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Graded semigroups

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Abstract

We systematically develop a theory of graded semigroups, that is, semigroups S partitioned by groups Γ, in a manner compatible with the multiplication on S. We define a smash product S#Γ, and show that when S has local units, the category S#Γ-Mod of sets admitting an S#Γ-action is isomorphic to the category S-Gr of graded sets admitting an appropriate S-action. We also show that when S is an inverse semigroup, it is strongly graded if and only if S-Gr is naturally equivalent to Sε-Mod, where Sε is the partition of S corresponding to the identity element ε of Γ. These results are analogous to well-known theorems of Cohen/Montgomery and Dade for graded rings. Moreover, we show that graded Morita equivalence implies Morita equivalence for semigroups with local units, evincing the wealth of information encoded by the grading of a semigroup. We also give a graded Vagner—Preston theorem, provide numerous examples of naturally-occurring graded semigroups, and explore connections between graded semigroups, graded rings, and graded groupoids. In particular, we introduce graded Rees matrix semigroups, and relate them to smash product semigroups. We pay special attention to graded graph inverse semigroups, and characterise those that produce strongly graded Leavitt path algebras.

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Acknowledgment

We are grateful to the referee for a very thoughtful review, and suggestions that have led to improvements in the exposition.

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Authors and Affiliations

  1. Centre for Research in Mathematics and Data Science Western Sydney University, Penrith, NSW, 2751, Australia

    Roozbeh Hazrat

  2. Department of Mathematics, University of Colorado, Colorado Springs, CO, 80918, USA

    Zachary Mesyan

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  1. Roozbeh Hazrat
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  2. Zachary Mesyan
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Correspondence to Roozbeh Hazrat.

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Hazrat, R., Mesyan, Z. Graded semigroups. Isr. J. Math. 253, 249–319 (2023). https://doi.org/10.1007/s11856-022-2361-z

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  • DOI: https://doi.org/10.1007/s11856-022-2361-z

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