# 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.

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

## Sources

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