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Cellular automata are systems evolving on lattices according to a local transition rule. In this paper we present an algebraic formalism for dealing with cellular automata whose local transition rule satisfies an additivity property. We discuss the phenomenon of self-replication and its connection with higher-order cellular automata and the state transition graph. 1.doi:10.1016/0024-3795(91)90116-e fatcat:qsbrldsvwvg4bbixolpmjrwom4