Definition:B-Algebra

Definition

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

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

 $(\text {AC})$ $:$ $\displaystyle \forall x, y \in X:$ $\displaystyle x \circ y \in X$ $(\text A 0)$ $:$ $\displaystyle \exists 0 \in X$ $(\text A 1)$ $:$ $\displaystyle \forall x \in X:$ $\displaystyle x \circ x = 0$ $(\text A 2)$ $:$ $\displaystyle \forall x \in X:$ $\displaystyle x \circ 0 = x$ $(\text A 3)$ $:$ $\displaystyle \forall x, y, z \in X:$ $\displaystyle \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} }$

Example

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.