Factorization invariants in half-factorial affine semigroups

Abstract

Let $\mathbb{N}\mathcal{A}$ be the monoid generated by $\mathcal{A}={\mathbf{a}}_1,...,{\mathbf{a}}_n\subseteq {\mathbb{Z}}^d.$ We introduce the homogeneous catenary degree of $\mathbb{N}\mathcal{A}$ as the smallest $N\in \mathbb{N}$ with the following property: for each $\mathbf{a}\in \mathbb{N}\mathcal{A}$ and any two factorizations $\mathbf{u},\mathbf{v}$ of $\mathbf{a}$, there exists factorizations $\mathbf{u}={\mathbf{w}}_1,...,{\mathbf{w}}_t=\mathbf{v}$ of $\mathbf{a}$ such that, for every $k,\mathrm{d}({\mathbf{w}}_k,{\mathbf{w}}_{k+1})\le N,$ where $\mathrm{d}$ is the usual distance between factorizations, and the length of ${\mathbf{w}}_k,|{\mathbf{w}}_k|,$ is less than or equal to $\max |\mathbf{u}|,|\mathbf{v}|.$ We prove that the homogeneous catenary degree of $\mathbb{N}\mathcal{A}$ improves the monotone catenary degree as upper bound for the ordinary catenary degree, and we show that it can be effectively computed. We also prove that for half-factorial monoids, the tame degree and the $\omega$-primality coincide, and that all possible catenary degrees of the elements of an affine semigroup of this kind occur as the catenary degree of one of its Betti elements.

8 pages, 1 figure

References in corpus (2)

• Uniquely presented finitely generated commutative monoids
• Simplicial complexes and minimal free resolution of monomial algebras

Cited by in corpus (10)

• numericalsgps, a GAP package for numerical semigroups
• The catenary degree of Krull monoids II
• On factorization invariants and Hilbert functions
• Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid
• Numerical semigroups on compound sequences
• Minimal presentations of shifted numerical monoids
• On the set of catenary degrees of finitely generated cancellative commutative monoids
• Realizable sets of catenary degrees of numerical monoids
• The geometry of double nested Hilbert schemes of points on curves
• Factorization in Finitely-Presented Monoids