Abstract
A group tessellation automaton (TA) is one whose state alphabet consists of the elements of a group, and whose local transformation is defined in terms of the group's composition rule. If the domain and range of a group TA mapping are restricted to finite configurations, the mapping is one-to-one. On the set of all configurations (including infinite ones), group mappings are onto, but not one-to-one. Questions about the existence of predecessors and successors of configurations, are undecidable for general TA, are constructively solvable for group TA. It is shown that every finite configuration of an arbitrary group TA is self-reproducing. A method of calculating successive generations of abelian group TA configurations is presented, and then used to demonstrate a self-reproduction theorem for abelian group TA. The arithmetical numbers, a generalization of binomial coefficients, are used to derive this result.