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

TL;DR

This paper investigates the algebraic structure of additive invariants in a broad class of reversible, one-dimensional cellular automata, revealing an order-preserving property under a specific binary operation.

Contribution

It introduces a binary operation on Takesue-type reversible cellular automata that preserves the inclusion relation of their additive invariants.

Findings

Additive invariants form an algebraic structure for these automata.

The binary operation $igvee$ preserves the order of invariants.

The set of invariants is related through inclusion under this operation.

Abstract

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.