{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2N7YCO5UYZ2LESQANPXBSQPNFY","short_pith_number":"pith:2N7YCO5U","canonical_record":{"source":{"id":"1503.05531","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-03-18T18:53:11Z","cross_cats_sorted":[],"title_canon_sha256":"f9029a07f60810a5cfb984466dad53f854c0f181800eb316e6ccc15773b25a34","abstract_canon_sha256":"14a469e048006bef8f29eac5b37f3262fadf44c5dad745d473fce05ca945e001"},"schema_version":"1.0"},"canonical_sha256":"d37f813bb4c674b24a006bee1941ed2e1bafbfbc3cc081d48105bbb57098fd20","source":{"kind":"arxiv","id":"1503.05531","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.05531","created_at":"2026-05-18T01:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"1503.05531v1","created_at":"2026-05-18T01:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.05531","created_at":"2026-05-18T01:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"2N7YCO5UYZ2L","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2N7YCO5UYZ2LESQA","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2N7YCO5U","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2N7YCO5UYZ2LESQANPXBSQPNFY","target":"record","payload":{"canonical_record":{"source":{"id":"1503.05531","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-03-18T18:53:11Z","cross_cats_sorted":[],"title_canon_sha256":"f9029a07f60810a5cfb984466dad53f854c0f181800eb316e6ccc15773b25a34","abstract_canon_sha256":"14a469e048006bef8f29eac5b37f3262fadf44c5dad745d473fce05ca945e001"},"schema_version":"1.0"},"canonical_sha256":"d37f813bb4c674b24a006bee1941ed2e1bafbfbc3cc081d48105bbb57098fd20","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:49.006982Z","signature_b64":"A+xK6cv0Dzlw8sATVjB0RwwKN5KgbMnC1H6zGqC9SO2MaYhgaE5/VY1HIr/nBtGtfo/PlOHKrkz7x+rAh8PoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d37f813bb4c674b24a006bee1941ed2e1bafbfbc3cc081d48105bbb57098fd20","last_reissued_at":"2026-05-18T01:09:49.006417Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:49.006417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.05531","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xtVipDlF/WTXZmG+zLzFFr2GMOeSXQDHO/I61vbcwW4+4jiYU+dVE/zQSnXkSNA32WrhhWwzSvTZ38KKuWwyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:21:26.766951Z"},"content_sha256":"546f78befa84618d123d8f68f45a8fc2d407f7995edd9381033b4fa155c04488","schema_version":"1.0","event_id":"sha256:546f78befa84618d123d8f68f45a8fc2d407f7995edd9381033b4fa155c04488"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2N7YCO5UYZ2LESQANPXBSQPNFY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cofinite subsets and double negation topologies on locales of filters and ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Jos\\'e Manuel Garc\\'ia-Calcines, Luis Espa\\~nol, M. Carmen M\\'inguez","submitted_at":"2015-03-18T18:53:11Z","abstract_excerpt":"We study the role of the filter $c\\mathcal{K}(X)$ of cofinite subsets of $X$ in the locale $\\mathcal{F}ilt(X)$ of all filters on $X$, by means of the double negation topology of $\\mathcal{F}ilt(X)$, and an essential locale morphism $\\mathcal{P}(X)^{op}\\to\\mathcal{F}ilt(X)$. Moreover, in the case $X=\\mathbb{N}$, we characterise cofinite subsets by means of the double negation topology on the monoid $\\mathbb{M}$ of the maps $\\mathbb{N}\\to \\mathbb{N}$ with finite fibers, or on the submonoid $\\mathbb{E}\\subseteq \\mathbb{M}$ of the monotone and injective maps $\\mathbb{N}\\to\\mathbb{N}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05531","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PJducq2n/pGnpVcVga83bijlNpd/oYhGMyZ+GtCpRnTdA6TM+eEYky4KZWbgFH44p238I9uptr2qgv4mW6qKBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:21:26.767300Z"},"content_sha256":"61f68585af5bbd07901460b3eb39f69de68f516a62669f83e95608c028073981","schema_version":"1.0","event_id":"sha256:61f68585af5bbd07901460b3eb39f69de68f516a62669f83e95608c028073981"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2N7YCO5UYZ2LESQANPXBSQPNFY/bundle.json","state_url":"https://pith.science/pith/2N7YCO5UYZ2LESQANPXBSQPNFY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2N7YCO5UYZ2LESQANPXBSQPNFY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T23:21:26Z","links":{"resolver":"https://pith.science/pith/2N7YCO5UYZ2LESQANPXBSQPNFY","bundle":"https://pith.science/pith/2N7YCO5UYZ2LESQANPXBSQPNFY/bundle.json","state":"https://pith.science/pith/2N7YCO5UYZ2LESQANPXBSQPNFY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2N7YCO5UYZ2LESQANPXBSQPNFY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2N7YCO5UYZ2LESQANPXBSQPNFY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"14a469e048006bef8f29eac5b37f3262fadf44c5dad745d473fce05ca945e001","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-03-18T18:53:11Z","title_canon_sha256":"f9029a07f60810a5cfb984466dad53f854c0f181800eb316e6ccc15773b25a34"},"schema_version":"1.0","source":{"id":"1503.05531","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.05531","created_at":"2026-05-18T01:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"1503.05531v1","created_at":"2026-05-18T01:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.05531","created_at":"2026-05-18T01:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"2N7YCO5UYZ2L","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2N7YCO5UYZ2LESQA","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2N7YCO5U","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:61f68585af5bbd07901460b3eb39f69de68f516a62669f83e95608c028073981","target":"graph","created_at":"2026-05-18T01:09:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the role of the filter $c\\mathcal{K}(X)$ of cofinite subsets of $X$ in the locale $\\mathcal{F}ilt(X)$ of all filters on $X$, by means of the double negation topology of $\\mathcal{F}ilt(X)$, and an essential locale morphism $\\mathcal{P}(X)^{op}\\to\\mathcal{F}ilt(X)$. Moreover, in the case $X=\\mathbb{N}$, we characterise cofinite subsets by means of the double negation topology on the monoid $\\mathbb{M}$ of the maps $\\mathbb{N}\\to \\mathbb{N}$ with finite fibers, or on the submonoid $\\mathbb{E}\\subseteq \\mathbb{M}$ of the monotone and injective maps $\\mathbb{N}\\to\\mathbb{N}$.","authors_text":"Jos\\'e Manuel Garc\\'ia-Calcines, Luis Espa\\~nol, M. Carmen M\\'inguez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-03-18T18:53:11Z","title":"Cofinite subsets and double negation topologies on locales of filters and ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05531","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:546f78befa84618d123d8f68f45a8fc2d407f7995edd9381033b4fa155c04488","target":"record","created_at":"2026-05-18T01:09:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"14a469e048006bef8f29eac5b37f3262fadf44c5dad745d473fce05ca945e001","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2015-03-18T18:53:11Z","title_canon_sha256":"f9029a07f60810a5cfb984466dad53f854c0f181800eb316e6ccc15773b25a34"},"schema_version":"1.0","source":{"id":"1503.05531","kind":"arxiv","version":1}},"canonical_sha256":"d37f813bb4c674b24a006bee1941ed2e1bafbfbc3cc081d48105bbb57098fd20","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d37f813bb4c674b24a006bee1941ed2e1bafbfbc3cc081d48105bbb57098fd20","first_computed_at":"2026-05-18T01:09:49.006417Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:49.006417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A+xK6cv0Dzlw8sATVjB0RwwKN5KgbMnC1H6zGqC9SO2MaYhgaE5/VY1HIr/nBtGtfo/PlOHKrkz7x+rAh8PoDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:49.006982Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.05531","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:546f78befa84618d123d8f68f45a8fc2d407f7995edd9381033b4fa155c04488","sha256:61f68585af5bbd07901460b3eb39f69de68f516a62669f83e95608c028073981"],"state_sha256":"b4fb05b70f964e947bcfe9fc8a3fae75275607df3b79d1d0227b5675ac047fe8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TiPvFUe6dAlgl1FIQZB7kks2lTOc0lg/oxRGyVZxs8Hyrm2Ptd94+H6/sWCfkvNk6VjTPoDvQr6g4KowRcAoDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:21:26.769413Z","bundle_sha256":"fcf52fcf81f72c5405e05fcfd32b48b415cc78eec80193f0751cbefb100c2ee2"}}