Isolated factorizations and their applications in simplicial affine semigroups

TL;DR

This paper introduces the concept of isolated factorizations in commutative monoids, explores their properties in simplicial affine semigroups, and characterizes various classes of these semigroups with applications to Betti elements and minimal presentations.

Contribution

It generalizes the notion of isolated factorizations to simplicial affine semigroups and characterizes classes like Betti sorted and Betti divisible semigroups, extending previous results.

Findings

Bounds for the number of isolated factorizations in simplicial affine semigroups

Characterization of complete intersection simplicial affine semigroups with one Betti element

Complete classification of Betti divisible numerical semigroups

Abstract

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators.