Axiom:B-Algebra Axioms

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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.

Also see