The tight groupoid of an inverse semigroup
  In this work we present algebraic conditions on an inverse semigroup S (with zero) which imply that its associated tight groupoid G_{tight)(S) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient. 
Ruy Exel and Enrique Pardo
math.OA math.DS