Revisiting regular sequences in light of rational base numeration systems

Regular sequences generalize the extensively studied automatic sequences. Let S be an abstract numeration system. When language L is prefix-closed and regular, a sequence said to S-regular if module generated by its S-kernel finitely generated. In this paper, we give new characterization of such in terms underlying tree T(L) whose nodes are words L. We may decorate these interest following breadth-first enumeration. For regular L, prove that only decorated linear, i.e., decoration node depends linearly on decorations fixed number ancestors. Next, introduce study rational base system, known highly non-regular. motivate discuss our definition pq-regular linear. first few properties sequences, provide examples them, propose method for guessing pq-regularity. Then relationship between pq-automatic finally present graph-directed linear representation sequence. Our permits us highlight places where regularity plays predominant role.