Axiom:Group Axioms/Left

Definition

A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions:

$(\text G 0)$	$:$	Closure Axiom	$\ds \forall a, b \in G:$	$\ds a \circ b \in G $
$(\text G 1)$	$:$	Associativity Axiom	$\ds \forall a, b, c \in G:$	$\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c $
$(\text G_{\text L} 2)$	$:$	Left Identity Axiom	$\ds \exists e \in G: \forall a \in G:$	$\ds e \circ a = a $
$(\text G_{\text L} 3)$	$:$	Left Inverse Axiom	$\ds \forall a \in G: \exists b \in G:$	$\ds b \circ a = e $

\((\text G 0)\)	$:$	Closure Axiom	\(\ds \forall a, b \in G:\)	\(\ds a \circ b \in G \)
\((\text G 1)\)	$:$	Associativity Axiom	\(\ds \forall a, b, c \in G:\)	\(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)
\((\text G_{\text L} 2)\)	$:$	Left Identity Axiom	\(\ds \exists e \in G: \forall a \in G:\)	\(\ds e \circ a = a \)
\((\text G_{\text L} 3)\)	$:$	Left Inverse Axiom	\(\ds \forall a \in G: \exists b \in G:\)	\(\ds b \circ a = e \)