Inside factorial monoids and the cale monoid of a single Diophantine equation

TL;DR

This paper provides a structure theorem for inside factorial domains and analyzes the monoid of nonnegative solutions to specific linear Diophantine equations, revealing its simplicial affine semigroup structure.

Contribution

It introduces a new structure theorem for inside factorial domains and characterizes the solution monoid of certain linear Diophantine equations as a simplicial affine semigroup.

Findings

The solution set forms a simplicial full affine semigroup.

The monoid can be described via extremal rays and Apéry sets.

Provides a structural understanding of solutions to specific Diophantine equations.

Abstract

We give a structure theorem for inside factorial domains. As an example we study the monoid of nonnegative integer solutions of equations of the form $a_1x_1+\cdots +a_{r-1}x_{r-1}=a_rx_r$, with $a_1,\ldots,a_r$ positive integers. This set is isomorphic to a simplicial full affine semigroup, and thus it can be described in terms of its extremal rays and the Ap\'ery sets with respect to the extremal rays.