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The piecewise testable separability problem asks, given two input languages, whether there exists a piecewise testable language that contains the first input language and is disjoint from the second. We prove a general characterisation of piecewise testable separability on languages in a well-quasiorder, in terms of ideals of the ordering. This subsumes the known characterisations in the case of finite words. In the case of finite ranked trees ordered by homeomorphic embedding, we show usingdoi:10.4230/lipics.icalp.2016.97 dblp:conf/icalp/Goubault-Larrecq16 fatcat:6kzs33bamfgvlhuor72x7o4nra