Does non-monotonicity imply non-satiation always? Why or why not? I understand that monotonic preferences imply non-satiation. But I am not sure 100% if non-monotonic functions always have satiation. An intuitive and mathematical explanation would be very helpful.
It depends on the function. A non-monotonic function with satiation:
$$
U(x_1,x_2) = 0.
$$
A non-monotonic function without (global) satiation:
$$
U(x_1,x_2) = \left \lfloor{x_1}\right \rfloor  + \left \lfloor{x_2}\right \rfloor .
$$