Dual purities and interrelation between the \(P\)-purities and the purities given by systems of equations (Q1334395)

Let \(P\) be a unary predicate defined on the class of all the groups ``closed'' to subgroups and factor groups. A subgroup \(H\) is called \(P\)- pure in a group \(G\) if every morphism from \(H\) to an arbitrary group \(A\) is extendible to \(G\) if \(P(A)\) is true. Using the \(P\)-purity one defines \(P\)-pure exactness and \(P\)-pure projectivity. A \(P\)-purity is called dual if any group \(G\) for which \(P\) holds is \(P\)-pure-projective. Theorem. A group \(G\) such that every element is contained in a subgroup for which \(P\) is true is \(P\)-pure-projective for a dual \(P\)-purity iff \(G\) is a retract of a coproduct of groups satisfying \(P\). The second part of the paper deals with interrelations between \(P\)- purities and purities given by systems of equations such as: if \(H\) is a \(P\)-pure subgroup of \(G\) and \(P(H)\) is true then \(H\) is absolutely pure in \(G\) (i.e. if every system of equations over \(H\) which has solutions in \(G\) has also solutions in \(H\)).