Definition:Opposite Magma

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Let $S$ be a set.

Let $\struct {S, \circ}$ and $\struct {S, *}$ be magmas on $S$.

$\struct {S, *}$ is the opposite magma of $\struct {S, \circ}$ if and only if:

$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_2 * x_1 = x_3$

The operation $*$ is sometimes referred to as the opposite law of $\circ$.

Also known as

This concept was introduced with this name in the books by Nicolas Bourbaki.

Other sources refer to $\struct {S, *}$, as defined here, as the $(1-2)$ parastrophe of $\struct {S, \circ}$.

Also see

  • Results about parastrophes can be found here.