Multidimensional generalized automatic sequences and shape-symmetric morphic words

Numeration Systems Definition An abstract numeration system is a triple S = (L,Σ, <) where L is a regular language over a totally ordered alphabet (Σ, <). Enumerating the words of L with respect to the genealogical ordering induced by < gives a one-to-one correspondence repS : N → L valS = rep −1 S : L → N. Example L = a∗, Σ = {a} n 0 1 2 3 4 · · · rep (n) ε a aa aaa aaaa · · · Abstract Numeration SystemsNumeration Systems Example L = {a, b}∗, Σ = {a, b}, a < b n 0 1 2 3 4 5 6 7 · · · rep (n) ε a b aa ab ba bb aaa · · · Example L = a∗b∗, Σ = {a, b}, a < b n 0 1 2 3 4 5 6 · · · rep (n) ε a b aa ab bb aaa · · ·