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On zigzag maps and the path category of an inverse semigroup

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Abstract

We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz–Krieger \(C^*\)-algebras of singly aligned categories include the tight \(C^*\)-algebras of all countable inverse semigroups, up to Morita equivalence.

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Authors and Affiliations

  1. Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

    Allan Donsig

  2. Department of Mathematics and Statistics, California State University, Long Beach, 1250 Bellflower Blvd, Long Beach, CA, 90840, USA

    Jennifer Gensler

  3. Department of Mathematics, Taylor University, 236 W Reade Ave, Upland, IN, 46989, USA

    Hannah King

  4. Department of Mathematics, The University of Texas at Tyler, 3900 University Boulevard, Tyler, TX, 75799, USA

    David Milan

  5. Department of Mathematics, Rice University, 6100 Main St, Houston, TX, 77005, USA

    Ronen Wdowinski

Authors
  1. Allan Donsig
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  2. Jennifer Gensler
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  3. Hannah King
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  4. David Milan
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  5. Ronen Wdowinski
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Corresponding author

Correspondence to David Milan.

Additional information

Communicated by Mark V. Lawson.

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The second through fifth authors were supported by an NSF Grant (DMS-1659221).

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Donsig, A., Gensler, J., King, H. et al. On zigzag maps and the path category of an inverse semigroup. Semigroup Forum 100, 790–805 (2020). https://doi.org/10.1007/s00233-019-10031-2

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  • DOI: https://doi.org/10.1007/s00233-019-10031-2

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