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We show the powers $\\omega^{(n)}$ of $\\omega$ in the group of divisors $\\mathrm{Div}({\\cal R}_{\\mathbb{K}}[H])$ is identical with the ordinal powers of $\\omega$, describe the $\\mathbb{K}$-vector space basis of $\\omega^{(n)}$ for $n\\in\\mathbb{Z}$. Further, we show that the fiber cones $\\bigoplus_{n\\geq 0}\\omega^n/\\mathfrak{m}\\omega^n$ and $\\bigoplus_{n\\geq0}(\\omega^{(-1)})^n"},"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":"1804.01046","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-04-03T15:52:43Z","cross_cats_sorted":[],"title_canon_sha256":"0253439f4603d669161e3e9bf7c7b3271816f564d6ad5f44e07aaa6f76a4c0a1","abstract_canon_sha256":"e094e773e0328d3bbf59179a007de514da70a2e9ed9d52566dbf29e633c9b4da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:55.622616Z","signature_b64":"kxnX5IqTFVrzduDZ7W2PWvAgB8GVoNLCFt45N2VWfU1VublAVdqPy/6Av73vIpC01sUZGfYMPHZE/wmFGm6yCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62adc7268899e5c1e68904c75be70acb5481d478d3cf39ecac8a01f2ac730490","last_reissued_at":"2026-05-17T23:48:55.622087Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:55.622087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Mitsuhiro Miyazaki","submitted_at":"2018-04-03T15:52:43Z","abstract_excerpt":"Let ${\\cal R}_{\\mathbb{K}}[H]$ be the Hibi ring over a field $\\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $\\omega$ the canonical ideal of ${\\cal R}_{\\mathbb{K}}[H]$. We show the powers $\\omega^{(n)}$ of $\\omega$ in the group of divisors $\\mathrm{Div}({\\cal R}_{\\mathbb{K}}[H])$ is identical with the ordinal powers of $\\omega$, describe the $\\mathbb{K}$-vector space basis of $\\omega^{(n)}$ for $n\\in\\mathbb{Z}$. Further, we show that the fiber cones $\\bigoplus_{n\\geq 0}\\omega^n/\\mathfrak{m}\\omega^n$ and $\\bigoplus_{n\\geq0}(\\omega^{(-1)})^n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01046","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":"1804.01046","created_at":"2026-05-17T23:48:55.622163+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.01046v3","created_at":"2026-05-17T23:48:55.622163+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01046","created_at":"2026-05-17T23:48:55.622163+00:00"},{"alias_kind":"pith_short_12","alias_value":"MKW4OJUITHS4","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_16","alias_value":"MKW4OJUITHS4DZUJ","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_8","alias_value":"MKW4OJUI","created_at":"2026-05-18T12:32:37.024351+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/MKW4OJUITHS4DZUJATDVXZYKZN","json":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN.json","graph_json":"https://pith.science/api/pith-number/MKW4OJUITHS4DZUJATDVXZYKZN/graph.json","events_json":"https://pith.science/api/pith-number/MKW4OJUITHS4DZUJATDVXZYKZN/events.json","paper":"https://pith.science/paper/MKW4OJUI"},"agent_actions":{"view_html":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN","download_json":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN.json","view_paper":"https://pith.science/paper/MKW4OJUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.01046&json=true","fetch_graph":"https://pith.science/api/pith-number/MKW4OJUITHS4DZUJATDVXZYKZN/graph.json","fetch_events":"https://pith.science/api/pith-number/MKW4OJUITHS4DZUJATDVXZYKZN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN/action/storage_attestation","attest_author":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN/action/author_attestation","sign_citation":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN/action/citation_signature","submit_replication":"https://pith.science/pith/MKW4OJUITHS4DZUJATDVXZYKZN/action/replication_record"}},"created_at":"2026-05-17T23:48:55.622163+00:00","updated_at":"2026-05-17T23:48:55.622163+00:00"}