The Bi-LF Positive Conjecture Holds

Abstract

We prove that a semidomain has length-factorial additive and multiplicative monoids if and only if it is isomorphic to the semiring of nonnegative integers or is a unique factorization domain. Consequently, bi-length-factoriality and bi-factoriality coincide for all semidomains, settling the Bi-LF Positive Conjecture. The main step shows that additive length-factoriality and multiplicative factoriality force additive factoriality, without an additive reducedness assumption. Multiplicative factoriality forces every semidomain with a nonzero additive unit to be a unique factorization domain. In the additively reduced case, a uniform multiplier for lattice points in a finitely generated cone yields cancellation of divisibility across sums, which leads to an obstruction to additive atoms dividing positive integers and completes the structural argument. We also consider related questions on quasi-factoriality and classify the rational and transcendental monogenic positive semirings whose multiplicative monoids are quasi-$n$-factorial for a prescribed integer $n \ge 2$.