we characterize the rational subsets of a finite group and discuss the relations to integral cayley graphs.

Although they can be topologically different, two distinct transducers may actually recognize the same rational relation. Being able to test the equivalence of transducers allows to implement such operations as incremental minimization and iterative composition. This paper presents an algorithm for testing the equivalence of deterministic weighted finite-state transducers, and outlines an imple...

We prove that there exist some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Sim92]. Then we show that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class Σα ( respectively Π 0 α). Furthermore one cannot decide whether a given infinitary rational relation is a Borel ...

In a recent paper, this author studied invariant ideals in abelian group algebras under the action of certain infinite, locally finite, quasi-simple groups. While the main result was reasonably definitive, there are nevertheless certain natural extensions that should be considered. One approach to a proof of such extensions is to improve the basic tools that were used in the original work. Howe...

We introduce α-synchronous relations for a rational number α. We show that if a rational relation is both αand α-synchronous for two different numbers α and α, then it is recognizable. We give a synchronization algorithm for α-synchronous transducers. We also prove the closure under boolean operations and composition of α-synchronous relations.

An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang’s conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techni...

We state a simple condition on a rational subset X of a free monoid B for the existence of a sequential function that is a one-to-one mapping of some free monoid A onto X. As a by-product we obtain new sequential bijections of a free monoid onto another.

We consider the set Rω(Γ, D) of infinite real traces, over a dependence alphabet (Γ, D) with no isolated letter, equipped with the topology induced by the prefix metric. We then prove that all rational languages of infinite real traces are analytic sets. We reprove also that there exist some rational languages of infinite real traces which are analytic but non Borel sets, and even Σ11-complete,...

We give in this paper an example of infinitary rational relation, accepted by a 2-tape Büchi automaton, which is Π 3 -complete in the Borel hierarchy. Moreover the example of infinitary rational relation given in this paper has a very simple structure and can be easily described by its sections.