Fine hierarchies via Priestley duality

In applications of the fine hierarchies their characterizations in terms of the so called alternating trees are of principal importance. Also, in many cases a suitable version of many–one reducibility exists that fits a given fine hierarchy. With a use of Priestley duality we obtain a surprising result that suitable versions of alternating trees and of m-reducibilities may be found for any given fine hierarchy, i.e. the methods of alternating trees andm-reducibilities are quite general, which is of some methodological interest.Along with the hierarchies of sets, we consider also more general hierarchies ofk-partitions and in this context propose some new notions and establish new results, in particular extend the above-mentioned results for hierarchies of sets.