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I’m starting to study model theory from the book Model Theory: An Introduction by David Marker. Our professor gave us some notes which contain the phrase 'constructible group', and I don't know the definition.

I have searched the internet, and I found the following definition on the website Groupprops: "A group (possibly with additional structures and relations) is said to be constructible if it is definable in a pure algebraically closed field." But I also don’t know the meaning of the word 'definable' here.

I do know the definition of constructible set in a topological space: it is a finite union of locally closed sets.

What is a 'constructible group' in model theory? How is it related to the notion of a constructible set in topology?

I don’t have much information about logic, so I would also appreciate it if someone could suggest some good references for logic and model theory.

Alex Kruckman
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mmm
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    Your question in its original form was badly organized and sloppily written. I totally rewrote it for you - I hope I maintained the intention of the question. It's unlikely that people will keep rewriting your questions for you, so I strongly encourage you to pay more attention to style, organization, spelling, and grammar when you post here. Your questions will be much better received if they look like you care. – Alex Kruckman Jan 15 '18 at 15:31
  • Also, I should say that I think this is a good question, and I'm glad you posted it here, since it will probably help others with the same confusion. But have you considered just asking your professor to explain the concept? One of the benefits of taking courses, in contrast to studying on your own from books, is that you can ask for clarifications! – Alex Kruckman Jan 15 '18 at 15:36
  • Indeed: what did your professor answer when you asked him/her about this? – Mariano Suárez-Álvarez Jan 15 '18 at 19:55
  • For the next time I will learn how to write my questions hear but thank you for your editing. I didn’t asked him yet because Our next lecture will be tomorrow but I will ask him just I wanted to learn more about this structure before asking him. – mmm Jan 15 '18 at 20:36

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Definition: Let $M$ be an $\mathcal{L}$-structure. A group $(G,e,\bullet,^{-1})$ is definable in $M$ if:

  1. $G$ is an $\mathcal{L}$-definable subset of $M^n$ for some $n$.
  2. $e$ is an $\mathcal{L}$-definable element of $G$.
  3. $\bullet\colon G\times G\to G$ is an $\mathcal{L}$-definable function.
  4. $^{-1}\colon G\to G$ is an $\mathcal{L}$-definable function.

Usually "definable" means "definable with parameters from $M$". And you can also check that if $\bullet$ is definable, then $e$ and $^{-1}$ are too, so really you just need to check that $G$ and $\bullet$ are definable.

There's nothing special about groups here. You can easily generalize to say what it means for an arbitrary $\mathcal{L}'$-structure to be definable in $M$, for any language $\mathcal{L}'$.

Now just as the definition in Groupprops says...

Definition: A group $(G,e,\bullet,^{-1})$ is constructible if it is definable in an algebraically closed field $(K,0,1,+,-,\times)$.

As a very simple example, the multiplicative group $K^*$ is constructible, since its domain is definable by $x \neq 0$, and the graph of $\bullet$ is definable by $x_1\times x_2 = y$.

As a slightly more complicated example, you could check that $\mathrm{GL}_n(K)$ is constructible. (Hint: the domain is the subset of $K^{n^2}$ consisting of all $n\times n$ matrices with nonzero determinant.)

So why do we use the word constructible here? Well, quantifier elimination gives a very simple description of all the definable sets in an algebraically closed field. Any quantifier-free formula can be put into disjunctive normal form as a finite disjunction of finite conjunctions of polynomial equations and inequations. So any definable set can be described as a finite union of finite intersections of basic closed sets and basic open sets in the Zariski topology. And it is an exercise in general topology to show that these are exactly the constructible sets in the Zariski topology. In fact, one way of stating quantifier elimination for algebraically closed fields is as Chevalley's theorem: the coordinate projection of a constructible set in the Zariski topology is constructible.

The conclusion is that in an algebraically closed field $K$, "definable subset of $K^n$" and "constructible subset of $K^n$ in the Zariski topology" coincide. So a group is definable in $K$ if and only if its domain is a constructible subset of $K^n$ for some $n$ and the graph of the group multiplication is a constructible subset of $(K^{n})^3$. Thus the name "constructible group".

Alex Kruckman
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    +1. To the OP, this answer assumes that you are familiar with the usual notion of "definable (with parameters) set" in model theory; if not, this old answer of mine may be helpful. – Noah Schweber Jan 15 '18 at 19:43
  • Right, and the notion of definable function: A function $A\to B$ (where $A$ and $B$ are definable sets) is definable if its graph is definable as a subset of $A\times B$! The notions of definable set and definable function are introduced in the first chapter of Marker's book. – Alex Kruckman Jan 15 '18 at 19:50