# Definition:Commutative B-Algebra

From ProofWiki

## Definition

Let $\left({X, \circ}\right)$ be a $B$-algebra.

Then $\left({X, \circ}\right)$ is said to be **$0$-commutative** (or just **commutative**) if and only if:

- $\forall x, y \in X: x \circ (0 \circ y) = y \circ (0 \circ x)$

## Note

Note the independent properties of $\left({X, \circ}\right)$ being **$0$-commutative** and $\circ$ being commutative.

To demonstrate consider the $B$-algebra $\left({\R, -}\right)$ where $-$ denotes conventional subtraction.

$\left({\R, -}\right)$ *is* **0-commutative** but $-$ is *not* commutative.

## Sources

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