# Definition:B-Algebra

## Definition

Let $\left({X, \circ}\right)$ be an algebraic structure.

Then $\left({X, \circ}\right)$ is a $B$-algebra if and only if:

 $(AC)$ $:$ $\displaystyle \forall x, y \in X:$ $\displaystyle x \circ y \in X$ $(A0)$ $:$ $\displaystyle \exists 0 \in X$ $(A1)$ $:$ $\displaystyle \forall x \in X:$ $\displaystyle x \circ x = 0$ $(A2)$ $:$ $\displaystyle \forall x \in X:$ $\displaystyle x \circ 0 = x$ $(A3)$ $:$ $\displaystyle \forall x,y,z \in X:$ $\displaystyle \left({x \circ y}\right) \circ z = x \circ \left({z \circ \left({0 \circ y}\right)}\right)$

## Example

This is a $B$-algebra for the finite set $\left\{{0, 1, 2, 3, 4, 5}\right\}$:

$\begin{array}{c|cccccc} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 2 & 1 & 3 & 4 & 5 \\ 1 & 1 & 0 & 2 & 4 & 5 & 3 \\ 2 & 2 & 1 & 0 & 5 & 3 & 4 \\ 3 & 3 & 4 & 5 & 0 & 2 & 1 \\ 4 & 4 & 5 & 3 & 1 & 0 & 2 \\ 5 & 5 & 3 & 4 & 2 & 1 & 0 \\ \end{array}$

## Also see

• Results about $B$-algebras can be found here.