I was reading Tent and Ziegler's A Course In Model Theory, and in chapter 5 they explain how to define the algebraic closure operator on the universe of a structure, and how to modify it to strongly minimal sets to define a pregeometry. To paraphrase their explanations, here are the following definitions from the book. Let $\mathcal{M}$ be a structure with universe $M$.

  1. An $\mathcal{L}(A)$-formula $\phi(x)$ (i.e an $\mathcal{L}$-formula with parameters from $A$) is algebraic if $\phi(\mathcal{M}) :=\{a \in M:\mathcal{M}\vDash \phi(a)\}$ is finite. An element $a \in M$ is algebraic over $A$ if it satisfies some algebraic $\mathcal{L}(A)$-formula. The algebriac closure of $A$, denoted $acl(A)$, is the set of all elements algebraic over $A$.
  2. An $\mathcal{L}(M)$-formula $\phi(\underline{x})$ (where $\underline{x}$ is just a $k$-tuple of variables) is a minimal formula in $\mathcal{M}$ if it is not algebraic, and for every $\mathcal{L}(M)$-formula $\psi(\underline{x})$ ($\underline{x}$ also a $k$-tuple), $\phi(\mathcal{M})\land\psi(\mathcal{M})$ is either finite or cofinite in $\phi(\mathcal{M})$ (cofinite means the complement of this set [in $\phi(\mathcal{M})$] is finite). A minimal formula $\phi(\underline{x})$ in $\mathcal{M}$ is strongly minimal if it is minimal in every elementary extensions of $\mathcal{M}$.
  3. (This was defined as part of theorem 5.7.5) Let $\phi(x)$ be a strongly minimal formula. Define the closure operator $Cl:\mathcal{P}(\phi(\mathcal{M})) \to \mathcal{P}(\phi(\mathcal{M}))$ (where $\mathcal{P}(\cdot)$ is the power set operator) by $$Cl(A)=acl^\mathcal{M}(A) \cap \phi(\mathcal{M})$$ Tent and Ziegler proceeds to prove that $(\phi(\mathcal{M}), Cl)$ defines a pregeometry.

I fully understand every definition here and how $(\phi(\mathcal{M}), Cl)$ defines a pregeometry (I followed the proof in David Marker's Model Theory: An Introduction). However, I would like to see how to formulate definition 1 and 3 above to strongly minimal formulas of more than one variable.

Question: How do you generalise definitions 1 and 3 to more than one variable (2 is already generalised)?

For example (for definition 1)

$\mathcal{L}(A)$-formula $\phi(\underline{x})$ is algebraic if $\phi(\mathcal{M}):=\{\underline{a} \in M^k:\mathcal{M}\vDash \phi(\underline{a})\}$. A $k$-tuple $\underline{a} \in M^k$ is algebraic over $A$ if it satisfies some algebraic $\mathcal{L}(A)$-formula of $k$ free variables. I think there are several the $acl(\cdot)$ operator can be defined:

  • $acl^1(A)=\{x \in M: x \ \mbox{is a coordinate of some algebraic tuple over} \ A\}$
  • $acl^2(A)=\{\underline{x} \in M: \underline{x} \ \mbox{is a tuple (of any length) algebraic over} \ A\}$
  • Or a series of $acl_k(A)$ definitions:$$acl_k(A)=\{\underline{x} \in M^k: \ \underline{x} \ \mbox{is a $k$-tuple algebraic over}\ A\}$$

I'm inclined to go with the $acl_k$ series, as then I can define the closure operator given in definition with the following:

Let $\phi(\underline{x})$ be a strongly minimal $\mathcal{L}(M)$-formula of $\mathcal{M}$ of $n$ free variables, and for any $k$ and any $A \subseteq M^k$, let $\underline{A}=\{a \in M: a \ \mbox{is a coordinate of some tuple} \ \underline{a} \in A\}$. Then define the closure operator $Cl_n^\phi:\mathcal{P}(\phi(\mathcal{M}))\to \mathcal{P}(\phi(\mathcal{M}))$ by $$Cl_n^\phi(A)=acl^n(\underline{A})\cap \phi(\mathcal{M})$$

Though I am not 100% sure if I have made a mistake or not, I think that this definition still gives that $(\phi(\mathcal{M}), Cl^\phi_n)$ defines a pregeometry by following the proof given by David Marker.

Question: is this a standard generalised definition?


1 Answer 1


The key observation here is that a tuple $\overline{b} = (b_1,\dots,b_k)$ is algebraic over $A$ if and only if each of its components $b_i$ is algebraic over $A$.

Proof: Suppose each $b_i$ is algebraic over $A$, witnessed by the $L(A)$-formula $\varphi_i(x_i)$. Then $\bigwedge_{i=1}^k \varphi_i(x_i)$ is an algebraic formula over $A$ satisfied by $\overline{b}$. Conversely, suppose $\overline{b}$ is algebraic over $A$, witnessed by the $L(A)$-formula $\psi(x_1,\dots,x_k)$. Then $\exists x_2,\dots,x_k\, \psi(x_1,\dots,x_k)$ is an algebraic formula over $A$ satisfied by $b_1$, and similarly for the other $b_i$.

It follows that your definitions $\text{acl}^1$, $\text{acl}^2$, and $(\text{acl}_k)_{k\in \omega}$ are not actually so different, and model-theorists typically abuse notation by writing them all as $\text{acl}$, with the precise meaning depending on the context. For example, we can express all of your operators in terms of the standard $\text{acl}$ operator: If $$\text{acl}(A) = \{x\in M\mid x\text{ is algebraic over }A\},$$ then we have \begin{align*} \text{acl}^1(A)&=\text{acl}(A) \\ \text{acl}_k(A) &= (\text{acl}(A))^k, \text{ for all }k\\ \text{acl}^2(A)&=\bigcup_{n\in\omega} (\text{acl}(A))^n\\ \end{align*}

Similarly, we usually don't have to be too careful about parameter sets. If $A$ is a set of $k$-tuples, and $\underline{A}$ is the set of elements of these tuples (as you defined), then any formula with parameters from $A$ is clearly also a formula with parameters from $\underline{A}$, and conversely any formula with parameters from $\underline{A}$ is equivalent to a formula with parameters from $A$ (anytime you use a parameter from a $k$-tuple, just throw in the rest of that $k$-tuple as extra parameters that the formula doesn't mention).

The upshot is that your definition of the closure operator on $\varphi(\mathcal{M})$ when $\varphi(\overline{x})$ is a strongly minimal set in multiple variables is correct, though most people would happily abuse notation by writing this as $$\text{acl}(A)\cap \varphi(\mathcal{M}).$$ And it makes $\varphi(\mathcal{M})$ into a pregeometry, by exactly the usual argument.

Besides convenience, another explanation for why abuse of notatation involving elements and tuples is so common is that any theory $T$ (let's assume $T$ has a single sort called $S$ for simplicity) is bi-interpretable with a multi-sorted theory $T'$ obtained by adding a new sort $S_n$ for each $n>1$, together with an $n$-ary function $f_n\colon S^n\to S_n$ and axioms asserting that the $f_n$ are bijections. In $T'$, each $k$-tuple from $S$ is canonically identified with an element of $S_k$, and more generally each $m$-tuple from $\prod_{i=1}^m S_{k_i}$ is canonically identified with an element of $S_{k_1+\dots+k_m}$. We can then happily do model theory in this theory $T'$ without having to worry about the distinction between tuples and elements.

Note that $T'$ is actually a reduct of $T^{\text{eq}}$: we obtain the sort $S_n$ and the bijection $f_n$ as the quotient of $S^n$ by the trivial definable equivalence relation $\bigwedge_{i=1}^n (x_i = y_i)$. So if you're comfortable with $T^{\text{eq}}$, you should already be comfortable with $T'$.

  • $\begingroup$ I never knew that there was this much abuse of notation going on - it has confused me quite a bit. This has really cleared up my intuition - thanks very much $\endgroup$ Apr 5, 2019 at 18:00

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