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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"},"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":"2607.09677","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-12T03:04:14Z","cross_cats_sorted":[],"title_canon_sha256":"4de7c97f726fabea601368cd6f82c9657d0a707df541b17b065c154c8f0a6051","abstract_canon_sha256":"33f5b6c2f99cb50e9a863839a4269be25b0c56e81ae5cb2e8cacc5ba9bd35fe7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T00:18:30.049961Z","signature_b64":"WOO2VPoJ2XT7izkokfNwtyE4VKQgGBa/ql+Vmozidu7r77aIp1s9viifNmTjWq2amqyaAm4Soru6GYoHrl8kBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"771eca63847065104bcb870510045210c1318b299709eab821dc86bfec6dee77","last_reissued_at":"2026-07-14T00:18:30.049200Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T00:18:30.049200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathbf{M}_{n+1}(S)$. 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