In this paper, we explore the refined semilattice of left $C$-wrpp semigroups, and show that a left $C$-wrpp semigroup $S$ is a refined semilattice of left-${\cal R}$ cancellative stripes if and only if it is a spined product of a $C$-wrpp component and a left regular band. It is a generalization of the refined semilattice decomposition of left $C$-rpp semigroups.
