When you name an element in an uncountably categorical theory....

Question

When you name an element in an uncountably categorical theory $T$ does it remains uncountably categorical?

In other words, given a finite elementary map $f:M\to N$ between models of an uncountably categorical theory $T$ of equal uncountable cardinality, is there an isomorphism $h:M\to N$ that extends $f$?

(The language is countable.)

Answer 1

Here's another way to see the answer: The Baldwin-Lachlan theorem says that a theory $T$ in a countable language is uncountably categorical if and only if it is $\omega$-stable and has no Vaughtian pairs, and both of these conditions are clearly preserved under naming countably many constants.

Of course, tomasz's method is better, since it shows that you can name $\kappa$ many constants, and the resulting theory will still be $\lambda$-categorical for all $\lambda > \kappa$.

Answer 2

For the first question the answer is yes. Notice that a categorical structure is automatically saturated and use a standard back-and-forth argument.

For the second question the answer is no, not even if $M,N$ are isomorphic. Just consider the theory of one equivalence relation with two infinite classes.