  We investigate properties of convergent and forward-closed string rewriting
systems in the context of the syntactic criteria introduced in
\cite{LynchMorawska} by Christopher Lynch and Barbara Morawska (we call these
$LM$-Systems). Since a string rewriting system can be viewed as a
term-rewriting system over a signature of purely monadic function symbols, we
adapt their definition to the string rewriting case. We prove that the
subterm-collapse problem for convergent and forward-closed string rewriting
systems is effectively solvable. Therefore, there exists a decision procedure
that verifies if such a system is an $LM$-System. We use the same construction
to prove that the \emph{cap problem} from the field of cryptographic protocol
analysis, which is undecidable for general $LM$-systems, is decidable when
restricted to the string rewriting case.
