Definition:Opposite Magma
(Redirected from Definition:(1-2) Parastrophe)
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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)