{"paper":{"title":"Two applications of monoid actions to cross-sections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alan J. Cain, Tara Brough, Victor Maltcev","submitted_at":"2018-03-28T17:34:01Z","abstract_excerpt":"Using a construction that builds a monoid from a monoid action, this paper exhibits an example of a direct product of monoids that admits a prefix-closed regular cross-section, but one of whose factors does not admit a regular cross-section; this answers negatively an open question from the theory of Markov monoids. The same construction is then used to show that for any full trios $\\mathfrak{C}$ and $\\mathfrak{D}$ such that $\\mathfrak{C}$ is not a subclass of $\\mathfrak{D}$, there is a monoid with a cross-section in $\\mathfrak{C}$ but no cross-section in $\\mathfrak{D}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10747","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"}