  We study expansive dynamical systems in the setting of distributive lattices
and their automorphisms, the usual notion of expansiveness for a homeomorphism
of a compact metric space being the particular case when the lattice is the
topology of the phase space ordered by inclusion and the automorphism the one
induced by the homeomorphism, mapping open sets to open sets. We prove in this
context generalizations of Ma\~n\'e's Theorem and Utz's Theorem about the
finite dimensionality of the phase space of an expansive system, and the
finiteness of a space supporting a positively expansive homeomorphism,
respectively. We also discuss the notion of entropy in this setting calculating
it for the non-Hausdorff shifts.
