
Constructing Double Magma on Groups Using Commutation Operations
A magma $(M,\star)$ is a nonempty set with a binary
operation. A double magma $(M, \star, \bullet)$ is a
nonempty set with two binary operations satisfying the
interchange law,
$(w \star x) \bullet (y\star z)=(w\bullet y)\star(x \bullet
z)$. We call a double magma proper if the two operations
are distinct and commutative if the operations are commutative.
A double semigroup, first introduced by Kock,
is a double magma for which both operations are associative.
Given a nontrivial group $G$ we define a system of two magma
$(G,\star,\bullet)$ using the commutator operations $x \star
y = [x,y](=x^{1}y^{1}xy)$ and $x\bullet y = [y,x]$. We show
that $(G,\star,\bullet)$ is a double magma if and only if $G$
satisfies the commutator laws $[x,y;x,z]=1$ and $[w,x;y,z]^{2}=1$.
We note that the first law defines the class of 3metabelian
groups. If both these laws hold in $G$, the double magma is proper
if and only if there exist $x_0,y_0 \in G$ for which $[x_0,y_0]^2
\not= 1$. This double magma is a double semigroup if and only
if $G$ is nilpotent of class two. We construct a specific example
of a proper double semigroup based on the dihedral group of order
16. In addition we comment on a similar construction for rings
using Lie commutators.
Keywords:double magma, double semigroups, 3metabelian Categories:20E10, 20M99 