On whether reducts and axioms commute.

Question

Consider theories $T$ and $T'$, where $T$ is a subset of $T'$, in the same language $L$. Also, let $L'$ be a subset of the language $L$. We can either take the reduct of $T$ or the reduct of $T'$ with $L'$. Would it be the case that the reduct of $T$ is a subset of the reduct of $T'$?

Answer 1

If $T$ is a theory, the reduct of $T$ to the language $L'$ is $T\cap \mathrm{Sent}(L')$, where $\mathrm{Sent}(L')$ is the set of $L'$-sentences.

If $T\subseteq T'$, then $(T\cap \mathrm{Sent}(L'))\subseteq (T'\cap \mathrm{Sent}(L'))$.

(This is just elementary set theory: If $X\subseteq Y$, then for all $Z$, $(X\cap Z)\subseteq (Y\cap Z)$.)

Answer 2

Yes, since the reducts of $T$ and $T'$ are still the same sets, with some of the logical structure removed. Any embedding $\pi: T \to T'$ will still be an embedding between the reducts.

Misread the question. @AlexKruckman's answer is correct.