Pith Number

pith:EJT3GHJZ

pith:2026:EJT3GHJZA74GB733ONCCDVQ4ZO

not attested not anchored not stored refs resolved

Commutative algebras of series

Lorenzo Clemente

Polynomial product rules completely characterise all bilinear associative commutative products on noncommuting series and make P-automaton equivalence decidable.

arxiv:2601.19809 v2 · 2026-01-27 · cs.FL

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\usepackage{pith}
\pithnumber{EJT3GHJZA74GB733ONCCDVQ4ZO}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

35 extracted · 35 resolved · 0 Pith anchors

[1] Henning Basold, Helle Hvid Hansen, Jean-Éric Pin, and Jan Rutten

[2] doi:10.1017/s0960129517000159 2017 · doi:10.1017/s0960129517000159

[3] 12 Jean-Baptiste Courtois and Sylvain Schmitz 2017 · doi:10.1109/lics.2017.8005101

[4] 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 1990 · doi:10.1016/s0195-6698(13)80035-2

[5] J. Berstel and C. Reutenauer. 2010.Noncommutative rational series with applications. CUP.isbn: 0521190223 2010

Receipt and verification

First computed	2026-05-18T02:45:05.802305Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2267b31d3907f860ff7b734421d61ccb8195630e722805b6dd03795f72fe6dcb

Aliases

arxiv: 2601.19809 · arxiv_version: 2601.19809v2 · doi: 10.48550/arxiv.2601.19809 · pith_short_12: EJT3GHJZA74G · pith_short_16: EJT3GHJZA74GB733 · pith_short_8: EJT3GHJZ

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/EJT3GHJZA74GB733ONCCDVQ4ZO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 2267b31d3907f860ff7b734421d61ccb8195630e722805b6dd03795f72fe6dcb

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "99e71d10483066709f8e3eaedb50c14b5283d6f3dc52580f6e392e18be77c0cc",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.FL",
    "submitted_at": "2026-01-27T17:09:53Z",
    "title_canon_sha256": "797e9097ed806376b4dae80d9856cfcb24a1cd18c5e870175030ac47377b1864"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2601.19809",
    "kind": "arxiv",
    "version": 2
  }
}