# Definition:B-Algebra

## Contents

## 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.

## Sources

- 2002: J. Neggers and Hee Sik Kim:
*On B-Algebras*(*Matematički Vesnik***Vol. 54**: 21 – 29)