Trait mathru::algebra::abstr::AbelianGroup
source · pub trait AbelianGroup<O: Operator>: Group<O> { }Expand description
An Abelian group is a commutative group.
A Group is a triple $(\mathbb{A}, \circ, e)$, composed by a set $\mathbb{A}$ and a binary inner operation $\circ$ and the element $e \in \mathbb{A}$
§Definition
\circ: \mathbb{A} \times \mathbb{A} \rightarrow \mathbb{A} , (x, y) \mapsto x \circ y
- Closure $\forall x, y \in \mathbb{A},: x \circ y \in \mathbb{A}$
- associativity
$\forall x, y, z \in \mathbb{A}$: $x \circ (y \circ z) = (x \circ y) \circ z$ 3. $e$ neutral element(identity)
$\forall x \in \mathbb{A}$: $x \circ e = e \circ x = x$ - Inverse element $x^{-1} \in \mathbb{A}: x^{-1} \circ x = x \circ x^{-1} = e$
- Commutativity $\forall x, y, \in \mathbb{A}: x \circ y = y \circ x$
Object Safety§
This trait is not object safe.