Definition:B-Algebra

Definition

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

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

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

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

• $B$-Algebra Induces Group
• Group Induces $B$-Algebra

• 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)