The Quotient Algebra of Labeled Forests Modulo h-Equivalence

We introduce and study some natural operations on the structure 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 quotient-algebra modulo the so called h-equivalence 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.