  We perform a study on the struck quark to the $\Lambda$-hyperon fragmentation
processes by taking into account the anti-quark fragmentations and intermediate
decays from other hyperons. We concentrate on how the longitudinally polarized
quark fragments to the longitudinally polarized $\Lambda$, how unpolarized
quark and anti-quark fragment to the unpolarized $\Lambda$, and how quark and
anti-quark fragment to the $\Lambda$ through the intermediate decay processes.
We calculate the effective fragmentation functions in the light-cone SU(6)
quark-spectator-diquark model via the Gribov-Lipatov relation, with the
Melosh-Wigner rotation effect also included. The calculated results are in
reasonable agreement with the HERMES semi-inclusive $ep$ experimental data and
the OPAL and ALEPH $e^{+}e^{-}$ annihilation experimental data.
