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:

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


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.


Sources