Definition:Commutative B-Algebra
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This theorem requires a proof.
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.
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.
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Work In Progress: 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. (discuss) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. When this has been done, the template should be removed from the code. |
$\struct {\R, -}$ is 0-commutative but $-$ is not commutative.
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.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
- 2002: J. Neggers and Hee Sik Kim: On B-Algebras (Matematički Vesnik Vol. 54: pp. 21 – 29)