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 \(\Inverse{a}\).