Abstract:
This thesis presents a survey of the Invariant Basis Number (IBN) property within the context of Leavitt Path Algebras (LPAs), which lie at the intersection of ring theory, graph theory, and noncommutative algebra. Originating from W.G. Leavitt's work on rings lacking the IBN property, LPAs are constructed from directed graphs and exhibit rich structural properties linked to their underlying graphs.
The primary focus is investigating the conditions under which an LPA possesses the IBN property. We explore the fundamental structure, definitions, and key examples of LPAs, contrasting them with related concepts like Cohn path algebras. Utilizing monoid-theoretic techniques (specifically the graph monoid M_E and its group completion) and matrix-based formulations derived from the graph's incidence matrix (following the work of T.G. Nam and N.T. Phuc), we establish explicit criteria for determining if L_k(E) satisfies IBN property.
These criteria are then applied to analyze the IBN property for LPAs associated with specific graph constructions arising from finite groups. We examine Cayley graphs, demonstrating that their LPAs have IBN if and only if the generating set for the group contains a single element. Furthermore, we investigate the power graphs of cyclic groups of prime power order (Z_{p^m}).
This work aims to provide a self-contained exploration of the IBN property for Leavitt path algebras, combining theoretical developments with concrete examples and graphical constructions, primarily based on the findings presented in the paper by Nam and Phuc (2019) titled The structure of Leavitt path algebras and the Invariant Basis Number property.
