  The aim of this work is to generalize Johnson's techniques in order to apply
them to establish a bijective correspondence between $S$-derivations and
continuous derivations on $M_a(S,\omega),$ where $S$ is a locally compact
foundation semigroup with identity $e$, and $\omega$ is a weight function on
$S$, and apply it to find a necessary condition for amenability of weighted
group algebras.
