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It embeds into its universal group $\\Upsilon$ $\\pm$ (P), the interval group of P , which is also the universal group of the homotopy groupoid of the chain complex of P. We prove the following results: $\\bullet$ The monoid $\\Upsilon$(P) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-sem"},"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":"1712.02787","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-07T09:26:21Z","cross_cats_sorted":[],"title_canon_sha256":"5922075b72674868bbaa3159bd72416730a31a58e131695308c4dcf44500e630","abstract_canon_sha256":"2f2d3d52c63ef82e583ecc5036564fe2418fe36dc5a94304a0d76f8ab4bbbf8e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:52.165946Z","signature_b64":"R8aKC95QkOKWU7DZH8DEJiS2ryROIxQJbpUis/hzRVA6S3c9gY89IbZuE+T+XcDK7OQha4I8Ls7oL5rWcdopDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b57fcafc2304bd1110f8f7d018d01bcc11dcf9898173fc1d6854330fb87fc82","last_reissued_at":"2026-05-18T00:15:52.165220Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:52.165220Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gcd-monoids arising from homotopy groupoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Friedrich Wehrung (LMNO)","submitted_at":"2017-12-07T09:26:21Z","abstract_excerpt":"The interval monoid $\\Upsilon$(P) of a poset P is defined by generators [x, y], where x $\\le$ y in P , and relations [x, x] = 1, [x, z] = [x, y] $\\times$ [y, z] for x $\\le$ y $\\le$ z. It embeds into its universal group $\\Upsilon$ $\\pm$ (P), the interval group of P , which is also the universal group of the homotopy groupoid of the chain complex of P. We prove the following results: $\\bullet$ The monoid $\\Upsilon$(P) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-sem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02787","kind":"arxiv","version":2},"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":"1712.02787","created_at":"2026-05-18T00:15:52.165327+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.02787v2","created_at":"2026-05-18T00:15:52.165327+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.02787","created_at":"2026-05-18T00:15:52.165327+00:00"},{"alias_kind":"pith_short_12","alias_value":"FNL7ZL6CGBF5","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_16","alias_value":"FNL7ZL6CGBF5CEIP","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_8","alias_value":"FNL7ZL6C","created_at":"2026-05-18T12:31:15.632608+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/FNL7ZL6CGBF5CEIPR56QDDIBXT","json":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT.json","graph_json":"https://pith.science/api/pith-number/FNL7ZL6CGBF5CEIPR56QDDIBXT/graph.json","events_json":"https://pith.science/api/pith-number/FNL7ZL6CGBF5CEIPR56QDDIBXT/events.json","paper":"https://pith.science/paper/FNL7ZL6C"},"agent_actions":{"view_html":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT","download_json":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT.json","view_paper":"https://pith.science/paper/FNL7ZL6C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.02787&json=true","fetch_graph":"https://pith.science/api/pith-number/FNL7ZL6CGBF5CEIPR56QDDIBXT/graph.json","fetch_events":"https://pith.science/api/pith-number/FNL7ZL6CGBF5CEIPR56QDDIBXT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT/action/storage_attestation","attest_author":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT/action/author_attestation","sign_citation":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT/action/citation_signature","submit_replication":"https://pith.science/pith/FNL7ZL6CGBF5CEIPR56QDDIBXT/action/replication_record"}},"created_at":"2026-05-18T00:15:52.165327+00:00","updated_at":"2026-05-18T00:15:52.165327+00:00"}