Row-finite equivalents exist only for row-countable graphs. (Q2895451)

The desingularization process introduced in a paper by \textit{G. Abrams} and \textit{G. Aranda Pino} [Houston J. Math. 34, No. 2, 423-442 (2008; Zbl 1169.16007)] in 2008 (see also the book in 2005 by \textit{I. Raeburn} [Graph algebras. CBMS Reg. Conf. Ser. Math. 103. Providence: AMS (2005; Zbl 1079.46002)]) states that for a graph \(E\) such that each vertex emits at most countably many edges, there exists a row-finite graph \(F\) such that for any field \(K\), the Leavitt path algebras \(L_K(E)\) and \(L_K(F)\) are Morita equivalent. This desingularization process has been successfully used in some works on Leavitt path algebras of not necessarily row-finite graphs.NEWLINENEWLINE In the paper under review the authors prove that for an arbitrary graph \(E\) the following conditions are equivalent: (1) There is a row-finite graph \(F\) such that \(L_K(E)\) and \(L_K(F)\) are Morita equivalent. (2) \(E\) contains no vertex \(v\) such that \(v\) emits uncountably many edges.NEWLINENEWLINE Furthermore the previous result is refined by the authors in the final theorem of the paper: For an unrestricted graph \(E\) the following conditions are equivalent: (1) \(E\) admits a row-finite-equivalent. (2) \(E\) admits a row-finite graph \(F\) with no sinks such that \(L_K(E)\) and \(L_K(F)\) are Morita equivalent. (3) \(E\) is row-countable.NEWLINENEWLINEFor the entire collection see [Zbl 1232.16001].