The signature of rational languages

After having set in detail the framework of signature, we study and characterise the signatures of rational languages. Using a known construction from numeration system theory, we show that these signatures form a special subclass of morphic words. We then use this framework to give an alternative definition to morphic numeration systems (also called Dumont-Thomas numeration systems). We finally highlight that the classes of morphic numeration systems and of (prefix-closed) rational abstract numeration systems are essentially the same.