I know from "Numerical Methods for Conservation Laws" by Randall J. LeVeque that there is an implication chain of properties of methods for conservation laws:
monotone $\Rightarrow$ $L^1$-contractive $\Rightarrow$ TVD $\Rightarrow$ monotonicity preserving
For the conservation of mass, positivity preservation is most important, but how does it fit into this chain? Does monotonicity preservation imply positivity preservation or vice-versa? A lot of people seem to use both terms as equivalent, but I'm not so sure that's true...