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Protasova","submitted_at":"2016-03-29T10:28:53Z","abstract_excerpt":"Given two ordinal $\\lambda$ and $\\gamma$, let $f:[0,\\lambda) \\rightarrow [0,\\gamma)$ be a function such that, for each $\\alpha<\\gamma$, $\\sup\\{f(t): t\\in[0, \\alpha]\\}<\\gamma.$ We define a mapping $d_{f}: [0,\\lambda)\\times [0,\\lambda) \\longrightarrow [0,\\gamma)$ by the rule: if $x<y$ then $d_{f}(x,y)= d_{f}(y,x)= \\sup\\{f(t): t\\in(x,y]\\}$, $d(x,x)=0$. The pair $([0,\\lambda), d_{f})$ is called a $\\gamma-$comb defined by $f$. We show that each cellular ordinal ballean can be represented as a $\\gamma-$comb. In {\\it General Asymptology}, cellular ordinal balleans play a part of ultrametric spaces."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1603.08713","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-03-29T10:28:53Z","cross_cats_sorted":[],"title_canon_sha256":"08eedc7e9138a5cc4ed786a33388d8d00fe279f10c4a89dd14f526a5c921839c","abstract_canon_sha256":"4f74c682903923fe4aea4c6caeb040bcb6bb0906e03d0d5181db01e54a26683d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:05.523010Z","signature_b64":"TgpgWoqavs/A1+rvoJDIMaNqyucrV9MEViZk2U5tHXAf/kYaQnlMakz9k9XvSfPgLWhjAlfqmO3wJdBsE+grDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d71e70bb871517f3eadd4867efd5a031e4dd8792a509f92ab55c626bdf19620","last_reissued_at":"2026-05-18T01:18:05.522415Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:05.522415Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The comb-like representations of cellular ordinal balleans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"I.V. Protasov, K.D. Protasova","submitted_at":"2016-03-29T10:28:53Z","abstract_excerpt":"Given two ordinal $\\lambda$ and $\\gamma$, let $f:[0,\\lambda) \\rightarrow [0,\\gamma)$ be a function such that, for each $\\alpha<\\gamma$, $\\sup\\{f(t): t\\in[0, \\alpha]\\}<\\gamma.$ We define a mapping $d_{f}: [0,\\lambda)\\times [0,\\lambda) \\longrightarrow [0,\\gamma)$ by the rule: if $x<y$ then $d_{f}(x,y)= d_{f}(y,x)= \\sup\\{f(t): t\\in(x,y]\\}$, $d(x,x)=0$. The pair $([0,\\lambda), d_{f})$ is called a $\\gamma-$comb defined by $f$. We show that each cellular ordinal ballean can be represented as a $\\gamma-$comb. 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