Row-finite equivalents exist only for row-countable graphs
Abstract
If E is a not-necessarily row-finite graph, such that each vertex of E emits at most countably many edges, then a desingularization F of E can be constructed (see e.g. (1) G. Abrams, G. Aranda Pino, Leavitt path algebras of arbitrary graphs, Houston J. Math 34(2) (2008), 423-442, or (2) I. Raeburn, "Graph algebras". CBMS Regional Conference Series in Mathematics 103, Conference Board of the Mathematical Sciences, Washington, DC, 2005, ISBN 0-8218-3660-9). The desingularization process has been effectively used to establish various characteristics of the Leavitt path algebras of not-necessarily row-finite graphs. Such a desingularization F of E has the properties that: (1) F is row-finite, and (2) the Leavitt path algebras L(E) and L(F) are Morita equivalent. We show here that for an arbitrary graph E, a graph F having properties (1) and (2) exists (we call such a graph F a row-finite equivalent of E) if and only if E is row-countable; that is, E contains no vertex v for which v emits uncountably many edges.