Definition:B-Algebra

From ProofWiki
Jump to: navigation, search

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