{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CYEWLRWANSE3IRL77FRUCGJUCP","short_pith_number":"pith:CYEWLRWA","canonical_record":{"source":{"id":"1208.3233","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-15T21:47:07Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"3781126a94af09985ef530960885e1b9cd69373c51f7ce1f555faa2fbc1795fb","abstract_canon_sha256":"ac4a1ada1529f19c8257966b550477fda776a186d9f459a606b6fb8079b076d4"},"schema_version":"1.0"},"canonical_sha256":"160965c6c06c89b4457ff96341193413f441060e89c6ffe8929ce3bc36d92c3c","source":{"kind":"arxiv","id":"1208.3233","version":7},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.3233","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1208.3233v7","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.3233","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"CYEWLRWANSE3","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CYEWLRWANSE3IRL7","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CYEWLRWA","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CYEWLRWANSE3IRL77FRUCGJUCP","target":"record","payload":{"canonical_record":{"source":{"id":"1208.3233","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-15T21:47:07Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"3781126a94af09985ef530960885e1b9cd69373c51f7ce1f555faa2fbc1795fb","abstract_canon_sha256":"ac4a1ada1529f19c8257966b550477fda776a186d9f459a606b6fb8079b076d4"},"schema_version":"1.0"},"canonical_sha256":"160965c6c06c89b4457ff96341193413f441060e89c6ffe8929ce3bc36d92c3c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:21.202577Z","signature_b64":"Al/qj+owxwlOFIZebiEOgqDjr4dQchB7eN7PUJX6kMyCHygWcjZxwK7gQs9ZeBG4zJKvTiTR1NvSjEGUB8RWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"160965c6c06c89b4457ff96341193413f441060e89c6ffe8929ce3bc36d92c3c","last_reissued_at":"2026-05-18T02:28:21.201951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:21.201951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.3233","source_version":7,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IGVtBZ5rp0INUjV30O2wYdIzm8Twzm+ylg+dsIoAPUrEI7dHS+Loc8VWC6N+NFYuXsegfKjOb4cJ7i99AZlIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T13:44:44.539036Z"},"content_sha256":"0ee97665a621e11099fc8a5d05fe3445a79d29a9d4c895ee1e8de051eb7d566a","schema_version":"1.0","event_id":"sha256:0ee97665a621e11099fc8a5d05fe3445a79d29a9d4c895ee1e8de051eb7d566a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CYEWLRWANSE3IRL77FRUCGJUCP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Small doubling in ordered semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Salvatore Tringali","submitted_at":"2012-08-15T21:47:07Z","abstract_excerpt":"Let $\\mathbb{A} = (A, \\cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\\mathbb{A}$ is linearly orderable if there exists a total order $\\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \\in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\\mathbb{A}$, then $|S^2| \\ge 3|S|-2$. On the road to this goal, we also prove a number of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3233","kind":"arxiv","version":7},"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":"2026-05-18T02:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7fi9Ti47C0OybgJaA1CoXsoHC/2o7wSCTzzIqu+QU9n58XbymBTjd3zEdVeyl8A2wJum+Z8rP+lj+q5XtlrQCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T13:44:44.539395Z"},"content_sha256":"be4716fa18e7614526e474f0ee07a6f085645f402ffab0d1d46c02bb6c770c33","schema_version":"1.0","event_id":"sha256:be4716fa18e7614526e474f0ee07a6f085645f402ffab0d1d46c02bb6c770c33"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CYEWLRWANSE3IRL77FRUCGJUCP/bundle.json","state_url":"https://pith.science/pith/CYEWLRWANSE3IRL77FRUCGJUCP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CYEWLRWANSE3IRL77FRUCGJUCP/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":"2026-07-01T13:44:44Z","links":{"resolver":"https://pith.science/pith/CYEWLRWANSE3IRL77FRUCGJUCP","bundle":"https://pith.science/pith/CYEWLRWANSE3IRL77FRUCGJUCP/bundle.json","state":"https://pith.science/pith/CYEWLRWANSE3IRL77FRUCGJUCP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CYEWLRWANSE3IRL77FRUCGJUCP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CYEWLRWANSE3IRL77FRUCGJUCP","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":"ac4a1ada1529f19c8257966b550477fda776a186d9f459a606b6fb8079b076d4","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-15T21:47:07Z","title_canon_sha256":"3781126a94af09985ef530960885e1b9cd69373c51f7ce1f555faa2fbc1795fb"},"schema_version":"1.0","source":{"id":"1208.3233","kind":"arxiv","version":7}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.3233","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1208.3233v7","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.3233","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"CYEWLRWANSE3","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CYEWLRWANSE3IRL7","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CYEWLRWA","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:be4716fa18e7614526e474f0ee07a6f085645f402ffab0d1d46c02bb6c770c33","target":"graph","created_at":"2026-05-18T02:28:21Z","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":"Let $\\mathbb{A} = (A, \\cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that $\\mathbb{A}$ is linearly orderable if there exists a total order $\\le$ on $A$ such that $xz < yz$ and $zx < zy$ for all $x,y,z \\in A$ with $x < y$. In particular, we find that if $S$ is a finite subset of $A$ generating a non-abelian subsemigroup of $\\mathbb{A}$, then $|S^2| \\ge 3|S|-2$. On the road to this goal, we also prove a number of","authors_text":"Salvatore Tringali","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-15T21:47:07Z","title":"Small doubling in ordered semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3233","kind":"arxiv","version":7},"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":"sha256:0ee97665a621e11099fc8a5d05fe3445a79d29a9d4c895ee1e8de051eb7d566a","target":"record","created_at":"2026-05-18T02:28:21Z","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":"ac4a1ada1529f19c8257966b550477fda776a186d9f459a606b6fb8079b076d4","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-15T21:47:07Z","title_canon_sha256":"3781126a94af09985ef530960885e1b9cd69373c51f7ce1f555faa2fbc1795fb"},"schema_version":"1.0","source":{"id":"1208.3233","kind":"arxiv","version":7}},"canonical_sha256":"160965c6c06c89b4457ff96341193413f441060e89c6ffe8929ce3bc36d92c3c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"160965c6c06c89b4457ff96341193413f441060e89c6ffe8929ce3bc36d92c3c","first_computed_at":"2026-05-18T02:28:21.201951Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:21.201951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Al/qj+owxwlOFIZebiEOgqDjr4dQchB7eN7PUJX6kMyCHygWcjZxwK7gQs9ZeBG4zJKvTiTR1NvSjEGUB8RWBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:21.202577Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.3233","source_kind":"arxiv","source_version":7}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ee97665a621e11099fc8a5d05fe3445a79d29a9d4c895ee1e8de051eb7d566a","sha256:be4716fa18e7614526e474f0ee07a6f085645f402ffab0d1d46c02bb6c770c33"],"state_sha256":"334cae17ff1920e654c442460654772b371241a30e909014dabc6dac7c5a656c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K4mTe6vPbG44/Ta8+dJEqGNXSt+w2qudUe8IKIlwHGPcDvYcm2iKf1Nyh5XLeoefk/k3Jojy2SbwgSRm6Ab6Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T13:44:44.543198Z","bundle_sha256":"d34aa2121715c80e9b929d484a0586e08c007b6b0e47add200dac2f1679502bc"}}