# A note on cellular automata

## TL;DR

This paper introduces a new algebraic perspective on cellular automata by defining a monoid structure on uniformly continuous functions, providing an alternative proof of the Curtis-Hedlund theorem.

## Contribution

It establishes an isomorphism between uniformly continuous functions and cellular automata monoids, offering a novel definition and proof approach.

## Key findings

- Monoid structure on uniformly continuous functions is isomorphic to cellular automata monoid.
- Provides a new, simpler proof of the Curtis-Hedlund theorem.
- Offers an alternative algebraic characterization of cellular automata.

## Abstract

For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.