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The tight groupoid of an inverse semigroup

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Abstract

In this work we present algebraic conditions on an inverse semigroup \(\mathcal {S}\) (with zero) which imply that its associated tight groupoid \(\mathcal {G}_\mathrm{tight}(\mathcal {S})\) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.

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Notes

  1. A relevant exception is the universal C*-algebra generated by a single partial isometry.

  2. The inverse also looks nicer in the alternative notation: \([y,t,x]^{-1}=[x,t^*,y]\).

  3. An inverse semigroup \(\mathcal {S}\) is said to be E*-unitary if, whenever an element s in \(\mathcal {S}\) dominates a nonzero idempotent, then s must itself be idempotent. An alternative way to express this condition is that \(\mathcal {J}_s=\{0\}\), whenever s is not idempotent.

  4. Semi-lattices in which every tight filter is an ultra-filter have been called compactable semi-lattices [13, Theorem 2.5]. See also [15] for alternative definitions of this concept.

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Acknowledgments

Ruy Exel was partially supported by CNPq. Enrique Pardo was partially supported by PAI III Grants FQM-298 and P11-FQM-7156 of the Junta de Andalucía and by the DGI-MICINN and European Regional Development Fund, jointly, through Project MTM2011-28992-C02-02.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, SC, 88010-970, Brazil

    Ruy Exel

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510, Puerto Real, Cádiz, Spain

    Enrique Pardo

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  1. Ruy Exel
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Correspondence to Ruy Exel.

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Communicated by Markus Lohrey.

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Exel, R., Pardo, E. The tight groupoid of an inverse semigroup. Semigroup Forum 92, 274–303 (2016). https://doi.org/10.1007/s00233-015-9758-5

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