Idempotence criteria for lazy cellular automata

@misc{alcalaarroyo2026idempotence,
  title = {{Idempotence criteria for lazy cellular automata}},
  author = {Edgar Alcalá-Arroyo and Alonso Castillo-Ramirez and Maria G. Magaña-Chávez and Luguis de los Santos Baños},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11811},
  eprint = {2610.11811},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11811},
}

TY  - UNPB
TI  - Idempotence criteria for lazy cellular automata
AU  - Edgar Alcalá-Arroyo
AU  - Alonso Castillo-Ramirez
AU  - Maria G. Magaña-Chávez
AU  - Luguis de los Santos Baños
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.11811
UR  - https://arxiv.org/abs/2610.11811
AB  - A lazy cellular automaton τ:A^G→ A^G is determined by a finite neighborhood S⊆ G containing the group identity e, a pattern p∈ A^S, and a writing symbol a∈ A∖{p(e)}: its local rule is equal to the projection to e except when it reads the pattern p, in which case it writes a. For a nonempty subset T of the fiber S_a:= p^−1(a), we introduce compatibility conditions on p along the right translates St, t∈ T. We show that, over every group, the existence of a compatible set is necessary for τ to be non-idempotent, and that it is also sufficient whenever st = ts for all t ∈ S_a, s ∈ S_p(e). In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of S_a. Finally, for every dihedral group D_n with n≥ 4, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.
ER  -