Regular sequences and synchronized sequences in abstract numeration systems

The notion of b -regular sequences was extended to linear recurring bases by Allouche, Scheicher and Tichy in 2000, abstract numeration systems Maes Rigo 2002. Their definitions are based on a S -kernel that extends -kernel. However, these do not allow us generalize all the many characterizations sequences. In this paper, we present an alternative definition -kernel, hence sequences, which enables use recognizable formal series order most (if all) known systems. We then give two -automatic as particular Next, general method obtain various families enumerating -recognizable properties As example possible applications method, show that, provided addition is -recognizable, factor complexity sequence defines sequence. last part study -synchronized Along way, provide constructive proof, only weighted automata, obtained composition synchronized relation recognizable. consequence, shown be -regular. All our results presented arbitrary dimension d for semiring K .