{"paper":{"title":"Extending the scalars of minimizations","license":"","headline":"","cross_cats":["cs.DS","cs.SC"],"primary_cat":"math.CO","authors_text":"Eric Laugerotte (LIFAR EA2655), G\\'erard Duchamp (LIPN), Jean-Gabriel Luque (IGM-LabInfo)","submitted_at":"2006-07-18T07:06:59Z","abstract_excerpt":"In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of $\\Sigma^*$ where $\\Sigma$ is a set of symbols (or the alphabet). Now, given a semiring $(\\K,+,.)$, one can construct $\\K$-subsets of $\\Sigma^*$ in the sense of Eilenberg, that are alternatively called noncommutative formal power series for which a framework very similar to language theory has been constructed Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or $\\K$-automata). In order to get a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607411","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":""},"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"}