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      Semigroup actions on sets and the burnside ring

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      Author(s)
      Erdal, Mehmet Akif
      Ünlü, Özgün
      Date
      2018
      Source Title
      Applied Categorical Structures
      Print ISSN
      0927-2852
      Electronic ISSN
      1572-9095
      Publisher
      Springer Science
      Volume
      26
      Issue
      1
      Pages
      7 - 28
      Language
      English
      Type
      Article
      Item Usage Stats
      149
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      98
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      Abstract
      In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gr¨othendieck group.
      Keywords
      Semigroup actions
      Monoid actions
      Reverse actions
      Homotopical category
      Burnside ring
      16W22
      20M20
      20M35
      55U35
      Permalink
      http://hdl.handle.net/11693/48336
      Published Version (Please cite this version)
      https://doi.org/10.1007/s10485-016-9477-4
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      • Department of Mathematics 653
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