{"paper":{"title":"Characterizations of Cancellable Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.LO","authors_text":"Matthew Harrison-Trainor, Meng-Che \"Turbo\" Ho","submitted_at":"2018-09-19T13:43:53Z","abstract_excerpt":"An abelian group $A$ is said to be cancellable if whenever $A \\oplus G$ is isomorphic to $A \\oplus H$, $G$ is isomorphic to $H$. We show that the index set of cancellable rank 1 torsion-free abelian groups is $\\Pi^0_4$ $m$-complete, showing that the classification by Fuchs and Loonstra cannot be simplified. For arbitrary non-finitely generated groups, we show that the cancellation property is $\\Pi^1_1$ $m$-hard; we know of no upper bound, but we conjecture that it is $\\Pi^1_2$ $m$-complete."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07191","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}