arXiv subjects

Darren Han

Publications and source records attributed to Darren Han.

3 recordsLinked to original sources

arXiv2026

Maximal common divisors in monoid algebras

We say that a commutative monoid $M$ has the MCD property if every nonempty finite subset of $M$ has a maximal common divisor (MCD), and we say that $M$ has the MCD-finite property if every nonempty finite subset of $M$ has only finitely many MCDs up to associates. It is well known that every monoid that satisfies the ascending chain condition on principal ideals is an MCD monoid, while every finite factorization monoid and every pre-Schreier monoid is an MCD-finite monoid. In this paper, we study the MCD and MCD-finite properties in the setting of monoid algebras. After identifying a new class of rank-$1$ torsion-free MCD monoids, we investigate the ascent of the MCD property to monoid algebras over fields, proving that if a pre-Schreier monoid has the MCD property then its monoid algebras over any field also have the MCD property. Then we prove that, unlike for the case of polynomial extensions, the property of being atomic does not ascend to monoid algebras over fields when restricted to the class of MCD monoids. In the second part of the paper, we first identify a new class of rank-$1$ torsion-free MCD-finite monoids. Then we establish the ascent of the MCD-finite property to polynomial extensions. We conclude the paper proving that the q-GCD property (i.e., the condition that every nonempty finite subset has at most one MCD), which is a condition stronger than the MCD-finite property, does not ascend to monoid algebras over fields even when restricted to the class of rank-$1$ torsion-free monoids.

math.AC↗

arXiv2026

The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress

A complex semiring is a subset of the complex plane that is closed under the standard addition and multiplication of complex numbers and contains both $0$ and $1$. A complex semiring $S$ is called a bi-UFS if both its additive monoid $(S,+)$ and its multiplicative monoid $(S\setminus \{1\}, \cdot)$ are unique factorization monoids (UFM). The Bi-UF Positive Conjecture states that $\mathbb{N}_0$ is the only subsemiring of the nonnegative cone of the real line that is a bi-UFS. In this paper, we prove that no simple semiring extension of $\mathbb{N}_0$ by a quadratic algebraic number is a bi-UFS, identifying a natural class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. We also identify another class of complex semirings satisfying the statement of the Bi-UF Positive Conjecture. Then we extend the statement of the Bi-UF Positive Conjecture by motivated by a structural theorem we established for semidomains whose additive monoid are finite-rank free commutative monoids. Finally, we consider the bi-HF property, which is a relaxed version of the bi-UF property. We prove that $\mathbb{N}_0$ is the only positive rational semidomain having the bi-HF property, and we provide two methods to construct bi-HFS complex semirings that are distinct from $\mathbb{N}_0$.

math.GM↗

arXiv2024

Demazure product and hopping in type D

The Demazure product, also called the 0-Hecke product, is an associative operation on Coxeter groups with interesting properties and applications. In (Li et al 2024) it was shown that the Demazure product of two permutations can be described purely combinatorially: using only their one-line notation and not relying on reduced words. In this paper, we extend this to type D Coxeter groups.

math.CO↗