Comparability in the graph monoid

Let G 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 G-action instead. Starting from a directed graph E, one can define a cancellative commutative monoid MEG with a G-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 MEG is comparable, periodic, graphs such that every nonzero element of MEG is aperiodic, incomparable, graphs such that no nonzero element of MEG is periodic, and graphs such that no element of MEG is aperiodic. The Graded Classification Conjecture can be formulated to state that MEG is a complete invariant of the Leavitt path algebra LK(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 ME.G Our work also implies that some results of [11] hold without requiring the graph to be row-finite