Definition of prime model extension over a set

Question

Standard definition of prime model over a set is that: $M\vDash T$ is said to be a prime model extension of a set $A$ if $A\subset M$ and any partial elementary map $A\rightarrow N$ ($N\vDash$) extend elementary $M\rightarrow N$.

So far so good. But I am confused when I got to a standard exercise. First, it is not clear whether $A$ can be any sets, or must be a substructure. So if you are given a set $A$, any set $A$, can you ask the question of whether there are any prime model extension over $A$, even without having given interpretation to symbols? For example, is it meaningful to ask "is there any prime model extensions for the theory of algebraically closed field of characteristic 0 over the sphere $S^{5}$?"?

The exercise that cause me trouble here is that:

Let $L=\{<,U\}$ be a language (U unary, < binary relation), and $T$ be dense linear order without endpoints together with sentence that say that the set defined by $U$ and its complement are both dense. Let $M\vDash T$ and let $A=U^{M}$. Show that there are no prime model extensions of $A$.

Answer 1

Yes, it makes no sense to speak of the prime model of a theory $T$ over an arbitrary set. Part of the data is a specific embedding of the set in question in a model of $T$.

Let $T$ be a thoery. It is clear what is meant by a prime model of $T$: it is a model $M$ of $T$ with the property that it admits an elementary embedding in any other model of $T$.

Now suppose that $M$ is a model of $T$, in a language $L$, and $A$ is an arbitrary subset of $M$. Let $L_A$ be the language containing $L$, together with new constant symbols for every element of $A$. One can enrich $M$ to a structure for the language $L_A$ in the obvious way. Let $T_A$ be the theory of the resulting structure. By a prime model of $T$ over $A$, we really mean a prime model of the theory $T_A$ in the sense of the previous paragraph.

The data of a model of $T_A$ is equivalent to the data of a model $N$ of $T$, together with an injective map from $A$ to $N$, such that the following holds. The embedding of $A$ in $N$ gives us a map partially defined map from $N$ to $M$, by identifying the copies of $A$ inside these models, and this map is required to be partial elementary.

So the theory $T_A$ contains more information than the theory $T$ and the abstract set $A$ alone: it encodes the way in which $A$ sits inside your model.