On p/q-recognisable sets

Let p/q be a rational number. Numeration in base is defined by function that evaluates each finite word over A_p={0,1,...,p-1} to some We let N_p/q denote the image of this evaluation function. In particular, contains all nonnegative integers and literature on usually focuses set words are evaluated integers; it rather chaotic language which not context-free. On contrary, we study here subsets (N_p/q)^d p/q-recognisable, i.e. realised automata (A_p)^d. First, give characterisation these sets as those definable first-order logic, similar one given B\"uchi-Bruy\`ere Theorem for integer bases numeration systems. Second, show natural order relation modulo-q operator p/q-recognisable.