Effective construction of the syntactic algebra of a recognizable series on trees (Q911788)

We exhibit two different effective constructions of the syntactic algebra \({\mathfrak A}_ S\) associated to a recognizable formal series on trees S. The first method consists of a direct construction of \({\mathfrak V}_ S\) \((=\) a copy of \({\mathfrak A}_ S)\) which is the subspace \[ {\mathfrak V}_ S=<t^{-1}S/t\in T_{\Sigma}>\subseteq F^{P_{\Sigma}} \] with the natural algebra structure. The second method consists of considering an arbitrary surjective realization (\({\mathfrak A},\phi)\) of S, defining an appropriate ideal \({\mathfrak a}\) of \({\mathfrak A}\) and then constructing the quotient algebra \({\mathfrak A}/{\mathfrak a}\); this quotient is isomorphic to \({\mathfrak A}_ S\) and thus independent of the choice of (\({\mathfrak A},\phi)\).