### Abstract

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C^{*}-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units. © 2011 Elsevier B.V.

Original language | English |
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Pages (from-to) | 2262-2279 |

Number of pages | 18 |

Journal | Journal of Pure and Applied Algebra |

Volume | 215 |

Issue number | 9 |

DOIs | |

Publication status | Published - Sep 2011 |

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### Cite this

*Journal of Pure and Applied Algebra*,

*215*(9), 2262-2279. https://doi.org/10.1016/j.jpaa.2011.02.015

}

*Journal of Pure and Applied Algebra*, vol. 215, no. 9, pp. 2262-2279. https://doi.org/10.1016/j.jpaa.2011.02.015

**Characterizations of Morita equivalent inverse semigroups.** / Funk, J.; Lawson, M. V.; Steinberg, B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Characterizations of Morita equivalent inverse semigroups

AU - Funk, J.

AU - Lawson, M. V.

AU - Steinberg, B.

PY - 2011/9

Y1 - 2011/9

N2 - We prove that four different notions of Morita equivalence for inverse semigroups motivated by C*-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units. © 2011 Elsevier B.V.

AB - We prove that four different notions of Morita equivalence for inverse semigroups motivated by C*-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units. © 2011 Elsevier B.V.

UR - http://www.scopus.com/inward/record.url?scp=79953712288&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2011.02.015

DO - 10.1016/j.jpaa.2011.02.015

M3 - Article

VL - 215

SP - 2262

EP - 2279

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 9

ER -