{
  "id": "1911.00590",
  "version": "v1",
  "published": "2019-11-01T21:06:24.000Z",
  "updated": "2019-11-01T21:06:24.000Z",
  "title": "Graph inverse semigroups and Leavitt path algebras",
  "authors": [
    "John Meakin",
    "Zhengpan Wang"
  ],
  "comment": "39 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We study two classes of inverse semigroups built from directed graphs, namely graph inverse semigroups and a new class of semigroups that we refer to as Leavitt inverse semigroups. These semigroups are closely related to graph $C^*$-algebras and Leavitt path algebras. We provide a topological characterization of the universal groups of the local submonoids of these inverse semigroups. We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs. We describe the structure of graphs that admit a directed cover or directed immersion into a circle and we provide structural information about graph inverse semigroups of finite graphs that admit a directed cover onto a bouquet of circles. We also find necessary and sufficient conditions for a homomorphic image of a graph inverse semigroup to be another graph inverse semigroup. We find a presentation for the Leavitt inverse semigroup of a graph in terms of generators and relations. We describe the structure of the Leavitt inverse semigroup and the Leavitt path algebra of a graph that admits a directed immersion into a circle. We show that two graphs that have isomorphic Leavitt inverse semigroups have isomorphic Leavitt path algebras and we classify graphs that have isomorphic Leavitt inverse semigroups. As a consequence, we show that Leavitt path algebras are $0$-retracts of certain matrix algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-01T21:06:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20M18",
      "05C20"
    ],
    "keywords": [
      "graph inverse semigroup",
      "isomorphic leavitt inverse semigroups",
      "directed immersion",
      "isomorphic leavitt path algebras",
      "directed cover"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}