• Amenable Topological Groupoids in Java - Created by CodeByAidan
  • Definition
    • Groupoid:
      • Partiality:
      • Associativity:
    • Topological Groupoid:
    • Amenable Groupoid:
  • References
  • Installation

A groupoid is a generalization of a group. It is a set $G$ equipped with a partial binary operation $(\cdot)$ that satisfies the following properties for all elements $a, b, c$ in $G$:

The operation $a \cdot b$ is defined if and only if there exists an element $c$ such that $a \cdot b = c$.

$(a \cdot b) \cdot c = a \cdot (b \cdot c)$ whenever both sides are defined.
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:

$$ \begin{align*} \text{A groupoid } G \text{ is amenable if there exists a sequence of finite subsets } F_n \text{such that} \\ \text{for every open set } U \text{containing the unit space of } G, \text{the following limit holds:} \\ \end{align*} $$

$$ \lim_{n \to \infty} \frac{|U \cdot F_n - U|}{|F_n|} = 0 $$

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 $U \cdot F_n$ approximate the set $U$ in a specific way as $n$ approaches infinity.

• 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

System.out.println("Does the finite subset satisfy the Folner condition? " + satisfiesFolner);