Retracts and retractions of free algebras. I
Let \(V\) be a variety of linear algebras and \(P\) a projective \(V\)-algebra. Then \(P\) is a retract of a \(V\)-free algebra \(G\) with an embedding \(\theta:P\to G\) and a projection \(\pi:G\to P,\quad \theta\pi=1_P\). These morphisms induce morphisms \[ \phi(\pi,\theta):P\to F,\quad \psi(\pi,\theta):F\to P, \] where \(F\) is a free \(V\)-algebra whose basis coincides with a basis of the algebra \(P/P^2\) with trivial multiplication. The aim of the paper is a study of the morphisms \( \phi(\pi,\theta)\) and \(\psi(\pi,\theta)\). The paper presents some sufficient condition under which \( \phi(\pi,\theta)\) is an isomorphism.
scientific article; zbMATH DE number 1309955