Let X be a one-dimensional cellular automaton. A ‘power of X’ is another cellular automaton obtained by grouping several states of X into blocks and by considering as local transitions the ‘natural’ interactions between neighbor blocks. Based on this operation a preorder ⩽ on the set of one-dimensional cellular automata is introduced. We denote by (CA*, ⩽) the canonical order induced by ⩽. We prove that (CA*, ⩽) admits a global minimum and that very natural equivalence classes are located at the bottom of (CA*, ⩽). These classes remind us of the first two well-known Wolfram ones because they capture global (or dynamical) properties such as nilpotency or periodicity. Non-trivial properties such as the undecidability of ⩽ and the existence of bounded infinite chains are also proved. Finally, it is shown that (CA*, ⩽) admits no maximum. This result allows us to conclude that, in a ‘grouping sense’, there is no universal CA.
