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

SummaryIn this paper we exhibit two different effective constructions of the syntactic algebraAS associated to a recognizable formal series on treesS.The one method consists of a direct construction of (=a copy ofAS) 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 toAS and thus independent of the choice of (AS φ).