MathDB

Bi-UF Positive Conjecture

For every subsemiring S⊆R≥0S\subseteq\mathbb{R}_{\ge 0} with 1∈S1\in S, if the additive monoid (S,+)(S,+) and the nonzero multiplicative monoid (S∖{0},⋅)(S\setminus\{0\},\cdot) are both factorial monoids, then S=N0S=\mathbb{N}_0.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

commutative-algebrafactorization-theory

Progress summary

Refreshed
Claimed solved

An August 2026 preprint claims to prove the conjecture in full, but the result has not yet been independently verified.

The conjecture, posed by Baeth, Chapman, and Gotti, asserts that the natural-number semiring is the only positive semiring whose additive and multiplicative factorizations are both unique.

Known results

  • The conjecture was recorded as Conjecture 7.77.7 by Baeth, Chapman, and Gotti in 20212021.
  • A July 20262026 preprint proves the conjecture for finitely generated algebraic positive semidomains, including all monogenic cases and quadratic cases, but not arbitrary positive semirings.
  • The same preprint states that N0\mathbb{N}_{0} was the only known bi-UFS semidomain.

August 2026 claimed proof

Graia and Omar claim in an August 20262026 arXiv preprint that the conjecture holds for all positive semirings. The manuscript is unrefereed, and no independent verification, objection resolution, or referee report was found in the supplied sources.

Current status (as of August 2026): The conjecture has a published claim of a full proof, while the proof remains unverified; earlier restricted algebraic cases are established in an unrefereed preprint.

Sources

Solutions 0

No solutions have been posted yet.