New York Journal of Mathematics
Volume 26 (2020), 1375-1421


Roozbeh Hazrat and Lia Vas

Comparability in the graph monoid

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Published: November 25, 2020.
Keywords: Graph, group action, graph monoid, ordered abelian group, Grothendieck group.
Subject: 06F05, 05C25, 19A49, 06F20, 16E20, 20M32.

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 MEΓ 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 MEΓ is comparable, periodic, graphs such that every nonzero element of MEΓ is aperiodic, incomparable, graphs such that no nonzero element of MEΓ is periodic, and graphs such that no element of MEΓ is aperiodic.

The Graded Classification Conjecture can be formulated to state that MEΓ 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 MEΓ. 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.

Author information

Roozbeh Hazrat:
Centre for Research in Mathematics and Data Science
Western Sydney University


Lia Vas:
Department of Mathematics, Physics and Statistics
University of the Sciences
Philadelphia, PA 19104, USA