A groupoid is a generalization of a group. It is a set
The operation
A group is a special case of a groupoid where every element has an inverse.
A topological groupoid is a groupoid equipped with a topology that makes both the composition operation and the inversion operation continuous. This means that the groupoid operations interact nicely with the open sets in the topology.
The term "amenable" in the context of groupoids refers to a property that generalizes the concept of amenability from groups to groupoids. In the context of topological groupoids, a groupoid is amenable if it satisfies a certain version of the Følner condition. The Følner condition, in the context of groupoids, relates to the existence of approximations to the identity element within finite subsets of the groupoid.
The definition of an amenable topological groupoid can be expressed as follows:
In this notation:
-
$U \cdot F_n$ represents the set of all elements obtained by composing elements of$U$ with elements of$F_n$ . -
$\mathopen|U\mathclose|$ represents the cardinality of the set$X$ .
This limit essentially states that the sets
- nlab - amenable topological groupoid
- Aidan Sims, Dana P. Williams, Amenability for Fell bundles over groupoids (arXiv:1201.0792)
- Johannes Aastrup, Ryszard Nest, Elmar Schrohe, A Continuous Field of C-algebras and the Tangent Groupoid for Manifolds with Boundary (arXiv:math/0507317)
# Clone the repository
git clone https://github.com/CodeByAidan/Amenable-Topological-Groupoids
# Change directory
cd Amenable-Topological-Groupoids`
# Compile the program into the bin directory
javac -d bin src\com\CodeByAidan\topologicalgroupoid\*.java
# Compile Main.java to bin
cd "src" && javac -d ../bin Main.java && cd ..
# Run the program
java -cp bin Main
Note
Note: The result of Main.java in console will display a "?" instead of "ø" in IDE terminals typically. Please make sure that your console or terminal supports UTF-8 encoding to display special characters correctly. Alternatively, you can replace "ø" with "o" in the output message for simplicity
⬇️System.out.println("Does the finite subset satisfy the Folner condition? " + satisfiesFolner);