We introduce and study some natural operations on a structure of finite labeled forests, which is crucial in extending the difference hierarchy to the case of partitions. It is shown that the corresponding quotient algebra modulo the so-called h-equivalence is the simplest non-trivial semilattice with discrete closures. The algebra is also characterized as a free algebra in some quasivariety. Part of the results is generalized to countable labeled forests with finite chains.
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