We introduce and study some natural operations on the homomorphic
quasiorder of finite labeled forests which is of central
interest for extending the difference hierarchy to the case of
partitions. It is shown that the corresponding algebra is the simplest
nontrivial semilattice with discrete closures. The algebra is also
characterized as a free algebra in some quasivariety. Some of results
are generalized to countable labeled forests without infinite chains.