{"paper":{"title":"Semigroups uniquely determined by one-sided identity and zero sets","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Every right group with maximal subgroup size 2 and every commutative-rectangular band is uniquely determined by the one-sided identity and zero sets of its elements.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Julia Maddox","submitted_at":"2024-10-30T21:36:45Z","abstract_excerpt":"For a groupoid $S$ with elements $a$ and $b$, if $ba = a$, then $b$ is a left identity of $a$ and $a$ is a right zero of $b$. We define the left identity set of $a$ to be the set of all left identities of $a$ in $S$, and similarly for the right identity set of $a$ in $S$. We defined the left zero set of $a$ to be the set of all left zeroes of $a$ in $S$, and similarly for the right zero set of $a$. The one-sided identity and zero sets of a semigroup can be utilized in the determination of its maximal subgroups, maximal left and right zero subsemigroups, maximal left and right subgroups, and re"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove every right group with maximal subgroup size 2 is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements. We prove a commutative-rectangular band is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The structures under consideration satisfy the defining properties of a right group (every element has a right identity and right inverse) or of a commutative-rectangular band (every pair commutes or forms generalized inverses), which are invoked to establish that matching identity/zero sets force identical operations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Right groups with maximal subgroup size 2 and commutative-rectangular bands are stabilized semigroups uniquely determined by their one-sided identity and zero sets.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every right group with maximal subgroup size 2 and every commutative-rectangular band is uniquely determined by the one-sided identity and zero sets of its elements.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6c4064b2279bae56a649a85d1f304f8babb36f6ac70064f3da45c09acb2bd9cd"},"source":{"id":"2410.23473","kind":"arxiv","version":5},"verdict":{"id":"7c32f21a-1b04-45c7-bee7-f768b84309ea","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T19:20:44.951189Z","strongest_claim":"We prove every right group with maximal subgroup size 2 is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements. We prove a commutative-rectangular band is a stabilized semigroup with respect to the one-sided identity [zero] sets of its elements.","one_line_summary":"Right groups with maximal subgroup size 2 and commutative-rectangular bands are stabilized semigroups uniquely determined by their one-sided identity and zero sets.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The structures under consideration satisfy the defining properties of a right group (every element has a right identity and right inverse) or of a commutative-rectangular band (every pair commutes or forms generalized inverses), which are invoked to establish that matching identity/zero sets force identical operations.","pith_extraction_headline":"Every right group with maximal subgroup size 2 and every commutative-rectangular band is uniquely determined by the one-sided identity and zero sets of its elements."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.23473/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"}