Definition:Opposite Magma

Definition

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

• Definition:Parastrophe

• Definition:(1-3) Parastrophe
• Definition:(2-3) Parastrophe

• Opposite Group

• Results about parastrophes can be found here.

Sources

• 2015: W.A. Dudek: Parastrophes of Quasigroups (Quasigroups and Related Systems Vol. 23: pp. 221 – 230)