On the reversibility and the closed image property of linear cellular automata

When G is an arbitrary group and V is a finite-dimensional vector space, it is known that every bijective linear cellular automaton τ:VG→VG is reversible and that the image of every linear cellular automaton τ:VG→VG is closed in VG for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if G is a non-periodic group and V is an infinite-dimensional vector space, then there exist a linear cellular automaton τ1:VG→VG which is bijective but not reversible and a linear cellular automaton τ2:VG→VG whose image is not closed in VG for the prodiscrete topology.