What does the word "extend" mean in the context of model theory? Consider the following two problems:
(1) Let $L=\{E\}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many classes. Prove that there are infinitely many inequivalent complete theories extending $T$. 
(2) Prove that an ultrafilter on an infinite set is non-principal if  and only if it extends to the Frechet filter.
In both problems, I don't understand what the word "extend" means. 
Can someone explain to me? Thanks!

"Extend" just means "be a superset of" in this context.  So a theory $T'$ extends a theory $T$ if $T\subseteq T'$, and a filter $F'$ extends a filter $F$ if $F\subseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends".  Indeed, saying "$F$ extends to $F'$" would normally mean that $F'$ extends $F$, which is the opposite of the intended meaning here: it means to say that the ultrafilter extends the Frechet filter.)
