{"paper":{"title":"Product cones in dense pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Pantelis E. Eleftheriou","submitted_at":"2017-08-13T12:14:41Z","abstract_excerpt":"Let $\\mathcal M=\\langle M, <, +, \\dots\\rangle$ be an o-minimal expansion of an ordered group, and $P\\subseteq M$ a dense set such that certain tameness conditions hold. We introduce the notion of a `product cone' in $\\widetilde{\\mathcal M}=\\langle \\cal M, P\\rangle$, and prove: if $\\mathcal M$ expands a real closed field, then $\\widetilde{\\mathcal M}$ admits a product cone decomposition. If $\\mathcal M$ is linear, then it does not. In particular, we settle a question from [10]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.03894","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"}