On wild algebras and super-decomposable pure-injective modules

Assume that $k$ is an algebraically closed field and $A$ is a finite-dimensional wild $k$-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed $A$-modules is undefined and hence there exists a super-decomposable pure-injective $A$-module, if the base field $k$ is countable. Here we give a proof of a stronger result. Namely, we show that there exists a special family of pointed $A$-modules, called an independent pair of dense chains of pointed modules. This also yields that the width of the lattice of all pointed $A$-modules is undefined.