 

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. 


Abstract
Let Γ be the infinite cyclic group on a generator x. To avoid confusion when working with Zmodules which also have an additional Zaction, we consider the Zaction to be a Γaction instead.
Starting from a directed graph E, one can define a cancellative commutative monoid M_{E}^{Γ} with a Γaction 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 graphtheoretic 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 rowfinite. 

Acknowledgements
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
Australia
r.hazrat@westernsydney.edu.au
Lia Vas:
Department of Mathematics, Physics and Statistics
University of the Sciences
Philadelphia, PA 19104, USA
l.vas@usciences.edu

