\begin{definition}[Definition:B-Algebra]
Let $\struct {X, \circ}$ be an algebraic structure.
Then $\struct {X, \circ}$ is a '''$B$-algebra''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text {AC}
        | q = \forall x, y \in X
        | m = x \circ y \in X
}}
{{axiom | n = \text A 0
        | m = \exists 0 \in X
}}
{{axiom | n = \text A 1
        | q = \forall x \in X
        | m = x \circ x = 0
}}
{{axiom | n = \text A 2
        | q = \forall x \in X
        | m = x \circ 0 = x
}}
{{axiom | n = \text A 3
        | q = \forall x, y, z \in X
        | m = \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} }
}}
{{end-axiom}}
\end{definition}