{"paper":{"title":"The finite basis problem for matrix semirings $\\mathbf{M}_n(S_7)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jun Jiao, Miaomiao Ren","submitted_at":"2026-06-12T03:04:14Z","abstract_excerpt":"We first prove an embedding theorem for matrix semirings $\\mathbf{M}_n(S)$ over an additively idempotent semiring $S$: for all $n \\geq 2$, $\\mathbf{M}_n(S)$ embeds into $\\mathbf{M}_{n+1}(S)$. This yields an ascending chain of varieties $\\mathsf{V}(\\mathbf{M}_2(S)) \\leq \\mathsf{V}(\\mathbf{M}_3(S)) \\leq \\cdots$, which is strictly ascending when $S$ is the two-element distributive lattice.\n  We then show that every variety in the interval $[\\mathsf{V}(S_c(abc)), \\mathsf{V}(\\mathbf{M}_n(S_7))]$ is nonfinitely based (i.e., has no finite basis for its identities), where $S_c(abc)$ is an eight-elemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09677","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/2607.09677/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"}