{"paper":{"title":"Join irreducible semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.FL","math.RA"],"primary_cat":"math.GR","authors_text":"Benjamin Steinberg, Edmond W. H. Lee, John Rhodes","submitted_at":"2017-02-13T12:59:29Z","abstract_excerpt":"We begin a systematic study of finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups $S$ that generate join irreducible pseudovarieties are characterized as follows: whenever $S$ divides a direct product $A \\times B$ of finite semigroups, then $S$ divides either $A^n$ or $B^n$ for some $n \\geq 1$. We present a new operator ${ \\mathbf{V} \\mapsto \\mathbf{V}^\\mathsf{bar} }$ that preserves the property of join irreducibility, as does the dual operator, and show "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03753","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}