Surminimisation of Automata

We introduce the notion of surminimisation of a finite deterministic automaton; it consists in performing a transition relabelling before executing the minimisation and it produces an automaton smaller than a sole minimisation would. While the classical minimisation process preserves the accepted language, the surminimisation process preserves its underlying ordered tree only. Surminimisation induces on languages and on Abstract Rational Numeration Systems (ARNS) a transformation that we call label reduction. We prove that all positional numeration systems are label-irreducible and that an ARNS and its label reduction are very close, in the sense that converting the integer representations from one system into the other is done by a simple Mealy machine.