Equations in words and lengths with endomorphisms (Q1317588)

Let \(F\) (\(P\)) be the free group (semigroup) on two generators \(a,b\) and let \(\phi\) be the automorphism of \(F\) (endomorphism of \(P\)) extending \(\phi(a) = ab\), \(\phi(b) = b\). The author studies the decidability of systems of equations in the first order theory of \(F\) (\(P\)) enriched with the endomorphism \(\phi\) and word lengths. The author constructs a system of equations \(\Psi(x,x_ 1,\dots,x_ n)\) of the form \(w = 1\;\&\;\&_{(i,j)\in B} | x_ i| = | x_ j|\;\&\;\phi(x_ 1) = x_ 2\) such that for an arbitrary positive integer \(p\) the solvability of the system \(\Psi(a^ p,x_ 1,\dots, x_ n)\) in \(F\) is undecidable. Similar results are proved for the free semigroup \(P\).