{"paper":{"title":"Universal homogeneous two-sorted ultrametric spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings.","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam Barto\\v{s}, Aleksandra Kwiatkowska, Maciej Malicki, Wies{\\l}aw Kubi\\'s","submitted_at":"2026-05-13T14:41:48Z","abstract_excerpt":"We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining isometries and linear order embeddings. We show that the class of all finite two-sorted ultrametric spaces with dc-embeddings is Fra\\\"iss\\'e, and that the limit is the countable rational Urysohn ultrametric space $\\mathbb{U}$. The space $\\mathbb{U}$ is dc-universal for all countable ultrametric spaces, and its Cauchy completion $\\overline{\\mathbb{U}}$ is dc-uni"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and the limit is the countable rational Urysohn ultrametric space U. The space U is dc-universal for all countable ultrametric spaces, and its Cauchy completion is dc-universal for all separable ultrametric spaces.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the class of finite two-sorted ultrametric spaces equipped with distance-carrying embeddings satisfies the hereditary, joint embedding, and amalgamation properties required for the Fraïssé theorem to apply.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The countable rational Urysohn ultrametric space U is the Fraïssé limit of finite two-sorted ultrametric spaces under distance-carrying embeddings and is dc-universal for countable ultrametric spaces, with its completion universal for separable ones.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eecf9f950813d5d6c7b95772be719592f0f4d09f8fc4ec254288e16c7574853e"},"source":{"id":"2605.13608","kind":"arxiv","version":1},"verdict":{"id":"c1734e2e-a1f6-4927-80a0-dbc179ce4924","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:51:44.248421Z","strongest_claim":"The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé, and the limit is the countable rational Urysohn ultrametric space U. The space U is dc-universal for all countable ultrametric spaces, and its Cauchy completion is dc-universal for all separable ultrametric spaces.","one_line_summary":"The countable rational Urysohn ultrametric space U is the Fraïssé limit of finite two-sorted ultrametric spaces under distance-carrying embeddings and is dc-universal for countable ultrametric spaces, with its completion universal for separable ones.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the class of finite two-sorted ultrametric spaces equipped with distance-carrying embeddings satisfies the hereditary, joint embedding, and amalgamation properties required for the Fraïssé theorem to apply.","pith_extraction_headline":"Treating ultrametric spaces as two-sorted structures with ordered distances yields a countable homogeneous universal space under distance-carrying embeddings."},"references":{"count":33,"sample":[{"doi":"","year":2026,"title":"A. Bartoš, W. Kubiś , Hereditarily indecomposable continua as generic mathematical structures , Selecta Math. (N.S.) 32 (2026), no. 1, Paper No. 14","work_id":"6b153337-e890-493c-9cb4-65566f827110","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"A. Bartoš, W. Kubiś, A. Kwiatkowska, M. Malicki , Generic dc-auto\\-morphisms of two-sorted ultrametric spaces , preprint, 2026","work_id":"4f380068-be15-4e42-9bd4-9a8f03b7da38","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"R. Camerlo, A. Marcone, L. Motto Ros , Isometry groups of Polish ultrametric space , arXiv:2508.08480","work_id":"96e65b86-ae09-4cd5-9c55-ebba89541c77","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"On homogeneous ultrametric spaces","work_id":"1d79085c-400e-4f7c-8127-0d61ec1c4ea6","ref_index":4,"cited_arxiv_id":"1509.04346","is_internal_anchor":true},{"doi":"","year":1984,"title":"F. Delon , Espaces ultram\\'etriques , J. Symbolic Logic 49 (1984), no. 2, 405--424","work_id":"2420d06a-6f5b-4e2d-8363-8d0c71c3473f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":33,"snapshot_sha256":"3a22d8af1fa0024415d2dc0b17bed478776ea37f6e2d7723611b08b16dbb2fa2","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2ab77da2ea35f12066d59495abff4d7f7c7b247eededd65f3c473d87fdcf5c7b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}