# Definition:Commutative B-Algebra

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## Definition

Let $\struct {X, \circ}$ be a $B$-algebra.

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

- $\forall x, y \in X: x \circ \paren {0 \circ y} = y \circ \paren {0 \circ x}$

## Note

Note the independent properties of $\struct {X, \circ}$ being **$0$-commutative** and $\circ$ being commutative.

To demonstrate consider the $B$-algebra $\struct {\R, -}$ where $-$ denotes real subtraction.

Work In ProgressIn particular: A page establishing $\R$ with subtraction is a $B$-algebra.Note that it is not only $\R$ that forms a $B$-algebra with subtraction - so does any of the standard number sets, if I'm not mistaken, so that will need to be taken into account. In fact, we now have Group Induces B-Algebra which shows that this is a general rule for ALL groups. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

$\struct {\R, -}$ *is* **0-commutative** but $-$ is *not* commutative.

This theorem requires a proof.In particular: Another page establishing the non-equivalence of 0-commutativity and commutativity, rather than hiding it in this "notes" section.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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