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In an additive factorial monoid each element can be represented as a linearcombination of irreducible elements (atoms) with uniquely determinedcoefficients running over all natural numbers.  In this paper we develop for a wide class of non-factorial monoids a conceptof stratification for atoms which allows to represent each element as a linearcombination of atoms where the coefficients are uniquely determined whenrestricted in a particular way. This wide class includes inside factorialmonoids and in particular simplicial affine semigroups. In the latter case thequestion of uniqueness is related to a problem studied by E. B. Elliott in apaper from 1903.  For the monoid of all nonnegative solutions of a certain linear Diophantineequation 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 astratification is always possible, which answers to Elliott's problem also forthe cases he left open.  The results in this paper on the stratification of atoms for monoids ingeneral may be seen also as an answer to a "generalized Elliott problem".

Factorization in monoids by stratification of atoms and the Elliott Problem