Rational sets in virtually abelian groups: languages and growth

In this paper, we generalise and unify the results and methods used by Benson, Liardet, Evetts, and Evetts & Levine to show that rational sets in a virtually abelian group G have rational (relative) growth series with respect to any generating set for G. We prove equivalences between the structures used in the literature and establish the rationality of important classes of sets in G: definable sets, algebraic sets, conjugacy representatives, and coset representatives (of any fixed subgroup), among others. Furthermore, we show that any rational set, when written as words over the generating set of G, has several EDT0L representations.