The boolean hierarchy of NP-partitions

The quotient algebra of labeled forests modulo h-equivalence

“…We will characterize the corresponding algebra in the class of so-called semilattices with discrete closures, introduced in our previous work on complete numberings and fine hierarchies of the arithmetical sets and functions (see [6,7]). We also extend the main results of this paper, as well as some of the results in [1,4,5], to the countable case, which yields a generalization of the classical Hausdorff difference hierarchy to the case of k-partitions.…”

“…Can the notion of the Boolean hierarchy be naturally extended to arbitrary k in place of 2? A partial answer to this question was suggested in [4,5] by introducing a hierarchy RBH k (L), which was called the Boolean hierarchy of k-partitions over L, because elements of k M are in a natural bijective correspondence with k-partitions of M (i.e., pairwise disjoint sets A 0 , . .…”

Hierarchies of Δ02‐measurable k ‐partitions

“…In this section we recall some definitions and facts for the so called homomorphic preorder [8,9,13,14,21,23]. Most posets considered here are assumed to be (at most) countable and without infinite chains.…”

“…In this section we consider the Boolean hierarchy of k-partitions, which was studied for the case of finite k-posets in [13,14,21] and for the countable case in [23]. Let P = (P ; ≤) be a countable poset without infinite chains, X be a space, and L be a σ-base in X (see Section 1).…”

“…In Sections 4 and 5 we recall definitions and some properties of two well-known generalizations of the Hausdorff hierarchy from [19,11,10,6] which we call here the limit-hierarchy and the level-hierarchy. After recalling some auxiliary facts in Section 6 we proceed in Section 7 with discussing the so called Boolean hierarchy of k-partitions [14,13,21,23] which is another generalization of the Hausdorff hierarchy refining the limit-and level-hierarchies. In Section 8 we characterize the structure of Wadge degrees of ∆ 0 2 -measurable k-partitions of the Baire space; this extends a result in [9,21].…”