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