Algebraic Properties of the Block Transformation on Cellular Automata

By grouping several sites together into one a cellular automaton can be transformed into another with more states and a smaller neighbor hood if the neighborhood has just two sites we can think of the resulting CA rule as a binary operation We show that if the blocked rule satis es an identity which holds for a broad class of algebras then the underly ing rule must have essentially the same structure and must depend only on its leftmost and rightmost inputs roughly speaking that the block transformation cannot turn a non linear rule into a linear one