  Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice
{\Gamma} acts on the left strictly conservatively. The following rigidity
results are obtained: morphisms, factors and joinings defined apriori only in
the measurable category are in fact algebraically constrained. Arguing in an
elementary fashion we manage to classify all the measurable {\Phi} commuting
with the {\Gamma}-action: assuming ergodicity, we find they are algebraically
defined.
