Axiom:B-Algebra Axioms

Definition

Let $\struct {X, \circ}$ be an algebraic structure.

$\struct {X, \circ}$ is a $B$-algebra if and only if the following conditions are satisfied:

$(\text {AC})$	$:$	$\ds \forall x, y \in X:$	$\ds x \circ y \in X $
$(\text A 0)$	$:$		$\ds \exists 0 \in X $
$(\text A 1)$	$:$	$\ds \forall x \in X:$	$\ds x \circ x = 0 $
$(\text A 2)$	$:$	$\ds \forall x \in X:$	$\ds x \circ 0 = x $
$(\text A 3)$	$:$	$\ds \forall x, y, z \in X:$	$\ds \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} } $

These criteria are called the $B$-algebra axioms.

\((\text {AC})\)	$:$	\(\ds \forall x, y \in X:\)	\(\ds x \circ y \in X \)
\((\text A 0)\)	$:$		\(\ds \exists 0 \in X \)
\((\text A 1)\)	$:$	\(\ds \forall x \in X:\)	\(\ds x \circ x = 0 \)
\((\text A 2)\)	$:$	\(\ds \forall x \in X:\)	\(\ds x \circ 0 = x \)
\((\text A 3)\)	$:$	\(\ds \forall x, y, z \in X:\)	\(\ds \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} } \)