Can you build a new model of a theory out of the set of all complete 1-types?
Marker in Model Theory: An Introduction on page 117 says the following:
Complete types tell us what possible first-order properties an element
can have in an elementary extension.
That's pretty cool, and it provides a nice motivation for types.
The first thing it makes me wonder is whether we can use the complete 1-types themselves as building material to make a new structure … and if we do whether that structure will be the model of the theory we started out with.
Here's my attempt to build such a thing.
I'm restricting attention to theories with no non-nullary functions, i.e. all function symbols are constant symbols.
Let $L$ be a language with only constants and relations.
Let $T$ be an $L$-theory.
Let $M \models T$.
Let $A$ be a subset of the domain of $M$.
I'm looking at $S^M_1(A)$, which I'll call $S^M(A)$ for brevity.
I'm going to try to build a new model $M'$.
Let the domain of $M'$ be $S^M(A)$.
For every constant in $L$ and fresh constant from $c_A$ (created implicitly by talking about parameters), we choose the type $p(x)$ consisting of precisely the formulas $\varphi(x)$ that hold in $M$ as the interpretation for the constant in question.
Let $R$ be a relation symbol in $T$.
Suppose $R$ is unary.
$M' \models R(p)$ holds if and only if, for every finite set of formulas $q(x)$ in $p(x)$, $M \models \exists x \mathop. q(x) \land R(x)$.
And in general, $M' \models R(p_1 \cdots p_n)$ if and only if, for every sequence of finite sets of formulas $q_1(x) \cdots q_n(x)$ where it holds for all $1 \le i \le n$ that $q_i(x) \subset p_i(x)$, it holds that $M \models \exists x_1 \cdots x_n \mathop. q_1(x_1) \land \cdots \land q_n(x_n) \land R(x_1 \cdots x_n)$.
I'm pretty sure that $M'$ gives me back a well-defined relational structure, but it isn't clear to me whether it's actually a model of $T$ or not.
Best Answer
No, looking at the $1$-types is not in general enough.
One way to see this is to observe that there are "interesting" structures whose automorphism group acts $1$-transitively (so a fortiori there is only one $1$-type). For example, every group is bi-interpretable (with parameters) with such a structure: given a group $\mathcal{G}=(G;*,e,{}^{-1})$ consider its torsor reduct $$\mathcal{T}_\mathcal{G}=(G; (a,b,c)\mapsto a*b^{-1}*c).$$ The torsor reduct is to the original group as affine space is to a vector space; basically, we "forget the origin" in a controlled way. Incidentally, with a bit more work you can show the same result for arbitrary rings - see this answer of Matt F. (which while phrased for the rationals in particular, works for all rings).