{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:OMMFVDLVLZIQP5RPN5KYMOF6FV","short_pith_number":"pith:OMMFVDLV","canonical_record":{"source":{"id":"math/0604454","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2006-04-20T17:45:56Z","cross_cats_sorted":[],"title_canon_sha256":"305e9e0b4e1ef7fb6ec11c321a83b2b8fd2fe8f19beb01530972b0ebf5745225","abstract_canon_sha256":"1445f85e14033066773429090fce8ffd20fc1ec024e18af2965f514d99c66942"},"schema_version":"1.0"},"canonical_sha256":"73185a8d755e5107f62f6f558638be2d448cbbbf8acc6de01e4fda0b9ac987bc","source":{"kind":"arxiv","id":"math/0604454","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0604454","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"math/0604454v2","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0604454","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"OMMFVDLVLZIQ","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"OMMFVDLVLZIQP5RP","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"OMMFVDLV","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:OMMFVDLVLZIQP5RPN5KYMOF6FV","target":"record","payload":{"canonical_record":{"source":{"id":"math/0604454","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2006-04-20T17:45:56Z","cross_cats_sorted":[],"title_canon_sha256":"305e9e0b4e1ef7fb6ec11c321a83b2b8fd2fe8f19beb01530972b0ebf5745225","abstract_canon_sha256":"1445f85e14033066773429090fce8ffd20fc1ec024e18af2965f514d99c66942"},"schema_version":"1.0"},"canonical_sha256":"73185a8d755e5107f62f6f558638be2d448cbbbf8acc6de01e4fda0b9ac987bc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:07.769481Z","signature_b64":"PDiz88CzlSjXaLefx0WnSiQRBQGYF+cEKfnKA9elQbMZixIzr+jGpSW6cqE7zlFi72Lxe/CLdken50XSzGdPDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73185a8d755e5107f62f6f558638be2d448cbbbf8acc6de01e4fda0b9ac987bc","last_reissued_at":"2026-05-18T03:02:07.768757Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:07.768757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0604454","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VXSEE85z4P/mPcYhYJACPNexylTnFEZv3+IdxOeVynNwuCxEEb4POUHgkq0919LsPsYjNYtS1KFlY6rDe+/sCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T00:44:24.176388Z"},"content_sha256":"9e81fde3fd1c05de14793dd0a18458b835a1ebd65879f041c54c370a70520ae5","schema_version":"1.0","event_id":"sha256:9e81fde3fd1c05de14793dd0a18458b835a1ebd65879f041c54c370a70520ae5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:OMMFVDLVLZIQP5RPN5KYMOF6FV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generators, extremals and bases of max cones","license":"","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Hans Schneider, Peter Butkovic, Sergei Sergeev","submitted_at":"2006-04-20T17:45:56Z","abstract_excerpt":"Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in ${{\\mathbb R}}_+^n$. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0604454","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kgh9WEmP6l2uHgvnq4DtH6YRRK74026NRhokgKccdrw8LSASkc2KBd4me43wZtKdg1d8eLHHp/3Iiv2CAzTOCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T00:44:24.176734Z"},"content_sha256":"2591ecfbbf640d9014f130cb3a4cdb074c8ef48e46c95f1488682196a26103fa","schema_version":"1.0","event_id":"sha256:2591ecfbbf640d9014f130cb3a4cdb074c8ef48e46c95f1488682196a26103fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV/bundle.json","state_url":"https://pith.science/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV/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":"2026-06-22T00:44:24Z","links":{"resolver":"https://pith.science/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV","bundle":"https://pith.science/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV/bundle.json","state":"https://pith.science/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OMMFVDLVLZIQP5RPN5KYMOF6FV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:OMMFVDLVLZIQP5RPN5KYMOF6FV","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":"1445f85e14033066773429090fce8ffd20fc1ec024e18af2965f514d99c66942","cross_cats_sorted":[],"license":"","primary_cat":"math.RA","submitted_at":"2006-04-20T17:45:56Z","title_canon_sha256":"305e9e0b4e1ef7fb6ec11c321a83b2b8fd2fe8f19beb01530972b0ebf5745225"},"schema_version":"1.0","source":{"id":"math/0604454","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0604454","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"arxiv_version","alias_value":"math/0604454v2","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0604454","created_at":"2026-05-18T03:02:07Z"},{"alias_kind":"pith_short_12","alias_value":"OMMFVDLVLZIQ","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"OMMFVDLVLZIQP5RP","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"OMMFVDLV","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:2591ecfbbf640d9014f130cb3a4cdb074c8ef48e46c95f1488682196a26103fa","target":"graph","created_at":"2026-05-18T03:02:07Z","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":"Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in ${{\\mathbb R}}_+^n$. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dep","authors_text":"Hans Schneider, Peter Butkovic, Sergei Sergeev","cross_cats":[],"headline":"","license":"","primary_cat":"math.RA","submitted_at":"2006-04-20T17:45:56Z","title":"Generators, extremals and bases of max cones"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0604454","kind":"arxiv","version":2},"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":"sha256:9e81fde3fd1c05de14793dd0a18458b835a1ebd65879f041c54c370a70520ae5","target":"record","created_at":"2026-05-18T03:02:07Z","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":"1445f85e14033066773429090fce8ffd20fc1ec024e18af2965f514d99c66942","cross_cats_sorted":[],"license":"","primary_cat":"math.RA","submitted_at":"2006-04-20T17:45:56Z","title_canon_sha256":"305e9e0b4e1ef7fb6ec11c321a83b2b8fd2fe8f19beb01530972b0ebf5745225"},"schema_version":"1.0","source":{"id":"math/0604454","kind":"arxiv","version":2}},"canonical_sha256":"73185a8d755e5107f62f6f558638be2d448cbbbf8acc6de01e4fda0b9ac987bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73185a8d755e5107f62f6f558638be2d448cbbbf8acc6de01e4fda0b9ac987bc","first_computed_at":"2026-05-18T03:02:07.768757Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:07.768757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PDiz88CzlSjXaLefx0WnSiQRBQGYF+cEKfnKA9elQbMZixIzr+jGpSW6cqE7zlFi72Lxe/CLdken50XSzGdPDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:07.769481Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0604454","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9e81fde3fd1c05de14793dd0a18458b835a1ebd65879f041c54c370a70520ae5","sha256:2591ecfbbf640d9014f130cb3a4cdb074c8ef48e46c95f1488682196a26103fa"],"state_sha256":"36a3269bd8ebfbeff515cdb24b06b7a9c0099018b12612fbef9bd5c308fefd38"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QBBf3BiKHlpVX7GdpZehjzMuBgTpDDGBdN+HAnLQNGb6KRB3LAMvzAMXTsdBBnfqnaMhnFDkmiY8r9M4y5RFBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T00:44:24.178894Z","bundle_sha256":"5bcd7792a1c8707695d6c868733f3b812aad70ac9547b00d7e44f64961aa03a6"}}