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 algebra

associated to a recognizable formal series on treesS.

The one method consists of a direct construction of

(=a copy of

) which is the subspace

with the natural algebra structure.

We first determine a basis

$$S\tau _1^{ - 1} ,...,S\tau _m^{ - 1} $$

of the subspace

$$F_S = \left\langle {{{S\tau ^{ - 1} } \mathord{\left/ {\vphantom {{S\tau ^{ - 1} } {\tau \in P_\Sigma }}} \right. \kern-\nulldelimiterspace} {\tau \in P_\Sigma }}} \right\rangle \subseteq F^{T_\Sigma } $$

and then, using the junction isomorphism

we obtain a basis for

.

The second method consists of considering an arbitrary surjective realization (

, φ) ofS, defining an appropriate ideal ℬ of

and then constructing the quotient algebra

; this quotient is isomorphic to

and thus independent of the choice of (

φ).