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Two ordinals $\\alpha$ and $\\beta$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $\\alpha$ and $\\beta$ respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if $\\alpha$ and $\\beta$ are two countable ordinals, with $\\alpha \\leq \\beta$, then $\\alpha$ and $\\beta$ are orthogonal if and only if either $\\omega + 1\\leq \\alpha$ or $\\alpha =\\omega$ and $\\beta < \\omega \\beta$."},"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":"1407.0940","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-07-03T15:03:06Z","cross_cats_sorted":[],"title_canon_sha256":"911c4a2c933b45058aabce1904cdc5213801e2cb3f274d4f25c0ae93a11c1437","abstract_canon_sha256":"63bda8843756e5a099d73427c3fd9906add11ffcfe5afd66a8b8606d1939cbe4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:25.112346Z","signature_b64":"M+0VTLIj87BiYyuJjDth/FZmbe56VEr/+3qLCH5uPSuP8dBRCPDy2bZDL2I8ijG96iX7jtAEmf/mKRynZpuWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bf3364f1f460a1ac399d784d5228e79083100b758f949152c8d96a47faa54d3","last_reissued_at":"2026-05-18T02:48:25.111740Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:25.111740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pairs of orthogonal countable ordinals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Claude Laflamme, Imed Zaguia, Maurice Pouzet, Nobert Sauer","submitted_at":"2014-07-03T15:03:06Z","abstract_excerpt":"We characterize pairs of orthogonal countable ordinals. 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We prove that if $\\alpha$ and $\\beta$ are two countable ordinals, with $\\alpha \\leq \\beta$, then $\\alpha$ and $\\beta$ are orthogonal if and only if either $\\omega + 1\\leq \\alpha$ or $\\alpha =\\omega$ and $\\beta < \\omega \\beta$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0940","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.0940","created_at":"2026-05-18T02:48:25.111844+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.0940v1","created_at":"2026-05-18T02:48:25.111844+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0940","created_at":"2026-05-18T02:48:25.111844+00:00"},{"alias_kind":"pith_short_12","alias_value":"PPZTMTY7IYFB","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"PPZTMTY7IYFBVQ4Z","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"PPZTMTY7","created_at":"2026-05-18T12:28:43.426989+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE","json":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE.json","graph_json":"https://pith.science/api/pith-number/PPZTMTY7IYFBVQ4Z26CNKIUOPE/graph.json","events_json":"https://pith.science/api/pith-number/PPZTMTY7IYFBVQ4Z26CNKIUOPE/events.json","paper":"https://pith.science/paper/PPZTMTY7"},"agent_actions":{"view_html":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE","download_json":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE.json","view_paper":"https://pith.science/paper/PPZTMTY7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.0940&json=true","fetch_graph":"https://pith.science/api/pith-number/PPZTMTY7IYFBVQ4Z26CNKIUOPE/graph.json","fetch_events":"https://pith.science/api/pith-number/PPZTMTY7IYFBVQ4Z26CNKIUOPE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE/action/storage_attestation","attest_author":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE/action/author_attestation","sign_citation":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE/action/citation_signature","submit_replication":"https://pith.science/pith/PPZTMTY7IYFBVQ4Z26CNKIUOPE/action/replication_record"}},"created_at":"2026-05-18T02:48:25.111844+00:00","updated_at":"2026-05-18T02:48:25.111844+00:00"}