This clear, well-organized paper develops two different constructions of a syntactic algebra associated with a recognizable formal series on trees. The paper begins with a brief and useful description of the general properties of algebras and syntactic trees. In the first construction, the author directly builds a subspace with a finite dimension generated by a finite list. The author demonstrates finiteness in a clear way, using induction. In the second construction, referred to as minimization, the author uses the kernel of a function (defined on the vector space associated with a family of multilinear functions), which is an ideal, to build a quotient algebra isomorphic to the searched syntactic algebra.