{"paper":{"title":"Commutative algebras of series","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Polynomial product rules completely characterise all bilinear associative commutative products on noncommuting series and make P-automaton equivalence decidable.","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Lorenzo Clemente","submitted_at":"2026-01-27T17:09:53Z","abstract_excerpt":"We consider a large family of product operations of formal power series in noncommuting indeterminates, the classes of automata they define, and the respective equivalence problems. A $P$-product of series is defined coinductively by a polynomial product rule $P$, which gives a recursive recipe to build the product of two series as a function of the series themselves and their derivatives.\n  The first main result of the paper is a complete and decidable characterisation of all product rules $P$ giving rise to $P$-products which are bilinear, associative, and commutative. The characterisation s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The first main result is a complete and decidable characterisation of all product rules P giving rise to P-products which are bilinear, associative, and commutative. The equivalence problem for P-automata is decidable for P-products satisfying our characterisation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The P-product is defined coinductively by a polynomial product rule P on formal power series in noncommuting indeterminates, with the characterisation relying on this recursive structure.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A complete decidable characterization of polynomial rules P yielding bilinear associative commutative P-products on series is given, together with decidability of equivalence for the associated P-automata.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Polynomial product rules completely characterise all bilinear associative commutative products on noncommuting series and make P-automaton equivalence decidable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"54b7bb9fdc1ccaf937a1035638cc37e0b420dd10d90b1e5f971dd55cdd7fad5e"},"source":{"id":"2601.19809","kind":"arxiv","version":2},"verdict":{"id":"d558589e-34ab-4024-8d5c-aaef1a645b3c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T10:37:17.695037Z","strongest_claim":"The first main result is a complete and decidable characterisation of all product rules P giving rise to P-products which are bilinear, associative, and commutative. The equivalence problem for P-automata is decidable for P-products satisfying our characterisation.","one_line_summary":"A complete decidable characterization of polynomial rules P yielding bilinear associative commutative P-products on series is given, together with decidability of equivalence for the associated P-automata.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The P-product is defined coinductively by a polynomial product rule P on formal power series in noncommuting indeterminates, with the characterisation relying on this recursive structure.","pith_extraction_headline":"Polynomial product rules completely characterise all bilinear associative commutative products on noncommuting series and make P-automaton equivalence decidable."},"references":{"count":35,"sample":[{"doi":"","year":null,"title":"Henning Basold, Helle Hvid Hansen, Jean-Éric Pin, and Jan Rutten","work_id":"645b54f6-b8eb-4569-a7e3-1ba73229e16d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1017/s0960129517000159","year":2017,"title":"doi:10.1017/s0960129517000159","work_id":"02bf3322-dbcf-437c-8a99-78331d39f6bc","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1109/lics.2017.8005101","year":2017,"title":"12 Jean-Baptiste Courtois and Sylvain Schmitz","work_id":"07ef88bb-2cea-4f60-92bc-0d5a9907e221","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/s0195-6698(13)80035-2","year":1990,"title":"François Bergeron and Christophe Reutenauer. 1990. Combinatorial resolution of systems of differential equations iii: a special class of differentially algebraic series.European Journal of Combinatori","work_id":"4a740a72-77bb-401b-ac85-6038857ea710","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"J. Berstel and C. Reutenauer. 2010.Noncommutative rational series with applications. CUP.isbn: 0521190223","work_id":"aab71db5-4e06-4001-8ad2-2e0b49349355","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":35,"snapshot_sha256":"dce6930db99197ce66998cd0fccc0c09e68487ab872333af57ace0afaf427f88","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"}