{"paper":{"title":"A syntactic approach to the MacNeille completion of $\\bold\\Lambda^{\\ast}$, the free monoid over an ordered alphabet $\\bold \\Lambda$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hans-J\\\"urgen Bandelt, Maurice Pouzet","submitted_at":"2017-12-22T15:29:07Z","abstract_excerpt":"Let $\\Lambda^{\\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $\\Lambda$, which is equipped with some (partial) order. This ordering lifts to $\\Lambda^{\\ast}$, where it extends the divisibility ordering of words. The MacNeille completion of $\\Lambda^{\\ast}$ constitutes a complete lattice ordered monoid and is realized by the system of \"closed\" lower sets in $\\Lambda^*$ (ordered by inclusion) or its isomorphic copy formed of the \"closed\" upper sets (ordered by reverse inclusion). Under some additional hypothesis on $\\Lambda$, one can easily identify the closed "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08516","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"}