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:

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


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