\mathcal{L}_{\sigma})=\bigg{\{}R^{(n)}_{p}:0<n<\omega,\,\,p(x,\overline{y})\in\mathcal{S}_{n}^{\sigma}\bigg{\}}.

where, for each $p\in\mathcal{S}_{n}^{\sigma}$ , $R_{p}^{(n)}$ means $R_{p}$ is an $n$ -ary relation symbol.
Now, let $\mathcal{M}$ be an $\aleph_{0}$ -categorical expansion of $\Gamma(S)$ , and let $e\in\Gamma(S)$ such that $\pi(e)=\sigma$ and $\mathtt{pred}(e)=b$ . Then we define $\mathcal{M}(e)$ to be the $\mathcal{L}_{\sigma}$ -structure with universe $M(e)$ and interpretations

R^{\mathcal{M}(e)}_{p}=\bigg{\{}\overline{a}\in M(e)^{n}:\mathcal{M}\vDash p(b,\overline{a})\bigg{\}}

for all $0<n<\omega$ and $p(x,\overline{y})\in\mathcal{S}_{n}^{\sigma}$ .

Also, we define

G(e)=\left\{g{\upharpoonright}M(e):g\in Aut(\mathcal{M}/\mathtt{pred}(e))\right\}

which is a closed subgroup of $\mathit{Sym}(M(e))$ .

Definition 3.1.2.

Let $\mathcal{M}$ be an expansion of $\Gamma(S)$ . We say that $\mathcal{M}$ is a nil-interaction coordinatized structure (or NIC structure) if all of the following conditions hold:

N1.

$\mathcal{M}$ is $\aleph_{0}$ -categorical, eliminates quantifiers, and has $\operatorname{acl}=\mathrm{tcl}$ .
N2.

For each single element $e\in\Gamma(S)$ , $\operatorname{tp}_{\Gamma}(e)\vDash\operatorname{tp}^{\mathcal{M}}(e)$ .

N3.

For $C\subset_{\textrm{fin}}\Gamma(S)$ and $e\in\Gamma(S)$ , if $\pi(e)\in I(\Gamma)$ , then

\operatorname{tp}_{\Gamma}\big{(}e/C\big{)}\cup\operatorname{tp}^{\mathcal{M}(e)}\big{(}e/\mathrm{tcl}(C)\cap M(e)\big{)}\vDash\operatorname{tp}^{\mathcal{M}}\big{(}e/C\big{)}.

The next proposition shows that for each infinity-node $\sigma$ of $\Gamma$ , there is an $\aleph_{0}$ -categorical theory $T_{\sigma}$ such that $\mathcal{M}(a)\vDash T_{\sigma}$ whenever $\pi(a)=\sigma$ . It shows that, up to isomorphism, the set of components of an NIC structure is finite, which we count as desirable.

Proposition 3.1.3.

Let $\Gamma$ be a tree plan and let $\mathcal{M}$ be a NIC structure expanding $\Gamma(S)$ . Let $\sigma\in I(\Gamma)$ . Then there is an $\aleph_{0}$ -categorical theory $T_{\sigma}$ such that $\mathcal{M}(e)\vDash T_{\sigma}$ whenever $\pi(e)=\sigma$ . Moreover, for any $g\in\mathrm{Aut}(\mathcal{M})$ , $g{\restriction}M(e)$ is an isomorphism of $\mathcal{M}(e)$ onto $\mathcal{M}(g(e))$ .

Proof.

Suppose $\pi(e)=\sigma\in I(\Gamma)$ and $g\in\mathrm{Aut}(\mathcal{M})$ . Then $g{\restriction}M(e)$ is a bijection of $M(e)$ onto $M(g(e))$ . Also, for all $n\in\mathbb{N}^{+}$ , for all $p(x,\overline{y})\in\mathcal{S}_{n}^{\sigma}$ (see Definition 3.1.1), and for all $\overline{a}\in M(e)^{n}$ we have

	$\displaystyle\overline{a}\in R_{p}^{\mathcal{M}(e)}$	$\displaystyle\Leftrightarrow\mathcal{M}\vDash p(\mathtt{pred}(e),\overline{a})$
		$\displaystyle\Leftrightarrow\mathcal{M}\vDash p(\mathtt{pred}(g(e)),g\overline{a})$
		$\displaystyle\Leftrightarrow g\overline{a}\in R_{p}^{\mathcal{M}(g(e))}.$

Therefore $\mathcal{M}(e)\cong\mathcal{M}(g(e))$ via $g{\restriction}M(e)$ . Suppose $\pi(e)=\pi(e^{\prime})=\sigma\in I(\Gamma)$ and $\varphi\in Th\left(\mathcal{M}(e)\right)$ . Let $\theta(x,e)$ define $M(e)$ in $\mathcal{M}$ , and let $\varphi^{\theta(x,e)}\in\operatorname{tp}^{\mathcal{M}}(e)$ be a formula that $\varphi(\mathcal{M}(e))=\varphi^{\theta(x,e)}(\mathcal{M})$ (i.e. for any $e^{\prime\prime}$ , $\mathcal{M}\vDash\varphi^{\theta(x,e^{\prime\prime})}(e^{\prime\prime})$ if and only if $\mathcal{M}(e^{\prime\prime})\vDash\varphi$ ). Since $\operatorname{tp}_{\Gamma}(e)=\operatorname{tp}_{\Gamma}(e^{\prime})$ , $\operatorname{tp}^{\mathcal{M}}(e)=\operatorname{tp}^{\mathcal{M}}(e^{\prime})$ by N2. So $\varphi^{\theta(x,e)}\in\operatorname{tp}^{\mathcal{M}}(e^{\prime})$ . This follows that $\varphi\in Th\left(\mathcal{M}(e^{\prime})\right)$ . We set $T_{\sigma}=Th(\mathcal{M}(e))$ for any $e$ such that $\pi(e)=\sigma$ . Since $\mathcal{M}$ is $\aleph_{0}$ -categorical by N1, $T_{\sigma}$ is $\aleph_{0}$ -categorical as well. ∎

3.2. Existence of NIC structures

In this subsection, we provide an example which shows that nil-interaction coordinatized structures exist. We start with a tree plan $\Gamma$ . For each $\sigma\in I(\Gamma)$ , we fix an $\aleph_{0}$ -categorical theory $T_{\sigma}$ in a relational language $\mathcal{L}_{\sigma}$ . We extend our language $\mathcal{L}_{\Gamma}$ to $\mathcal{L}^{*}$ by adding an $(n+1)$ -ary predicate symbol $\hat{R}$ for each $n$ -ary relation symbol $R\in\mathrm{sig}(\mathcal{L}_{\sigma})$ . Roughly speaking, we define a nil-interaction coordinatized structure as a $\mathcal{L}^{*}$ -structure as follows: $\Gamma(S)$ is the underlying set, and for each infinity-node $a\in\Gamma(S)$ with $\pi(a)=\sigma\in I(\Gamma)$ we identify $M(a)$ with a model $\mathcal{N}_{\sigma}$ of $T_{\sigma}$ .

3.2.1. Construction

The ingredients for constructing a NIC structure $\mathcal{M}$ are the following:

•

A tree plan $\Gamma$ , and a countably infinite set $S$ .
•

For each $\sigma\in I(\Gamma)$ , let $T_{\sigma}$ be a theory in a relational language $\mathcal{L}_{\sigma}$ . We assume that $T_{\sigma}$ is $\aleph_{0}$ -categorical, algebraically trivial, eliminates quantifiers, and has $|S_{1}(T_{\sigma})|=1$ .

We also assume that if $\sigma_{1},\sigma_{2}$ are distinct, then $\mathrm{sig}(\mathcal{L}_{\sigma_{1}})\cap\mathrm{sig}(\mathcal{L}_{\sigma_{2}})=\varnothing$ .
•

For each $\sigma\in I(\Gamma)$ , fix a bijection $f_{\sigma}:S\to N_{\sigma}$ , where $\mathcal{N}_{\sigma}$ is a fixed countable model of $T_{\sigma}$ .

We then define

\mathrm{sig}(\mathcal{L}^{*}):=\mathrm{sig}(\mathcal{L}_{\Gamma})\cup\bigcup_{\sigma\in I(\Gamma)}\left\{\hat{R}^{(n+1)}:R^{(n)}\in\mathrm{sig}(\mathcal{L}_{\sigma})\right\}.

Finally, $\mathcal{M}$ is the $\mathcal{L}^{*}$ -structure such that:

•

$\mathcal{M}{\upharpoonright}\mathcal{L}_{\Gamma}=\Gamma(S)$

•

For each $\sigma\in I(\Gamma)$ , $a\in\Gamma(S)$ such that $\pi(a)=\sigma=\sigma_{0}{{}^{\smallfrown}}k$ , for each $R^{(n)}\in\mathrm{sig}(\mathcal{L}_{\sigma})$ ,

\hat{R}^{\mathcal{M}}=\left\{\left(b,b{{}^{\smallfrown}}(k,s_{0}),...,b{{}^{\smallfrown}}(k,s_{n-1})\right):\pi(b)=\sigma_{0},\,(f_{\sigma}(s_{0}),...,f_{\sigma}(s_{n-1}))\in R^{\mathcal{N}_{\sigma}}\right\}.

Note that if $\pi(b)=\sigma_{0},\,\pi(e)=\sigma=\sigma_{0}{{}^{\smallfrown}}k\in I(\Gamma)$ and $\mathtt{pred}(e)=b$ , then $M(e)=\left\{b{{}^{\smallfrown}}(k,s):s\in S\right\}$ .

3.2.2. Verification

In this subsection, we verify that the structure obtained from the construction just presented satisfies N1, N2 and N3. Throughout this subsection, $\mathcal{L}_{\sigma},T_{\sigma},\mathcal{N}_{\sigma}$ (for $\sigma\in I(\Gamma)$ ) and $\mathcal{M}$ are as in that construction.

Remark 3.2.1.

For $\sigma\in I(\Gamma)$ , $a\in M$ such that $\pi(a)=\sigma$ , $b=\mathtt{pred}(a)$ so that $a=b{{}^{\smallfrown}}(k,s^{*})$ for some $s^{*}\in S$ , and $g\in Aut(\mathcal{N}_{\sigma})$ , we define

X_{a/b}=\bigcup_{e\in M(a)}\left\{c:e\leq c\right\},\,\,\,\,Y_{a/b}=\Gamma(S)\setminus X_{a/b},

and $g_{a/b}:\Gamma(S)\to\Gamma(S)$ by setting $g_{a/b}(y)=y$ if $y\in Y_{a/b}$ , and with a slight abuse of notation

g_{a/b}(b{{}^{\smallfrown}}(k,s){{}^{\smallfrown}}c)=b{{}^{\smallfrown}}(k,g(s)){{}^{\smallfrown}}c

whenever $b{{}^{\smallfrown}}(k,s){{}^{\smallfrown}}c\in X_{a/b}$ . We also notice that $Aut(\mathcal{M})$ is the topological closure of the subgroup of $\mathit{Sym}(\Gamma(S))$ generated by

\left\{g_{a/b}\,\,:\begin{array}[]{l}\sigma\in I(\Gamma),\,\,\pi(a)=\sigma,\\ b=\mathtt{pred}(a),\,\,g\in Aut(\mathcal{N}_{\sigma})\end{array}\right\}.

Notation 3.2.2.

Let $a\in\Gamma(S)$ such that $\pi(a)=\sigma\in I(\Gamma)$ , and let $\sigma=\sigma_{0}{{}^{\smallfrown}}k,\,b=\mathtt{pred}(a)$ . We write $\xi_{a}:\mathcal{M}(a)\longrightarrow\mathcal{N}_{\sigma}$ to denote the map $f_{\sigma}\circ h$ where $h:M(a)\longrightarrow S$ is given by $b{{}^{\smallfrown}}(k,s)\mapsto s$ .

The following lemma has a key role to verify that $\mathcal{M}$ satisfies N1 and N2.

Lemma 3.2.3.

Let $A\subset_{\textrm{fin}}\Gamma(S)$ such that $A=\mathrm{tcl}(A)$ , and let $e,e^{\prime}\in\Gamma(S)$ such that $e\neq e^{\prime}$ . Let $\operatorname{tp}_{\Gamma}(e/A)=\operatorname{tp}_{\Gamma}(e^{\prime}/A)$ and $\operatorname{tp}^{\mathcal{M}(b)}(b/A_{0})=\operatorname{tp}^{\mathcal{M}(b)}(b^{\prime}/A_{0})$ where $b\leq e,\,b^{\prime}\leq e^{\prime},$ $\mathtt{pred}(b)=e\sqcap e^{\prime}=\mathtt{pred}(b^{\prime})$ and $A_{0}=A\cap M(b)$ . Then there is an automorphism $g\in\mathrm{Aut}(\mathcal{M}/A)$ such that $g(e)=e^{\prime}$ . In particular, $\operatorname{tp}^{\mathcal{M}}(e/A)=\operatorname{tp}^{\mathcal{M}}(e^{\prime}/A)$ .

Proof.

Let $e\sqcap e^{\prime}=e_{0}$ . There is $n\in\mathbb{N}^{+}$ such that

e\sqcap e^{\prime}=e_{0}<b=e_{1}<e_{2}<\dots<e_{n}=e

and

e\sqcap e^{\prime}=e_{0}<b^{\prime}=e^{\prime}_{1}<e^{\prime}_{2}<\dots<e^{\prime}_{n}=e^{\prime}.

We prove the statement by induction on $n$ , the length of the path from $e_{0}$ to $e$ . If $n=1$ , then $e=b$ and $e^{\prime}=b^{\prime}$ . Let $\pi(e_{0})=\sigma_{0}$ , $\pi(e)=\pi(e^{\prime})=\sigma=\sigma_{0}{{}^{\smallfrown}}k$ for some $k<\omega$ . Let $e=e_{0}{{}^{\smallfrown}}(k,s)$ and $e^{\prime}=e_{0}{{}^{\smallfrown}}(k,s^{\prime})$ for some $s,s^{\prime}\in S$ . Since $\operatorname{tp}^{\mathcal{M}(b)}(b/A_{0})=\operatorname{tp}^{\mathcal{M}(b)}(b^{\prime}/A_{0})$ and $\mathcal{M}(b)$ is ultra-homogeneous, there is an automorphism $h\in\mathrm{Aut}(\mathcal{N}_{\sigma})$ such that $\xi_{e}^{-1}\circ h\circ\xi_{e}(e)=e^{\prime}$ and $\left(\xi_{e}^{-1}\circ h\circ\xi_{e}(e)\right){\restriction}A_{0}=id_{A_{0}}$ (see Notation 3.2.2). We get $\hat{h}=f_{\sigma}^{-1}\circ h\circ f_{\sigma}\in\mathit{Sym}(S)$ such that $\hat{h}(s)=s^{\prime}$ and for each $e_{0}{{}^{\smallfrown}}(k,t)\in A_{0}$ , $\hat{h}(t)=t$ . We define a bijection $v:{\uparrow}M(b)\longrightarrow{\uparrow}M(b)$ by

v\left(e_{0}{{}^{\smallfrown}}(k,s){{}^{\smallfrown}}c\right)=e_{0}{{}^{\smallfrown}}(k,\hat{h}(s)){{}^{\smallfrown}}c.

We set $g=id_{Z}\cup v$ where $Z=\Gamma(S)\setminus{\uparrow}M(b)$ . If $a\in A\cap{\uparrow}M(b)$ , then some ancestor of $a$ is in $A_{0}$ , so $a=e_{0}{{}^{\smallfrown}}(k,s){{}^{\smallfrown}}c$ where $\hat{h}(s)=s$ and thus $g(a)=e_{0}{{}^{\smallfrown}}(k,s){{}^{\smallfrown}}c=a$ . Suppose the statement holds for the case that the length of the path from $e_{0}$ to $e$ is $\leq n$ , and suppose the length of the path from $e_{0}$ to $e$ equals $n+1$ . Since $\operatorname{tp}_{\Gamma}(e/A)=\operatorname{tp}_{\Gamma}(e^{\prime}/A)$ , $\operatorname{tp}_{\Gamma}(\mathtt{pred}(e)/A)=\operatorname{tp}_{\Gamma}(\mathtt{pred}(e^{\prime})/A)$ . So $\mathtt{pred}(e)$ and $\mathtt{pred}(e^{\prime})$ satisfy the assumptions of the statement. By the induction hypothesis, there is $g_{0}\in\mathrm{Aut}(\mathcal{M}/A)$ such that $g_{0}(\mathtt{pred}(e))=\mathtt{pred}(e^{\prime})$ . A similar argument like the case $n=1$ will work here. More precisely, since $\mathtt{pred}(g_{0}(e))=g_{0}(\mathtt{pred}(e))=\mathtt{pred}(e^{\prime})$ , $g_{0}(e)\sqcap e^{\prime}=\mathtt{pred}(e^{\prime})$ . So $g_{0}(e)$ and $e^{\prime}$ satisfy the hypotheses for the case $n=1$ . Let $\pi(e^{\prime})=\pi(g_{0}(e))=\tau$ . Since $|S_{1}(T_{\tau})|=1$ , there is an automorphism $f\in\mathrm{Aut}(\mathcal{N}_{\tau})$ such that $\xi_{e^{\prime}}^{-1}\circ f\circ\xi_{e^{\prime}}\left(g_{0}(e)\right)=e^{\prime}$ . So by an argument like the case $n=1$ , we get $g_{1}\in\mathrm{Aut}(\mathcal{M}/A)$ with $g_{1}\left(g_{0}(e)\right)=e^{\prime}$ , and we set $g=g_{1}\circ g_{0}$ . ∎

Corollary 3.2.4.

$\mathcal{M}$ satisfies N1.

Proof.

Let $T=Th(\mathcal{M})$ . To prove $\aleph_{0}$ -categoricity, we show that for all $n\in\mathbb{N}^{+}$ , $|S_{n}(T)|$ is finite. For $n=1$ , it follows from Lemma 3.2.3 for $A=\varnothing$ , and the fact that $Th(\Gamma(S))$ is $\aleph_{0}$ -categorical (see Lemma 2.1.7) and $|S_{1}(T_{\tau})|=1$ for all $\tau$ . Assume $|S_{k}(T)|<\aleph_{0}$ for all $k\leq n$ . We want to show that $|S_{n+1}(T)|<\aleph_{0}$ . For this, it is enough to show that for every $\overline{a}\in\Gamma(S)^{n}$ , the set $\left\{\operatorname{tp}^{\mathcal{M}}(\overline{a},b):b\in\Gamma(S)\right\}$ is finite. Suppose, towards a contradiction, that there are $\overline{a}\in M^{n}$ and $\{b_{i}:i<\omega\}\subset\Gamma(S)$ , such that $\operatorname{tp}^{\mathcal{M}}(\overline{a},b_{i})\neq\operatorname{tp}^{\mathcal{M}}(\overline{a},b_{j})$ whenever $i<j<\omega$ . Since $Th(\Gamma)$ is $\aleph_{0}$ -categorical, the pigeonhole principle allows us to assume that $\operatorname{tp}_{\Gamma}(\overline{a},b_{i})=\operatorname{tp}_{\Gamma}(\overline{a},b_{j})$ for all $i,j<\omega$ . So we may assume that $\operatorname{tp}_{\Gamma}(b_{i}/\mathrm{tcl}(\overline{a}))=\operatorname{tp}_{\Gamma}(b_{j}/\mathrm{tcl}(\overline{a}))$ for all $i,j<\omega$ . Since $T_{\sigma}$ is algebraically trivial (for each $\sigma\in I(\Gamma)$ ) and $\overline{a}$ is finite, $\mathrm{tcl}(\overline{a})$ is finite as well. By the pigeonhole principle, there are finitely many types over $\mathrm{tcl}(\overline{a})\cap M(b)$ in $\mathcal{M}(b)$ , for each $b\in\Gamma(S)$ . So for all but finitely many $i<j<\omega$ , the pairs $(b_{i},b_{j})\in\Gamma(S)^{2}$ satisfy the assumptions of the Lemma 3.2.3, so we may apply Lemma 3.2.3 to get an automorphism $g_{i,j}\in\mathrm{Aut}(\mathcal{M}/\mathrm{tcl}(\overline{a}))$ such that $g_{i,j}(b_{i})=b_{j}$ . This implies that for all but finitely many $i<j<\omega$ , $\operatorname{tp}^{\mathcal{M}}(\overline{a},b_{i})=\operatorname{tp}^{\mathcal{M}}(\overline{a},b_{j})$ which is a contradiction. Thus $|S_{n+1}(T)|<\aleph_{0}$ .

Since $\mathcal{M}$ is $\aleph_{0}$ -categorical, it eliminates quantifiers whenever $\mathcal{M}$ is ultra-homogeneous. So it is enough to show that $\mathcal{M}$ is ultra-homogeneous. Since $\mathtt{pred}$ is in the signature, every substructure of $\mathcal{M}$ will be closed downwards (see Definition 2.1.5).

Claim.

Suppose $f:A\longrightarrow\mathcal{M}$ is an $\mathcal{L}^{*}$ -embedding where $A$ is a finite substructure of $\mathcal{M}$ . Let $b\in M\setminus A$ such that $\mathtt{pred}(b)\in A$ . Then there is an $\mathcal{L}^{*}$ -embedding $f^{b}:Ab\longrightarrow\mathcal{M}$ such that $f\subset f^{b}$ .

Proof.

Suppose $\pi(b)\notin I(\Gamma)$ . Let $E=\{e\in\Gamma(S):\mathtt{pred}(e)=f(\mathtt{pred}(b))\}$ . Since $\mathtt{pred}(b)\in A$ and $f$ is an $\mathcal{L}^{*}$ -embedding, $E\neq\varnothing$ . We choose $c\in E$ such that $\pi(c)=\pi(b)$ . We set $f^{b}(b)=c$ . Now suppose $\pi(b)=\sigma\in I(\Gamma)$ . Let $A_{0}=A\cap M(b)$ and $p(x)=\operatorname{tp}^{\mathcal{M}(b)}(b/A_{0})$ . Since $T_{\sigma}$ is $\aleph_{0}$ -categorical, there is formula, say $\varphi(x,\overline{a}_{0})$ , that isolates $p(x)$ where $\overline{a}_{0}$ is an enumeration of $A_{0}$ . Let $d\in\Gamma(S)$ such that $\operatorname{tp}_{\Gamma}(d)=\operatorname{tp}_{\Gamma}(b)$ and $\mathtt{pred}(d)=f(\mathtt{pred}(b))$ . We need to show that $\varphi(x,f\overline{a}_{0})$ is realized in $\mathcal{M}(d)$ . Since $f{\restriction}A_{0}:A_{0}\longrightarrow f[A_{0}]$ is an $\mathcal{L}_{\sigma}$ -embedding (partial elementary map) and $\mathcal{M}(b)\vDash\exists x\varphi(x,\overline{a}_{0})$ , $\mathcal{M}(d)\vDash\exists x\varphi(x,f\overline{a}_{0})$ . We choose $c\in\Gamma(S)$ such that $\mathcal{M}(d)\vDash\varphi(c,f\overline{a}_{0})$ . Finally we set $f^{b}(b)=c$ . ∎

Now let an $\mathcal{L}^{*}$ -embedding $f:A\longrightarrow\mathcal{M}$ be given, where $A$ is a finite substructure of $\mathcal{M}$ . Let $x_{0},x_{1},x_{2},x_{3},\dots$ be an enumeration of $\Gamma(S)\setminus A$ . We construct an increasing sequence as follows.

•

Stage $0$ : set $f_{0}=f$ and $B_{0}=A$ .
•

Stage $2n+1$ : If $x_{2n}\in B_{2n}$ , then we set $f_{2n+1}=f_{2n}$ and $B_{2n+1}=B_{2n}$ . Otherwise, we may assume that

$[x_{2n}\wedge B_{2n}]=y_{0}<y_{1}<\dots<y_{k}=x_{2n}$

with $\mathtt{pred}(y_{i})=y_{i-1}$ for each $i=1,\dots,k$ . Then we set

$f_{2n+1}=\big{(}(f_{2n}^{y_{1}})^{y_{2}}\dots\big{)}^{y_{k}}\,\,\,\,\,\text{ and }\,\,\,\,\,B_{2n+1}={\downarrow}(B_{2n}x_{2n}).$
•

Stage $2n+2$ : If $x_{2n}\in\mathrm{img}(f_{2n+1})$ , then we set $f_{2n+2}=f_{2n+1}$ and $B_{2n+2}=B_{2n+1}$ . If $x_{2n}\notin\mathrm{img}(f_{2n+1})$ , then we consider $f_{2n+1}^{-1}$ which is a partial isomorphism whose domain does not contain $x_{2n+1}$ . Assume

$[x_{2n}\wedge\mathrm{dom}(f_{2n+1}^{-1})]=z_{0}<z_{1}<\dots<z_{\ell}=x_{2n}$

with $\mathtt{pred}(z_{i})=z_{i-1}$ for each $i=1,\dots,\ell$ . Let $g=\big{(}((f_{2n+1}^{-1})^{z_{1}})^{z_{2}}\dots\big{)}^{z_{\ell}}$ . Then we set

$f_{2n+2}=g^{-1}\,\,\,\,\,\text{ and }\,\,\,\,\,B_{2n+2}={\downarrow}\mathrm{dom}(g^{-1}).$

Finally, we set $\hat{f}:=\bigcup_{n<\omega}f_{n}$ which is an isomorphism from $\bigcup_{n<\omega}B_{n}=\Gamma(S)$ onto itself, and $f\subset\hat{f}$ .

Now we show that $\operatorname{acl}=\mathrm{tcl}.$ We know that $\mathrm{tcl}\subseteq\operatorname{dcl}\subseteq\operatorname{acl}$ . We will show that for any $C\subset_{\textrm{fin}}\Gamma(S)$ , $\operatorname{acl}(C)\subseteq\mathrm{tcl}(C)$ . Let $C=\mathrm{tcl}(C)\subset_{\textrm{fin}}\Gamma(S)$ be given, and let $b\notin C$ . We will show that $b\notin\operatorname{acl}(C)$ . Let

[b\wedge C]=b_{0}<b_{1}<\dots<b_{n-1}<b_{n}=b

such that $\mathtt{pred}(b_{i})=b_{i-1}$ , for each $i=1,\dots,n$ . Since $C=\mathrm{tcl}(C)$ , $\pi(b_{1})=\sigma\in I(\Gamma)$ . Let $C_{1}=C\cap M(b_{1})$ . Since by hypothesis $T_{\sigma}$ is algebraically trivial, there is a set $E=\{e_{j}:j<\omega\}\subseteq M(b_{1})$ such that for all $j<k<\omega$ , $e_{j}\neq e_{k}$ and $\operatorname{tp}(e_{j}/C_{1})=\operatorname{tp}(b_{1}/C_{1}).$ For each $i<\omega$ , $e_{i}$ and $b$ satisfy the assumptions of Lemma 3.2.3. So for each $i<\omega$ , there is an automorphism $g_{i}\in\mathrm{Aut}(\mathcal{M}/C)$ such that $g_{i}(b_{1})=e_{i}$ . Since $b$ has distinct images under each of these automorphisms, $\left\{g_{i}(b):i<\omega\right\}$ is an infinite set of realizations of $\operatorname{tp}(b/C)$ . Thus $b\notin\operatorname{acl}(C)$ . ∎

Corollary 3.2.5.

$\mathcal{M}$ satisfies $N2$ .

Proof.

Let $e^{\prime}\vDash\operatorname{tp}_{\Gamma}(e)$ , $b\leq e$ and $b^{\prime}\leq e^{\prime}$ such that $\mathtt{pred}(b)=e\sqcap e^{\prime}=\mathtt{pred}(b^{\prime})$ . We claim that $\operatorname{tp}^{\mathcal{M}(b)}(b)=\operatorname{tp}^{\mathcal{M}(b)}(b^{\prime})$ . First we note that $\pi(b)=\pi(b^{\prime})\in I(\Gamma)$ . Let $\pi(b)=\pi(b^{\prime})=\sigma$ . Since $|S_{1}(T_{\sigma})|=1$ , $\operatorname{tp}^{\mathcal{M}(b)}(b)=\operatorname{tp}^{\mathcal{M}(b)}(b^{\prime})$ . Hence by Lemma 3.2.3, $e^{\prime}\vDash\operatorname{tp}^{\mathcal{M}}(e)$ . ∎

Corollary 3.2.6.

$\mathcal{M}$ satisfies N3.

Proof.

Let $C\subset_{\textrm{fin}}\Gamma(S)$ and $e\in\Gamma(S)$ such that $\pi(e)\in I(\Gamma)$ . Let $e^{\prime}\in\Gamma(S)$ such that

e^{\prime}\vDash\operatorname{tp}_{\Gamma}\left(e/\mathrm{tcl}(C)\right)\cup\operatorname{tp}^{\mathcal{M}(e)}\left(e/M(e)\cap\mathrm{tcl}(C)\right).

Suppose $M(e)\cap\mathrm{tcl}(C)=\{e_{0},\dots e_{k-1}\}$ and set $\pi(e)=\sigma\in I(\Gamma)$ . Also, suppose $e^{\prime}\in M(e)\setminus\{e_{0},\dots e_{k-1}\}$ , note that if not, then $e=e^{\prime}$ so there is nothing to do. Since

e^{\prime}\vDash\operatorname{tp}^{\mathcal{M}(e)}\left(e/M(e)\cap\mathrm{tcl}(C)\right),

there is $h\in\mathrm{Aut}(\mathcal{N}_{\sigma})$ such that $h$ fixes $\xi_{e}[M(e)\cap\mathrm{tcl}(C)]$ (see Notation 3.2.2) pointwise, and $h(\xi_{e}(e))=\xi_{e}(e^{\prime})$ . Then by Lemma 3.2.3, there is an automorphism $g\in\mathrm{Aut}(\mathcal{M}/\mathrm{tcl}(C))$ such that $g(e)=e^{\prime}$ . Therefore $e^{\prime}\vDash\operatorname{tp}^{\mathcal{M}}(e/\mathrm{tcl}(C))$ .

∎

Theorem 3.2.7.

$\mathcal{M}$ is a NIC structure.

Proof.

Combine Corollary 3.2.4, Corollary 3.2.5 and Corollary 3.2.6. ∎

Here we prove a proposition that is interesting by itself, but it does not have any role in the rest of the paper.

Proposition 3.2.8.

Suppose $\overline{a}\in\Gamma(S)^{n}$ for some $n$ , and $\overline{a}=\mathrm{tcl}(\overline{a})$ . Let $X=\left\{i:\lambda(\pi(a_{i}))=\infty\right\}$ . Let $\left\{I_{0},...,I_{k-1}\right\}$ be the coarsest partition of $X$ such that for each $j<k$ , for any $i_{1},i_{2}\in I_{j}$ , we have $\pi(a_{i_{1}})=\pi(a_{i_{2}})$ and $\mathtt{pred}(a_{i_{1}})=\mathtt{pred}(a_{i_{2}})$ . (For each $j<k$ , let $i^{*}(j)\in I_{j}$ and $\sigma_{j}=\pi(\mathtt{pred}(a_{i^{*}(j)}))$ .) Then, abusing notation a little bit,

\operatorname{tp}^{\mathcal{M}}(\overline{a})\,\,\equiv\,\,\operatorname{tp}_{\Gamma}(\overline{a})\wedge\bigwedge_{j<k}{\operatorname{tp}^{\mathcal{M}(a_{i^{*}(j)})}}{\left(\overline{a}_{{\upharpoonright}I_{j}}\right)}

where $\overline{a}_{{\upharpoonright}I_{j}}=(a_{i})_{i\in I_{j}}$ for each $j<k$ . More precisely, let

\widetilde{\operatorname{tp}}(\overline{a}):=\operatorname{tp}_{\Gamma}(\overline{a})\cup\bigcup_{j<k}\left\{\varphi(x_{i_{j,0}},...,x_{i_{j,m_{j}-1}}):\varphi(v_{0},...,v_{m_{j}-1})\in{\operatorname{tp}^{\mathcal{M}(a_{i^{*}(j)})}}{\left(a_{i_{j,0}},...,a_{i_{j,m_{j}-1}}\right)}\right\}.

For every $\overline{a},\overline{a}^{\prime}\in\Gamma(S)^{n}$ , if $\widetilde{\operatorname{tp}}(\overline{a})=\widetilde{\operatorname{tp}}(\overline{a}^{\prime})$ , then there is an automorphism $g\in\mathrm{Aut}(\mathcal{M})$ such that $g\overline{a}=\overline{a}^{\prime}$ .

Proof.

By induction on $n$ we show that for any $\overline{a},\overline{a}^{\prime}\in\Gamma(S)^{n}$ , if $\widetilde{\operatorname{tp}}(\overline{a})=\widetilde{\operatorname{tp}}(\overline{a}^{\prime})$ , then there is an automorphism $g\in\mathrm{Aut}(\mathcal{M})$ such that $g\overline{a}=\overline{a}^{\prime}$ . The base case, $n=1$ , follows from Lemma 3.2.3. Assume that the statement holds for any tuple of length $\leq n$ . Let $\overline{a},\overline{a}^{\prime}\in\Gamma(S)^{n},\,\,e,e^{\prime}\in\Gamma(S)$ . Suppose $\widetilde{\operatorname{tp}}(\overline{a},e)=\widetilde{\operatorname{tp}}(\overline{a}^{\prime},e^{\prime})$ . We will show that there is an automorphism $g\in\mathrm{Aut}(\mathcal{M})$ such that $g\overline{a}=\overline{a}^{\prime}$ and $g(e)=e^{\prime}$ . We have $\widetilde{\operatorname{tp}}(\overline{a})=\widetilde{\operatorname{tp}}(\overline{a}^{\prime})$ . By the induction hypothesis, we get $g_{0}\in\mathrm{Aut}(\mathcal{M})$ such that $g_{0}\overline{a}^{\prime}=\overline{a}$ . So $\operatorname{tp}^{\mathcal{M}}(\overline{a}^{\prime},e^{\prime})=\operatorname{tp}^{\mathcal{M}}(\overline{a},g_{0}(e^{\prime}))$ and $\widetilde{\operatorname{tp}}(\overline{a},e)=\widetilde{\operatorname{tp}}(\overline{a},g_{0}(e^{\prime}))$ . It is enough to show that there is an automorphism $g\in\mathrm{Aut}(\mathcal{M})$ such that $g\overline{a}^{\prime}=\overline{a}$ and $g(e)=g_{0}(e^{\prime})$ . Since $\operatorname{tp}^{\mathcal{M}}(e/\overline{a})\vDash\operatorname{tp}^{\mathcal{M}}(e/\mathrm{tcl}(\overline{a}))$ and $\widetilde{\operatorname{tp}}(e/\overline{a})\vDash\widetilde{\operatorname{tp}}(e/\mathrm{tcl}(\overline{a}))$ , we can replace $\overline{a}$ with $\mathrm{tcl}(\overline{a})$ . Now there are two possibilities. One possibility is that $e$ and $g_{0}(e^{\prime})$ are both in one of the components that already meets $\overline{a}$ or not. In each case, we just need to apply Lemma 3.2.3 (similar to proof of Corollary 3.2.6).

∎

4. Lifting MS-measurability for coordinatization

In this section, we state and prove our main theorem which says how MS-measurability can be lifted from components to the nil-interaction coordinatized structure. Also, in this section we provide a criterion for MS-measurability of $\aleph_{0}$ -categorical structures. This criterion helps us to verify that the function $h$ , constructed in Subsection 4.3, is MS-measurable.

4.1. Definitions and the statement of the main theorem

First we recall the following definition.

Definition 4.1.1.

Let $\mathcal{M}$ be any structure. For each positive $n<\omega$ , $\mathsf{Def}^{n}(\mathcal{M})$ denotes the set of all definable sets $X\subseteq M^{n}$ , where definable means “definable with parameters”. We note that in $\mathsf{Def}^{n}(\mathcal{M})$ , the formulas involved do not matter – only the sets themselves are considered. We also define $\mathsf{Def}(\mathcal{M})=\bigcup_{n}\mathsf{Def}^{n}(\mathcal{M})$ .

Here we recall the definition of MS-measurable structures from [9].

Definition 4.1.2.

An infinite $\mathcal{L}$ -structure $\mathcal{M}$ is called MS-measurable if there is a function

h=(\dim,\mathrm{meas}):\mathsf{Def}(\mathcal{M})\longrightarrow(\mathbb{N}{\times}\mathbb{R}^{>0})\cup\{(0,0)\}

such that the following conditions hold.

MS1.

For each $\mathcal{L}$ -formula $\varphi(\overline{x},\overline{y})$ , there is a finite set $D\subset\mathbb{N}\times\mathbb{R}^{>0}\cup\{(0,0)\}$ , such that for all $\overline{a}\in M^{m}$ , we have $h(\varphi(\mathcal{M},\overline{a}))\in D$ .
MS2.

If $\varphi(\mathcal{M},\overline{a})$ is finite, then $h(\varphi(\mathcal{M},\overline{a}))=(0,|\varphi(\mathcal{M},\overline{a})|)$ .
MS3.

For every $\mathcal{L}$ -formula $\varphi(\overline{x},\overline{y})$ and all $(d,\mu)\in D$ , the set $\{\overline{a}\in M^{m}:h(\varphi(\mathcal{M},\overline{a}))=(d,\mu)\}$ is $\varnothing$ -definable.

MS4.

(Fubini) Suppose $f:X\longrightarrow Y$ is a definable surjective function, where $X,Y\in\textrm{Def}(\mathcal{M})$ . Let $D=\left\{h\left(f^{-1}[\overline{b}]\right):\overline{b}\in Y\right\}$ which is finite by MS1. For each $(d,\mu)\in D$ , let $Y(d,\mu)=\left\{\overline{b}\in Y:h\left(f^{-1}[\overline{b}]\right)=(d,\mu)\right\}$ , which is definable by MS3. Then, $\left\{Y(d,\mu):(d,\mu)\in D\right\}$ is a partition of $Y$ , and

	$\displaystyle\dim(X)$	$\displaystyle=\max\big{\{}d+\dim\left(Y(d,\mu)\right):(d,\mu)\in D\big{\}},$
	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\sum\left\{\mu\cdot\mathrm{meas}\left(Y(d,\mu)\right):\begin{array}[]{l}(d,\mu)\in D,\\ d+\dim\left(Y(d,\mu)\right)=\dim(X)\end{array}\right\}.$

Here we express the statement of the Main Theorem.

Main Theorem (4.1).

Let $\Gamma$ be a tree plan and let $\mathcal{M}$ be a NIC structure. Suppose that for each $\sigma\in I(\Gamma)$ , $T_{\sigma}$ is MS-measurable via $(\dim_{\sigma},\mathrm{meas}_{\sigma})$ such that $\dim_{\sigma}(x=x)=1$ . Then $\mathcal{M}$ is MS-measurable via $(\dim,\mathrm{meas})$ such that for all $e\in\Gamma(S)$ with $\sigma=\pi(e)\in I(\Gamma)$ and all tree-closed sets $B\subset_{\textrm{fin}}\Gamma(S)$ ,

\dim(e/B)=\big{|}\big{\{}x:\pi(x)\in I(\Gamma)\textnormal{ and }[e\wedge B]<x\leq e\big{\}}\big{|}

and

\mathrm{meas}\left(e/B\cup\{\mathtt{pred}(e)\}\right)=\mathrm{meas}_{\sigma}\left(e/B\cap M(e)\right).

The proof of the Main Theorem will appear in Subsection 4.3.

We note that the assumption that $\dim_{\sigma}(x=x)=1$ , for each $\sigma\in I(\Gamma)$ , is just a convenience for the exposition. One can eliminate it at the expense of more arduous bookkeeping.

4.2. MS-measurability for $\aleph_{0}$ -categorical theories

In this subsection, we provide a criterion for MS-measurability among $\aleph_{0}$ -categorical theories. As mentioned before, this criterion is used in Subsection 4.3.

Theorem 4.2.1.

Let $\mathcal{M}$ be a countably infinite $\aleph_{0}$ -categorical structure. The following are equivalent.

(1)

$\mathcal{M}$ is MS-measurable.

(2)

There is a function

h=(\dim,\mathrm{meas}):\mathsf{Def}(\mathcal{M})\longrightarrow(\mathbb{N}{\times}\mathbb{R}^{>0})\cup\{(0,0)\}

satisfying the following conditions:

cMS1.

For all $C\subset_{\textrm{fin}}M$ and $\overline{a}\in M^{n}$ , for all $g\in Aut(\mathcal{M})$ , $h(g\overline{a}/gC)=h(\overline{a}/C)$ .

cMS2.

For any $C\subset_{\textrm{fin}}M$ and any $C$ -definable set $X\subseteq M^{n}$ , if

X=\operatorname{tp}(\overline{a}_{0}/C)\uplus\cdots\uplus\operatorname{tp}(\overline{a}_{k-1}/C),

then

	$\displaystyle\dim(X)$	$\displaystyle=\max_{i<k}\dim(\overline{a}_{i}/C),$
	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\sum\big{\{}\mathrm{meas}(\overline{a}_{i}/C):\dim(\overline{a}_{i}/C)=\dim(X)\big{\}}.$

cMS3.

For all $C\subset_{\textrm{fin}}M$ and $\overline{a}\in\operatorname{acl}(C)^{n}$ , $\dim(\overline{a}/C)=0$ and

$\mathrm{meas}(\overline{a}/C)=\big{|}\big{\{}g\overline{a}:g\in Aut(\mathcal{M}/C)\big{\}}\big{|}.$

cMS4.

For any $C\subset_{\textrm{fin}}M$ , $\overline{a}\in M^{n}$ , and $\overline{b}\in M^{m}$ , if $\overline{b}\subseteq\operatorname{dcl}(C\overline{a})$ and $X=\operatorname{tp}(\overline{a}/C)$ , then

	$\displaystyle\dim(X)$	$\displaystyle=\dim(\overline{b}/C)+\dim(\overline{a}/C\overline{b})$
	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(\overline{a}/C\overline{b}).$

Proof.

The proof of $2\Rightarrow 1$ consists of Lemma 4.2.3 through Lemma 4.2.7. The proof of $1\Rightarrow 2$ consists of Lemma 4.2.8 through Lemma 4.2.11. ∎

Let $\mathcal{M}$ be a countably infinite $\aleph_{0}$ -categorical structure. Set $T=Th(\mathcal{M})$ . Let $h$ be as in item 2 of Theorem 4.2.1. Lemmas 4.2.3 through 4.2.7 all together show that $\mathcal{M}$ is MS-measurable via $h$ .

The following easy Lemma shows that $h$ is also automorphism-invariant for definable sets and it is used in Lemma 4.2.3 and Lemma 4.2.5.

Lemma 4.2.2.

Suppose $C\subset_{\textrm{fin}}\Gamma(S)$ and $\overline{a}\equiv_{C}\overline{a}^{\prime}$ . Suppose cMS1 and cMS2 hold. Then $h(\varphi(\overline{x},\overline{a}))=h(\varphi(x,\overline{a}^{\prime}))$ .

Proof.

First we decompose $\varphi(\mathcal{M},\overline{a})$ into a disjoint union of complete types, then we apply cMS2 and cMS1. ∎

Lemma 4.2.3.

MS1 holds.

Proof.

Let $\varphi(\overline{x},\overline{y})\in\mathcal{L}$ , where $\overline{y}=(y_{0},...,y_{m-1})$ . For each $p(\overline{y})\in S_{m}(T)$ , let $\overline{a}_{p}$ be a realization of $p$ . Then, by cMS1, Lemma 4.2.2 and the Ryll-Nardzewski Theorem,

\left\{h(\varphi(\mathcal{M},\overline{a})):\overline{a}\in M^{m}\right\}=\left\{h(\varphi(\mathcal{M},\overline{a}_{p})):p\in S_{m}(T)\right\}

which is finite. ∎

Lemma 4.2.4.

MS2 holds.

Proof.

Let $\varphi(\overline{x},\overline{c})\in\mathcal{L}(M)$ where $\overline{x}=(x_{0},...,x_{n-1})$ , and suppose $\varphi(\mathcal{M},\overline{c})$ is finite. Let $p_{0},...,p_{k-1}$ be an enumeration of $\left\{p\in S_{n}(\overline{c}):p(\mathcal{M})\cap X\neq\varnothing\right\}$ , so that $X$ is the disjoint union of the solution sets of $p_{0},...,p_{k-1}$ . By cMS3, $\dim(p_{i})=0$ for each $i$ , and then by cMS2, $\dim(X)=\max_{i}\dim(p_{i})=0$ . Also, by cMS3, we have

\mathrm{meas}(X)=\sum_{i<k}\mathrm{meas}(p_{i})=\sum_{i<k}|X\cap p_{i}(\mathcal{M})|=|X|

as required. ∎

Lemma 4.2.5.

MS3 holds.

Proof.

Let $\varphi(\overline{x},\overline{y})\in\mathcal{L}$ , where $\overline{x}=(x_{0},...,x_{n-1})$ and $\overline{y}=(y_{0},...,y_{m-1})$ , and let $(d,\mu)\in\mathbb{N}\times\mathbb{R}_{>0}$ . For each $p(\overline{y})\in S_{m}(T)$ , let $\overline{a}_{p}$ be a realization of $p$ , and let $\theta_{p}(\overline{y})\in p$ be an isolating formula. Let

V=\left\{p\in S_{m}(T):h(\varphi(\mathcal{M},\overline{a}_{p}))=(d,\mu)\right\}

and $\theta(\overline{y})=\bigvee_{p\in V}\theta_{p}(\overline{y})$ . Then,

\theta(\mathcal{M})=\left\{\overline{a}\in M^{m}:h(\varphi(\mathcal{M},\overline{a}))=(d,\mu)\right\}.

by Lemma 4.2.2. ∎

Lemma 4.2.6 (MS4 Special Case).

Let $C\subset_{\textrm{fin}}M$ , and let $f:X\to Y$ be a surjective $C$ -definable function, where $Y$ is the solution set of $\operatorname{tp}(\overline{b}/C)$ for some $\overline{b}\in M^{m}$ . Then, MS4 holds for this $f$ – i.e.

	$\displaystyle\dim(X)$	$\displaystyle=\dim(Y)+\dim(f^{-1}[\overline{b}])$
	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\mathrm{meas}(Y)\cdot\mathrm{meas}(f^{-1}[\overline{b}]).$

Proof.

We may assume that $f^{-1}[\overline{b}]=\operatorname{tp}(\overline{a}_{0}/C\overline{b})\uplus\cdots\uplus\operatorname{tp}(\overline{a}_{k-1}/C\overline{b})$ . For each $i<k$ , we define

X_{i}=\left\{\overline{a}\in X:\operatorname{tp}(\overline{a},f(\overline{a})/C)=\operatorname{tp}(\overline{a}_{i},\overline{b}/C)\right\}

which is just the solution set of $\operatorname{tp}(\overline{a}_{i}/C)$ , and we observe that $X=X_{0}\uplus\cdots\uplus X_{k-1}$ . By cMS2, we have $\dim(X)=\max_{i<k}\dim(\overline{a}_{i}/C)$ ; let

I=\left\{i<k:\dim(X_{i})=\dim(X)\right\}=\left\{i<k:\dim(\overline{a}_{i}/C\overline{b})+\dim(\overline{b}/C)=\dim(X)\right\}.

We observe that

	$\displaystyle\dim(f^{-1}[\overline{b}])$	$\displaystyle=\max_{i<k}\dim(\overline{a}_{i}/C\overline{b})=\dim(\overline{a}_{i}/C\overline{b})\emph{\,\,\,\,\,(some/any $i\in I$)}$
	$\displaystyle\mathrm{meas}(f^{-1}[\overline{b}])$	$\displaystyle=\sum_{i\in I}\mathrm{meas}(\overline{a}_{i}/C\overline{b}).$

Now, by cMS2 again, we have $\mathrm{meas}(X)=\sum_{i\in I}\mathrm{meas}(X_{i})$ .

	$\displaystyle\dim(X)$	$\displaystyle=\max_{i<k}\dim(\overline{a}_{i}/C)$
		$\displaystyle{=}\max_{i<k}\left(\dim(\overline{b}/C)+\dim(\overline{a}_{i}/C\overline{b})\right)$
		$\displaystyle=\dim(\overline{b}/C)+\max_{i<k}\dim(\overline{a}_{i}/C\overline{b})$
		$\displaystyle=\dim(Y)+\dim(f^{-1}[\overline{b}]).$

and

	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\sum_{i\in I}\mathrm{meas}(\overline{a}_{i}/C)$
		$\displaystyle{=}\sum_{i\in I}\left(\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(\overline{a}_{i}/C\overline{b})\right)$
		$\displaystyle=\mathrm{meas}(\overline{b}/C)\cdot\sum_{i\in I}\mathrm{meas}(\overline{a}_{i}/C\overline{b})$
		$\displaystyle=\mathrm{meas}(Y)\cdot\mathrm{meas}(f^{-1}[\overline{b}]).$

which completes the proof of the lemma. ∎

Lemma 4.2.7.

MS4 holds.

Proof.

Let $C\subset_{\textrm{fin}}M$ , and let $f:X\to Y$ be a surjective $C$ -definable function. We may assume that $Y=\operatorname{tp}(\overline{b}_{0}/C)\uplus\cdots\uplus\operatorname{tp}(\overline{b}_{k-1}/C)$ . For each $i<k$ , let $X_{i}=\left\{\overline{a}\in X:f(\overline{a})\equiv_{C}\overline{b}_{i}\right\}$ . By cMS2, we have $\dim(X)=\max_{i<k}\dim(X_{i})$ and $\mathrm{meas}(X)=\sum_{i\in I}\mathrm{meas}(X_{i})$ , where $I=\left\{i:\dim(X_{i})=\dim(X)\right\}$ . By Lemma 4.2.6, we find that

	$\displaystyle\dim(X)$	$\displaystyle=\max_{i<k}(\overline{a}_{i}/C)=\dim(\overline{b}_{i}/C)+\dim(f^{-1}[\overline{b}_{i}])\emph{\,\,\,\,\,(some/any $i\in I$)}$
	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\sum_{i\in I}\mathrm{meas}(\overline{b}_{i}/C)\cdot\mathrm{meas}(f^{-1}[\overline{b}_{i}]).$

as desired. ∎

Let $\mathcal{M}$ be a countably infinite $\aleph_{0}$ -categorical MS-measurable structure via $h=(\dim,\mathrm{meas})$ . Lemmas 4.2.8 through 4.2.11 all together show that $h$ satisfies cMS1, …, cMS4.

Lemma 4.2.8.

cMS1 holds.

Proof.

Let $C\subset_{\textrm{fin}}M$ and $\overline{a}\in M^{n}$ , and let $g\in\mathrm{Aut}(\mathcal{M})$ . Let $\varphi(\overline{x},\overline{c})$ be a formula that isolates $\operatorname{tp}(\overline{a}/C)$ , and let $\overline{c}$ enumerate $C$ . Let $h(\overline{a}/C)=h(\varphi(\mathcal{M},\overline{c}))=(d,\mu)$ . By MS3, there is an $\mathcal{L}$ -formula (i.e. without parameters) $\theta(\overline{x})$ which defines $\left\{\overline{b}\in M^{m}:h(\varphi(\mathcal{M},\overline{b})=(d,\mu))\right\}.$ Since $\overline{c}\in\theta(\mathcal{M})$ and $g\in\mathrm{Aut}(\mathcal{M})$ , $g\overline{c}\in\theta(\mathcal{M})$ . So $h(\varphi(\mathcal{M},g\overline{c}))=h(g\overline{a}/gC)=(d,\mu)$ .

∎

Lemma 4.2.9.

cMS2 holds.

Proof.

Let $C\subset_{\textrm{fin}}M$ and $X\subseteq M^{n}$ a $C$ -definable set. Let $X=\operatorname{tp}(\overline{a}_{0}/C)\uplus\cdots\uplus\operatorname{tp}(\overline{a}_{k-1}/C)$ . We proceed by induction on $k$ . For $k=2$ , we define $f:\operatorname{tp}(a_{0}/C)\uplus\operatorname{tp}(a_{1}/C)\longrightarrow\{\overline{a}_{0},\overline{a}_{1}\}$ by $f(\overline{x}):=\overline{a}_{0}$ if $\overline{x}\in\operatorname{tp}(\overline{a}_{0}/C)$ ; $f(\overline{x}):=\overline{a}_{1}$ if $\overline{x}\in\operatorname{tp}(\overline{a}_{1}/C)$ . Note that $f$ is definable with parameters from $C\overline{a}_{0}\overline{a}_{1}$ . We observe that $f^{-1}[\overline{a}_{i}]=\operatorname{tp}(\overline{a}_{i}/C)$ . Case 1: if $h\big{(}f^{-1}(\overline{a}_{0})\big{)}=h\big{(}f^{-1}(a_{1})\big{)}$ , then the result follows from MS. Case 2: if $h\big{(}f^{-1}(\overline{a}_{0})\big{)}\neq h\big{(}f^{-1}(a_{1})\big{)}$ , then $Y_{i}:=\big{\{}\overline{a}\in\{\overline{a}_{0},\overline{a}_{1}\}:h(f^{-1}[\overline{a}])=\big{(}\dim(\overline{a}_{i}/C),\mathrm{meas}(\overline{a}_{i}/C)\big{)}\big{\}}=\{\overline{a}_{i}\}$ for $i<2$ . By MS4,

	$\displaystyle\dim(X)$	$\displaystyle=\max\left\{\dim(\overline{a}_{i}/C)+\dim\big{(}Y_{i}\big{)}:i<2\right\}$
		$\displaystyle=\max\left\{\dim(\overline{a}_{i}/C)+\dim\left(\{\overline{a}_{i}\}\right):i<2\right\}$
		$\displaystyle=\max_{i<2}\dim(\overline{a}_{i}/C)$

and

	$\displaystyle\mathrm{meas}(X)$	$\displaystyle=\sum\left\{\mathrm{meas}(\overline{a}_{i}/C)\cdot\mathrm{meas}\big{(}Y_{i}\big{)}:\begin{array}[]{l}i<2,\,\,\text{and}\\ \dim(\overline{a}_{i}/C)+\dim\big{(}Y_{i}\big{)}=\dim(X)\end{array}\right\}$
		$\displaystyle=\sum\left\{\mathrm{meas}(\overline{a}_{i}/C):\begin{array}[]{l}i<2,\,\,\text{and}\\ \dim(\overline{a}_{i}/C)=\dim(X)\end{array}\right\}.$

Now, suppose both statements hold for $k=\ell$ . Let $X=\operatorname{tp}(\overline{a}_{0}/C)\uplus\dots\uplus\operatorname{tp}(\overline{a}_{\ell}/C)$ . We first apply the induction hypothesis to $\operatorname{tp}(\overline{a}_{0}/C),\dots,\operatorname{tp}(\overline{a}_{\ell-1}/C)$ , and conclude the statements by applying the case $n=2$ to $\operatorname{tp}(\overline{a}_{0}/C)\uplus\dots\uplus\operatorname{tp}(\overline{a}_{\ell-1}/C)$ and $\operatorname{tp}(\overline{a}_{\ell}/C)$ . ∎

Lemma 4.2.10.

cMS3 holds.

Proof.

Let $C\subset_{\textrm{fin}}M$ and $\overline{a}\in\operatorname{acl}(C)^{n}$ . Let $p(\overline{x})=\operatorname{tp}(\overline{a}/C)$ . Since $\overline{a}$ is algebraic over $C$ , there are finitely many, say $\overline{a}_{0},\dots,\overline{a}_{k-1}$ , realizations for $p(\overline{x})$ . By MS2, $h(\overline{a}/C)=(0,|p(\mathcal{M})|)$ . Clearly $\{g\overline{a}:g\in\mathrm{Aut}(\mathcal{M}/C)\}\subseteq p(\mathcal{M})$ . On the other hand by $\aleph_{0}$ -categoricity, for each $i<k$ , there is $g_{i}\in\mathrm{Aut}(\mathcal{M}/C)$ such that $g_{i}\overline{a}=\overline{a}_{i}$ . Hence $h(\overline{a}/C)=(0,|\{g\overline{a}:g\in\mathrm{Aut}(\mathcal{M}/C)\}|)$ . ∎

Lemma 4.2.11.

cMS4 holds.

Proof.

Let $C\subset_{\textrm{fin}}M,\overline{a}\in M^{n}$ and $\overline{b}\subseteq\operatorname{dcl}(C\overline{a})$ . Let $X=\operatorname{tp}(\overline{a}/C)$ and $Y=\operatorname{tp}(\overline{b}/C)$ . Let $\varphi(\overline{x},\overline{c})$ and $\theta(\overline{y},\overline{c})$ be the formulas that isolate $\operatorname{tp}(\overline{a}/C)$ and $\operatorname{tp}(\overline{b}/C)$ respectively where $\overline{c}$ enumerates $C$ . Let $\psi(\overline{y},\overline{c},\overline{a})$ witness that $\overline{b}\subseteq\operatorname{dcl}(C\overline{a})$ . We define $f:X\longrightarrow Y$ with $\overline{e}\mapsto f(\overline{e})$ such that

f\overline{e}\vDash\varphi(\overline{e},\overline{c})\,\wedge\,\psi(\overline{y},\overline{c},\overline{e}).

We observe that $f$ is a $C$ -definable surjective function onto $Y$ , for each $\overline{b}^{\prime}\in Y$ we have $f^{-1}[\overline{b}^{\prime}]=\{\overline{a}^{\prime}:\overline{a}^{\prime}\overline{b}^{\prime}=_{C}\overline{a}\overline{b}\}$ and so by MS1, $h(f^{-1}[\overline{b}])=h(\overline{a}/C\overline{b})$ . By MS4,

\dim(X)=\dim(Y)+\dim(f^{-1}[\overline{b}])=\dim(\overline{b}/C)+\dim(\overline{a}/C\overline{b}),

and

\mathrm{meas}(X)=\mathrm{meas}(Y)\cdot\mathrm{meas}(f^{-1}[\overline{b}])=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(\overline{a}/C\overline{b}).

as desired. ∎

4.3. Proof of the Main Theorem Main Theorem

The proof of the Main Theorem consists of three parts, including construction of $h$ (Subsection 4.3.1), well-defindness (Subsection 4.3.2) and verification (Subsection 4.3.3).

4.3.1. Construction of $h=(\dim,\mathrm{meas})$

We define the function $h:\textrm{Def}(\mathcal{M})\longrightarrow(\mathbb{N}\times\mathbb{R}^{>0})\cup\{(0,0)\},$ with $h(X)=\left(\dim(X),\mathrm{meas}(X)\right)$ as follows.

First, we consider complete 1-types over closed sets. Consider $\operatorname{tp}^{\mathcal{M}}(a/B)$ where $\mathrm{tcl}(B)=B$ .

•

If $a\in B$ , then $\dim(a/B)=0$ and $\mathrm{meas}(a/B)=1$ .

•

If $a\notin B$ , then there is a $k>0$ such that $a=e_{k}>e_{k-1}>\dots>e_{1}>e_{0}=[a\wedge B]$ where $\mathtt{pred}(e_{i})=e_{i-1}$ for each $i=1,\dots,k$ . Let $\sigma_{i}=\pi(e_{i})$ for each $i$ . Note that if $\sigma_{i}\notin I(\Gamma)$ , then we define $(\dim_{\sigma_{i}},\mathrm{meas}_{\sigma_{i}})\big{(}e_{i}/\mathrm{tcl}(Be_{0},\dots,e_{i-1})\cap M(e_{i})\big{)}=(0,1)$ simply because $M(e_{i})=\{e_{i}\}$ .

Then we define

	$\displaystyle\dim(a/B)$	$\displaystyle=\sum_{i=1}^{k}\dim_{\sigma_{i}}\big{(}e_{i}/\mathrm{tcl}(Be_{0},\dots,e_{i-1})\cap M(e_{i})\big{)}$
		$\displaystyle=\big{\|}\big{\{}i\in\{1,\dots,k\}\,\,:\,\,\pi(e_{i})\in I(\Gamma)\big{\}}\big{\|}$

\mathrm{meas}(a/B)=\prod_{i=1}^{k}\mathrm{meas}_{\sigma_{i}}\big{(}e_{i}/\mathrm{tcl}(Be_{0},\dots,e_{i-1})\cap M(e_{i})\big{)}.

II.

Suppose we have defined $(\dim,\mathrm{meas})$ for complete ${\leq}n$ -types over closed sets. Let $B\subset_{\textrm{fin}}\Gamma(S)$ be a closed subset, $\overline{a}\in\Gamma(S)^{n}$ , and $e\in\Gamma(S)$ . We want to define $\dim(e\overline{a}/B)$ and $\mathrm{meas}(e\overline{a}/B)$ . We take the Lascar (in)equality as a pattern to follow, setting

\dim(e\overline{a}/B)=\dim\big{(}\overline{a}/B\big{)}+\dim\big{(}e/\mathrm{tcl}(B\overline{a})\big{)}

and

\mathrm{meas}(e\overline{a}/B)=\mathrm{meas}\big{(}\overline{a}/B\big{)}\cdot\mathrm{meas}\big{(}e/\mathrm{tcl}(B\overline{a})\big{)}.

III.

So far we have defined $(\dim,\mathrm{meas})$ for complete types over closed sets. Now, let $X$ be an arbitrary definable set over $C$ . Let $p_{0}(\overline{x}),\dots,p_{k-1}(\overline{x})\in S_{n}\left(\mathrm{tcl}(C)\right)$ such that

$X=p_{0}(\mathcal{M})\uplus\dots\uplus p_{k-1}(\mathcal{M}).$

Then we define

$\dim(X):=\max\big{\{}\dim(p_{i})\,:\,i<k\big{\}}.$

We set $I:=\big{\{}i\,:\,\dim(p_{i})=\dim(X)\big{\}}$ , and then we define

$\mathrm{meas}(X):=\sum_{i\in I}\mathrm{meas}(p_{i}).$

4.3.2. Well-definedness of $h=(\dim,\mathrm{meas})$

For a given definable set $X\subseteq M^{n}$ , there are many different finite sets $C\subset_{\textrm{fin}}M$ over which $X$ is definable. Depending on which set of parameters we choose, it is possible that different values for $h(X)$ may arise. So, in this subsection we show that the function $h=(\dim,\mathrm{meas})$ , as defined above, is independent of the choice of which set $C$ provides the parameters for defining $X$ .

First we prove a general fact about $\aleph_{0}$ -categorical MS-measurable structures which is needed in Lemma 4.3.2.

Fact 4.3.1.

Suppose $\mathcal{N}$ is MS-measurable via $(\dim,\mathrm{meas})$ and $\aleph_{0}$ -categorical. Let $B\subseteq C\subset_{\textrm{fin}}N$ and $a\in N\setminus B$ . Then there is some $a^{\prime}\in N$ such that $\operatorname{tp}(a^{\prime}/B)=\operatorname{tp}(a/B)$ and $\dim(a^{\prime}/C)=\dim(a/B)$ . Moreover,

\mathrm{meas}(a/B)=\sum\bigg{\{}\mathrm{meas}(a^{\prime}/C):a^{\prime}\equiv_{B}a\text{ and }\dim(a^{\prime}/C)=\dim(a/B)\bigg{\}}.

Proof.

Let $B\subseteq C\subset_{\textrm{fin}}N$ , $a\in N\setminus B$ , and let $q(x)=\operatorname{tp}(a/B)$ . By $\aleph_{0}$ -categoricity, there are finitely many complete types over $C$ , say $p_{0}(x)=\operatorname{tp}(a_{0}/C),\dots,p_{n-1}(x)=\operatorname{tp}(a_{n-1}/C)$ , that extend $q$ . So $q(x)=p_{0}(x)\uplus\dots\uplus p_{n-1}(x)$ . By cMS2, $\dim(a/B)=\max_{i<n}\dim(a_{i}/C)$ . Suppose that this maximum is attained by $i<n$ . Set $a^{\prime}=a_{i}$ . So $A=\left\{\operatorname{tp}(a^{\prime}):a^{\prime}\in N,\,a^{\prime}\equiv_{B}a\,\text{ and }\dim(a^{\prime}/C)=\dim(a/B)\right\}$ is non-empty. By cMS2, $\mathrm{meas}(a/B)=\sum_{a^{\prime}\in A}\mathrm{meas}(a^{\prime}/C)$ . ∎

Lemma 4.3.2.

Let $B\subseteq C\subset_{\textrm{fin}}\Gamma(S)=M$ be closed sets, and let $a\in\Gamma(S)\setminus B$ . Let $q=\operatorname{tp}^{\mathcal{M}}(a/B)$ , and let $p_{0},\dots,p_{m-1}$ be the complete extensions of $q$ over $C$ . Let $I=\left\{i<m:\dim(p_{i})=\dim(q)\right\}$ . Then

(1)

There is an element $a^{*}\vDash q$ such that $\dim(a^{*}/C)=\dim(a/B)$ — i.e. $I$ is non-empty.
(2)

$\mathrm{meas}(q)=\sum_{i\in I}\mathrm{meas}(p_{i})$ .

Proof.

For item 1, we may assume that $[a\wedge B]=e_{0}<e_{1}<\dots<e_{k}=a$ where $k>0$ and $\mathtt{pred}(e_{\ell})=e_{\ell-1}$ for each $\ell=1,\dots,k$ . Since $B$ is closed and $a\notin B$ , we know that $e_{1}$ is an infinity-node, say $\pi(e_{1})=\sigma\in I(\Gamma)$ . Let $B_{1}=M(e_{1})\cap B$ and $C_{1}=M(e_{1})\cap C$ . Since $\dim_{\sigma}(x=x)=1$ and $\operatorname{acl}$ is trivial in $\mathcal{M}(e_{1})$ and $e_{1}\notin B_{1}$ , we know that $\dim_{\sigma}(e_{1}/B_{1})=1$ . By Fact 4.3.1, there is an element $e_{1}^{*}\in M(e_{1})$ such that $\operatorname{tp}^{\mathcal{M}(e_{1})}(e_{1}/B_{1})=\operatorname{tp}^{\mathcal{M}(e_{1})}(e_{1}^{*}/B_{1})$ and $\dim_{\sigma}(e_{1}^{*}/C_{1})=\dim_{\sigma}(e_{1}/B_{1})=1$ . Subsequently, by $\aleph_{0}$ -categoricity of $T_{\sigma}$ there is an automorphism $g\in\mathrm{Aut}\big{(}\mathcal{M}(e_{1})/B_{1}\big{)}$ such that $g(e_{1})=e_{1}^{*}$ . By N3, there is $h\in\mathrm{Aut}(\mathcal{M}/B)$ such that $h{\restriction}_{M(e_{1})}=g$ . We choose $a^{*}=h(a)$ . Now we verify that $\dim(a^{*}/C)=\dim(a/B)$ . Suppose $e_{i}^{*}=h(e_{i})$ for each $i=1,\dots k$ . Then

[a\wedge B]=[a^{*}\wedge C]=e_{0}<e^{*}_{1}<\dots<e^{*}_{k}=a^{*}.

By definitoin of $\dim$ we can write

	$\displaystyle\dim(a/B)$	$\displaystyle=\sum_{i=1}^{k}\dim_{\sigma_{i}}\big{(}e_{i}/\mathrm{tcl}(Be_{0},\dots,e_{i-1})\cap M(e_{i})\big{)}$
		$\displaystyle=\big{\|}\left\{i\in\{1,\dots k\}:\pi(e_{i})\in I(\Gamma)\right\}\big{\|}$
		$\displaystyle=\big{\|}\left\{i\in\{1,\dots k\}:\pi(e^{*}_{i})\in I(\Gamma)\right\}\big{\|}\,\,\,\,\text{ (since $h$ is an automorphism)}$
		$\displaystyle=\sum_{i=1}^{k}\dim_{\sigma_{i}}\big{(}e^{}_{i}/\mathrm{tcl}(Ce_{0},e_{1}^{},\dots,e^{}_{i-1})\cap M(e^{}_{i})\big{)}$
		$\displaystyle=\dim(a^{*}/C).$

For item 2, let $a_{i}$ be a realization of $p_{i}$ for each $i<m$ , and let $h_{i}\in\mathrm{Aut}(\mathcal{M}/B)$ be an automorphism such that $h_{i}(a)=a_{i}$ . For each $i<m$ , let $e^{*}_{1i}=h_{i}(e_{1})$ .

Claim.

For each $i<m$ , $\dim(a_{i}/C)=\dim(a/B)$ if and only if $\dim_{\sigma}(e_{1i}^{*}/C_{1})=\dim_{\sigma}(e_{1}/B_{1})$ .

Proof.

Suppose $\dim_{\sigma}(e_{1i}^{*}/C_{1})=\dim_{\sigma}(e_{1}/B_{1})$ . Then

	$\displaystyle\dim(a_{i}/C)$	$\displaystyle=\sum_{j=1}^{k}\dim_{\sigma_{j}}(h_{i}(e_{j})/\mathrm{tcl}(Ce_{0}e^{*}_{1,j},\dots,h_{i}(e_{j-1})\cap M(h_{i}(e_{j})))$
		$\displaystyle=\sum_{j=1}^{k}\dim_{\sigma}(e_{j}/\mathrm{tcl}(Be_{0},\dots,e_{j-1})\cap M(e_{j}))$
		$\displaystyle=\dim(a/B).$

Coversely, suppose $\dim(a_{i}/C)=\dim(a/B)$ for some $i<m$ . Since the correspomdimg summands of the above equation should be the same, $\dim_{\sigma}(e_{1i}^{*}/C_{1})=\dim_{\sigma}(e_{1}/B_{1})$ . ∎

From this, it follows that $\mathrm{meas}_{\sigma}(e_{1}/B_{1})=\sum_{i\in I}\mathrm{meas}_{\sigma}(e^{*}_{1i}/C_{1})$ . For each $i<m$ and for each $j<k$ , let $h_{i}(e_{j})=e_{j,i}^{*}$ . By definition of $\mathrm{meas}$ we have

	$\displaystyle\mathrm{meas}(a/B)$	$\displaystyle=\prod_{i\in I}\mathrm{meas}_{\sigma_{i}}(e_{i}/\mathrm{tcl}(Be_{0},\dots,e_{i-1})\cap M(e_{i}))$
		$\displaystyle=\bigg{(}\sum_{i\in I}\mathrm{meas}_{\sigma_{1}}\big{(}e^{*}_{1,i}/C_{1}\big{)}\bigg{)}\prod_{j=2}^{k}\mathrm{meas}_{\sigma_{j}}\big{(}e_{j}/\mathrm{tcl}(Be_{0},\dots,e_{j-1})\cap M(e_{j})\big{)}.$

Now, we distribute over the summation, and we note that for $i\in I$ and $j\geq 2$

	$\displaystyle\mathrm{meas}_{\sigma_{j}}\big{(}e_{j}/\mathrm{tcl}(Be_{0},\dots$	$\displaystyle,e_{j-1})\cap M(e_{j})\big{)}$
		$\displaystyle=\mathrm{meas}_{\sigma_{j}}\big{(}e^{}_{j,i}/\mathrm{tcl}(Ce^{}_{0,i},\dots,e^{}_{(j-1),i})\cap M(e^{}_{j,i})\big{)}.$

Since neither $B$ nor $C$ meet $M(e_{j})$ and $M(e^{*}_{j,i})$ ,

\displaystyle\sum_{i\in I}\prod_{j=1}^{k}\mathrm{meas}_{\sigma_{j}}\bigg{(}e^{*}_{j,i}/\mathrm{tcl}(Ce^{*}_{0,i},\dots,e^{*}_{j-1,i})\cap M(e^{*}_{j,i})\bigg{)}=\sum_{i\in I}\mathrm{meas}(a_{i}/C).

∎

In order to extend Lemma 4.3.2 to $n$ -types, for arbitrary $n$ , we need to know that permuting the indices does not matter. In the following proposition we show that $h$ is permutation/coordinate invariant. This is used in the proof of Lemma 4.3.9 as well.

Proposition 4.3.3 (Permutation/coordinate invariance).

For all closed subsets $C\subset_{\textrm{fin}}\Gamma(S)$ , for all $\overline{a}=(a_{0},\dots,a_{n-1})\in\Gamma(S)^{n}$ , and for all $\sigma\in\mathit{Sym}(n)$ , we have

(1)

$\dim\big{(}a_{\sigma(0)},\dots,a_{\sigma(n-1)}/C\big{)}=\dim(\overline{a}/C)$ .
(2)

$\mathrm{meas}\big{(}a_{\sigma(0)},\dots,a_{\sigma(n-1)}/C\big{)}=\mathrm{meas}(\overline{a}/C)$ .

Proof.

Since $\mathit{Sym}(n)$ is generated by adjacent transposition, it is enough to show that for any closed set $C\subset_{\textrm{fin}}\Gamma(S)$ , for any $\overline{a},\overline{b}\in\Gamma(S)^{<\omega}$ and $e,e^{\prime}\in\Gamma(S)$

	$\displaystyle\dim(\overline{a}ee^{\prime}\overline{b}/C)$	$\displaystyle=\dim(\overline{a}e^{\prime}e\overline{b}/C)$
	$\displaystyle\mathrm{meas}(\overline{a}ee^{\prime}\overline{b}/C)$	$\displaystyle=\mathrm{meas}(\overline{a}e^{\prime}e\overline{b}/C).$

Since

	$\displaystyle\dim(\overline{a}ee^{\prime}\overline{b}/C)$	$\displaystyle=\dim(\overline{b}/C)+\dim(\overline{a}ee^{\prime}/\mathrm{tcl}(C\overline{b}))$
		$\displaystyle=\dim(\overline{b}/C)+\dim(ee^{\prime}/\mathrm{tcl}(C\overline{b}))+\dim(\overline{a}/\mathrm{tcl}(C\overline{b}ee^{\prime})),$
	$\displaystyle\dim(\overline{a}e^{\prime}e\overline{b}/C)$	$\displaystyle=\dim(\overline{b}/C)+\dim(\overline{a}e^{\prime}e/\mathrm{tcl}(C\overline{b}))$
		$\displaystyle=\dim(\overline{b}/C)+\dim(e^{\prime}e/\mathrm{tcl}(C\overline{b}))+\dim(\overline{a}/\mathrm{tcl}(C\overline{b}e^{\prime}e)),$

and

	$\displaystyle\mathrm{meas}(\overline{a}ee^{\prime}\overline{b}/C)$	$\displaystyle=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(\overline{a}ee^{\prime}/\mathrm{tcl}(C\overline{b}))$
		$\displaystyle=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(ee^{\prime}/\mathrm{tcl}(C\overline{b}))\cdot\mathrm{meas}(\overline{a}/\mathrm{tcl}(C\overline{b}ee^{\prime})),$
	$\displaystyle\mathrm{meas}(\overline{a}e^{\prime}e\overline{b}/C)$	$\displaystyle=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(\overline{a}e^{\prime}e/\mathrm{tcl}(C\overline{b}))$
		$\displaystyle=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(e^{\prime}e/\mathrm{tcl}(C\overline{b}))\cdot\mathrm{meas}(\overline{a}/\mathrm{tcl}(C\overline{b}e^{\prime}e)),$

it is enough to show that for any closed set $C\subset_{\textrm{fin}}\Gamma(S)$ and any $e,e^{\prime}\in\Gamma(S)$ ,

	$\displaystyle\dim(ee^{\prime}/C)=\dim(e^{\prime}e/C),$
	$\displaystyle\mathrm{meas}(ee^{\prime}/C)=\mathrm{meas}(e^{\prime}e/C).$

We consider the following cases: $e$ and $e^{\prime}$ are comparable, $e$ and $e^{\prime}$ are incomparable and meet $C$ at the same point, $e$ and $e^{\prime}$ are incomparable and meet $C$ at different points.

case 1.

Suppose $[e\wedge C]\leq e^{\prime}<e$ . Then we have

	$\displaystyle\dim(ee^{\prime}/C)$	$\displaystyle=\dim(e^{\prime}/C)+\dim(e/\mathrm{tcl}(Ce^{\prime})),$
	$\displaystyle\dim(e^{\prime}e/C)$	$\displaystyle=\dim(e/C)+\dim(e^{\prime}/\mathrm{tcl}(Ce))$
		$\displaystyle=\dim(e/C)\hskip 113.81102pt(\text{since $e^{\prime}\in\mathrm{tcl}(Ce)$})$

So it suffices to show that $\dim(e/C)=\dim(e^{\prime}/C)+\dim\left(e/\mathrm{tcl}(Ce^{\prime})\right).$ Let

[e\wedge C]=e_{0}<e_{1}<\dots<e_{k-1}<e_{k}=e

where $\mathtt{pred}(e_{i})=e_{i-1}$ for $i=1,\dots,k$ , and let for some $\ell<k$ , $e^{\prime}=e_{\ell}$ . Then

	$\displaystyle\dim(e^{\prime}/C)$	$\displaystyle=\sum\limits_{i=1}^{\ell}\dim_{\sigma_{i}}\left(e_{i}/\mathrm{tcl}(Ce_{0},\dots,e_{i-1})\cap M(e_{i})\right),$
	$\displaystyle\dim(e/C)$	$\displaystyle=\sum\limits_{j=1}^{k}\dim_{\sigma_{j}}\left(e_{j}/\mathrm{tcl}(Ce_{0},\dots,e_{j-1})\cap M(e_{j})\right)$

So we have to show that

\dim\left(e/\mathrm{tcl}(Ce^{\prime})\right)=\sum\limits_{i=\ell+1}^{k}\dim_{\sigma_{i}}\left(e_{i}/\mathrm{tcl}(Ce_{0},\dots,e_{i-1})\cap M(e_{i})\right).

For each $\ell+1\leq i\leq k$ , if $e_{i}\in\mathrm{tcl}(Ce_{0},\dots,e_{i-1})$ , then

\dim_{\sigma}\left(e_{i}/\mathrm{tcl}(Ce_{0},\dots,e_{i-1})\cap M(e_{i})\right)=0.

On the other hand, for each $i$ , $e_{i}\notin\mathrm{tcl}\left(Ce_{0},\dots,e_{i-1}\right)$ if and only if $\pi(e_{i})\in I(\Gamma)$ . So

	$\displaystyle\sum\limits_{i=\ell+1}^{k}\dim_{\sigma_{i}}\left(e_{i}/\mathrm{tcl}(Ce_{0},\dots,e_{i-1})\cap M(e_{i})\right)$	$\displaystyle=\big{\|}\left\{x:\pi(x)\in I(\Gamma),e^{\prime}<x\leq e\right\}\big{\|}$
		$\displaystyle=\dim(e/\mathrm{tcl}(Ce^{\prime})).$

as we desire.

case 2.

Suppose $[e\wedge C]=[e^{\prime}\wedge C]\leq[e\wedge e^{\prime}]$ . Let $b=[e\wedge e^{\prime}]$ . Then

	$\displaystyle\dim(ee^{\prime}/C)$	$\displaystyle=\dim(e^{\prime}/C)+\dim(e/\mathrm{tcl}(Ce^{\prime}))$
		$\displaystyle=\dim(e^{\prime}/C)+\dim(e/\mathrm{tcl}(Cb)).$

and

	$\displaystyle\dim(e^{\prime}e/C)$	$\displaystyle=\dim(e/C)+\dim(e^{\prime}/\mathrm{tcl}(Ce))$
		$\displaystyle=\dim(e/C)+\dim(e^{\prime}/\mathrm{tcl}(Cb)).$

So it is enough to show that

(1)

\dim(e^{\prime}/C)+\dim(e/\mathrm{tcl}(Cb))=\dim(e/C)+\dim(e^{\prime}/\mathrm{tcl}(Cb)).

We have

	$\displaystyle\dim(e/C)$	$\displaystyle=\dim(e/\mathrm{tcl}(Cb))+\dim(b/C),$
	$\displaystyle\dim(e^{\prime}/C)$	$\displaystyle=\dim(e^{\prime}/\mathrm{tcl}(Cb))+\dim(b/C).$

and by replacing them in equation (1) we have

	$\displaystyle\dim(e^{\prime}/\mathrm{tcl}(Cb))+\dim(b/C)+\dim(e/\mathrm{tcl}(Cb))$
	$\displaystyle=$
	$\displaystyle\dim(e/\mathrm{tcl}(Cb))+\dim(b/C)+\dim(e^{\prime}/\mathrm{tcl}(Cb)).$

This completes case 2.

case 3.

Suppose $[e\wedge e^{\prime}]<[e\wedge C]$ , $[e\wedge e^{\prime}]<[e^{\prime}\wedge C]$ and $[e\wedge C]\neq[e^{\prime}\wedge C]$ . In this situation we observe that

	$\displaystyle\dim(e/\mathrm{tcl}(Ce^{\prime}))$	$\displaystyle=\dim(e/C),$
	$\displaystyle\dim(e^{\prime}/\mathrm{tcl}(Ce))$	$\displaystyle=\dim(e^{\prime}/C).$

So we calculate

	$\displaystyle\dim(ee^{\prime}/C)$	$\displaystyle=\dim(e^{\prime}/C)+\dim(e/\mathrm{tcl}(Ce^{\prime}))$
		$\displaystyle=\dim(e^{\prime}/C)+\dim(e/C)$
		$\displaystyle=\dim(e/C))+\dim(e^{\prime}/\mathrm{tcl}(Ce))$
		$\displaystyle=\dim(e^{\prime}e/C).$

as we desire.

We can do a similar calculation for $\mathrm{meas}$ as well. ∎

Lemma 4.3.4.

Let $B\subseteq C\subset_{\textrm{fin}}\Gamma(S)$ be closed sets. Then for every $n\in\mathbb{N}^{+}$ , for every $\overline{a}\in(\Gamma(S)\setminus B)^{n}$ , there is a finite list $\overline{a}_{0},\dots,\overline{a}_{m-1}\in\Gamma(S)^{n}$ such that:

(1)

$\operatorname{tp}(\overline{a}_{j}/B)=\operatorname{tp}(\overline{a}/B)$ and $\dim(\overline{a}_{j}/C)=\dim(\overline{a}/B)$ for each $j<m$ .
(2)

For any $\overline{a}^{\prime}\in\Gamma(S)^{n}$ , if $\operatorname{tp}(\overline{a}^{\prime}/B)=\operatorname{tp}(\overline{a}/B)$ and $\dim(\overline{a}^{\prime}/C)=\dim(\overline{a}/B)$ , then $\operatorname{tp}(\overline{a}^{\prime}/C)=\operatorname{tp}(\overline{a}_{j}/C)$ for exactly one $j<m$ .

Moreover, given these conditions, we also find that

\mathrm{meas}(\overline{a}/B)=\sum_{j<m}\mathrm{meas}(\overline{a}_{j}/C).

Proof.

We proceed by induction on $n=1,2,3,\dots$ . The base case, $n=1$ , follows from Lemma 4.3.2. Inductively, we assume that the lemma has been proven for $n$ -tuples and any closed sets $B\subseteq C\subset_{\textrm{fin}}\Gamma(S)$ . Now, let $B\subseteq C\subset_{\textrm{fin}}\Gamma(S)$ be closed sets, and let $\overline{a}\in(\Gamma(S)\setminus B)^{n}$ and $y\in\Gamma(S)$ ; we prove the statement of the lemma for the $(n+1)$ -tuple $\overline{a}y$ . By the induction hypothesis, let $\overline{a}_{0},\dots,\overline{a}_{m-1}\in\Gamma(S)^{n}$ be as in the statement of the lemma for $\overline{a},B,C$ , and for each $i<m$ , let $g_{i}\in\mathrm{Aut}(\mathcal{M}/B)$ be such that $g_{i}\overline{a}=\overline{a}_{i}$ .

Now, for each $i<m$ , we apply Lemma 4.3.2 with $q(x)=\operatorname{tp}(g_{i}(y)/\mathrm{tcl}(B\overline{a}_{i}))$ and with $\mathrm{tcl}(C\overline{a}_{i})$ in place of $C$ to obtain elements $y_{i,0},\dots,y_{i,k_{i}-1}$ such that

•

$y_{i,\ell}\vDash q$ and $\dim\big{(}y_{i,\ell}/\mathrm{tcl}(C\overline{a}_{i})\big{)}=\dim\big{(}g_{i}(y)/\mathrm{tcl}(B\overline{a}_{i})\big{)}$ for all $\ell<k_{i}$ .
•

For any $z\in\Gamma(S)$ , if $z\vDash q$ and $\dim(z/\mathrm{tcl}(C\overline{a}_{i}))=\dim(g_{i}(y)/\mathrm{tcl}(B\overline{a}_{i}))$ , then $\operatorname{tp}(z/\mathrm{tcl}(C\overline{a}_{i}))=\operatorname{tp}(y_{i,\ell}/\mathrm{tcl}(C\overline{a}_{i}))$ for exactly one $\ell<k_{i}$ .

The next claim verifies that $\{\overline{a}_{i}y_{i,\ell}:i<m,\,\,\ell<k_{i}\}$ works for $\overline{a}y$ .

Claim.

(1)

$\operatorname{tp}(\overline{a}_{i}y_{i,\ell}/B)=\operatorname{tp}(\overline{a}y/B)$ and $\dim(\overline{a}_{i}y_{i,\ell}/C)=\dim(\overline{a}y/B)$ for all $i<m$ and $\ell<k_{i}$ .
(2)

For any $\overline{e}\in\Gamma(S)^{n+1}$ , if $\operatorname{tp}(\overline{e}/B)=\operatorname{tp}(\overline{a}y/B)$ and $\dim(\overline{e}/C)=\dim(\overline{a}y/B)$ , then $\operatorname{tp}(\overline{e}/C)=\operatorname{tp}(\overline{a}_{i}y_{i,\ell}/C)$ for exactly one pair of numbers $i<m$ and $\ell<k_{i}$ .

Proof.

For item (1), suppose $i<m$ and $\ell<k_{i}$ , then we can write:

	$\displaystyle\dim(\overline{a}y/B)$	$\displaystyle=\dim(y\overline{a}/B)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{by Proposition \ref{permutation_inv}})$
		$\displaystyle=\dim(\overline{a}/B)+\dim(y/\mathrm{tcl}(B\overline{a}))\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{by definition of $\dim$})$
		$\displaystyle=\dim(\overline{a}_{i}/C)+\dim(y_{i,\ell}/\mathrm{tcl}(C\overline{a}_{i}))\,\,\,\,\,\,(\text{by I.H.})$
		$\displaystyle=\dim(y_{i,\ell}\overline{a}_{i}/C)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{by definition of $\dim$})$
		$\displaystyle=\dim(\overline{a}_{i}y_{i,\ell}/C).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{by Proposition \ref{permutation_inv}})$

Also, since by the induction hypothesis $\overline{a}\equiv_{B}\overline{a}_{i}$ and $\operatorname{tp}(y_{i,\ell}/\mathrm{tcl}(B\overline{a}_{i}))=\operatorname{tp}(g_{i}(y)/\mathrm{tcl}(B\overline{a}_{i}))$ , $\operatorname{tp}(\overline{a}_{i}y_{i,\ell}/B)=\operatorname{tp}(\overline{a}y/B)$ .

For item (2), we note that there is $g\in\mathrm{Aut}(\mathcal{M}/B)$ such that $g(\overline{e})=\overline{a}y$ . So by abuse of notation we may assume that $\overline{e}=\overline{e}^{\prime}e_{0}$ such that $\overline{e}^{\prime}\in\Gamma(S)^{n}$ , $e_{0}\in\Gamma(S)$ , $\overline{e}^{\prime}\equiv_{B}\overline{a}$ , and $e_{0}\equiv_{\mathrm{tcl}(B\overline{a})}y$ . Since $\dim(\overline{e}^{\prime}e_{0}/C)=\dim(\overline{a}y/B)$ , by definition of $\dim$ we have $\dim(\overline{e}^{\prime}/C)=\dim(\overline{a}/B)$ and $\dim(e_{0}/\mathrm{tcl}(C\overline{e}^{\prime}))=\dim(y/\mathrm{tcl}(B\overline{a}))$ . Since $\overline{e}^{\prime}\equiv_{B}\overline{a}$ and $\dim(\overline{e}^{\prime}/C)=\dim(\overline{a}/B)$ , by the induction hypothesis there is $i<m$ such that $\overline{e}^{\prime}\equiv_{C}\overline{a}_{i}$ . Also by the induction hypothesis, there is $\ell<k_{i}$ such that $e_{0}\equiv_{\mathrm{tcl}(C\overline{a}_{i})}y_{i,\ell}$ . Thus $\overline{e}^{\prime}e_{0}\equiv_{C}\overline{a}_{i}y_{i,\ell}$ . ∎

∎

Proposition 4.3.5.

The function $h=(\dim,\mathrm{meas})$ , defined in the proof of Theorem Main Theorem, is well-defined. More precisely, let $X\in\mathsf{Def}(\mathcal{M})$ . Let $B_{1}\subset_{\textrm{fin}}\Gamma(S)$ such that $X$ is $B_{1}$ -definable. Let $B_{2}\subset_{\textrm{fin}}\Gamma(S)$ such that $X$ is $B_{2}$ -definable as well. Let $p_{0},\dots,p_{k-1}\in S_{n}(B_{1})$ and $q_{0},\dots,q_{\ell-1}\in S_{n}(B_{2})$ such that $X=p_{0}(\mathcal{M})\uplus\dots\uplus p_{k-1}(\mathcal{M})$ and $X=q_{0}(\mathcal{M})\uplus\dots\uplus q_{\ell-1}(\mathcal{M}).$ Let $I=\big{\{}i<k:\dim(p_{i})=\max\{\dim(p_{t}):t<k\}\big{\}}$ and $J=\big{\{}j<\ell:\dim(q_{j})=\max\{\dim(p_{t}):t<\ell\}\big{\}}.$ Then for all $i\in I,\,j\in J$

\dim(p_{i})=\dim(q_{j}),\text{ and }\,\sum\limits_{i\in I}\mathrm{meas}(p_{i})=\sum\limits_{j\in J}\mathrm{meas}(q_{j}).

Proof.

Let $C=\mathrm{tcl}(B_{1}B_{2})$ . It suffices to apply Lemma 4.3.4 to each $p_{i}$ and $q_{j}$ . More precisely, let $p_{i}=\operatorname{tp}(\overline{a}_{i}/B_{1})$ for each $i<k$ , and let $q_{j}=\operatorname{tp}(\overline{b}_{j}/B_{2})$ for each $j<\ell$ . By Lemma 4.3.4, for each $i<k$ and each $j<\ell$ there are $\{\overline{a}_{i,s}:s<t_{i}\}$ and $\{\overline{b}_{j,s}:s<m_{j}\}$ such that $\operatorname{tp}(\overline{a}_{i,s}/B_{1})=\operatorname{tp}(\overline{a}_{i}/B_{1})$ , $\dim(\overline{a}_{i,s}/C)=\dim(\overline{a}_{i}/B_{1})$ for each $s<t_{i}$ , and $\operatorname{tp}(\overline{b}_{j,s}/B_{2})=\operatorname{tp}(\overline{b}_{j}/B_{2})$ , $\dim(\overline{b}_{j,s}/C)=\dim(\overline{b}_{j}/B_{2})$ for each $s<m_{j}$ . Suppose $i\in I$ and $j\in J$ , then

	$\displaystyle\dim(p_{i})$	$\displaystyle=\dim(\overline{a}_{i}/B_{1})=\dim(\overline{a}_{i,s}/C)\,\,\,\,\,\text{ for each }s<t_{i},$
	$\displaystyle\dim(q_{j})$	$\displaystyle=\dim(\overline{b}_{j}/B_{2})=\dim(\overline{b}_{j,r}/C)\,\,\,\,\,\text{ for each }r<m_{j}.$

We have to show that $\dim(\overline{a}_{i,s}/C)=\dim(\overline{b}_{j,r}/C)$ for all $s<t_{i},\,\,r<m_{j}$ . They are equal because they are the maximum values among dimensions of types over $B_{1}B_{2}$ as well. We can do a similar calculation for $\mathrm{meas}$ as well. ∎

4.3.3. Verification of $h=(\dim,\mathrm{meas})$

Here we show that the function defined in Subsection 4.3.1 works, that is, $\mathcal{M}$ is MS-measurable via $h$ . By Theorem 4.2.1, we just need to verify that $h$ satisfies conditions $\textrm{cMS}1,\dots,\textrm{cMS}4$ .

Lemma 4.3.6.

cMS1 holds.

Proof.

It is enough to prove this for complete 1-types over a closed set. Let $C\subset_{\textrm{fin}}\Gamma(S)$ be closed and $a\in\Gamma(S)$ . Let $g\in\mathrm{Aut}(\mathcal{M})$ . Suppose

[a\wedge C]=e_{0}<e_{1}<\dots<e_{n-1}<e_{n}=a

such that $\mathtt{pred}(e_{i})=e_{i-1}$ for each $i=1,\dots,n$ . Suppose

J=\left\{1\leq j\leq n:\pi(e_{j})\in I(\Gamma)\right\}.

Then by construction of $h$ we have

\dim(a/C)=|J|,

and

\mathrm{meas}(a/C)=\prod_{i=1}^{n}\mathrm{meas}_{\sigma_{i}}(e_{i}/\mathrm{tcl}(Ce_{0},\dots,e_{i-1})\cap M(e_{0}))

On the other hand we have $g\left([a\wedge C]\right)=[g(a)\wedge g[C]]$ ,

[g(a)\wedge g[C]]=g(e_{0})<g(e_{1})<\dots<g(e_{n-1})<g(e_{n})=g(a)

and $\mathtt{pred}(g(e_{i}))=g(e_{i-1})$ for each $i=1,\dots n$ . Let

J^{\prime}=\left\{1\leq j\leq n:\pi(g(e_{j}))\in I(\Gamma)\right\}.

Since the image of any infinity-node under an automorphism $g\in\mathrm{Aut}(\mathcal{M})$ is an infinity-node, $|J|=|J^{\prime}|$ . Thus

\dim(a/C)=|J|=|J^{\prime}|=\dim(g(a)/g[C]),

and

	$\displaystyle\mathrm{meas}(a/C)$	$\displaystyle=\prod_{i=1}^{n}\mathrm{meas}_{\sigma_{i}}(e_{i}/\mathrm{tcl}(Ce_{0},\dots,e_{i-1})\cap M(e_{0}))$
		$\displaystyle=\prod_{i=1}^{n}\mathrm{meas}_{\sigma_{i}}(g(e_{i})/\mathrm{tcl}(Cg(e_{0}),\dots,g(e_{i-1}))\cap M(e_{0}))\,\,\,\,$
		$\displaystyle=\mathrm{meas}(g(a)/g[C]).$

∎

Lemma 4.3.7.

cMS2 and cMS3 hold.

Proof.

This follows from part III of the construction of $h$ . ∎

Lemma 4.3.8.

cMS3 holds.

Proof.

This follows from parts I and II of the construction of $h$ , and $\operatorname{acl}=\mathrm{tcl}$ . ∎

Lemma 4.3.9.

cMS4 holds.

Proof.

Let $C\subset_{\textrm{fin}}M,\,\overline{a}\in M^{n},\,\overline{b}\in M^{m},\,$ and $\overline{b}\subseteq\operatorname{dcl}(C\overline{a})=\mathrm{tcl}(C\overline{a})$ . Let $X=\operatorname{tp}(\overline{a}/C)$ . By a simple induction on $n$ and using part II of the construction of $h$ , we have

\dim(\overline{a}\overline{b}/C)=\dim(\overline{b}/C)+\dim(\overline{a}/C\overline{b})

and

\mathrm{meas}(\overline{a}\overline{b}/C)=\mathrm{meas}(\overline{b}/C)\cdot\mathrm{meas}(\overline{a}/C\overline{b}).

So, it suffices to show that

\dim(X)=\dim(\overline{a}\overline{b}/C),

\mathrm{meas}(X)=\mathrm{meas}(\overline{a}\overline{b}/C).

We calculate:

$\displaystyle\dim(\overline{a}\overline{b}/C)$	$\displaystyle=\dim(\overline{b}\overline{a}/C)$	$\displaystyle(\text{by Proposition \ref{permutation_inv}})$
	$\displaystyle=\dim(\overline{a}/C)+\dim(\overline{b}/C\overline{a})$
	$\displaystyle=\dim(\overline{a}/C)$	$\displaystyle(\text{since }\overline{b}\subseteq\mathrm{tcl}(C\overline{a}))$
	$\displaystyle=\dim(X).$

and

$\displaystyle\mathrm{meas}(\overline{a}\overline{b}/C)$	$\displaystyle=\mathrm{meas}(\overline{b}\overline{a}/C)$	$\displaystyle(\text{by Proposition \ref{permutation_inv}})$
	$\displaystyle=\mathrm{meas}(\overline{a}/C)\cdot\mathrm{meas}(\overline{b}/C\overline{a})$
	$\displaystyle=\mathrm{meas}(\overline{a}/C)$	$\displaystyle(\text{since }\overline{b}\subseteq\mathrm{tcl}(C\overline{a}))$
	$\displaystyle=\mathrm{meas}(X).$

as desired. ∎

Acknowledgment

The author would like to thank Cameron Hill for all of his support and very helpful guidance. The author also would like to thank Alex Kruckman for his reading of the first draft and for the helpful comments.

References

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[3] M. Djordjevic. Finite satisfiability and $\aleph_{0}$ -categorical structures with trivial dependence. Journal of Symbolic Logic, 71(3):810–830, 2006.
[4] R. Elwes. Dimension and measure in finite first order structures. PhD thesis, University of Leeds, 2005.
[5] R. Elwes. Asymptotic classes of finite structures. Journal of Symbolic Logic, 72(2):418–438, 2007.
[6] C. D. Hill. On 0,1-laws and asymptotics of definable sets in geometric Fraïssé classes. Fundamenta Mathematicae, 239(3):201 – 219, 2017.
[7] C. D. Hill. Pseudo-finite $\aleph_{0}$ -categorical structures via coordinatization. preprint, 202X.
[8] A. Kruckman. Disjoint n-amalgamation and pseudofinite countably categorical theories. Notre Dame J. Formal Log., 60:139–160, 2019.
[9] D. Macpherson and C. Steinhorn. One-dimensional asymptotic classes of finite structures. Transactions of the American Mathematical Society, 360(1):411–448, January 2008.
[10] D. Marker. Model Theory: An Introduction, volume 217 of Graduate Texts in Mathematics. Springer, 2002.

MS-measurability via coordinatization

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

1.1. Notation and conventions

2. Tree structures

2.1. Tree plans

Definition 2.1.1.

Definition 2.1.2.

Definition 2.1.3.

Observation 2.1.4.

Definition 2.1.5.

Definition 2.1.6.

Lemma 2.1.7.

Proof.

3. Coordinatization

3.1. Nil-interaction coordinatized structures

Definition 3.1.1.

Definition 3.1.2.

Proposition 3.1.3.

Proof.

3.2. Existence of NIC structures

3.2.1. Construction

3.2.2. Verification

Remark 3.2.1.

Notation 3.2.2.

Lemma 3.2.3.

Proof.

Corollary 3.2.4.

Proof.

Claim.

Proof.

Corollary 3.2.5.

Proof.

Corollary 3.2.6.

Proof.

Theorem 3.2.7.

Proof.

Proposition 3.2.8.

Proof.

4. Lifting MS-measurability for coordinatization

4.1. Definitions and the statement of the main theorem

Definition 4.1.1.

Definition 4.1.2.

Main Theorem (4.1).

4.2. MS-measurability for ℵ0\aleph_{0}-categorical theories

Theorem 4.2.1.

Proof.

Lemma 4.2.2.

Proof.

Lemma 4.2.3.

Proof.

Lemma 4.2.4.

Proof.

Lemma 4.2.5.

Proof.

Lemma 4.2.6 (MS4 Special Case).

Proof.

Lemma 4.2.7.

Proof.

Lemma 4.2.8.

Proof.

Lemma 4.2.9.

Proof.

Lemma 4.2.10.

Proof.

Lemma 4.2.11.

Proof.

4.3. Proof of the Main Theorem Main Theorem

4.3.1. Construction of h=(dim,meas)h=(\dim,\mathrm{meas})

4.3.2. Well-definedness of h=(dim,meas)h=(\dim,\mathrm{meas})

Fact 4.3.1.

Proof.

Lemma 4.3.2.

Proof.

Claim.

Proof.

Proposition 4.3.3 (Permutation/coordinate invariance).

Proof.

4.2. MS-measurability for $\aleph_{0}$ -categorical theories

4.3.1. Construction of $h=(\dim,\mathrm{meas})$

4.3.2. Well-definedness of $h=(\dim,\mathrm{meas})$

4.3.3. Verification of $h=(\dim,\mathrm{meas})$