# Definition:Opposite Magma

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## 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

- Results about
**parastrophes**can be found**here**.

## Sources

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