{"paper":{"title":"Relative uniform completion of a vector lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The relative uniform completion of a vector lattice X inside a larger uniformly complete Z is the intersection of all uniformly complete sublattices of Z containing X.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Eugene Bilokopytov, Vladimir G. Troitsky","submitted_at":"2026-01-13T22:34:19Z","abstract_excerpt":"In the paper, we revisit several approaches to the concept of uniform completion $X^{\\mathrm{ru}}$ of a vector lattice $X$. We show that many of these approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly complete vector lattice $Z$ then $X^{\\mathrm{ru}}$ may be viewed as the intersection of all uniformly complete sublattices of $Z$ containing $X$. $X^{\\mathrm{ru}}$ may also be constructed via a transfinite process of taking uniform adherences in $Z$ with regulators coming from the previous adherences. If, in addition, $X$ is majorizing in $Z$ then $X^{\\mathrm{"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If X is a sublattice of a uniformly complete vector lattice Z then X^ru may be viewed as the intersection of all uniformly complete sublattices of Z containing X. X^ru may also be characterized via a universal property: every positive operator from X to a uniformly complete vector lattice extends uniquely to X^ru.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructions assume X is a vector lattice that can be embedded as a sublattice into some uniformly complete vector lattice Z, together with the standard definitions of uniform completeness and positive operators.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Multiple constructions of the relative uniform completion of a vector lattice coincide and it satisfies a universal property for extending positive operators to uniformly complete targets.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The relative uniform completion of a vector lattice X inside a larger uniformly complete Z is the intersection of all uniformly complete sublattices of Z containing X.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d8e49b9c7b6e1bcdedd5d7f55fd55307ebe66a3d38f9b9a4bb00d1df843dc3a6"},"source":{"id":"2601.09015","kind":"arxiv","version":2},"verdict":{"id":"d3e805e9-c02d-458b-b94d-5c1dd6cba459","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T14:06:01.427736Z","strongest_claim":"If X is a sublattice of a uniformly complete vector lattice Z then X^ru may be viewed as the intersection of all uniformly complete sublattices of Z containing X. X^ru may also be characterized via a universal property: every positive operator from X to a uniformly complete vector lattice extends uniquely to X^ru.","one_line_summary":"Multiple constructions of the relative uniform completion of a vector lattice coincide and it satisfies a universal property for extending positive operators to uniformly complete targets.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructions assume X is a vector lattice that can be embedded as a sublattice into some uniformly complete vector lattice Z, together with the standard definitions of uniform completeness and positive operators.","pith_extraction_headline":"The relative uniform completion of a vector lattice X inside a larger uniformly complete Z is the intersection of all uniformly complete sublattices of Z containing X."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"5e02614a5f3f202af8fb00ab594134787b907bb8810c613a9237c169b689be1a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}