Consider an elementary class $\mathcal{K}$. It is quite common in model theory that a structure $K$ in $\mathcal K$ comes with a closure operator $$\text{cl}: \mathcal{P}(K) \to \mathcal{P}(K), $$ which establishes a pregeometry on $K$.

Any pregeometry yields a notion of dimension, say:

$$\text{dim} (K) = \min \{|A|: A \subset K \text{ and } \text{cl}(A) = K\}$$ I am interested in some natural properties shared by dimensions induced by pregeometries.

What kind of properties I am looking for? An example is the following,

Suppose $K = \bigcup K_i$ (non redundant increasing chain) and that its dimesion is infinite, is it true that $\text{dim}(K) = \sum_i \text{dim}(K_i)$?

I already know that there are many results like "trivial geometries are modular" but this is not the kind of result I am looking for. I am looking for structural properties of dimension because I am interested in giving an axiomatic definition of dimension.

  • $\begingroup$ It's not clear to me exactly what setup you have in mind for the axiomatic definition of dimension. Are you considering $\dim$ as a function from $\mathcal{K}$ to cardinals? It's often more useful in model theory to assign dimensions to subsets of a model (e.g. the monster model), rather than to models. $\endgroup$ – Alex Kruckman Dec 27 '17 at 20:46
  • $\begingroup$ Yes, it is what I have in mind, I did not say not to bias your answers. Moreover, maybe a functor is too much, I don't care about what it should do on morphisms. $\endgroup$ – Ivan Di Liberti Dec 27 '17 at 20:46
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    $\begingroup$ And regarding the highlighted question, are you assuming that the union is an increasing union of a chain of models? Otherwise, the answer is no, since any model (of a first-order theory in a countable language) can be written as a union of countable elementary submodels. $\endgroup$ – Alex Kruckman Dec 27 '17 at 20:48
  • $\begingroup$ Absolutely yes. $\endgroup$ – Ivan Di Liberti Dec 27 '17 at 20:48
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    $\begingroup$ And are you asking only about unions along chains indexed by $\omega$? Otherwise, the answer is no, since structures of dimension $\aleph_1$ can be written as (uncountable) unions of structures of dimension $\aleph_0$. $\endgroup$ – Alex Kruckman Dec 28 '17 at 2:56

Here's an example showing that in general, for pregeometries arising in model theory, you can't characterize the dimension of a union of a chain of models just in terms of the dimensions of the models. In other words, it matters how the models embed into each other.

Consider the theory of a single equivalence relation $E$ with infinitely many infinite classes, and define $\dim(M) = |M/E|$.

First union: Let $M_0$ be the unique countable model of this theory up to isomorphism. For every countable ordinal $\alpha$, let $M_{\alpha+1}$ be the elementary extension of $M_\alpha$ obtained by adding a single new equivalence class with countably many elements. For a limit ordinal $\lambda$, let $M_\lambda = \bigcup_{\alpha<\lambda} M_\alpha$. Then $\dim(M_\alpha) = \aleph_0$ for all $\alpha<\omega_1$, but $\dim(M_{\omega_1}) = \aleph_1$,

Second union: Let $N_0 = M_0$, and pick an equivalence class $C$. This time, for every $\alpha$, let $N_{\alpha+1}$ be the elementary extension of $N_\alpha$ obtained by adding a single new element to $C$. As before, take unions as limit stages. Then $\dim(N_\alpha) = \aleph_0$ for all $\alpha<\omega_1$, and also $\dim(N_{\omega_1}) = \aleph_0$.

Now I need to convince you that this dimension function arises in a standard way from a pregeometry studied in model theory. In stability theory, there's the notion of a regular type: a stationary type which is orthogonal to all of its forking extensions. The key point is that if $p(x)$ is a regular type (let's say over $\emptyset$ for simplicity), then forking dependence gives rise to a pregometry on the realizations of $p$ via the closure operator $\mathrm{cl}(A) = \{b\models p(x)\mid \text{tp}(b/A) \text{ forks over }\emptyset\}$.

In my example, the theory is stable, the unique $1$-type is a regular type, and forking is governed by the equivalence relation $E$, so we get a pregometry on the whole model with closure operator $\mathrm{cl}(A) = \bigcup_{a\in A} [a]_E$, where $[a]_E$ is the $E$-class of $a$. And the induced dimension function is $\dim(M) = |M/E|$.

Well, maybe you don't like this kind of pregeometry, and you only want to consider the kind you meet more often in model theory, namely pregeometries induced by the $\text{acl}$ operator. That's fine, but then the dimensions are only interesting for models that are at most the size of the language (so only countable models if the language is countable).

Indeed, suppose $T$ is a theory such that $\text{acl}$ induces a pregometry on every model of $T$, and let $M\models T$ with $|M| > |L|$. Since $|\text{acl}(A)| = \max(|A|,|L|)$ for all $A\subseteq M$, any basis for $M$ must have cardinality $|M|$, and $\dim(M) = |M|$.

In this case, the answer to your question about unions is an easy yes.

Added in edit: You might also decide that you're only interested in closure operators with the property that when $A\subseteq M\prec N$, $\text{cl}(A)$ in $M$ equals $\text{cl}(A)$ in $N$, i.e. closures don't grow in elementary extensions. This is the case for $\text{cl} = \text{acl}$, and it would salvage the proof in your answer that $\dim$ takes unions of chains to sums, since if $N$ is a proper elementary extension of $M$, the closure of a basis for $M$ is contained in $M$, and we need at least one new element to form a basis for $N$. But we actually don't get anything beyond $\text{acl}$ under this assumption.

Indeed, suppose $\text{cl}$ satisfies the condition above, and look at $A\subseteq M$. Embed $M$ in a large monster model $\mathbb{M}$. Then $\text{cl}_M(A) = \text{cl}_{\mathbb{M}}(A)$. In fact, for any $A\subseteq N\prec \mathbb{M}$, we have $\text{cl}_N(A) = \text{cl}_{\mathbb{M}}(A)$, so $\text{cl}_{\mathbb{M}}(A)\subseteq N$. But $\bigcap\{N\mid A\subseteq N\prec \mathbb{M}\} = \text{acl}(A)$, so $\text{cl}(A)\subseteq \text{acl}(A)$.

If you're interested in axiomatizing dimension functions, you might want to look this paper, which gives a number of equivalent axiom systems for infinite matroids. In particular, look at the axioms in terms of rank functions. Their rank functions take values in $\mathbb{N}\cup \{\infty\}$, but you might as well be in this situation if you're thinking about $\text{acl}$ pregometries ($\dim(M) = \infty$ means $\dim(M) = |M|$).

  • $\begingroup$ Thanks! This is very interesting! Can you give a look to my proof to spot the mistake? $\endgroup$ – Ivan Di Liberti Dec 28 '17 at 19:06

The following is valid when $A\subseteq M\prec N$, $\text{cl}(A)$ in $M$ equals $\text{cl}(A)$ in $N$, i.e. closures don't grow in elementary extensions.

The answer to my questions looks to me to be yes.

Let $K = \bigcup K_i$ We know that $K_i$ has a basis $A_i$.

Step 1. We can suppose that $A_i \subset A_{i+1}$. In fact, if it is not the case, consider the $\text{cl}(A_i)$ inside $K_{i+1}$. We through in $A_i$ elements untill it gets a basis $A_{i+1}^*$ of $K_i$ and it must be the case that $|A_{i+1}^*| = |A_{i+1}|$ because of exachange property. By transfinite induction we can replace $\{A_i\}$ so that they are an increasing chain.

Step 2. $\bigcup A_i$ is a basis of $K$. In fact $$K = \bigcup K_i = \bigcup \text{cl} (A_i) \subset \text{cl} (\bigcup A_i)$$ because of monotonicity.

Step 3. $|\bigcup A_i| = \sum |A_i|$ as soon as they are infinite.

Is this correct? Am I really using that $A_i \subset A_{i+1}$?

  • $\begingroup$ Oh, you posted this while I was typing my answer. I think your argument works, except in Step 3, where we don't know that $A_i$ is properly contained in $A_{i+1}$. $\endgroup$ – Alex Kruckman Dec 28 '17 at 19:06
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    $\begingroup$ I've added a paragraph to my answer. $\endgroup$ – Alex Kruckman Dec 28 '17 at 19:12

Marcel van de Vel's "Theory of Convex Structures" has material on dimension theory. Currently I do not have access to the book and its been a very long time since I read it. Here is a copy of the table of contents. The second and third from the last items may be of interest.

Table of Contents

Introduction. List of Frequent Symbols. I. Abstract Convex Structures. Basic concepts. The Hull operator. Half-spaces and separation. Interval spaces. Base-point orders. Modular spaces. Bryant-Webster spaces. II. Convex Invariants. Classical convex invariants. Invariants and product spaces. Invariants in other constructions. Infinite combinatorics. Tverberg numbers. III. Topological Convex Structures. Topology and convexity on the same set. Continuity of the Hull operator. Uniform convex structures. Topo-convex separation. Intrinsic topology. IV. Miscellaneous. Embedding Bryant-Webster spaces into vector spaces. Extremality, pseudo-boundary and pseudo-interior. Continuous selection. Dimension theory. Dimension and convex invariants. Fixed points.


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