# Definition:B-Algebra

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## Definition

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

Then $\struct {X, \circ}$ is a **$B$-algebra** if and only if:

\((\text {AC})\) | $:$ | \(\ds \forall x, y \in X:\) | \(\ds x \circ y \in X \) | |||||

\((\text A 0)\) | $:$ | \(\ds \exists 0 \in X \) | ||||||

\((\text A 1)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds x \circ x = 0 \) | |||||

\((\text A 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds x \circ 0 = x \) | |||||

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

## Examples

### $B$-Algebra Induced by $S_3$

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

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

## Also see

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

## Linguistic Note

A literature search has failed to reveal the origin or derivation of the term **$B$-algebra**.

Hence the reason for why it is called that is still under investigation.

## Sources

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