Every semigroup of order at most five is globally determined
Abstract
The power semigroup P(S) of a semigroup S is the set of nonempty subsets of S under A · B={ab: a ∈ A, b ∈ B}, and S is globally determined when P(S)congP(T) forces Scong T. Whether every finite semigroup is globally determined is a question of Schein and, independently, Tamura, recorded as Problem 5.1 of the recent edition of Lyapin's notebook of open problems; it is open, notwithstanding a 1987 announcement without proof that has propagated through the literature as a theorem. We settle the question for all semigroups of order at most five, with the second semigroup ranging over all finite semigroups: if |S₁| ≤ 5, S₂ is finite and P(S₁)congP(S₂), then |S₁|=|S₂| and S₁cong S₂. The second order is not assumed but forced, because |P(S)|=2^(|S|)-1. The proof combines an isomorph-free enumeration of the 2 133 semigroups of order at most five with a single individualization–refinement invariant of a finite magma, proved to be an isomorphism invariant, and checked once per order to be injective on the power semigroups of the catalogue. The whole argument, including the finite searches, is machine-checked in Lean 4; separately, an independent implementation in C carries the census through order six.
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Source snapshot 2026-08-30 15:34 UTC
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be3ddec362c5be0f21d2dbef2bcac54af4584ebc178488de30be18d737688589
Claim ledger
Stated results
T1candidate2026-08-23
Headline: Tamura–Schein holds for every semigroup of order at most 5, with the second semigroup ranging over all finite semigroups
T2routine2026-08-23
The invariant is an isomorphism invariant, and separates each order's catalogue
T3known data2026-08-23
The enumeration is a proved transversal, and reproduces A027851 at orders 1–5
T4routine2026-08-23
Negative controls: the wrong OEIS anchor on both sides, the insufficiency of plain refinement, the non-injectivity of the invariant, the bridge, and the too-strong conclusion
T5routine2026-08-22
Order 6 externally: all 28,634 semigroups of order 6 are globally determined
This ledger entry is reported in prose and is not bound to a Lean theorem.T6routine2026-08-30
The order-6 machinery: a one-pass preimage, a tabulated power table, and isomorphism-invariance at every setting
T7candidate2026-08-30
Tamura-Schein holds at order 6, in the kernel, at uniform settings
T8known data2026-08-30
Exactly 28,634 semigroups of order 6 up to isomorphism – A027851(6), machine-checked
T9routine2026-08-30
Order-6 negative controls: both wrong OEIS anchors, the failure of the family's own invariant, the failure of refinement alone, and the too-strong conclusion
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- P(S) is the semigroup of nonempty subsets of S under A · B = ab: a ∈ A, b ∈ B. S is globally determined when P(S) ≅ P(T) forces S ≅ T, for every semigroup T.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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