{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:PXJWYSDZ5QFA4YJJNLD2NP7YNN","short_pith_number":"pith:PXJWYSDZ","schema_version":"1.0","canonical_sha256":"7dd36c4879ec0a0e61296ac7a6bff86b6e8b836546f6ad5b4909c95b03e08022","source":{"kind":"arxiv","id":"2411.15081","version":1},"attestation_state":"computed","paper":{"title":"Transformation Semigroups Which Are Disjoint Union of Symmetric Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Kritsada Sangkhanan, Utsithon Chaichompoo","submitted_at":"2024-11-22T17:17:54Z","abstract_excerpt":"Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by\n  $$\n  T_{E^*}(X)=\\{\\alpha\\in T(X):\\text{for all}\\ x,y\\in X, (x,y)\\in E\\Leftrightarrow (x\\alpha,y\\alpha)\\in E\\}.\n  $$\n  We have the regular part of $T_{E^*}(X)$, denoted by $\\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by\n  $$\n  Q_{E^*}(X)=\\{\\alpha\\in T_{E^*}(X):|A\\alpha|=1\\ \\text{and}\\ A\\cap X\\alpha\\neq\\emptyset\\ \\text{for all}\\ A\\in X/E\\}.\n  $$\n  T"},"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":"2411.15081","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2024-11-22T17:17:54Z","cross_cats_sorted":[],"title_canon_sha256":"60aa90f6c3ce3be545286b7ca61f6145b8020170c5b95580663eeac33089fbc4","abstract_canon_sha256":"59d2de3883643ad519c202d3cd0d3982ab2f3ebe670cccb6dbc5368c2c953c67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:39:16.072757Z","signature_b64":"eLuyXu9siSGmEK/1plkvabfjsINiHwkHt5gMQsuZvQYVOW3TzBIsBRkm11oFfZFlN2ZYHnZt8eg9Vgan3cHRBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dd36c4879ec0a0e61296ac7a6bff86b6e8b836546f6ad5b4909c95b03e08022","last_reissued_at":"2026-07-05T09:39:16.072335Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:39:16.072335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transformation Semigroups Which Are Disjoint Union of Symmetric Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Kritsada Sangkhanan, Utsithon Chaichompoo","submitted_at":"2024-11-22T17:17:54Z","abstract_excerpt":"Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by\n  $$\n  T_{E^*}(X)=\\{\\alpha\\in T(X):\\text{for all}\\ x,y\\in X, (x,y)\\in E\\Leftrightarrow (x\\alpha,y\\alpha)\\in E\\}.\n  $$\n  We have the regular part of $T_{E^*}(X)$, denoted by $\\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by\n  $$\n  Q_{E^*}(X)=\\{\\alpha\\in T_{E^*}(X):|A\\alpha|=1\\ \\text{and}\\ A\\cap X\\alpha\\neq\\emptyset\\ \\text{for all}\\ A\\in X/E\\}.\n  $$\n  T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15081","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15081/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2411.15081","created_at":"2026-07-05T09:39:16.072392+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.15081v1","created_at":"2026-07-05T09:39:16.072392+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.15081","created_at":"2026-07-05T09:39:16.072392+00:00"},{"alias_kind":"pith_short_12","alias_value":"PXJWYSDZ5QFA","created_at":"2026-07-05T09:39:16.072392+00:00"},{"alias_kind":"pith_short_16","alias_value":"PXJWYSDZ5QFA4YJJ","created_at":"2026-07-05T09:39:16.072392+00:00"},{"alias_kind":"pith_short_8","alias_value":"PXJWYSDZ","created_at":"2026-07-05T09:39:16.072392+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/PXJWYSDZ5QFA4YJJNLD2NP7YNN","json":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN.json","graph_json":"https://pith.science/api/pith-number/PXJWYSDZ5QFA4YJJNLD2NP7YNN/graph.json","events_json":"https://pith.science/api/pith-number/PXJWYSDZ5QFA4YJJNLD2NP7YNN/events.json","paper":"https://pith.science/paper/PXJWYSDZ"},"agent_actions":{"view_html":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN","download_json":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN.json","view_paper":"https://pith.science/paper/PXJWYSDZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.15081&json=true","fetch_graph":"https://pith.science/api/pith-number/PXJWYSDZ5QFA4YJJNLD2NP7YNN/graph.json","fetch_events":"https://pith.science/api/pith-number/PXJWYSDZ5QFA4YJJNLD2NP7YNN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN/action/storage_attestation","attest_author":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN/action/author_attestation","sign_citation":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN/action/citation_signature","submit_replication":"https://pith.science/pith/PXJWYSDZ5QFA4YJJNLD2NP7YNN/action/replication_record"}},"created_at":"2026-07-05T09:39:16.072392+00:00","updated_at":"2026-07-05T09:39:16.072392+00:00"}