{"paper":{"title":"The finite basis problem for additively idempotent semirings that relate to S_7","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Marcel Jackson, Miaomiao Ren, Xianzhong Zhao, Zidong Gao","submitted_at":"2025-01-31T11:21:21Z","abstract_excerpt":"The $3$-element additively idempotent semiring $S_7$ is a nonnitely based algebra of the smallest possible order. In this paper we study the nite basis problem for some additively idempotent semirings that relate to $S_7$. We present a su cient condition under which an additively idempotent semiring variety is nonnitely based and as applications, show that some additively idempotent semiring varieties that contain $S_7$ are also nonnitely based. We then consider the subdirectly irreducible members of the variety $\\mathsf{V}(S_7)$ generated by $S_7$. We show that $\\mathsf{V}(S_7)$ contains exac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.19049","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/2501.19049/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"}