{"paper":{"title":"Relative uniform convergence and Archimedean property in pre-ordered vector spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Quotienting a pre-ordered vector space by the intersection of the ru-closure of its positive wedge with the negative produces an Archimedeanization.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Eduard Emelyanov","submitted_at":"2026-02-18T12:49:47Z","abstract_excerpt":"It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is an Archimedeanization of $X$, where $W$ is the closure of the positive wedge $X_+$ in ru-topology, $A=W\\cap(-W)$, and $[W]$ is the quotient set of $W$ in $X/A$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"It is proved that, for a pre-ordered vector space X, the quotient space (X/A,[W]) is an Archimedeanization of X, where W is the closure of the positive wedge X+ in ru-topology, A=W∩(-W), and [W] is the quotient set of W in X/A.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The relative uniform topology is well-defined on the pre-ordered space and the closure W forms a wedge such that the quotient order is compatible and Archimedean.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For a pre-ordered vector space X, the quotient (X/A, [W]) with W the ru-closure of X+ and A = W ∩ (-W) is an Archimedeanization of X.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quotienting a pre-ordered vector space by the intersection of the ru-closure of its positive wedge with the negative produces an Archimedeanization.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"91dba510823e476a03b147f472ba9d0a4adca6740e05a6fd807f85fd1b61cd98"},"source":{"id":"2602.16419","kind":"arxiv","version":7},"verdict":{"id":"cfc1463b-5c2f-4b84-bb64-fb0ee6fca5c3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T21:15:22.383965Z","strongest_claim":"It is proved that, for a pre-ordered vector space X, the quotient space (X/A,[W]) is an Archimedeanization of X, where W is the closure of the positive wedge X+ in ru-topology, A=W∩(-W), and [W] is the quotient set of W in X/A.","one_line_summary":"For a pre-ordered vector space X, the quotient (X/A, [W]) with W the ru-closure of X+ and A = W ∩ (-W) is an Archimedeanization of X.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The relative uniform topology is well-defined on the pre-ordered space and the closure W forms a wedge such that the quotient order is compatible and Archimedean.","pith_extraction_headline":"Quotienting a pre-ordered vector space by the intersection of the ru-closure of its positive wedge with the negative produces an Archimedeanization."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.16419/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ea90787b412f567cb540cabcf3dfa58052935f229277f9e3f6c5f395a6dd8288"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}