Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Cofinite Graphs and Groupoids and their Profinite Completions

  • Acharyya, Amrita (Department of Mathematics and Statistics, University of Toledo, Main Campus) ;
  • Corson, Jon M. (Department of Mathematics, University of Alabama, Tuscaloosa) ;
  • Das, Bikash (Department of Mathematics, University of North Georgia, Gainesville Campus)
  • Received : 2016.02.08
  • Accepted : 2018.04.01
  • Published : 2018.06.23
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Abstract

Cofinite graphs and cofinite groupoids are defined in a unified way extending the notion of cofinite group introduced by Hartley. These objects have in common an underlying structure of a directed graph endowed with a certain type of uniform structure, called a cofinite uniformity. Much of the theory of cofinite directed graphs turns out to be completely analogous to that of cofinite groups. For instance, the completion of a directed graph Γ with respect to a cofinite uniformity is a profinite directed graph and the cofinite structures on Γ determine and distinguish all the profinite directed graphs that contain Γ as a dense sub-directed graph. The completion of the underlying directed graph of a cofinite graph or cofinite groupoid is observed to often admit a natural structure of a profinite graph or profinite groupoid, respectively.

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References

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