Given two NFW $A$ and $B$, we regarded $A$ and $B$ as Büchi automata.

We can show that the containment property is not preserved in Büchi automata. That is, we can construct a example: $L(A) \subseteq L(B)$ when A and B are both NFW, but $L_\omega(A) \nsubseteq L_\omega(B)$ when A and B are both Büchi automata.

My question is : Are there any interesting properties is preserved in Büchi automata? Conversely, given two Büchi automata, are there any interesting properties is preserved in NFW?