Definition 1.1 (absolute). If $M\subseteq N$ and $M,N$ are $L$-structures and $\phi$ an $L$-formula, then we say $\phi$ is absolute between $M$ and $N$ if for all $x_1,\dots ,x_m\in M$,

$$M\models \phi (x_1,\dots ,x_n)⟺N\models \phi (x_1,\dots ,x_n).$$

If the $\Rightarrow$ direction holds, then we say “upwards absolute”, and if the $\Leftarrow$ direction holds, then we say “downwards absolute”.