Axioms

Definition 1 (associativity) For all \({a},{b},{c}\) in \(\Set\), one has \(\Operation{(\Operation{a}{b})}{c}=\Operation{a}{(\Operation{b}{c})}\).

Definition 2 (identity) There exists an element \(\Identity\) in \(\Set\) such that, for every \({a}\) in \(\Set\), one has \(\Operation{\Identity}{a}={a}\) and \(\Operation{a}{\Identity}={a}\). Such an element is unique and is called the identity element.

Definition 3 (inverse) For each \({a}\) in \(\Set\), there exists an element \({b}\) in \(\Set\) such that \(\Operation{a}{b}=\Identity\) and \(\Operation{b}{a}=\Identity\), where \(\Identity\) is the identity element.

For each \({a}\), the element \({b}\) is unique and is called the inverse of \({b}\) and is denoted \(\Inverse{a}\).

Outlinks

• Følner sequence
• Actions
• Furstenberg’s Correspondence Principle
• Recurrence and Ergodic Theorems
• A Short Proof of a Generalised Conjecture of Erdős for Amenable Groups
• Are we doing enough to address the shared social challenges of cross-cultural and cross-neurotype interactions at the University of Manchester?