Indecomposable local maps of tessellation automata

Indecomposable local maps of one-dimensional tessellation automata are studied. The main results of this paper are the following. (1) For any alphabet ∑ containing two or more symbols and for anyn≥ 1, there exist indecomposable scope-n local maps over ∑. (2) If ∑ is a finite field of prime order, then a linear scope-n local map over ∑ is indecomposable if and only if its associated polynomial is an irreducible polynomial of degreen − 1 over ∑, except for a trivial case. (3) Result (2) is no longer true if ∑ is a finite field whose order is not prime.