Preservation of structure in model theory

Solution 1:

Strictly speaking the latter (narrower) version is correct, and the term "reflects" is used for the dual notion. So, for example, surjective homomorphisms preserve but do not necessarily reflect universally-quantified equations, and general homomorphisms preserve but do not necessarily reflect relations.

That said, the conflation of "preserves" and "preserves and reflects" is fairly common (unfortunately, I would argue).

uncertain orthogonality of discrete Fourier transform on the ring of integers modulo some number

In a finite poset $P$, a down-set $I$ is meet-irreducible $\iff I=P\backslash\uparrow x$ for some $x\in P$

Determining the CDF of an absolutely continuous random vector, given its density.

How do I combine $-\ln(a)-\ln(b)$?

The induced map $\overline{\pi}: \operatorname{Hom}(Z_2,Z) \to\operatorname{Hom}(Z_2,Z_2)$?

Maybe a Monty Hall kind of problem?

Solve $(y+1)dx+(x+1)dy=0$

Prove that for all $c \in [0,\infty)$ there exists $x \in \mathbb{R}$ such that $xe^x=c$ by the intermediate value theorem

Reference for an estimate on the sum of powers $\sum_{k=1}^4 | x_k |^{2k}$ if $\sum_{k=1}^4 | x_k |^2 =1$?

$\int^{2\pi}_0 i\sin(x\sin \theta -n\theta) d\theta$

What is known about sums of the form $\sum_{n=-\infty}^{\infty} \operatorname{sinc} (n^{p})$?