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We show that every monoid which satisfies $u_n = v_n$ for each positive $n$ and generates the variety containing the bicyclic monoid is nonfinitely based.\n  This implies that the monoid of 2 by 2 upper triangular tropical matrices over the tropical semiring is nonfinitely based."},"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":"1509.01707","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-05T15:05:45Z","cross_cats_sorted":[],"title_canon_sha256":"092da857229efeaea34b1826fa70a2df9e9c0d6344bfb38b3ccd2f22fbd43103","abstract_canon_sha256":"0c36f65197bba8aaac91259d2cdb507f09197382b3b24924663d20f0ec109097"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:57.847179Z","signature_b64":"djokE54ixtDrbawm+8ERYN4qqU54XOmdLHD/OLDMZxiu3xrQOQhki9t1XDRk3oNWkCZLwrkhoSZqEN3XFmGuBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5023a90d3139255d16cfc65546667a0c037364ff11752a34c71653c5debe348","last_reissued_at":"2026-05-18T01:04:57.846541Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:57.846541Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The finite basis problem for the monoid of 2 by 2 upper triangular tropical matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Olga Sapir, Xun Hu, Yanfeng Luo, Yuzhu Chen","submitted_at":"2015-09-05T15:05:45Z","abstract_excerpt":"For each positive $n$, let $u_n = v_n$ denote the identity obtained from the Adjan identity $(xy) (yx) (xy) (xy) (yx) = (xy) (yx) (yx) (xy) (yx)$ by substituting $(xy) \\rightarrow (x_1 x_2 \\dots x_n)$ and $(yx) \\rightarrow (x_n \\dots x_2 x_1)$. We show that every monoid which satisfies $u_n = v_n$ for each positive $n$ and generates the variety containing the bicyclic monoid is nonfinitely based.\n  This implies that the monoid of 2 by 2 upper triangular tropical matrices over the tropical semiring is nonfinitely based."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01707","kind":"arxiv","version":3},"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":"1509.01707","created_at":"2026-05-18T01:04:57.846647+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.01707v3","created_at":"2026-05-18T01:04:57.846647+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01707","created_at":"2026-05-18T01:04:57.846647+00:00"},{"alias_kind":"pith_short_12","alias_value":"YUBDVEGTCOJF","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"YUBDVEGTCOJFLULM","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"YUBDVEGT","created_at":"2026-05-18T12:29:52.810259+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/YUBDVEGTCOJFLULM7RSVIZTHUD","json":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD.json","graph_json":"https://pith.science/api/pith-number/YUBDVEGTCOJFLULM7RSVIZTHUD/graph.json","events_json":"https://pith.science/api/pith-number/YUBDVEGTCOJFLULM7RSVIZTHUD/events.json","paper":"https://pith.science/paper/YUBDVEGT"},"agent_actions":{"view_html":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD","download_json":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD.json","view_paper":"https://pith.science/paper/YUBDVEGT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.01707&json=true","fetch_graph":"https://pith.science/api/pith-number/YUBDVEGTCOJFLULM7RSVIZTHUD/graph.json","fetch_events":"https://pith.science/api/pith-number/YUBDVEGTCOJFLULM7RSVIZTHUD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD/action/storage_attestation","attest_author":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD/action/author_attestation","sign_citation":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD/action/citation_signature","submit_replication":"https://pith.science/pith/YUBDVEGTCOJFLULM7RSVIZTHUD/action/replication_record"}},"created_at":"2026-05-18T01:04:57.846647+00:00","updated_at":"2026-05-18T01:04:57.846647+00:00"}