Definition:Commutative B-Algebra

From ProofWiki
Jump to: navigation, search

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