Linear automaton transformations

TLDR: For a finite commutative ring with unit the authors determine which linear transformations M: RN—+RN can be realized by finite automata.

Abstract: Let R be a nonempty set, let N consist of all non-negative rational integers, and denote by RN the set of all functions on N to R. If R is a ring, a map M: R"—>P^ is linear if M(rxfx+r2f2)=rx(Mfx) +r2(Mf2) for rx, r2 in R, fx, f2 in RN. For a finite commutative ring with unit we determine which linear transformations M: RN—+RN can be realized by finite automata. More precisely, let A, B he finite nonempty sets. A map M: AN—>BN is an automaton transformation if there exists a finite set Q, maps Mq: A X£>—><2, Mb: A XQ-*B, elements h in B, q in Q such that corresponding to each/ in AN there exists an h in QN satisfying