  We consider some questions concerning the monotonicity properties of entropy
and mean entropy of states on translationally invariant systems (classical
lattice, quantum lattice and quantum continuous). By taking the property of
strong subadditivity, which for quantum systems was proven rather late in the
historical development, as one of four primary axioms (the other three being
simply positivity, subadditivity and translational invariance) we are able to
obtain results, some new, some proved in a new way, which appear to complement
in an interesting way results proved around thirty years ago on limiting mean
entropy and related questions. In particular, we prove that as the sizes of
boxes in Z^n or R^n increase in the sense of set inclusion, (1) their mean
entropy decreases monotonically and (2) their entropy increases monotonically.
Our proof of (2) uses the notion of "m-point correlation entropies" which we
introduce and which generalize the notion of "index of correlation" (see e.g.
R. Horodecki, Phys. Lett. A 187 p145 1994). We mention a number of further
results and questions concerning monotonicity of mean entropy for more general
shapes than boxes and for more general translationally invariant (/homogeneous)
lattices and spaces than Z^n or R^n.
