Factorization in monoids by stratification of atoms and the Elliott Problem

TL;DR

This paper introduces a stratification method for atoms in a broad class of non-factorial monoids, enabling unique element representation and addressing Elliott's longstanding problem in affine semigroups.

Contribution

It develops a new stratification concept for atoms in non-factorial monoids, including simplicial affine semigroups, providing solutions to Elliott's problem in two dimensions.

Findings

Stratification always possible in two-dimensional simplicial affine semigroups.

Addresses Elliott's problem for cases previously left open.

Provides a generalized approach to the Elliott problem in monoids.

Abstract

In an additive factorial monoid each element can be represented as a linear combination of irreducible elements (atoms) with uniquely determined coefficients running over all natural numbers. In this paper we develop for a wide class of non-factorial monoids a concept of stratification for atoms which allows to represent each element as a linear combination of atoms where the coefficients are uniquely determined when restricted in a particular way. This wide class includes inside factorial monoids and in particular simplicial affine semigroups. In the latter case the question of uniqueness is related to a problem studied by E. B. Elliott in a paper from 1903. For the monoid of all nonnegative solutions of a certain linear Diophantine equation in three variables, Elliott considers "simple sets of solutions" (atoms of the monoid) and looks for a method that gives "every set once only". We show that for simplicial affine semigroups in two dimensions a stratification is always possible, which answers to Elliott's problem also for the cases he left open. The results in this paper on the stratification of atoms for monoids in general may be seen also as an answer to a "generalized Elliott problem".