{"paper":{"title":"Maximal-Hull $z$-Ideals, Congruence Closures, and Coherent Frames of Commutative Semirings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Amartya Goswami, Pronay Biswas, Pubali Sengupta, Sujit Kumar Sardar","submitted_at":"2026-07-08T12:06:34Z","abstract_excerpt":"We develop a spectral theory of $z$-ideals for commutative semirings. The lattice $\\mathsf{ZId}(S)$ of $z$-ideals is a \\emph{coherent frame} for every commutative semiring $S$ -- unconditionally, without cancellativity, subtractivity, or Noetherian hypothesis -- so the prime spectrum $\\mathsf{Spec}_z(S)$ is spectral. Under an explicit finite-type hypothesis on the canonical congruence-generated closure~$g$, the lattice $\\mathsf{Id}_{g}(S)$ of $g$-closed ideals is likewise a coherent frame, and $\\mathsf{Spec}_g(S)$ is spectral and homeomorphic to the space of prime $g$-congruences. These frame "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07319","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/2607.07319/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"}