Let G be a group and let A finite set with at least two elements. cellular automaton (CA) over AG is function τ:AG→AG defined via memory S⊆G local μ:AS→A. The goal of this paper to introduce the definition generalized (GCA) τ:AG→AH, where H another arbitrary group, homomorphism ϕ:H→G. Our preserves essence CA, as we prove analogous versions three key results in theory CA: Curtis-Hedlund Theorem...

Finite automata have been used effectively in recent years to define infinite groups. The twomain lines of research have as their most representative objects the class of automatic groups (in-cluding word-hyperbolic groups as a particular case) and automata groups (singled out among themore general self-similar groups).The first approach implements in the language of automat...

Pin showed that every p-state automaton (p a prime) containing a cyclic permutation and a non-permutation has a synchronizing word of length at most (p− 1). In this paper we consider permutation automata with the property that adding any non-permutation will lead to a synchronizing word and establish bounds on the lengths of such synchronizing words. In particular, we show that permutation grou...

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products and abelian groups: after introducing enriched automata (i.e., with extra labels), we obtain an explicit bijection between a certain type such automata, which - it happens in group is computable finitely generated case. This approach provides neat geometric description (even non gener...