{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:AUFSEMSFHXMDXG66TR4NPZUSZY","short_pith_number":"pith:AUFSEMSF","canonical_record":{"source":{"id":"1704.02666","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-04-09T22:25:47Z","cross_cats_sorted":[],"title_canon_sha256":"f5b0db1880a6e8a2cae646c7d98149a0b1b2e5ea7f8cfd03964f83be5f2a3bd3","abstract_canon_sha256":"4549433ceecf4dd122f96144e5bf6f0e3a7b52d66237ab7a5c44daffe91e3f36"},"schema_version":"1.0"},"canonical_sha256":"050b2232453dd83b9bde9c78d7e692ce36eedfff11f3804b77ae44513b529fec","source":{"kind":"arxiv","id":"1704.02666","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02666","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02666v1","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02666","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"pith_short_12","alias_value":"AUFSEMSFHXMD","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"AUFSEMSFHXMDXG66","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"AUFSEMSF","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:AUFSEMSFHXMDXG66TR4NPZUSZY","target":"record","payload":{"canonical_record":{"source":{"id":"1704.02666","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-04-09T22:25:47Z","cross_cats_sorted":[],"title_canon_sha256":"f5b0db1880a6e8a2cae646c7d98149a0b1b2e5ea7f8cfd03964f83be5f2a3bd3","abstract_canon_sha256":"4549433ceecf4dd122f96144e5bf6f0e3a7b52d66237ab7a5c44daffe91e3f36"},"schema_version":"1.0"},"canonical_sha256":"050b2232453dd83b9bde9c78d7e692ce36eedfff11f3804b77ae44513b529fec","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:56.539620Z","signature_b64":"0MCqKKljH5UeoPFlbSimT7DRl37Sst49SSz/JcKRTrdGTcAZCI0GzcfBQP69GdLC6NEthWA1w7uCzTddXD5FBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"050b2232453dd83b9bde9c78d7e692ce36eedfff11f3804b77ae44513b529fec","last_reissued_at":"2026-05-18T00:20:56.539125Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:56.539125Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.02666","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":"2026-05-18T00:20:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MkLMKpiIyrrPRYKC+cGlF7OFhmBVFQzR4jtrIm8nqQrchlQNwaYF589kI4Qf+0xCc2eUSxq8+TVjGy/FKVywBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:31:28.681626Z"},"content_sha256":"96ddc91463f448ff732642f0877a595648fd71dd1ea9a203847d56d1134564da","schema_version":"1.0","event_id":"sha256:96ddc91463f448ff732642f0877a595648fd71dd1ea9a203847d56d1134564da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:AUFSEMSFHXMDXG66TR4NPZUSZY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ordered groups as a tensor category","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Dale Rolfsen","submitted_at":"2017-04-09T22:25:47Z","abstract_excerpt":"It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms. With this functor interpreted as a tensor product this category becomes a tensor (or monoidal) category. Moreover, if $O(G)$ denotes the space of orderings of the group $G$ with the natural topology, then for fixed groups $F$ and $G$ our construction can be considered a function $O(F) \\times O(G) \\to O(F * G)$. We sh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02666","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":"2026-05-18T00:20:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AdESmrix2VZptZiYVDtMtI5R5pK7s21zkMQQqV19Zf+4/0EZb3MLRmv0bGqSN/CPWcL5U+7hnOMonmP9ftqpCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:31:28.682260Z"},"content_sha256":"080187d3c8d02bed2d0709cec93df6d8f06c7a3bcc60ead7a3f58021824e4ed6","schema_version":"1.0","event_id":"sha256:080187d3c8d02bed2d0709cec93df6d8f06c7a3bcc60ead7a3f58021824e4ed6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AUFSEMSFHXMDXG66TR4NPZUSZY/bundle.json","state_url":"https://pith.science/pith/AUFSEMSFHXMDXG66TR4NPZUSZY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AUFSEMSFHXMDXG66TR4NPZUSZY/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":"2026-05-27T05:31:28Z","links":{"resolver":"https://pith.science/pith/AUFSEMSFHXMDXG66TR4NPZUSZY","bundle":"https://pith.science/pith/AUFSEMSFHXMDXG66TR4NPZUSZY/bundle.json","state":"https://pith.science/pith/AUFSEMSFHXMDXG66TR4NPZUSZY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AUFSEMSFHXMDXG66TR4NPZUSZY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:AUFSEMSFHXMDXG66TR4NPZUSZY","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":"4549433ceecf4dd122f96144e5bf6f0e3a7b52d66237ab7a5c44daffe91e3f36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-04-09T22:25:47Z","title_canon_sha256":"f5b0db1880a6e8a2cae646c7d98149a0b1b2e5ea7f8cfd03964f83be5f2a3bd3"},"schema_version":"1.0","source":{"id":"1704.02666","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02666","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02666v1","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02666","created_at":"2026-05-18T00:20:56Z"},{"alias_kind":"pith_short_12","alias_value":"AUFSEMSFHXMD","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"AUFSEMSFHXMDXG66","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"AUFSEMSF","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:080187d3c8d02bed2d0709cec93df6d8f06c7a3bcc60ead7a3f58021824e4ed6","target":"graph","created_at":"2026-05-18T00:20:56Z","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":"It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms. With this functor interpreted as a tensor product this category becomes a tensor (or monoidal) category. Moreover, if $O(G)$ denotes the space of orderings of the group $G$ with the natural topology, then for fixed groups $F$ and $G$ our construction can be considered a function $O(F) \\times O(G) \\to O(F * G)$. We sh","authors_text":"Dale Rolfsen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-04-09T22:25:47Z","title":"Ordered groups as a tensor category"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02666","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":"sha256:96ddc91463f448ff732642f0877a595648fd71dd1ea9a203847d56d1134564da","target":"record","created_at":"2026-05-18T00:20:56Z","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":"4549433ceecf4dd122f96144e5bf6f0e3a7b52d66237ab7a5c44daffe91e3f36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-04-09T22:25:47Z","title_canon_sha256":"f5b0db1880a6e8a2cae646c7d98149a0b1b2e5ea7f8cfd03964f83be5f2a3bd3"},"schema_version":"1.0","source":{"id":"1704.02666","kind":"arxiv","version":1}},"canonical_sha256":"050b2232453dd83b9bde9c78d7e692ce36eedfff11f3804b77ae44513b529fec","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"050b2232453dd83b9bde9c78d7e692ce36eedfff11f3804b77ae44513b529fec","first_computed_at":"2026-05-18T00:20:56.539125Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:56.539125Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0MCqKKljH5UeoPFlbSimT7DRl37Sst49SSz/JcKRTrdGTcAZCI0GzcfBQP69GdLC6NEthWA1w7uCzTddXD5FBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:56.539620Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.02666","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96ddc91463f448ff732642f0877a595648fd71dd1ea9a203847d56d1134564da","sha256:080187d3c8d02bed2d0709cec93df6d8f06c7a3bcc60ead7a3f58021824e4ed6"],"state_sha256":"ecce9154b252a5a5c0dc3f1b22086c4392cd496d379ea93c11007a319641dcca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iGUIaskXwCm56gOX+ZfiekavUOiBrKILCOX4mlbFq9vXWbSf+xORfuI09Ib+idluvrl3Z6y7d+mMvMLMGaQsCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:31:28.685804Z","bundle_sha256":"b927cbd42895cd887f307f64b6d1a794a0334bdce04cd5ada33ee75d05f5276f"}}