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

We show that, for a fairly large 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, we show that there is a binary operation on these automata $\vee$ such that $\psi(f)\subseteq \psi(f\vee g)$, where $\psi(f)$ denotes the set of additive invariants of $f$ and $\subseteq$ denotes the inclusion relation between real subspaces.