Let $X$ denote an algebraic structure. Then:
- Every subalgebra of $X$ satisfies each quasi-identity that is satisfied by $X$. In other words, taking subalgebras preserves quasi-identities.
- Every quotient of $X$ satisfies every existentially-quantified equation satisfied by $X$. In other words, taking quotients preserves existentially-quantified equations.
Question. Aside from the fact that taking quotient of a subalgebra of $X$ preserves identities, does this process preserve any other kinds of statements?