Finiteness property in Cantor real numeration systems

We consider a numeration system which is a common generalization of the positional systems introduced by Cantor and Renyi. Number representations are obtained using a composition of beta(k)-transformations for a given sequence of real bases B = (beta k)(k >= 1), beta(k) > 1. We focus on arithmetical properties of the set of numbers with finite B-expansion in case that B is an alternate base, i.e B is a periodic sequence. We provide necessary conditions for the so-called finiteness property. We further show a sufficient condition using rewriting rules on the language of representations. The proof is constructive and provides a method for performing addition of expansions in alternate bases. Finally, we give a family of alternate bases that satisfy this sufficient condition. Our work generalizes the results of Frougny and Solomyak obtained for the case when the base B is a constant sequence.