Let Γ be the infinite cyclic group on a generator x. To avoid confusion when working with Z-modules which also have an additional Z-action, we consider the Z-action to be a Γ-action instead.

Starting from a directed graph E, one can define a cancellative commutative monoid M_E^Γ with a Γ-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of M_E^Γ is comparable, periodic, graphs such that every nonzero element of M_E^Γ is aperiodic, incomparable, graphs such that no nonzero element of M_E^Γ is periodic, and graphs such that no element of M_E^Γ is aperiodic.

The Graded Classification Conjecture can be formulated to state that M_E^Γ is a complete invariant of the Leavitt path algebra L_K(E) of E over a field K. Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of E are well reflected by the structure of M_E^Γ. Our work also implies that some results of [11] hold without requiring the graph to be row-finite.

The authors are very grateful to the referee for a prompt, detailed and thoughtful report. The first author would also like to acknowledge Australian Research Council grant DP160101481.