The Bi-UFS Positive Conjecture for algebraic semidomains

Abstract

A semidomain is called bi-UFS if both its additive monoid and its nonzero multiplicative monoid are unique factorization monoids. The Bi-UFS Positive Conjecture predicts that the only positive semidomain with this property is the nonnegative integers. We prove this conjecture for finitely generated algebraic positive semidomains. In the cyclic case, we show that for every positive algebraic number $\alpha$, the semidomain $\mathbb{N}_0[\alpha]$ is bi-UFS if and only if $\alpha \in \mathbb{N}$, equivalently $\mathbb{N}_0[\alpha]=\mathbb{N}_0$. The proof separates the quadratic case, where an analysis of the least additive atom larger than $1$ leaves only the examples $\mathbb{N}_0[\sqrt 2]$ and $\mathbb{N}_0[(1+\sqrt 5)/2]$ to exclude, from the higher-degree case, where explicit multiplicative identities force the minimal polynomial into impossible forms. We then give a Perron-Frobenius argument showing that if $\alpha_1,\ldots,\alpha_n$ are positive algebraic numbers and $\mathbb{N}_0[\alpha_1,\ldots,\alpha_n]$ is bi-UFS then this semidomain is $\mathbb{N}_0$. Finally, we prove a reduction theorem for complex semidomains: every bi-UFS subsemidomain of $\mathbb{C}$ with finitely many additive atoms admits an isomorphic realization as a positive semidomain. Consequently, every finitely generated algebraic bi-UFS semidomain over $\mathbb{C}$ is isomorphic to $\mathbb{N}_0$.