Pith Number

pith:ONYACONC

pith:2026:ONYACONCR6TYFA2CK6WHTO7EZD

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A nonfinitely based additively idempotent semiring of order four

Mengya Yue, Miaomiao Ren

A 4-element additively idempotent semiring has no finite basis for its identities.

arxiv:2605.15493 v1 · 2026-05-15 · math.GR

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4 Citations open

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Claims

C1strongest claim

We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a 4-element semiring S_{(4,124)} ... Consequently, these semirings have no finite basis for their identities.

C2weakest assumption

The sufficient condition established in the paper is both valid and correctly verified for the 4-element semiring S_{(4,124)} whose additive reduct has two minimal elements and two coatoms.

C3one line summary

A 4-element additively idempotent semiring whose additive reduct has two minimal elements and two coatoms has no finite basis for its identities.

References

28 extracted · 28 resolved · 1 Pith anchors

[1] Alsulami, T., Jackson, M.: Finite models for positive combinatorial and exponential algebra. Bull. Lond. Math. Soc.57, 3380–3400 (2025) 2025

[2] Graduate Texts in Mathematics, vol 1998

[3] Dolinka I.: A nonfinitely based finite semiring. Internat. J. Algebra Comput.17(8), 1537–1551 (2007) 2007

[4] Algebra Universalis60, 19–35 (2009) 2009

[5] Dolinka, I., Gusev, S.V., Volkov, M.V.: The finite basis problem for the endomorphism semir- ings of finite semilattices. Bull. Belg. Math. Soc. Simon Stevin32(5), 657–674 (2025) 2025

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:01:01.505244Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

73700139a28fa782834257ac79bbe4c8f72f33a2372a41ad1f9a2bebcec8f6d0

Aliases

arxiv: 2605.15493 · arxiv_version: 2605.15493v1 · doi: 10.48550/arxiv.2605.15493 · pith_short_12: ONYACONCR6TY · pith_short_16: ONYACONCR6TYFA2C · pith_short_8: ONYACONC

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ONYACONCR6TYFA2CK6WHTO7EZD \
  | 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: 73700139a28fa782834257ac79bbe4c8f72f33a2372a41ad1f9a2bebcec8f6d0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "5df233b472020f5239bbe678544792bfb5d0307ed7b87419828ff2ee07e347f2",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GR",
    "submitted_at": "2026-05-15T00:17:46Z",
    "title_canon_sha256": "e5321545217e619007ee5c8654ee79e8c42e57dda85fef8700dd9fd92b3fc597"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15493",
    "kind": "arxiv",
    "version": 1
  }
}