Recognizable formal series on trees and cofree coalgebraic systems
Let \(\Omega\) be a collection of sets, \(\Omega=\{\Omega_n\mid n\geq 0\}\), and \(F\) a field. A linear space \(A\) over \(F\) is an \(\Omega\)-coalgebra if for any \(\omega\in\Omega_n\) there is given a linear mapping \(A\to A^{\otimes n}\) which is also denoted by \(\omega\). Here \(A^{\otimes 0}=F\). This notion is a dual one to the definition of a multilinear \(\Omega\)-algebra in the sense of A. G. Kurosh. By a tree \(M\) the authors mean a free \(\Omega\)-algebra on a set \(X\) (free \(\Omega\)-magma). The linear span \(FM\) of \(M\) is a free multilinear \(\Omega\)-algebra on \(X\). A tree series is a mapping \(f\colon M\to F\). A tree series is recognizable if there exists a linear expansion of \(f\) to an \(FM\) such that \(\ker f\) contains an ideal of \(FM\) of finite codimension. Let \(\text{Rec}(\Omega)\) be the space of all recognizable trees. It is observed that \(\text{Rec}(\Omega)\) is an \(\Omega\)-coalgebra. One of the main results states that \(\text{Rec}(\Omega)\) is the cofree \(\Omega\)-coalgebra on \((FX)^*\). Moreover certain subcoalgebras of \(\text{Rec}(\Omega)\) form a cofree \(\Omega\)-coalgebra on \(FX\).
scientific article; zbMATH DE number 1309128
scientific article