# Subsets of vertices give Morita equivalences of Leavitt path algebras

## TL;DR

This paper demonstrates that subsets of vertices in a directed graph induce Morita equivalences between subalgebras and ideals of associated Leavitt path algebras, and shows how graph contractions preserve Morita equivalence.

## Contribution

It introduces a method to obtain Morita equivalences via vertex subsets and extends graph contraction techniques to preserve algebraic properties.

## Key findings

- Subsets of vertices induce Morita equivalences in Leavitt path algebras.
- Graph contractions can produce Morita equivalent Leavitt path algebras.
- Examples include desingularisation and delaying of graphs.

## Abstract

We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.