Factorization in monoids by stratification of atoms and the Elliott problem

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  monoids. 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 monoids 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”. This  manuscript employs very much the concept of extraction monoid which, as many other tools, stems from discrete geometry but it is less  well-known.