A note on cellular automata

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 $\mathrm{C}\mathrm{A}(G,A)$. 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.