Effective construction of the syntactic algebra of a recognizable series on trees

Summary

In this paper we exhibit two different effective constructions of the syntactic algebraA _S associated to a recognizable formal series on treesS.

The one method consists of a direct construction of

(=a copy ofA _S ) which is the subspace

with the natural algebra structure.

We first determine a basis

of the subspace

and then, using the junction isomorphism

we obtain a basis for

.

The second method consists of considering an arbitrary surjective realization (A, φ) ofS, defining an appropriate ideal ℬ ofA and then constructing the quotient algebraA/ℬ this quotient is isomorphic toA _S and thus independent of the choice of (A _S φ).