# How does one generalise the definition of the closure operator on (strongly) minimal sets to define a pregeometry to more than one variable?

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?

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 $$T'$$.

• 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 Apr 5, 2019 at 18:00