An order-preserving property of additive invariants for Takesue-type reversible cellular automata

We show that, for a large and important class of reversible, one-dimensional cellular automata, the set of additive invariants exhibits an algebraic structure. More precisely, if f and g are one-dimensional, reversible cellular automata of the kind considered by Takesue (1989) [1], we show that the component-wise maximum ∨ on these automata is such that ψ(f)⊆ψ(f∨g), where ψ(f) denotes the set of additive invariants of f and ⊆ denotes the inclusion relation between real subspaces.