Near-rings of Cellular Automata
Tim Boykett
We investigate connections between near-rings and cellular automata. Near-rings are the nonlinear generalisation of rings. Collections of linear cellular automata are rings, the generalisation to near-rings allows us to look at nonlinear cellular automata in an algebraic setting, providing new tools.
We show that cellular automata with group structured state sets form a centralizer near-ring under composition and cell-wise addition. The property of being a unit is undecidable in certain near-rings of cellular automata, introducing a new type of undecidability into near-ring theory. Non continuous, infinite radius generalised cellular automata are investigated. The continuous near-ring of cellular automata with finite arity local functions is shown to be 2-primitive. The radical for cellular automata on a finite space group is shown to be large, in contrast to the case of torsion free space groups. The quotient near-ring is determined in this case.
Keywords: Cellular Automata, near-rings, algebra, reversibility, units, radical, semisimple, group alphabet
DOI: 10.32908/jca.v18.310718