Effective Projections on Group Shifts to Decide Properties of Group Cellular Automata

Many decision problems concerning cellular automata are known to be decidable in the case of algebraic automata, that is, when state set has an structure and automaton acts as a morphism. The most studied cases include finite fields, commutative rings groups. In this paper, we provide methods generalize these results broader group where is (possibly non-commutative) group. configuration space not even necessarily full shift but subshift — called subgroup on [Formula: see text], for any number text] dimensions. We show, particular, injectivity, surjectivity, equicontinuity, sensitivity nilpotency non-transitivity semi-decidable. Injectivity always implies jointly periodic points dense limit set. Moore direction Garden-of-Eden theorem holds all while Myhill fails some cases. proofs based effective projection operations shifts are, applied valid space-time diagrams automata. This allows one effectively construct traces sets A preliminary version work was presented at conference Mathematical Foundations Computer Science 2020.